Cfje fci&tatp
ottpt
3lnfoer0ftg of Jl3ort& Carolina
Cn&otocli tv f|e Malutk
Pfnlantltopsc §>ocietie0
MUSIC L! nn ^°v
.. $>^~Jk±l.
: I
TREATISE
OF
mv sic%,i
speculative, logical, and $iftoiicai+
By Alexander Malcolm;
Hail Sacred Art ! defended from above,
To crown our mortal 'Joys : Of thee we learn )
How happy Souls communicate their Raptures ;
For thourt the Language of the Blefl in Heaven:
— Divum hominumqi voluptas.
EDINBURGH,
(Printed for the A u t h o r. M D G CXXt
ON THE
Tower of MVS1CK,
tnfcrib'd to
Ml MALCOLM,
A s A
Monument of Friendftlipj
By Mr, MITCHELL,
HEN Nature yet in Emhrio\2cfi
Ere Things began to be;,
The Almighty froni eternal Da^
Spoke loud his de°p Decree:
The Voice was tuneful as his Love^
At which Creation fprurig^
And all th' rfngelick Hofts abovd
The Morning. Anthenr fun g;
\
5? a *
8; At
II.
At MuficKs fweet prevailing Call/
Thro' boundlefs Realms of Space,
The Atoms danc'd, obfequious all,
And, to compofe this wondrous Ball,
In Order took their Place.
How did the Piles of Matter part.
And huddled Nature from her Slumber ftart <
When, from the Mafs immenfely ffceep,
The Voice bid Order fudden leap,
To ufher in a World.
What heavenly Melody and Love
, Began in ev'ry Sphere to move ?
When Elements, that jarr'd before,
ytexe all afide diftinctly hurfd,
And Chaos reign 'd no more.
III.
Mufick the mighty Parent was,
Empower'd by G o d, the fovereign Caufe,
Mufick firft fpirited the lifelefs Wafte,
Sever'd the fullen, bulky Mafs,
And active Motion calfd from lazy Reft.
Summon'd by Mufick^' Form uprear'd her Head,
From Depths, where Life it felf lay dead,
While fudden Rays of everliving Light
Broke" from the Abyfs of ancient Night,
RdVeaFd the new-born Earth around and its fair
Influence fpreadi
Gob faw that all the Work was good ;
The Work, the EflM of Harmony, its won-
drous Offsprings flood.
_ ^ IV* Mufick
IV.
Mujtcky the beft of Arts divine}
Maintains the Tune it firft began;
And makes ev *n Oppofites combine
To be of Ufe to Man.
Difcords with tuneful Concords move
Thro' all the fpacious Frame ;
JBelozv is breath'd the Sound of Love,
While myftick Dances fhine Above^
And Mufictis Power to nether Worlds proclaim;
What various Globes in proper Spheres,
Perform their great Creator's Will ?
While never filent never ftil],
MeJodioufly they run,
Unhurt by Chance, or Length of Years; ]
Around the central Sun.
J
V.
The little perfect World, call'd Man,
In whom the Diapafon ends,
In his Contexture, fhews-a Plan
Of Harmony, that makes Amends,
By God-like Beauty that adorns his Race,'
For all the Spots on Nature's Face.
He boafts a pure, a tuneful Soul,
That rivals the celeftial Throng, 1
And can ev'n favage Beafts controul
With his inchanting Song.
Tho' diff 'rent Paffions ftruggle in his Mind J
JVhere Love and Hatred, Hope and Fear a ; e
joyn'd.
All, by a facred Guidance, tend
yTo one harmonious End,
a 3 VI. It$
VI.
Its great Original to prove.
And fhew it blefs'd us from above,
In creeping Winds, thro' Air it fweetly flotes'
And works ftrange Miracles by Notes.
Our beating Pulfes bear each bidden Part,
And ev'ry PaiTion of the mafter'd Heart
Is touched with Sympathy, and fpeaks the Worn
dcrs of the Art.
Now Love, in foft and whifpering Strains,
Thrills gently thro' the Veins,
And binds the Soul in filken Chains.
Then Rage and Fury fire the Blood,
'^.nd hurried Spirits, rifing high, ferment the
boiling Flood ;
Silent, anon, we fink, refign'd in Grief :
But ere our yielding Paifions quite fubfide,
Some fwelline Note calls back the ebbing
Tide,
And lifts us to Relief.
With Sounds we love, we joy, and we defpair,
fXhe folid Subftance hug, or grafp delufive Air,
VII.
In various Ways the Heart-fixings fhake,
And different Things they fpcak.
For, when the meaning Mailers ftrike the
Or Hautboys briskly move.
Our Souls, like Lightning, blaze with quick
Delire,
Or melt away in Love.
But when {he martial Trumpet 3 fwelling high, 1
Rolls,
Vll
Rolls its (hrill Clangor thro' the ecchoing
Sky;
If, anfwering hoarfe, the fallen Bruin's big
Beat
Does, in dead Notes, the lively Call repeat $
Bravely at once we break o'er Nature's Bounds,'
Snatch at grim Death, and look, unmov'd,
on Wounds.
Slumb'ring, our Souls lean o'er the trembling
Lute ;
Softly we mourn with the complaining Flute g
With the Violin laugh at our Foes,
By Turns with the Organ we bear on the Sky, 1
Whilft, exulting in Triumph on Mther we
Or, falling, grone upon the Harp, beneath |
Load of Woes.
Each Inilrument has magick Power
To enliven or deftroy,
To fink the Heart, and, in one Hour,:
Entrance, our Souls with Joy.
At ev'ry Touch, we lofeour ravifh'd Thoughts^
And Life, it felf, in quivering clings, hangs o'er
the varied Notes,
VIII.
How does the ftarting Treble raife
The Mind to rapt'rous Heights $
It leaves all Nature in Amaze,
And drowns us with Delights.
But, when the manly, the majeftick, Bafs
Appears with awfal Grace,
a 4 3Vha£
Till
What fofemti Thoughts are in the Mind in-
' fus'd?.
And how the Spirit's rous'd?
In flow-plac'd Triumph, we are led around.
And all the Scene with haughty Pomp is
crown'd ,
Till friendly Tenor gently flows,
Like fweet, meandring Streams,
And makes an Union, as it goes,
Betwixt the Two Extremes.
The blended Parts in That agree,
As Waters mingle in the Sea,
And yield a Compound of delightful Melody.
IX.
Strange is the FOi.ce of modulated Sound
T hat, like a Torrent, fweeps o'er ev'ry Mound I
It tunes the Heart at ev'ry Turn ,-
With ev'ry Moment gives new Paifions Birth r
Sometimes w r e take Delight to mourn,-
Sometimes enhance our Mirth.
It fpoths deep Sorrow in the Breaft t
It lul ■$ our waking Cares to Reft,
Fate's clouded Brow ferenes with Eafe,
And makes ev n Madnefs pleafe.
As much as Man can meaner Arts controul^
It manages his mafter'd Soul,
The moft invet'rate Spleen difarms,
And, like Aurelia^ charms :
T Aurelia ! dear diftinguifh'd Fair I
In whom the Graces center'd are I
JV^hofe hotes engage the Ear and Mind,
As Violets breathed on by the gentle Wind-
Whofe Beauty, Mufick in Difguife »
Attracts the gazing Eyes,
Thrills thro' theSoul,like Haywood s melting
Lines,
And, as it certain Conquer! makes, the favage
Soul refines,
X.
Mufick religious Thoughts infpires,
And kindles bright poetick Fires $
Fires ! fuch as great Hillarius raife
Triumphant in their Blaze I
Amidft the vulgar verfifying Throng,
His Genius, with Diftinction, ftiow,
And o'er our popular Metre lift his Song
High, as the Heav ns are arch'd o'er Orbs below.
As if the Man was pure Intelligence,
Mufick tranfports him o'er the Heights of
Senfe,
Thro' Chinks of Clay the Rays above lets in,
And makes Mortality divine.
Tho' Reafon's Bounds it ne'er defies,
Its Charms elude the Ken
Of heavy, grofs-ear'd Men,
Like Myfteries conceal'dfrom vulgar Eyes,
Others may that Diftra£tion call,
Which Mufick raifes in the Breaft,
To me 'tis Extafy and Triumph all.
The Foretaftes of the Raptures of the Bleft.
^ho knows not this, when Handel plays, 1
And Senefino fings ?
Our
Our Souls learn Rapture from their La^sj
Wlaile rival'd Angels fhow Amaze,
Arid drop their golden Wings.
XL
Still, God of Life,._entrance my Soul
With fuch Enthufiaftick Joys ,•
And, when grim Death,, with dire Con-
troul,
My Pleafures in this lower Orb deftroys,
Grant this Requeft whatever you deny.
For Love I bear to Melody,
That, round my Bed, a facred Choir
Of skilful Matters tune their Voice,
And, without Pain of agonizing Strife,
In Confort with the Lute confpire,
To untie the Bands of Life j
That, dying with the dying Sounds .
My Soul, well tun'd, may raife
And break o'er all the common Bounds
Of Minds, that grovel here below the Skies.
XII.
When Living die, and dead Men live,
And Order is again to Chaos huifd,
Thou, Melody •> Jfcpft ffifl furvive,
And triumph o'er the Ruins of the World*
A dreadful Trumpet never heard before,
By Angels never blown, till then.
Thro' all the Regions of the Air (hall rore
That Time- is now no more:
But lo ! a different Scene \
Eternity appears.
XI
Like Space unbounded and untold by Years,
High in the Seat of Happinefs divine
Shall Saints and Angels in full Chorus
jbyn.
In various Ways,
Seraphick Lays
The unceaiing Jubile {hall crown.
And, whilft Heav'n ecchoes with his Praifc,
The Almighty's felf /hall hear, and look,
delighted, down.
XIII.
Who would not wifh to have the Skill
Of tuning Inftruments at Will ?
Ye Pow'rs, who guide my A&ions, tell
Why I, in whom the Seeds of Mufick
dwell.
Who moft its Pow'r and Excellence admire
Whofe very Breaft, it f elf's, a Lyre^
Was never taught the heav'nly Art
Of modulating Sounds,
And can no more, in Confort, bear a Part
Than the wild Roe^ that o'er the Mountains
bounds ?
Could I live o'er my Youth again,
( But ah ! the Wifh how idly vain ! )
Inftead of poor deluding Hhime,
Which like a Syren murders Time,
Inftead of dull, fcholaftick Terms,
W T hich made me ftare and fancy Charms/
With Gordons brave Ambition fir'd,
Beyond the tow'ring Al$s 9 untir'd,
To
xii
To tune my Voice to his fweet Notes, Yd
roam >
Or fearch the Magazines of Sound,
Where MuficFs Treafures ly profound.
With M here at Home.
M 9 the dear, deferving Man, 7
Who taught in Nature's Laws,
To fpread his Country's Glory can
Praftife the Beauties of the Art, and (hew its
Grounds and Caufe.
# # #
TABLE
£111
VMS
'%&}ii&^&U&i&ii$^*^^
TABLE
O F
C ONTENTS.
HAP. I. Containing an Account of th«
Ohjedf and JEW of Mufick, and the
Nature of the Science in the Definiti-
on and Dwijion of it,
§ i. Of Sound: .77^ Gzz//<? of it; and the ®a*
rious Affections of it concerned in Mufick^ P. i»
§ 2. Containing the Definition and T>ivifion
of Mufick. p. 29.
G h a p. II. Of T u n e, or the Relation of
Acutenefs and Gravity in Sounds ; particularly
of the Caufe and Meafures of the Differences of
Tune.
§ 1. Containing forne neCeffary Definitions
find Explications, and the particular Method of
treating
XIV
treating this Branch of the Science concerning
Tune or Harmony. Pag. 34.
§ 2. Of the Caufe and Meafure of Tune ; or,
upon ivhai the Tune of a Sound depends ; and
how the relative Degrees and Differences of
Tune are determined and meafured. p. 42.
Chap. III. Of the Nature ofCoNcoRD and
Discord, as contained in the Caufes thereof.
§ i» Wherein the Reafons and Chara&erift-
icks of the fever al Differences of Concords and
Difcords are enquired into* p. 66»
{) 2. Explaining fome remarkable Appearan-
ces relating to this Subject, upon the preceeding
Grounds of Concord. p. 85*
• Chap. IV* Containing the H a r m o n i c a l
Arithmetic k*
(j 1. Definitions. p. 9 6*
§ 2. Of arithmetical and geometrical Pro-
portions . p. 102*
§3.Q/* harmonica! Proportion, p. iop.
§ 4, The Arithmetick of Ratios geometrical ;
or of the Competition and Refolution of Ra-
tios, p. 117.
§ 5, Containing an application of the pre-
ceeding Theory ^Proportion, to the Intervals
of Sound, p. 133*
Chap. V. Containing a more particular Con-
fideration of the Nature, Variety and Compofi-
tim of Concords; in Application of the
preceeding Theory.
§ 1. Of the Original Concords,^*
J&tfe and Dependence on each other > &c. p* *53*
§hOf
XV
§ £ Of Compound Concords'; and
o/^HaRMOnick Series; with fever al Ob~
fervations relating to both ample and compound
"Concords. Pag. 167.
CHAP.Vl.Of theGEOMETRlCAL P A R T of
Mulick Or how to divide right Lines,fb as their
Sections or Parts one with another, or with the
JVhole, (hall contain any given Interval.
§ 1. Of the more general Divifion of Chord%
p. 181.
§ 2. Of the harmonical Divifion of Chords.
p. 184.
<j 3. Containing further Reflections upon the
Divifion of Chords. p. 194.
Chap. VII. Of Harmony, explaining
the Nature and Variety of it, as it depends up-
on the various Combinations of concording
Sounds. p. 200.
Chap. VIII. Of Concinnous Inter-
vals, and the Scale of Music k.
§ 1. Of the Neceffity andUfe of concinnous
pifcords, and of their Original and Dependence
on the Concords. p. 217.
<) 2. Of the Ufe of Degrees in the ConftrutJi-
on of the Scale of M u s 1 c k. p. 229.
§ 3. Containing further Refle&ions upon the
Constitution of the Scale of Mufick $ and ex-
plaining the Names of 8ve, 5th, &c. which
Jjave been hitherto ufed without knowing all
their Meaning ; Jhemng alfo the proper Office
vf the Scale. p. 241.
, § 4. Of ^accidental Discords in the
$yftem of Musick. p. 253,
Chap,
Chap. Bt.OftheMoi)E of K i y in Mufich,
and a further Account of the true Office and
End of the Scale of Mufich
§ ■ i. Of the Mode or Key. Pag. 265.
§ 2. Of the Office of the Scale 0/ Muiick. p.
C h a p. X. Concerning the Scale of Mufick
limited to fixt Sounds, explaining the Defebls
of InftrumentS) and the Remedies thereof ;
wherein is taught the true Ufe and Original of
the Notes we commonly call Jloarp and flati
§ 1* Of the Defects of Interments, and of the
Remedy thereof in general, by the Means of
what we call Sharps and Fiats. p. 282.
§ 2. Of the true Proportions o/'^femitonick
Scale, and hozu far the Syftem is perfected by
it. p. 293.
§ 3» Of the common Method of tuning Spinets,
demonftr citing the Proportions that occur in it;
and the Pretence of a nicer Method confidered.
p. 30&
§ 4. A brief Recapitulation of the preceed-
ing. p. 31a.
Chap. XI. The Method and Art of
writing j^///f ^particularly how the Differences
of Tune are reprefented.
§ ii ^general Account of the Method, p.
323.
$ 2. A more particular Account of the Me-
thod ; where, of the Mature and Ufe of
Clefs. p. 330*
§ 3. Of the Reafon, Ufe, and Variety of ih*
Signatures of C l e £ s« p. 34 **
xvii
§ 4. Of Tranfpofitioril T. With refpeB to the
Clef, p. 361." And' 2. From one Key to another*
. PagT 365.
§ 5-. Qf Sol-fa-in g,withfome other particular
Remarks about the Names of Notes. p. 36$:
Appendix, Concerning Mr. Salmonlr Propo-
falfor reducing all Mufick to. one Clef. p. 378.'
Chap. XII. Of the Time or Duration of
Sounds in Mufich
§ 1. Of the Time /# general, anditsSuli-
divifion into abfolute and relative ; and partial*
larly of the Names, Signs, and Proportions or
relative Meafures of Notes^ as foTime. p. 385::
§ 2. Of the abfolute Time ; and the various
Modes or Conftitutions of Parts of a Piece of
Melody, on which the different Airs in Mufich
depend ; and particularly of the Difiinclion of
common and triple Time, and a Defcription of
the Chronometer for meafuring it. p. 393.'
(j 3. Concerning Refts or Paufes of Time ;
andjome other neceffarjy Marks in writing Mu-
fick. p. 409:
Chap. XIII. Containing the general Prin-
ciples and Rules of Harmonick Composition.
(j 1. Definitions. p. 414.
(j 2. Rules of Melody. ; p. 420.
§ 3. Of the Harmony of Concords, or fimple
Counterpoint. p. 422.
§ 4. Of theUfeofDHcords or Figurate Coun-
terpoint, p. 433.
§ 5. Of Modulation.
£hap. XIV. Of the AnciektMusick;
b § 1.0/
xviu
§ u Of the Name, with the various Defini-
tions and Divifions of the Science. Pag. 451.
§2. Of the Invention and Antiquity of Mu-
ficL \ p. 459.
§ 3. Of the Excellency and various Ufes of
Mufick. p. 474;
§ 4. Explaining the harmonick Principles
of the Ancient s> and their Scale of Muhck.
p. 495.
$ 5. A Short Hiftory of the Improvements in
Mufick. p. 552.
§ 6. the agcient and modern Mufick com-
IN-
ux
INTRODUCTION.
Have no fecret Hiftory to entertain
my Reader with, or rather to be
impertinent with, concerning the
Occafion of my ftudying, writing,
or publishing any. Thing upon this
Subject : If the Thing is well done, no mat-
ter how it came to pafs. And tho' it be fome-
■what unfashionable, I muft own it, I have no
Apology to make : My Lord Shafisburj^indeed^
aflures me, that the Generality of Readers are
not a little raifed by the Submiffion of a confe£
fing Author, and very ready on thefe Terms to
give him Abfolution, and receive him into their
good Grace and Favour - y whatever may be in
it, I have Nothing of this Kind wherewith to
bribe their Friendship ; being neither confcious
of LazinefS) Precipitancy ', or any other wilful
Vice y in the Management of this Work, that
fhould give me great Uneafinefs about k^ if
there be a Fault, it lies fome where elfej for,
tQ be plain, I have taken all the Pains I could.
b 2 X
SOB '■■
I h^t always thought it as impertinent for art
'Author to offer any Performance to the World,
with a flat Pretence of fufpe&ing it, as it is ridi-
culous to commend himfelf in a conceited and
faucy Manner ; there is certainly fomething juft
and reasonable, that lies betwixt thefe Ex-
tremes ; perhaps the beft Medium is to fay No-
thing at all ; but if one may fpeak, I think he
may with a very good Grace fay, he has de-
figned well and done his beflj the RefpecY due
to Mankind requires it, and as I can fincerely
profefs' this, I fhall have no Anxiety about the
Treatment my Book may nleet with. The
Criticks therefore may take their full Liberty :
I can lofe Nothing at their Hands, who examine
Things with a true Refpecl: to the real Service
of Mankind ; if they approve, I (hall rejoyce,
if not, I ftiall be the better for their judicious
Correction : And for thofe who may judge rafh-
ly thro* Pride or Ignorance, I fhall only pity
them.
But there is one common Place of Criticifm
I would beg Leave to confider a little. Some
Peop e, as foon as they hear of a new Book up-
on a known Subject, ask what Difcovery the
Author has made, or what he can fay, which
they don't know or cannot find elfewhere ? I
might defire thefe curious Gentlemen to read
and fee ; but that they may better underftand
my Pretences, and where to lay their Cenfures,
let them confider, there are Two Kinds of Dis-
coveries in Sciences i one is that of new Theo-
rems and Propofitions, the other is of the proper
" '- Met
XXI
Relation slzlA Connection of the Things already
found, and the eaiy Way of reprefenting them
to the Underftanding of others; the firft affords
the Materials, and the other the Form of thefe
intellectual Structures which we call Sciences:
How ufblefs the firft is without the other, needs
no Proof; and what an Odds, there may be in
tli6 Way of explaining and difpofmg the Parts
of any Subject, we have a Thoufand Demon-
strations in the numerous Writings upon every
Subject. An Author, who has made a Science
more intelligible, by a proper and d*ftin£t Ex-
plication of every {ingle Part, and a juft and na-
tural Method in the Connection of the Whole j
tho v he has faid Nothing, as to the Matter,
which was not before difcovered, is a real Be-
nefactor to Mankind : And if he has gathered
together in one&yftem, what, for want of know-
ing or not attending to their true Order and De- '
pendence, or whatever other Reafon, lay fcat-
tered in ieveral Treatifes, and perhaps added
many ufeful Reflections and Observations ; will
not this Author, do ye think, be acquitted of
the Charge of Plagiarifm^ before every reafo-
nable Judge ; and be reckoned uiflly more than
a mere Collector, and to have done fomething
new and ufeful ? If you appeal to a very wife
and learned Ancient, the Queftion is clearly
determined. — Etiamfi omnia a veteribus in-
vent a funt^ tamen erit hoc femper novum^ vfus
CJ difpofitio inventor urn ab aliis. Seneca _Ep.
64. How far this Character of a new Author
will be found in the following Treatise, de-
pends
xxii
pends upon the Ability and Equity of my Judg-*
es, and I leave it upon their Honour, i
But you muft have Patience to hear another
Thing, which Juftice demands of me in this
Place. It is, to inform you, that the 13 Ch.
of the following Book was communicated to me
by a Friend, whofe Modefly forbids me to
name. The fpeculative Part, and what elfe
there is, befides the Subjed of that Chapter-,
were more particularly my Study : But I found,
there would certainly be a Blank in the Work,
if at leaft the more general Principles of Com-
petition were not explained - y and whatever
Pains I had taken to underftand the Writers on
this Branch, yet for want of fuffrcient Pra&ice
in it, I durft not truft my own Judgment to ex^
tract out of them luch a Compend as would an-
fvver my Delign ; which I hope you will find ve-
ry happily fupplied, in what my Friend's Geni-
us and Generofity has afforded: And if I can
3udge any Thing about it, you have here not a
mere Compend of what any Body cKe has done,
but the firft Principles of harmonkk Competition
explained in a Manner peculiarly his own.
After fo long a perfonal Conference, you'll
perhaps expect I fcould fay fomcthing, in this
Introdii&iwn^to my Subject^ but this, I believe,
will be universally agreeable, the Experience of
fome Thoufand Years giving it fufficient Recom-
mendation jj and for any 'thing, dfc I have little
to fay in this Place ; The Contents you have
in the proceeding Table, and I fnall only make
this (hoif Traiifition'to the Book it felf.
The
xxiu
The Original and various Significations of
the Word M u s i c k, you'll find an Account of
it in the Beginning oiCh. 14. For, an hiftori-
cal Account of the ancient Miifick being one
Part of my Defign, I could not begin it better,
than with the various Ufe of the Name among
the Ancients. It (hall be enough therefore to
tell you here 5 that I take it in the common
Senfej for that Science, which confiders and ex-
plains thofe Properties and Relations of Sounds,
that make them capable of exciting the agree-
able Senfations, which the Experience of all
Mankind affures us to be a natural Effect of
certain Applications of them to the Ear. And,
for the fame Reafon, I forbear to fpeak in this
Place any Thing particularly of the Antiquity,
JExcellency^ and various Ufes^Sid Ends of M u-
si'CK, which I {hall at large confider in the fore-
mentioned Chap, according to the Sentiments
and Experience of the Ancients, and how far
the Experience of our Times agrees with that*
C R«'
CORRIGENDA.
T}Age 55. /. 3. for D readQ. p. 76* k 32. for
■*■ Two r. One. k 33. Fundamental, r. acute
Term. p. 77. /. 2. for 2. r. 1. acute Term, r. Fun-
damental, p. 125. /. 24. J by~. r. ~ by^-. p.
1146. /. 11. 2 : 5, r. 2 : 3. p. 158. /. 5. 3 r. 2.
^182. /. 7. may r* many. • p. 227. /. 18. in
harmonic al^ r ' inharmonic al, p. 250. /. 24. oth,
r. 6th. p. 256. /.i. c - c r. C - c. /. 5. D r.
d. p. 258. q/7£<? 7^/<?, /.3. AD. r, A d. /. .5-.
Bf,f.B £ I* 6. F D, r. Fd. /. 7. D C, r. Dc.
p. 295. /. 14. 1 r. b. p* 301. k 11. r. Plate 2.
Fig. 2. p. 319 k 26. Tune or r. human. /. 30.
J*?/? in. p. 329. /. 16. a r. or. />. 338. /. 14. c
r. e. p. 341. /. 11. a r. on ^. 356 /. 27. g$>, e^
r. ak,d^. ^7. 372. /. 20. rafting, f. rairmg. p. 401.
/. 26. at r. as. p. 424. /. 17. inr. the. p. 435. /*7.
■ the r. in the. /. 448. /. 16. this r. his., p. 452. /.
29, dele other, p. 45 8 < /. 22. are r. is. /. 23* leaft
jr. beft. f. 464. /. 15. re- r. reco-. ^ 465. /. 26.
their r . the* p. 466. /. 1 o. already r. afterwards.
p. <oy. k 31. dia-pafon r. of dia-pafon. ^7. 5*38./.
13. was f. were, p, 546. 1. 13 mentioning r. re-
peating, p. 549. /. 11. Feer r. Feet. /?. 550. /. 10.
Objects r. Subjects, p. 552. /. 21. next r. laft. p.
5 7 7. /. 20. r. concent um abfolutum p.t } "]%. /. 1. r.
aiifpicanti. p. 60%. k 12. r. fimilar. ^. 606. /. 260
moe f * more.
ADDENDA.
p Age 40%. I. 8. <i/ttr Bar. add or of any particular Note.
•* p. 411. /. 1 . <i/«?- Crotchets, <u&2 in the Inples $f-\
p. 41 3. : ^ /it *fo £wi : And if 1/ or $ is annexed to
tbefe t - ^5, k ilgnifies iejfer or greater, fo $% is 3d g.
and 3 1. p 485./. 11. after Memory, M % of
which we have a notable Example.
xxiii
Of the original and various Significations
of the Word Mu(jck T you'll have an Ac-
count in the Beginning of Chap. 14. For, an
hiitorical Account of the ancient Mufick being
one Part of my Defign, I could not begin it
better, than with the various Ufeof the Name
among the Ancients, It (hall be enough there-
fore to tell -you here, that I take it in the com-
mon Senfe, for that Science which considers and
explains thofe Properties and Relations of Sounds,
that make them capable of exciting the agree-
able Senfations, which the Experience of all
Mankind allures us to be a natural Effect of
certain Applications of them to the Ear.- And
for the fame Reafon I forbear to fpeak, in this
Place, any Thing particularly of the Antiquity y
Excellency, and various Z/jf'es and Ends of MU-
ficky which I (hall at large coniider in the fore-
mentioned Chapter, according to the Sentiments
and Experience of the Ancients, and how far
the Expedience of our Times agrees with that.
Corrigenda.
lAge $zA. lo.-read^ : 2. p.- 55. 1. 3. D. r*
p. p. 76. I 32. two r. one. L 33. funda-
mental r. acute Term. p. 77. L 2. 2. r. 1.
acute Term r. fundamental, p.. 125. 1. 24. r.
fv% j*'* P* I 4 < ^' } m ll - ft 2 : B' P* J 5& t 5- 3.
r. 2.. p. 227. ].. 1$. r. in harmonical (as one
Word) p. i$6 1. i. r. C-r- - 1. Sjv^TK r. d. p.
295. L 14. \t r. b, p. 301,1.11. r. Plate 2 Fig. 2,.
p. 319.
xxiv
p. 3x9. I 26. Tune or r. human. I. 36 *&-/* in.
p. 341. 1. 11. a r. or. p. 356, I. 27. g^, el/, r:
a^, d#. p. 435. 1. 7- ther. in the. p. 452. 1. 29.
^e?/t? other, p. 458. 1. 22. are r. is. J. 23. leaft r.
bed. p. 550. I. 10. Objects ri Subjects.
Prr/y excufe a few /matter JEj capes which the
Sen/e will ea/ily cornel.
Addenda,
PAge 408. I. 8. after Bar, add, or of any parti-
cular Note, p. 41 1. J. 1. after Crotchets, add^
in the Triples ~ ~ \. p. 413. add at the End-> and
if \f or $ is annexed to tiieie Figures, it fignifies
kjfer or greater ,'fo ^ is 3^ £, and 6^ is 6?£ J. p.
415. 1. 21. ^/>£r Example, add Plate 4. and mind,
/£#£ all the Examples of t lates 4, 5, & belong
to the 13 Chap. p. 485. J. 11. ^/t^r Memory, ^^,
we have a very o:d ana remarkable Proof of this
Virtue of Mufick.
N. B. In the Table of Examples Tage 258. the different Chara&ers of
Letters are neglected; but the Numbers of each Example will discover
what they ought to be, in Conformity to Fig, 5. P«w« 1. from whence they
are taken.
A 1 . B, See Page 50. at Lint 7. and confequentJy, &c, A wrong Con-
dition has here eicaped me, vz.. that lince the Chord palTes the Point O,
thcrefoie it ;s accelerated. 1 i wn the only Thing that follows from its
palTing that Point is, that the Chord in every Point d. (of a fingle vibra- '
tion) has more Foice than would retain it ti.eie: And the true Reafon of
Acceleration, is this, viz,* in the oitmoft Point U, it lias jnft as much.
Force as is equal to what would keep it theje .* This Foict isfuppofed not to
be deftroyed, but at the next Point d, to receive an Addition of us much,
as v.ould keep it in that , oint, and Id on tincigh every Point till ir pafs
the ftr.iight ' Line, and t! at it loies its Force by the fame Dcgiees^ from
whence follows the Law of Acceleration mentioned '
A*. B, See ' •><(• 6. Li.am\iU 3f- the 2 C \ 3^ 4.^, jtfcj and 6 lh Notes of
the Bafs ought to be each a Eegiee Io\ver 3
TREATISE
O F
CHAP. I.
Containing an Account of the Object and
End of M u s i c k, and the Nature of
the Science, in the Definition and Di~*
vifion. of it.
§ i. 0/SounD: The Ccmfeofitt, and the va^
rious Affeblions of it concerned in Mujich
USICK is a Science of Sounds!
whofe End is Pieafure. Sound
is the Objedf in general ,• or, to
fpeak with the Philofophers^ it
is the material Objebl* But it
is not the Bufinefs of Muficky
taken in a ftricl; and proper Senfe, to coniider
every Phenomenon and Property of Sound ; tKat
belongs to a more univerfal Philofophy : Yet,
that we may underftand what it i§ in Sounds
A upon
% A Treatise Chap. I.
upon which the Formality of Mufick depends*
Z e. whereby it is diftinguifhed from other Sci-
ences, of which Sound may alfo be the Object:
Or, What it is in Sounds that makes the par-
ticular and proper Object ot Mufick, whereby it
obtains its End -, we muft a little confider the
Nature of Sound.
Sound is a Word that ftands for every Per-
ception that comes by the Ear immediately.
And for the Nature of the Thing, it is now
generally agreed upon among Philofophers, and
alfo confirmed by Experience, to be the Effect
of the mutual Collifion, and confequent tre-
mulous Motion in Bodies communicated to the
circumambient Fluid of Air, and propagated
thro' it to the Organs of Hearing.
ATreatife that were defigned for explaining
the Nature of Sound univerfally, in all its
known and remarkable Phenomena, ■ fhould, no
doubt, examine very particularly every Thing
that belongs to the Caufe of it ,- Fir ft, The
Nature of that Kind of Motion in Bodies ( ex-
cited by their mutual Percuflion) which is com-
municated to the Air ,- then, how the Air re-
ceives and propagates that Motion to certain
Diftances : And, laftly, How that Motion is
received by the Ear, explaining the feveral
Parts of that Organ, and their Offices, that are
employed in Hearing. But as the Nature and
Deiign of what I propofe and have ejfajyed in
this Tre^tife, does not require fo large an Ac-
count of Sounds, I muft be content only to con-
sider fuch Fhmomena as belong properly to
Mufick*
§ i. of MUSIC K. 3
Mufick, or ferve for the better Underftanding of
it. In order to which I flialJ a little further en-
large the preceding general Account of the
Caufe of Sound. And,
Firfi, That Motion is neceffary in the Pro-
duction of Sounds is a Conclufion drawn from
all our Experience. Again, that Motion exifts,
firft among the fmall and infenfible Parts of fucli
Bodies as are Sonorous, or capable of So ni\
excited in them by mutual Coilifion and Percuf-
iion one againft another, which produces that
tremulous Motion fo obfervable in Bodies, efpe-
cially that have a free and clear Sound, as B^lls,
and the Strings of mufical Inftruments $ then,
this Motion is communicated to, or produces a
like Motion in the Air, or fuch Parts of it as
are apt to receive and propagate it t For no Mo-
tion of Bodies at Diftance can affect our Semes,
(or move the Parts of our Bodies) without the
Mediation of other Bodies, which receive thefs
Motions from the Sonorous Body,and communi-
cate them immediately to the Organs of Senfe'j
and no other than a Fluid can reasonably be fbp-
pofed. But we know this alfo by Experience;
for a Bell in the exhaufled Receiver of an Air*
pump can fcarcely be heard, which was loud
enough before the Air was drawn out. In the
loft Place, This Motion muft be communicated
to thofe Parts of the Ear that are the proper
and immediate Inftruments of Hearing. The Me-
chanifm of this noble Organ has ftili great Dif-
ficulties, which all the Induftry of the moft ca-
pable a§d curious Enquirers has not furmonnted s
4 A Treatise Chap. I.
The re are Queftions ftill unfolved about the Ufc
of fome Parts, and perhaps other neceffary Parts
never yet difcovered : But the moft important
Queftion among the Learned is about the laft
and immediate Inftrument of Hearing, or that
Part which laft receives the fonorous Motion,
and finiflies what is neceffary on the Part of
the Organ. Confult thefewith the Philofophers
and Anatomifts \ I fliall only tell you the com-
mon Opinion, in fuch general Terms as my De-
fign permits, thus : Next to the external vifible
Cavity or Paffage into the Ear, there is a Ca-
vity, of another Form, feparate from the former
by a thin Membrane, or Skin, which is called
the Tympan or Drum of the Ear, from the
Refemblance it has to that Inftrument : With-
in the Cavity of this Drum there is always Air,
like that external Air which is the Medium of
Sound. Now, the external Air makes its Im-
preflion firft on the Membrane of the Drum,
and this communicates the Motion to the in-
ternal Air, by which it is again communicated
to other Parts, till it reaches at laft to the au-
ditory Nerve,and there the Senfation is rniifned,
as far as Matter and Motion are concerned ;
and then the Mind^ by the Laws of its Union
with the Body, has that Idea we call Sound. It
is a curious Remark, that there are certain Parts
fitted for the bending and unbending of the
Drum of the Ear, in order, very probably, to
the perceiving Sounds that are raifed at greater
or lefler Diftances, or whofe Motions have dif-
ferent Degrees of Force, like what we are more
fenfible
§3 i/ of MUSIC K. i
fenfible of in the Eye, which by proper Mufcles
(which are Inftruments of Motion) we can move
outwards or inwards, and change the very Fi-
gure of, that we may better perceive very di-
* ftant or near Obje&s. But I have gone far e-
nough in this.
Lest what I have faid of the Caufe of Sound
be too general, particularly with refped: to the
Motion of the fonorous Body, which I call the
original Caufe, let us go a little farther with
it. That Motion in any Body, which is the
immediate Caufe of its founding, may be ow-
ing to two different Caufes ; one is, the mutual
Percnffion betwixt it and another Body, which is
the Cafe of Drums, Bells, and the Strings of.
muiical Inftruments, (jc. Another Caufe is, the
beating or dafhing of the fonorous BoSy and the
Air immediately againft one another,as in all Kind
of Wind-inftruments, Flutes,Trumpets, Hautboys,
&c. Now in all thefe Cafes,the Motion which is
• the Confequence of the mutual Percuffion be-
twixt the whole Bodies, and is the immediate
Caufe of the fonorous Motion which the Air
conveys to our Ears, is an invifible tremulous or
undulating Motion in the fmall and , infenfible
Parts of the Body. To explain this ;
All viiible Bodies are fuppofed to be compo-
fed of a Number of fmalj and infenfible Parts,
which are of the fame Nature in every Body,being
perfectly hard and incompreifible : Of thefe in-
finitely little Bodies are compofed others that
arefomethinggreater,butitiil infenfib!e,and thefe
are different, according to the different Figures
A3 and
6 ^Treatise Chap. I.
and Union of their component Parts: Thefe are
agaili fuppofed to conftitute other Bodies greater,
(which have greater Differences than the laft)
whofe different Combinations do, in the laft
Place, conftitute thofe grofs Bodies that are vifi-
ble and touchable. The firft and fmalleft Parts
are abfolutely hard • the others are compref-
fible, and are united in fuch a Manner, that be-
ing, by a fufficient external Impulfe, compreffed,
they reftore themfelves to their natural, or ordi-
nary, State : This Compreffion therefore hap-
pening upon the Shock or Impulfe made by on©
Body upon another, thefe fmall Parts or Parti-
cles, by their reftitutive Power (which we alfo
call elaftick Faculty) move to and again with
a very great Velocity or Swiftnefs, in a tremu-
lous and undulating Manner, fomething like the
vifible Motions of groffer Springs, as the
Chord of a mufical Inftrument ; and this is what
we may call the Sonorous Motion which is pro-
pagated to the Ear. But obferve that it is the
infenfible Motion of thefe Particles next to the
fmalleft, which is fuppofed to be the immediate
Caufe of Sound ; ' and of thefe, only thofe next
the Surface can communicate with the Air,*
their Motion is performed in very fmall Spaces,
and with extreme Velocity ; the Motion of the
Whole, or of the greater Parts being no further
concerned than as they contribute to the other.
And this is the Hypothecs upon which Monfieur
Per rank of the Royal Society in Fr ajice^exphins
the Nature sindPh^nc?nena of Sound,in his curious
fjreatife upon that Subject, JZJfais de Phjfequei
Tonu
I i- cf MUSIC K. 7
Tom. II. Du Bruit. How this Theory is fup-
ported I fhaJJ briefly fhew, while I confider a
few Applications of it.
Of thofe hard Bodies that found byPercuflion
of others, let us coniider a Bell : Strike it with
any other hard body, and while it founds we
can difcern a fenfible Tremor in the Surface,
which fpreads more fenfibly over the Whole, as
the Shock is greater. This Motion is not only
in the Parts next the Surface, but in all the
Parts thro' the whole Solidity, becaufe we can
perceive it alfo in the inner Surface of the Bell,
which mult be by Communication with thofe
Parts that are immediately touched by the
{hiking Body. And this is proven by the ceafing
of the Sound when the Bell is touched in any
other Part ; for this fhews the eafy and actual
Communication of the Motion. Now this is
plainly a Motion of the feveral fmall and in-
fenfible Parts changing their Situations with
refpecl to one another,which being fo many, and fo
clofely united, we cannot perceive their Motions
feparately and diftincl:ly,but only that Trembling
which we reckon to be the Effecl: of the Con-*
fufion of an infinite Number of little Particles
fo clofely joyned and moving in infinitely fmall
Spaces. Thus far any Bcdy will eafily go with
the Hypotheiis : But Monfieur Pert milt carries it
" farther, and affirms. That that vifible Motion
of the Parts is no otherwife the Caufe of the
Sound, than as it caufes the invifible Motion of
the yet fmaller Parts, (which he calls Particles^
to (diilinguifli them from the other which he
A 4 . calls
8 A Treatise Chap. I.
calls Parts., the leaft of all being with him Cor~
pufcks.) And this he endeavours to prove by
other Examples, as. of Chords and Wind-inftnn
ments. Let us confider them.
Take a Chord orStringofaMuficallnftrument,
ftretched to a fufficient Degree for Sounding; when
it is fixt at both Ends, we make it found by draw-
ing the Chord from its ftraight Pofition,and then
letting it go ; (which has the fame Effect as what
we properly call Perculfion ) the Parts by this
drawing, whereby the Whole is lengthned, be-^
ing put out of their natural State, or that which
they had in the ftraight Line, do by their E-
lafticity reftore themfelves, which caufes that
vibratory Motion of the Whole, whereby it
moves to and again beyond the ftraight Line,
in Vibrations gradually fmaller, till the Motion
ceafe, and the Chord recover its former Porti-
on. Now the fhorter the Chord is, and the
more it is ftretched in the ftraight Line, the
quicker thefe Vibrations are : But however quick
they are, Monfieur Perrauk denies them to be
the immediate Caufe of the Sound ; becaufe,
fays he, in a very long Chord, and not very
fmall, ftretched only fo far as that it may give
a diftincl: Sound, we can perceive with our Eye,
befides the Vibrations of the whole Chord,
a more confufed Tremor of the Parts, which is .
more difeernible towards the Middle of the
Chord, where the Parts vibrate in greater Spaces
in the Motion of the Whole ; this laft Moti-
tion of the Parts which is caufed by the firft
yibrations of the Whole, does again eceafion a
Motion
§. i. of MUSIC K. i
Motion in the leffer Parts or Particles, which
is the immediate Caufe of the Sound. And
this he endeavours to confirm by this Expe-
riment, viz. Take a long Chord (he fays he made
it with one of 30 Foot) and make it found; then
wait till the Sound quite ceafe, and then alio
the vifible Undulations of the whole Chord will
ceafe: If immediately upon this eeafing of the
Sound, you approach the Chord very foftly with
the Nail of your Finger, you'll perceive a tre-
mulous Motion in it, which is the remaining
fmall Vibrations of the whole Chord, and of
the Parts caufed by the Vibrations of the
Whole. Now thefe Vibrations of the Parts
are not the immediate Caufe of Sound; elfehow
comes it that while they are yet in Motion they
raife no Sound ? The Anfvver perhaps is this.
That the Motion is become too weak to make
the Sound to be heard at any great Diftance,
which might be heard were the Tympan of the
Ear as near as the Nail of the Finger, by which
we perceive the Motion. But to carry off this,
Mx.Perrault fays, That as foon as this fmall Mo-
tion is perceived, we (hall hear it found; which
is not occasioned by renewing or augmenting'
the greater Vibrations, becaufe the Finger is
not fuppofed to ftrike againft. the Chord, but
this againft the Finger, which ought rather to
ftop that Motion ; the Caufe of this renewed
Sound therefore is propably, That this weak
Motion of the Parts, which is not fufficient to
move the Particles ( whofe Motion is the Firft
|hat eeafes) receives fome Afliftance from the
Rafting
io ^Treatise Chap, fc
dafliing againft the Nail, whereby they are en-
abled to give the Particles that Motion which
is neceffary for producing the Sound. But left
it fhould ftill be thought, that this Encounter
with the Nail may as well be fuppofed to in-*
creafe the Motion of the Parts to a Degree fit
for founding, as to make them capable of moving
the Particles ; we may confider, That the Par-
ticles being at Reft in the Parts^and having each a
common Motion with the whole Part, may very
eaftly be fuppofed to receive a proper and partis
cular Motion by that Shock,- in the fame Man-
ner that Bodies which are relatively at Reft in,
a Ship, will be fhaked and moved by the Shock
of the Ship againft any Body th^t can any thing
confiderabiy oppofe its Motion. Now for as,
fimple as this Experiment appears to be, I am
afraid it cannot be fo eafily made as to give
perfect Satisfaction, becaufe we can hardly touch
a String with our Nail but it will found.
But Mr. Perrault finifhes the Proof of his Hypo-
thecs by the Phenomena of Wind-inftruments.
Take, for example, a Flute ; we make it found
by blowing into a long, broad, and thin Canal,
which ^conveys the Air thrown out of the Lungs,
till 'tis dafhed againft that thin folidPart which
we call the Tongue, or Wind-cutter, that is
oppofite to the lower Orifice of the forefaid Ca-
nal j by which Means the Particles of that
Tongue are compreffed, and by their reftitutive
Motion they communicate to the Air a Sonorous
Motion, which being immediately thrown a-?
gainft the inner concave Surface of the Flute,
and
§. i. ef MUSIC K. it
and moving its Particles, the Motion commu-
nicated to the Air, by all thefe Particles both
of the Tongue and inner Surface, makes up the
whole Sound of the Flute. #
Now to prove that only^ the very fmall
Particles of the inner Surface and Edge of
the Tongue are concerned in the Sound of
the Flute, we muft confider, That Flutes
of different Matter, as Metal, Wood, or
Bone, being of the fame Length and Bore, have
none, or very little fenfible Difference in their
Sound ; nor is this fenfibly altered by the diffe*
rent Thicknefs of the Flute betwixt the outer
and inner Surface; nor in the laft place, is the
Sound any way changed by touching the Flute,
even tho' it be hard preflfed, as it always hap-
pens in Bells and other hard Bodies that found
by mutual Percuffion. All this Mr. Perrault
accounts for by his Hypothecs, thus: He tells
us, That as the Corpufcles are the fame in
all Bodies, the Particles which they imme-
diately conftitute, have very fmall Differences
in their Nature and Form ; and that the fpe-
cifick Differences of vifible Bodies, depend on
the Differences of the Parts made up of thefe
Particles, and the various Connections of theie
Parts, which make them capable of different
Modifications of Motion. Now,hard Bodies that
found by mutual Percuffion one againft another,
owe their founding to the Vibrations of all their
Parts, and by thele to the infenfible Motions
of their Particles -> but according to the Diffe-
rences of the Parts and their Connections, which
make
\i A Treatise Chap. I,
make them, either Silver, or Brafs, or Wood,
Cjc. fo are the Differences of their Sounds. But
in Wind-inftruments (for example. Flutes) as
there are no fuck remarkable Differences anfwer-
ing to their Master, their Sound can only be
owing to the infenfible Motion of the Particles
of the Surface ; for thefe being very little diffe-
rent in all Bodies, if we fuppofe the Sound is
owing to their Motions only, it can have none,
or very fmall Differences : And becaufe we find
this true in Fa<5t,it makes the Hypothefis extreme-
ly probable. I have never indeed feen Flutes
of any Matter but Wood, except of the fmall
Kind we call Flageolets, of which I have feen
Ivoryo nes, whofe Sound has no remarkable Di£
ference from a wooden one ; and therefore I
imuft leave fo much of this Proof upon Monfieur
Perranlfs Credit. As to the other Part, which
is no lefs considerable, That no Compreffion of
the Flute can fer.Lbly change its Sound, 'tis cer-
tain, and every Body can eafily try it. To
. which we may .add. That Flutes of different-
Matter are founded with equal Eafe, which
could not well be if their Parts were to be
moved; for in different Bodies thefe are different^
ly moveable. But I muft make an End of this
Part, in which I think it is made plain enough.
That the Motion of a Body which caufes a
founding Motion in the Air, is not any Moti-
on which we can poifibly give to the whole
Body, wherein all the Parts are moved in one
common Direclion and Velocity ; but it is the
Motion of the feveral fmall and undiflinguifhable
Parts,
§. i. of MUSIC K. 13
Parts, which being compreffed by an external
Force, do, by their elaftick Power,reftore them-
felves, each by a Motion particular and proper
to it felf. But whether you'll diftinguifh Parts
and Particles as Mr. Perrault does , I leave to
your felves, my Defign not requiring any accu-
rate Determination of this Matter. And now to
come nearer to our Subject, I fhall next confider
the Differences and Affections of Sounds that are
any way concerned in Mufich
SO UN I) S are as various, or have as many
Differences, as the infinite Variety of Things
that concur in their Production ; which may be
reduced to thefe general Heads : ij?, The
Quantity, Conftitution, and Figure of the fono-
rous Body ; with the Manner of Percuflion, and
the confequent Velocity of the Vibrations of the
Parts of the Body ana the Air $ alfo their E-
quality and Uniformity, or Inequality and Irre-
gularnefs. idlj^ The Conftitution and State of
the fluid Medium through which the Motion is
propagated. zdl)\ The Difpofition of the Ear
that receives that Motion. And, ^thly^ The
Diftance of the Ear from the fonorous Body. To
which we may add, laftly^ the Consideration of
the Obftacles that interpofe betwixt the fonorous
Body and the Ear; with other adjacent Bodies
that, receiving an Imprelfion from the Fluid fo
moved, react upon it, and give new Modifica-
tion to the Motion, and confequently to the
Sound. Upon all thefe do our diiferent Percep-
tions of Sound depend.
The
14 A Treatise Chap. L
The Variety and Differences of Sounds,
owing to the various Degrees and Combinations
of the Conditions mentioned, are innumerable*
but to our prefentDefign we are to coniider the
following Diftin&ions.
I. SOUNDS^ come under a fpecifick Diftin*
£tion, according to the Kinds of Bodies from which
they proceed : Thus, Metal is eafily diftin^
guifhed from other Bodies by the Sound ; and a*
mong Metals there is great difference of Sounds,
as is difcernible, for Example, Betwixt Gold,
Silver, and Brafs. And for the Purpofe in
hand, a raoft notable Difference is that of ftring-
ed and Wind-inftruments ofMufick, of which
there are alfo Subdiviflons : Thefe Differences
depend, as has been faid, upon the different
Conftitutions of thefe Bodies ; but they are not
ftri&ly within the Confederation of Mufick, not
the Mathematical Part of it at leaft, tho* they
may be brought into the Practical $ of whicn
afterwards.
II. Experience teaches us, That fome
Sounds can be heard, by the fame Ear, at great-
er Diftances than others ; and when we are at
the fame Dilfance from two Sounds, I mean
from the fonorous Body or the Place where
the Sound firft rifes, we can determine (for we
learn it by Experience and Obfervation) which
of the Two will be heard fartheftrBy this Com-
parifon we have the Idea of a Difference whole
oppofite Terms are called L UD and LOW,
(or fir ong and weah) This Difference depends
both upon the Nature of different Bodies, and
UpOt*
§ x. of MUSIC K. i,
upon other accidental Circumftances, fuch
as their Figure' $ or the different Force in thfc
Percuflfion ; and frequently upon the Nature df
the circumjacent Bodies, that contribute to
the ftrengthning of the Sound, that is a Con-
junction of feveral Sounds fo united as to appear
only as one Sound : But as the Union of feve-
ral Sounds gives Occafion to another Diftinclion,
it fhall be confidered again, and we have on-
ly to obferve here that it is always the Caufe of
Loudnefs -, yet this Difference belongs not ftricl-
Iy to the Theory of Mufick, tho' it is brought
into the Practice, as that in the Firft Article.
III. There is an Affection or Property of Sound,
Whereby it is diftinguifhed into Acute, Jharp or
high; and Grave, flat or low. The Idea of
this Difference you'll get by comparing feveral
Sounds or Notes of a mufical Inftrument, or of a
human Voice ringing. Obferve the Term, Low,
is fometimes oppofcd to Loud, and fometimes
to acute, which yet are very different Things :
Loudnefs is very well meafured by the Diftancfe
or Sphere of Audibility, which makes the Na-
tion of it very clear. Acutenef is fo far diffe-
rent, that a Voice or Sound may afcend or rife
in Degree of Acutenefs, and yet lofe nothing of
its Loudnefs, which can eafily be demonftrated
upon any Inftrument, or even in the Voice ; and
particularly if we compare the Voice of a Boy
and a Man.
This Relation of Acutenef s and Gravity is
one of the principal Things concerned in Mu-
|ick 3 the ISature of which (hail be particularly
i6 A Treatise. Chap. I.
considered afterwards ,• and I (hall here obferve
that it depends altogether upon the Nature of
the ibnorous Body it felf, and the particular Fi-
gure and Quantity of it 5 and in fome Cafes up-
on the Part of the Body where it is ftruck. So
that, for Example, the Sounds of two Bells of
different Metals, and the fame Shape and Di~
menfions, being ftruck in the fame Place, will
differ as to Acutenefs and Gravity ,• and two
Bells of the fame Metal will differ in Acutenefs^
if they differ in Shape or in Magnitude, or be
ftruck in different Parts : So in Chords, all o-
ther Things being equal, if they differ either in
Matter, or Dimenfions, or the Degree of Ten-
fion, as being ftretched by different Weights,
they will alfo differ in Acutenefs.
But we muft carefully remark^That Acute-
nefs and Gravity alfo JLoudnefs and Lownefs
are but relative Things ,- fo that we cannot call
any Sound acute or loud, but with reipeel: to
another which is grave or low in reference to
the former ; and therefore the fame Sound may
be acute or grave ', alfo loud or low in different
Refpeds. Again, Thefe Relations are to be
found not only between the Sounds of different
Bodies, but alfo between different Sounds of the
fame Body ; for different Force in the Percu£
fion. will caufe a louder or lower Sound, and
ftriking the Body in different Parts will make
an acnter or graver Sound, ' as we have
remarkably demonftrated in a Bell, which as
the Stroke is greater gives a greater or louder
Sound, and being ftruck nearer the open Encf,
gives
§. i. of MU SICK. i7
gives the graver Sound. How tliefe Degrees
are meafured, we fliall learn again, only mind
that thefe Degrees of Acutenefs and Gravity
are alfo called different and difUnguifhablc Tones
or Tunes of a Voice or Sound j fo we fay one
Sound is in Tune with another when they are in
the fame Degree i Acute and Grave being but
Relations, We apply the Name oiTune to them
both;, to exprefs fomething that's conftant and
abfolute which is the Ground of the Relation •
in like manner as we apply the Name Magni*
tude both to the Things we call Great andJOittle^
which are but relative Idea's : Each of them have a
certain Magnitude, but only one of them is great
and the other little when they are compared; fo
of Two Sounds each has a Certain Tune Jout only
one is acute and tile other grave in Comparifon.
IV* T h e r e is a Diftinction of Sounds,\vhere^
by they are denominated long or Jhort - y which
relates to the Duration^ or continued, and fen-
fibly uninterrupted Exiftence of the Sound. This
is a Thing of very great Importance in Muficks
but to know how far, and in what refpect it
belongs to it 5 we muft diftinguifh betwixt the
natural and artificial Duration of Sound. I call
that the natural Duration or Continuity of
Sounds which is lefsormore in different Bodies,
owing to their different Conftitutions, whereby
one retains the Motion once received longer
than another does ; and confequently the Sound
continues longer ( tho' gradually weaker ) after
the external Impulfe ceafes ; fo Bells of diffe-
rent Metals^ all other Things being equal and
B alike
18 ^Treatise Chap. I;
alike,have differentContinuity of Sound after the
Stroke : And the fame is very remarkable in
Strings of different Matter : There is too a Dif-
ference in the fame Bell or String, according to
the Force of the Percuffion. This Continuity
is " fometimes owing to the fudden Reflection
of the Sound from the Surface of neighbouring
Bodies; which is, not fo properly the fame Sound
continued, as a new Sound fucceeding the Firfl
fo quickly as to appear to be only its Continu-
ation : But this Duration of Sound does not
properly belong to Mufick, wherefore let us
confider the other. The artificial Continuity
of Sound is, that which depends upon the conti-
nued Impulfe of the efficient Caufe upon the
funorous Body for a longer or iliorter Time.
Such are the Notes of a Voice,or any Wind-inftru-
ment, which are longer or fhorter as we conti-
nue to blow into them ; or, the Notes of a Vio-
lin and all ftring'd Inftruments that are ftruck
with a Bow, whofe Notes are made longer or
fhorter by Strokes of different lengths or Quick-
nefs of Motion^ for a long Stroke, if it is quick-
ly drawn, may make a Iliorter Note than afhort
Stroke drawn flowly. Now this kind of Conti-
nuity is properly the SuccefTi on of feveral Sounds,
or the Effect of feveral diftincl: Strokes,or repeated
Impulfes, upon the fonorous Body, fo quick that
Wq judge it to be one continued Sound, efpeci-
ally if it is continued in one Degree of Strength
and Loudnefs; but it muft alfo be continued in
one Degree of Tune, elfe it cannot be called
.one Note in Mufick. And this leads me natural-
h
§ t, of MUSICK. t 9
ly to confider the very old and notable Di-
ftincTion of a twofold Motion of Sound.,
thus.
Sound may move thro' various Degrees of
Acutenefs in a continual Flux, fo as not to reft
On any Degree for any affignable, or at leaft fen-
fible Time $ which the Ancients called the con-
tinuous Motion of Sound, proper only to Speak-
ing and Converfation. Or, 2do* it may pafs
from Degree to Degree, and make a fenfible
Stand at every Pitch, fo as every Degree (hall
be diftindj this they called the difcrete Or dif-
continued Motion of Sound, proper only to Mufick
or Singing. But that there may be no Obfcurity
here, confider , That as the Idea's of Motion ana
Diftance are infeparably conne&ed, fo they be-
long in a proper Senfe to Bodies and Space §
and whatever other Thing they are applied to,
it is in a figurative and metaphorical Senfe^ as
here to Sounds; yet the Application is very in-
telligible, as I (hall explain it. Voice or Sound
is confidered as one individual' Being, all other
Differences being neglected except that of A-*
cutenefs and Gravity ', which is not confidered
as conftituting different Sounds^ but different
States of the fame Sound; which is eafy to con-
ceive : And fo the feveral Degrees or Pitches
of Tune^ are confidered as feveral Places in
which a Voice may exift. And when we hear
a Sound fucceffively exifting in different Degrees
of 7\me 9 we conceive the Voice to have moved
from the one Place to the other j and then 'tis
eafy to conceive a Kind of Diftance between the
B 2 two
20 A Treatise Chap. I.
two Degrees or Places -, for as Bodies are faid
to be diftant, between which other Bodies may
be placed, fo two Sounds are faid to be at Di-
ftance, with refpecl: to Tune^ between which
other Degrees may be conceived, that lhall be
-acute with refpeci to the one, and grave with
refpecl: to the other. But when the Voice con-
-tinues in one Pitch, tho' there may be many
Interruptions and fenfible Refts whereby the
Sound doth end and begin again, yet there is-
no Motion in that Cafe, the Voice being all
the Time in one Place. Now this Motion, in
a {imple and proper Senfe, is nothing elfe but
the fuccefnve Exiftence of feveral Sounds differ-
ing in Tune. When the fuccefnve Degrees are
fo near,that like the Colours of a Rainbow, they
are as it were loft in one another, fo that in any
fenfible Diftance there is an indefinite Number
of Degrees, fuch kind of Succelfion is of no ufe
in Mufick ; but when it is fuch that the Ear is
Judge of every fingle Difference, and can com-
pare feveral Differences, and apply fome known
Meafure to them, there the Objecl of Mufick
does exift ; or when there is a Succeflion of feveral
Sounds diftincl: by fenfible Refts,tho' all in the lame
Tune, fuch a Succeffion belongs alfo to Mufick.
From this twofold Motion explain'd,
we fee a twofold Continuity of Sound,
both fubjecl to certain and determinate Mealures
of Duration > the one is that arifing from the
continuous Motion mentioned, which has no-
thing to do in Mufick ; the other is the Con-
tinuity or uninterrupted Exjftence of Sound in
one
§. i. of MUSIC K, it
one Degree of Tune, The Differences of
Sounds in this refpect, or the various Meafures
of long and Jhort, or, ( which is the fame, at
leaft a Confequence) /«?£/> and/Zoo?, in the fuc-
ceflive Degrees of Sound, while it moves in the
fecond Manner, make a principal and necelfary
Ingredient in Mufick; whofe Effect is not infe-
rior to any other Thing concerned in the
Practice ; and is what deferves to be very parti-
cularly confidered, tho' indeed it is not brought
under fo regular and determinate Rules as the
Differences of Tune,
V. Sounds are either Jimple or compound ;
but there is a twofold Simplicity and Compofi-
tion to be confidered here ; the Firft is the fame
with what we explain'd in the laft Article,and re-
lates to the Number of fucceffive Vibrations of the
Parts of the fonorous Body, and of the Air, which
come fo faft upon the Ear that we judge them
all to be one continued Sound, tho' it is really
a Compofition of feveral Sounds of fhorter Du-
ration. And our judging it to be one, is Very
well compared to the Judgment we make of
that apparent Circle of Fire, caufed by putting
the fired End of a Stick into a very quick cir-
cular Motion j for iuppofe the End of the Stick
in any Point of that Circle which it actually de-
fcribes, the Idea we receive of it there conti-
nues till the Impreffion is renewed by the fudden
Return ; and this being true of every Point, we
muft have the Idea of a Circle of Fire ; the on-
ly Difference is, that the End of the Stick has
a&ually exifted in every Point of the Circle,
B 3 whereas
%t A Treatise Chap. I.
whereas the Sound has had Interruptions, tho'
infenfible to us becaufe of their quick Succeffion ;
but the Things we compare are, the Succeflion
of the Sounds making a fenfible Continuity with
refpecl: to Time, and the Succeflion of the End
of the Stick in every Point of the Circle after a
whole Revolution ; for 'tis by this we judge it
to be a Circle, making a Continuity with refpecl
to Space. The Author of the Ehcidationes
Phyftc<£ upon B" Cartes Mufick, illuftrates it in
this Manner, fays he, As ftanding Corns are
bended by one Blaft of Wind, and before they
can recover themfelves the Wind has repeate4
the B/aft, fo that the Corn's ftanding in the fame
inclined Pofition for a certain Time, feems to
be the Effecl: of one (ingle Action of the Wind,
which is truly owing to feveral diflincl Opera-
tions ; in like Manner the fmall Branches (capil-
lamenta) of the auditory Nerve, refembling fo
many Stalks of Corn, being moved by one Vi-
bration of the Air,and this repeated before the
Nerve can recover its Situation,gives Occafion to
theMind to judge the wholeEffecl: to be oneSound.
The Nature of this kind of Compofition being
fo far explain'd, we are next to confider what
Simplicity in this Senfe is - 3 and I think it muft
be the Effect of one fmgle Vibration, or as
many Vibrations as are neceffary to raife in us
the Idea of Sound > but perhaps it may be a
Queftion, Whether we ever have, or if we can
raife fuch an Idea of Sound : There may be al-
fo another Queftion,Whether any Idea ofSound
can exift in the Mind for an indivifible- Space
' ; ' tf of
§ i. of MUSIC K. 23
of Time ; the Reafon of this Queftion is, That
if every Sound exifts for a finite Time, it can
be divided into Parts of a fhorter Duration, and
then there is no fuch Thing as an abfolute
Simplicity of this Kind, unlefs we take the No-
tion of it from the Action of the external Caufe
of Sound, viz. the Number of Vibrations necef-
fary to make Sound actually exift, without con-
sidering how long it exifts ,• but as it is not pro-
bable that we can ever a&ually produce this,
i. e. put a Body in a founding Motion, and flop
it precifely when there are as many Vibrations
finifhed as are abfolutely neceffary to make
Sound, we muft reckon the Simplicity of Sound,
confidered in this Manner, and with refpecl: to
Practice, a relative Thing; that being 'only
fimple to us which is the moft fimple, either
with refpecl: to the Duration or the Caufe,
that we ever hear ; But whether we confider fb
in the repeated Action of the Caufe or the con-
fequent Duration, which is the Subject of the
laft Article, there is ftill another Simplicity and
Compofition of Sounds very different from that,
and of gre it Importance in Mufick, which I fhall
next explain.
A fimple Sound is the Product of one Voice
or individual Body, as the Sound of one Flute
or one Man's Voice. A compound Sound con-
fifts of the Sounds of feveral diftinft Voices or
Bodies all united in the fame individual Time
and Meafure of Duration, i. e. all ftriking the
Ear together, whatever their other Differences
may be. But we muft here diftinguifh a natural
B 4 and
*4 ^Treatise Chap, h
and artificial Comfqfition;to underftand this, re-
member, That the Air being put into Motion by
any Body, communicates that Motion to other
Bodies; the naturalCompofition of Sounds is there-
f ore,that which proceeds from the manifold Re-
flexions of the Firft Sound, or that of the Body
which firft communicates founding Motion to
the Air, as the Flute or Violin in one's Hand $
thefe Reflexions, being many, according to the
Circumftances of the Place, or the Number,
Nature, and Situations of the circumjacent Bo-
dies, make Sounds more or lefs compound.
This is a Thing we know by common Expe-
rience ; we can have a hundred Proofs of it e-
very Day by ringing, or founding any mufical
Inftrument in different Places, either in the
Fields or within Doors; but thefe Reflexions
muft be fuch as returning very fuddenly don't
produce what we call an Eccho^ and have only
this Effect, to increafe the Sound, and make
an agreeable Refonance ; but (till in the fame
Tune with the original Note j or, if it be a
Compofition of different Degrees of Tune, they
are fuch as mix and unite, fo that the Whole
agrees with that Note. But this Compofition
is not under Rules of Art ; for tho' we learn by
Experience how to difpofe thefe Circumftances
that they may produce the defired Effect, yet
we neither know the Number or different Tunes
of the Sounds that enter into this Compofition y
and therefore they come not under the Mufi-
ciaa's Direction in what is hereafter called the
Compofition of Mufeck > his Care being only a-
bout
§. i, tfMUSICK. 15
bout the artificial Comfofition, or that Mixture
of fevera.1 Sounds, which being made by Art,
are feparable and diftinguifhable one from ano-
ther. So the diftincl; Sounds of feveral Voices
or Inftruments, or feveral Notes of the fame In-
ftrument, are called fimple Sounds, in Diftindtion
from the arti^ciedCompoJition,m which toanfwer
the End of Mufick, the Simples muft have fuck
an Agreement in all Relations, but principally
and above all in Acutenefs and Gravity, that
the Ear may receive the Mixture with Pleafure*
VI. There remains another Distinction of
Sounds neceffaryto be confidered, whereby they
are faid to befmooth and evenly,, or rough
andharJh;Sil{o clear or blunt , hoar/'e and obt life \
the Idea's of thefe Differences mult be fought
from Obfervations ; as to the Caufe of them,
they depend upon the Difpofition and State of the
fonorous Body, or the Circumftances of the
Place. Smooth and rough Sounds depend upon
the Body principally 5 We have a notable Ex-
ample of a rough and harjh Sound in Strings
that are unevenly and not of the fame Consti-
tution and Dimenfion throughout 5 and for this
Reafon that their Sounds are very grating, they
are called falfe Strings. I will let you in few Words
hear how Monfieur Perrault accounts for this.
He affirms that there is no fuch Thing as a limplo
Sound, and that the Sound of the fame Bell or
Chord is a Compound of the Sounds of the fe-
veral Parts of it ; fo that where the Parts are
homogeneous, and the Dimensions or Figure u-
uiform, there is always fuch a perfect Union
and
%6 ./f Treatise Chap. L
and Mixture of all thefe Sounds that make*
one uniform, fmooth and evenly Sound • and
the contrary produces Harfhnefs j for the
Likenefs of Parts and Figure makes an Uniformi-
ty of Vibrations, whereby a great Number of
fimilar and coincident Motions confpire to for-
tify and improve each other mutually, and u-
nite for the more effectual Production of the
fame Effect. He proves his Hypothefis by the
Phenomena of a RelJ, which differs in Tone ac-
cording to the Part you ftrike, and yet ftrike it
any where there is a Motion over all the Parts ;
he confiders therefore the Bell as compofed of an
infinite Number of Rings, which according to
their different Dimenfions have different Tones;
as Chords of different Lengths h&ve(c<eteris pa-
ribus) and when it is (truck, the Vibrations of
the Parts immediately (truck fpecify the Tone y
being fupported by a fufficient Number of con-
ionant Tones in other Parts : And to confirm
this, he relates a very remarkable Thing,- He
fays, He happen'd in a Place where a Bell foun-
ded a Fifth acuter than the Tone it ufed to
give in other Places ; whi6h in all Probability,
fays he, was owing to the accidental Difpofition
of the Place, that wasjurnifhed with fuch an
Adjuftment for reflecting that particular Tone
with Force, and fo unfit for reflecting others,
that it absolutely prevailed and determined the
Concord and total Sound to the Tone of that
Fifth. If we confider the Sound of a Violin,
and all (tring'd Inftruments, we have a plain
Demcnltration that every Note. is the Effect of
fever.
§ i. of MUSIC K. x 7
ieveral more fimple Sounds - 3 for there is not
only the Sound refulting from, the Motion of
the String, but alfo that of the Motion of the
Parts of the Inftrument ,- that this has a very
confiderable Effecl: in the total Sound is cer-
tain, becaufe we are very fenfible of the tre-
mulous Motion of the Parts of the Violin, and
efpecially becaufe the fame String upon different
Violins founds very differently, which can be
for no other Reafon but the different. Conftitu-
tion of the Parts of thefe Inftruments, which
being moved by Communication with the String
increafe the Sound, and make it more or lefs
agreeable, according to their different Natures :•
But Perraitlt affirms the fame of every String
in it felf without confidering the Inftrument ;
he fays. Every Part of the String has its parti-
cular Vibrations different from the grofs and
fenfible Vibrations of the Whole, and thefe are
the Caufes of different Motions ( and Sounds )
in the Particles '; which being mix'd and unite,
as was faid of the Sounds that compofe the
total Sound of a Bell, make an uniform and
evenly Compofition, wherein not only one
Tone prevails, but the Mixture is fmooth and
agreeable; but when the Parts are unevenly and
irregularly conftitute, the Sound is harfh and
theString from that called falfe. And therefore
fuch a String, or other Body having the like
Fault, has no certain and diftincl: Tone, being
a Compofition of feveral Tones that don't u-
nite and mix fo as to have one Predominant
that fpecifies the total Tone.
Again
z% A Treatise Chap. I.
Again for clear or hoarfe Sounds, they
depend upon Circumftances that are accidental
to the fonorous Body j fo a Man's Voice, or
the Sound of an Inftrument will be hollow and
hoarfe, if it is raifed within an empty Hogfhead,
which is clear and bright out of it ,• the Reafon
is very plainfy the Mixture of other and dif-
ferent Sounds raifed by Reflexion, that corrupt
and change the Species of the primitive and di-
rect Sound.
Now that Sounds may be fit for obtaining
the End of Mufick they ought to be/mooth ancf
clear ; efpecially the Firft, becaufe if they have
.not one certain and difcernible jTone^ capable
of being compared to others, and ftanding to
them in a certain Relation of jlcutenefs^
whofe Differences the Ear may be able to
judge of and meafure, they cannot poflibly an-
f wer the End of Mufick, and therefore, are no
Part of the Object of it,
But there are alfo Sounds which have a
certain Tbne, yet being exceflive either in A-
cutenefs or Gravity, bear not that juft Propor-
tion to the Capacity of the Organs of Hearing,
as to afford agreeable Senfations. Upon the
Whole then we (hall call that harmonick or
mufical Sounds which being clear and evenly
is agreeable to the Ear^ and gives a certain and
difcernible Tune ( hence alfo called tunable
Sound) which is the Subject of the whole Theo^
ry of Harmony.
Thus we have considered the Properties
and Affections of Sound that are any way ne-
ceffary
§ 2. of MUSIC K. ip
ceffary to the Subject in hand ; and of all
the Things mentioned, the Relation of Acute-
nefs and Gravity, or the Tune of Sounds, is the
principal Ingredient in Mufiek > the Diftinctnefs
and Determinatenefs of which Relation gives
found the Denomination of harmonical or
tnufical'.Next to which are the various Meafures
of Duration. There is nothing in Sounds with-
out thefe that can make Mufiek; sl juft Theory
whereof abftra&s from all other Things, to con-
lider the Relations of Sounds in the Meafures
of Tune and Duration ; tho' indeed in the
Practice other Differences are confidered ( of
which fomething more may be faid after-
wards ) but they are fo little, compared to the
other Two, and under fo very general and un-
certain Theory, that I don't find they have e-
ver been brought into the Definition of Mu-
fick.
(j 2. Containing the Definition and Divifion of
Mufiek.
WE may from what is already faid affirm;
That Mufiek has for its Object, in gene-
ral, Sound; and particularly, Sounds confidered
in their Relations of Tune and Duration, as
under that Formality they are capable of affor-
ding agreeable Senfations. I fhall therefore de-
fine Music k, ^Science that teaches hozo
S o v n d s, under certain Meafures o/Tune
and
30 ^Treatis£ Chap. f.
an d Time, may be produced'; and fo ordered
-or difpofed) as in C o n s o n a n c e ( i. e. joynt
founding ) or Succession^ or both^ they may
raife agreeable Senfations<
Pleasure, I have faid, is the immediate
End of Mufick i I fuppofe it therefore as a Prin-
ciple^ That the Objects propofed are capable,
being duly applied, to affect the Mind agreeably;
nor is it a precarious Principle ,- Experience
proves^ and we know by the infallible Teftimo-
ny of our Senfes 3 that fome Jimple Sounds
fucceed others upon the Ear with a politive
Pleafure, others difagreeably ; according to cer-
tain Relations of Tune and Time -, and fome
compound Sounds are agreeable, others offenfive
to the Ear ; and that there are Degrees and
Variety in this Pleafure, according to the va-*
rious Meafures of thefe Relations. For what
Pretences are made of the Application of Mu-
fick to fome other Purpofes than mere Pleafure
or Recreation, as thefe are obtain 'd chiefly by
Means of that Pleafure, they cannot be called
the immediate End of it.
From the Definition given, we have the
Science divided into thefe two general Parts.
Firftt The Knowledge of the Materia Mu-
sic a, or, how to produce Sounds, in fuch re-
lations of Tune and Time as fhall be agreeable
in Confonance or Succeffion^ or both. I don't
mean the actual producing of thefe Sounds by
an Inftrument or Voice, which is merely the
mechanical or effective Part $ But the Know-
ledge of the various Relations of Tune- andTme*
whic4
§2. of MUSIC K. 31
which are the effential Principles out of which
the Pleafure fought arifes,and upon which it de-
pends. This is the pure fpeculatwe Part of
Mufich Second How thefe Principles are to
be applied; or> how Sounds, in the Relations
that belong to Mufick(as thefe ate determined
in the Firft Part ) may be ordered, and varioufc
ly put together in Succeffion and Confonance fo
as to anfwer the End ; which Part we rightly
call The Art of Composition ; and it is
properly the practical Part of Mufich
Some have added a Third Part,#z.s.The Know-'
ledge of Instruments ; but as this depends
altogether upon the Firft, and is only an Appli-
cation orExprelfion of it,it could never be brought
regularly into the Definition; and fo can be no
Part of the Divifion of the Science; yet may it
deferve to be treated of, as a Consequent or
Dependent of it, and necelfary to be under-
ftood for the effective Part. As this has no
Share in my Defign, I /hall detain you but
while I fay, in a few Words, what I think fuch
a Treatife fhould contain. And imo 3 There
fhould be a Theory of Inftruments^ giving an
Account of their Frame and Conftru&ion, par-
ticularly, how, fuppofing them completely pro-
vided of all their y2pparatus£&c\i contains in it
the Principles of Mufick i. e. how the feveral
Degrees of Tune pertaining to Mufick are to
be found upon the Inftruments. The SecondPart
fiiould contain the Practice of Inftruments, in fuch
Directions as might be helpful for the dextrous and
nice handling of them,orthe elegant Performance
$i A Treatise Chap; %
oi Mufick i And here might be annex'd Rules
ior the right Ufe of the Voicei But after all, 1
believe thefeThings will before fuccefsfulfy done
by a living Inftru&or, I mean a skilful and ex-
perienced Matter, with the Ufe of his Voice or
Inftrument ; tho' I doubt not fuch might help
us too by Rules ; but I have done with this.
You muft next obferve with me. That as
the Art of common Writing is altogether difc
tinct from the Sciences to which it is fubfervient
by preferving what would otherwife be loft, and
communicating Thoughts at Diftance; fo there
is an Art of Writing proper to Mufickj which
teaches how, by a fit and convenient Way of re-
prefenting all theDegrees and Meafures of Sound,
efficient for directing in the executive Part one
who underftands how to ufe his Voice or In-
ftrument: The Artift when he has invented a
Compcfitionanfwering the Principles and End of
Mufick, may preferve it for his own Ufe, or
communicate it to another prelent or abfent.
To this I have very juftjy given a Place in the
following Work, as it is a Thing of a general
Concern to Mufick^ tho' no Part of the Science,
and merely a Handmaid to the Practice | and
particularly as the Knowledge of it is neceffary for
carrying on my Defign. I now return to the Di-
vifion above made, which I (hall follow in ex-
plaining this Science.
The Firft general Branch of this Subject
which is the contemplative Part, divides natu-
rally into thefe. Fir ft,the Knowledge of the Re-
lations and Meafures oiTune. And Secondly \ of
Time*
§. i. of MUSIC K. ff
Time. The Firft is properly what the Ancients
called Harmonica, or the Do&rine of Har-
mony 'in Sounds - y becaufe it contains an Expli-
cation of the Grounds,, with the various Mea-
Hires and Degrees of the Agreement ( Har-
mony) of Sounds in refpecl of their Tune. The
other they called Rythmica^ becaufe it treats
of the Numbers of Sounds or Notes with re-
fpecl: to Time^ containing an Explication of the
Meafures of long and floor £, or Jivift and Jlozti^
in the Succeffion of Sounds.
The Second general Branch, which is the
Practical Part, as naturally divides into
Two Parts anfwering to the Parts of the Firft :
That which anfwers to the Harmonica^ the
Ancients called Melopxia ; becaufe it contains
the Rules of making Songs with refpe& to
Tune and Harmony of Sounds ; tho' indeed we
have no Ground to believe that the Ancients
had any Thing like Compofition in Parts*
That which anfwers to the Rythmica^ they
called Rythmopceia, containing the Rules con-
cerning the Application of the Numbers and
Time. I fhall proceed according to this natu-
ral Division, and fo the Theory is to be firft
handled,
C CHA P.
34 -^Treatise Chap. II,
CHAP. II.
Of 'Tune, or the Relation of Acutenefs and
Gravity in Sounds - y particularly, of the
Caufe and Meafure of the Differences of
Tune.
§ i. Containing fome neceffary Definitions and
Explications, and the particular Method
of treating this Branch of the Science con-
. ceming Tune or Harmony.
I II ST, The Subject to be here explain-
ed is, That Property of Sounds which
I have called their Tune ; whereby they
come under the Relation of acute and grave
to one another : For as I have already obfer-
ved, there is no fuch Thing as Acutenefs and
Gravity in an abfolute Senfe, thefe being on-
ly the Names given to the Terms of the Rela-
tion j but when we confider the Ground of the
Relation which is the Tune of the Sound, we
may juftly affirm this to be fome thing
abfolute ; every Sound having its own proper
and peculiar Tn ne> which muft be under fome de-
terminate Meafure in the Nature of the Thing,
(but the Denominations of acute and grave re-
fpecl: always another Sound.) Therefore as to'
Tune <> we muft remark that the only Difference
can poffibly be betwixt one Tune and another,
is
§. i„ of MUSI CK. 3 y
is in their Degrees, which are naturally infinite ;
that is, we conceive there is fomething pofitive
in the Caufe of Sound which is capable of lefs
and more, and contains in it the Meafure of
the Degrees of Tune ; and becaufe we don't
fuppofe a leaft or greateft Quantity of this,
therefore we fay the Degrees depending
on thefe Meafures are infinite : But commonly
when we fpeak of thefe. Degrees, we call them
feveral Degrees of Acutenefs and Gravity, with-
out fuppoimg thefe Terms to exprefs any fixt
and determinate Thing ; but it implies fome
fuppofed Degree o£Tune, as a Term to which
we tacitely compare feveral otherDegrees; thus
we fuppofe any one given or determinate Mea-^
fure of Time, then we fuppofe a Sound to
move on either Side, and acquire on the one
greater Meafures of Tune-, and on the other
leffer, /. e. on the one Side to become gradual-
ly more acute, and on the other more grave
than the given Tune, and this in infinitum i
Why I afcribe the greater Meafure to Acute-
nefs will appear, when we fee upon what that
Meafure depends. Now tho' thefe Degrees are
infinite, yet with refpecl to us they are limited,
and we take fome middle Degree, within the
ordinary Compafs of the human Voice, which
we make the Term of Comparifon when we
fay of a Sound that it is very acute or very grave 3
or, as we commonly fpeak, very high or very-
low.
II. I f Two or more Sounds are compared
in the Relation we now treat of, they are ei~
C 2 ther
$6 ^Treatise Chap. II.
ther equal or unequal in the Degree of Tune :
Such as are equal ai e called Unifons with regard
to each other, as having one Tune ; the' une~
quai,bc'mg at Diftance one from another(as I have
already explained that Word ) conftitute what
we call an Interval in Mufick^ which is pro-
perly the Difference of Tune betwixt Two
Sounds Upon this Equality or Difference does
the whole Effect depend ; and in refpecl of this
we have thefe Relations again divided in-
to,
III. Concord and Difcord, Concord is
the Denomination of all thefe Relations that
are alwavsand of themfekes agreeable, whether
applied in Succefjion or Confonance ( by which
Word I always mean a mere founding together j)
that iS) If two fimple Sounds are in iiich a Re-
lation, or have fuch a Difference of Tune^ that
being founded together they make a Mixture
or compound Sound which theEar receives with
Plcafure, that is calledConeord;and whatever Two
Sounds make an agreeable Compound,they wilJ
always follow other agreeably. Difcord is the
Denomination of all the Relations orDifferences
of $up$ that have a contrary Effect.
I V. Concords are the eifential Principles
of Mufick j but their particular Diftinclions,
Degrees and Names, we muft expect in an-
other Place. Difcords have a more general and
very remarkable Diftin«5tion, which is proper to
be explained here; they are either concmnous
or inconcinnous Intervals ; the concmnous are
ilich as are apt or fit for^ Afufick, next to and
in
§r. of MUSIC K. , 37
in Combination with Concords ; and are neither
very agreeable nor very difagreeable in themfel ves ;
they are fiich Relations as have a good Effect
in Mufick only as, by their Oppoiition, they
heighten and illuftrate the more effential Prin-
ciples of the Pleafure we feek for ; or by their
Mixture and Combination with them, they pro-
duce a Variety neceffary to our being better
pleafedj and therefore are ftill called Difcord^
.as tlie Bitternefs of fome Things may help to
fet off the Sweetnefs of others, and yet ftill be
bitter : And therefore in the Definition of Con-
cord I have faid always and of themfelves a-
greeable^ becaufe the continuous could have no
good Effect without thefe, which might fubfift
without the other,tho' lefs perfectly. The other
Degrees of Z)ifcord that are never chofen in
Mufick come under the Name of inconcinnous
and have a greater Harfhnefs in them, tho' even
the greater! Difcord is not without its Ufe.
Again the concinnous come under a Diftinclion
with refpecl: to their Ufe, fome of them being
admitted only in Succejfion y and others only in
Confonance ; but enough of this here,
V. N o w to apply the Second and Third
Article obferve, Unifons Cannot poflibly have
any Variety, for there muft be Difference
where there is Variety,therefore Unifonance Row-
ing from a Relation of Equality which is in-
variable, there can be no Species or Diftinction
in it ; all Unifons are Concor <i, and in the Fiift
and moft perfecl Degree -, but an Interval de-
pending upon a Difference of Tunc or a Re- .
C q latioa
49 A Treatise Chap, II,
lame Parts or leffer Intervals, there may be a
Difference of the Order and Position of them
betwixt the Extremes.
IX. A moft remarkable DifUnction of Syftems
Is into concinnous and inconcinnous. How
thefe Words are applied to fimple Intervals we
have already feen j but to Syftems they are ap-
plied in a twofold Manner, thus. In every Syftem
that is concinnoufly divided, the Parts confide-*
red as fimple Intervals muft be concinnous in
the Senfe of Article Third ; but not only fo,
they muft be placed in a certain Order betwixt
the Extremes, that the Succeffion of Sounds
from one Extreme to the other, may be agree-
able, and have a good Effecl in Praaice. An
inconcinndus Syftem therefore is that where the
fimple Intervals are inconcinnous 7 or ill difpo-
fed betwixt the Extremes.
X. A Syftem is either particular^ or univer-
JMj containing within it every particular Syftem
that belongs to Mufick^ and is called, The
Scale of. Musick, which may be defined,
A Series of Sounds rifing or falling towards
A cute ness or Gravity from any given
Sounds to the great eft Biftance that is fit and
practicable^ thro' fuch intermediate Degree s^ as
niake the Succejfion moft agreeable and. ferfetl\
and in which we have all the concor ding In-*
iervals moft concinnoufly divided.
The right Competition -of fuch a Syftem is
of the greateft Importance in Mufick, becaufe
It will contain the whole Principles ; and fo the
'• . Task
§i. of MU SICK. 41
Task of this Part may be concluded in this,
viz. To explain the Nature, Conftitution and
Office of the Scale of Mufick ; for in doing
this, the whole fundamental Grounds and
Principles of Mufick will be explain'd ; which
I jliall go through in this Order, imo. Ifhall explain
upon what the Tune of a Sound depends,or at leaft
fomething which is infeparabiy conne&ed with
it$ and how from this the relative Degrees of
Tune, or the Intervals and Differences are de-
termined and meafured. tdo, I fliedl confider
the Nature of Concord and Difcord, to explain,
or at leaft fhow you what has been or may
be laid to explain the Grounds of their diffe-
rent Effects. 3 tio and qto. I fhall more parti-
ticularly confider the Variety of Concords, with
all their mutual Relations : In order to which
I fhall deliver as fuccinctly as I can the harmo-
nical Arithmetic^, teaching how mufical Inter-
vals are compounded and refolved, in order par-
ticularly to find their Differences and mutual
Relations, Connections with, and Dependencies
one on another. $to. 1 fhall explain what may be
called The geometric alV2.1t of the Theory,or,how
to exprefs theDegrees and Intervals of harmonick
Sound by the Sections and Diviflons of right
Lines. 6to. I fhall explain the Compofition
and Degrees of Harmony as that Term is al-
ready diftiiiguiflied from Concord, jmo. I fhall
confider the concinnous Difcords that belong
to Mufick ; and explain their Number and Ufej
how with the Concords they make up the ani-
verfal Syfteni) or conftitute what we call The
Scale
4* ^Treatise Chap. II.
Scale of Mufick) whofe Nature and Office I
fliall very particularly explain ; wherein there
will be feveral Things handled that are funda^
mental to the right underftandingof the^ftf£?/~
cal Part; particularly, 8#o. The Nature of
Modes and Keys in Mitfick ( fee the Words
explain'd in their proper Place:) And 9110, The
Confequences with refpect to Practice, that fol-
low from having a Scale of fix'd and determi-
nate Sounds upon Inftruments ; and how the
Defects arifing from this are corrected.
§ 2. Of the Caufe and Meafure of Tune ; or
upon what the Time of a Sound depends ; and
how the relative Degrees or Differences of
Tune are determined and meafured.
T was firft found by Experience, That
many Sounds differing in Tune, tho' the
Meafures of the Differences were not yet known,
raifed agreeable Senfations, when applied either
in Confonance or Snccejfion > and that there
were Degrees in this Pleaiiire. But while the
Meafures of thefe Differences were not known,
the Ear muft have been the only Director;
which tho' the infallible Judge of what's agree-
able to its felf ; yet perhaps not the beft Provi-
for: Reafon is a fuperior Faculty, andean make
ufe of former Experiences of Pleafure to con-
trive and invent new ones ; for, by examining
the Grounds and Caufes of Pleafure in one In-
ftance,
§ z, efMUSICK. 43
fiance, we may conclude with great Probabi-
lity, what Pleafure will arife from other Gau-
fes that have a Relation and Likenefs to the
former ; and tho' we may be miftaken, yet it
is plain, that Reafon, by making all the pro-
bable Conclufions it can, to be again exami-
ned by the Judgment of Senfe, will more rea-
dily difcover the agreeable and difagreeable,
than if we were left to make Experiments at
Random, without obferving any Order or Con-
ne&ion ? i. e. to End Things by Chance, And
particularly in the prefent Cafe, by difcovering
the Caufe of the Difference of Tune, or fome-
thing at leaft that is infeparably connected with
it, we have found a certain Way of meafiiring
-all their relative Degrees,- of making diftind:
Companions of the Intervals of Sound,* and in
a Word, we have by this Means found a per-
fect Art of railing the Pleafure, of which this
Relation of Sounds is capable, founded on a
rational and well ordered Theory, which
Senfe and Experience confirms. For unlefs we
could fix thefe Degrees of 'Tune, i. &•■ mea-
fure them, or rather their Relations, by
certain and determinate Quantities, they could
never be expreft upon Inliraments : If the Ear
were fufficient for this as to Concords, I may
fay, at leaft, that we fhould never otherwife
have had fo perfect an Art as we now have ;
becaufe, as I hope to make it appear, the Im-
provement is owing to the Knowledge of the
Numbers that exprefs thefe Relations : With-
out which, agailf, how could we know what
Pro-
48 -^Treatise* Chaa If.
ftretcht to D, or d, the elaftick Force is the fame
Thing, and in the fame Proportion at thefe
Points, whatever the bending Force is ; there-
Fore the Proportion is true.
Corollary. The Vibrations of the fame
Chord are all performed in equal Time; becanfe
in the Beginning of each Vibration, the refti-
tuent or moving Force, is as the Space to be
gone thro' ,* for it is as the half Space o D, but
Halfs are as the Wholes.
Scholium. In the preceedine Experiment
( which is Dr. Grave/ ande's ) the Vibrations
are taken very tVnall, that is, at the greateit
bending the Line o D is not above a Quarter
of an Inch, the Chord being Two Foot and a
Half long. And if the Propofition be but phy-
fically true with refpect to the very fmall Vi-
brations, it will fufficiently anfwer our Purpofe - 3
for indeed Chords while they found vibrate in
very fmall Spaces.
But again, as to the Cor o/Zr/ry, which is the
principal Thing we have ufe for, it will perhaps
be objecled^that I have only considered the Mo-
tion of the Point o or D, without proving that the
elaftick Force in the reft of the Points are alfo
proportional to the Diftances; but as the whol*
bending Force is immediately applied to one
Point, (tho' thereby it acls upon them all } the
reftitutive Force may be referred all to the fame
Point ; or, we may confider the whole Area
A B D, which is the Effect of the bending, as
the Space to be run thro 1 by the whole Body
or Chord A B D, and thefe Areas are as the
Lines
§. 2. of MUSIC K. 49
Lines o D, o d, viz. The Altitudes of different
Figures having the fame common Bale A B,
and a iimilar Curve A D B, and A d B ; for
itrictly fpeaking the Chord is a Curve in its
Vibrations ; and if we take A D, and D B for
ftraight Lines, as they are very nearly, and
without any fenfble Variation in fuch fma.ll Vi-
brations as we nqw fu.ppofe, then it will be
more plain that thefe Areas -are as the Lines
o D, odj and becanfe in this Way we confider
the Action upon, and Reaction of all the Points
of the Chord, therefore the Objection is remo-
ved.
But there remains one Thing more, mz4
That the Conclufion is drawn from the Forces
or Velocities in the feveral Points E^, d, as if
they were uniform thro' all the Space ; where-
as in the Nature of the Thing they are accele-
rated from D to o, and in the fame Proportion
retarded on the other Side of o : . The Anfvjer
to this is plainly, that fince the Acceleration is
of the lame Nature in all the Vibrations, it mud
be the fame Cafe with refpeft to the Time as
if the Motion were uniform.
Now from the Confideration of this Accele-
ration, there is another Demonft ration drawn
of the prcceeding Corollary ; and that I may
fhow it, let mejprfi prove that there nuift be
an Acceleration, and then explain the Nature
of it. Mr ft. Suppofe any one Vibration from
D to o, in that the Point D mult move into d^d^,
fucceifively, before it come to O ; and if there
were no Acceleration, but that the Point D, in
D every
44 ^Treatise Chap. II.
Progrefs were made in difcovering the Relati-
on's of Tune capable to pleafe ; for in all Proba-
bility it was with this, as much more of our
Knowledge, the firft Difcovery was by Acci-
dent, without any deliberate Enquiry, which
Men could never think of till fomething acci-
dental as to them made a Firft Difcovery -, nor
could we at this Day be reafonably fure that
fbme fuch Accident fliall not difcover to us a new
Concord^ unlefs we fatisfied our felves by what
we know of the Caufe of Acutenefs and Gra-
vity ^ and the mutual Relations of concording
Intervals., which I am now to explain.
According to the Method I have propofed
In this 2<ffi{}'y you muft expect in another Place,
an Account of the Firft Enquirers into theMea-
flires of Acutenefs and Gravity ; and here I go
on to explain it as our own Experience and Rea-
fon confirms to us.
This Affection of Sounds depends, as I
have already faid, altogether upon the fonorous
Body j which differs in Tune, imo. According
to the fpecifick Differences of the Matter ;
thus the Sound of a Piece of Gold is much gra-
ver than that of a Piece of Silver of the fame Shape
and Dimcnfions ■ and in this Cafe the Tones are
proportional to the fpecifick Gravities, (ceteris
paribus) i. e. the Weights of Two Pieces of the
fame Shape and Dimenfion. Or, 2do. Accord-
ing to the different Quantities of the fame fpeci-
fick Matter in Bodies of the fame Figure ; thus
a folid Sphere of Brafs one Foot Daimeter will
found acuter than one of the fame Brafs Two
Foot
§ 2. of MUSIC K. 4y
Foot Diameter; and here the Tones are pro-
portional to the Quantities of Matter, or the
abfolutc Weights.
But neither of thefe Experiments can rea-
fonably fatisfy the prefent Enquiry. There ap-
pears indeed no Reafon to doubt that the fame
Ratios of Weights (ceteris paribus) will always
produce Sounds with the fame Difference of
Tone, i. e* conftitute the fame Interval ;yet we
don't fee in thefe Experiments, the immedi-
ate Ground or Caufe of the Differences of
Tone t ; for tho' we find them connected with
the Weights, yet it is far from being obvious
how thefe influence the other ; fo that we can-
not refer the Degrees of Tone to thefe Quanti-
ties as the immediate Caufe ; for which Rea-
fon we fhould never find, in this Method of
determining thefe Degrees, anyExpIication of the
Grounds of Concord and Harmony ; which can
only be found in the Relations of the Motions
that are the Caufe , of Sound ; in thefe Motions
therefore muft we feek the true Meafures of
Tune ; and this we fhall find in the Vibrations
of Chords : For tho' we know that the Sound
is owing to the vibratory Motion of the Parts of
any Body, yet the Meafures of thefe Motions
are tolerably plain, only in the Cafe of
Chords.
I t has been already explained' ; that Sounds
are produced in Chords by their vibratory Moti-
ons j and thoaccording to what has been explai-
ned in the preceeding^C/^/p/tr, thefe fenfible Vi-
brations of the whole Chord are not the immedi-
ate
A T&eatis£ Chap. IL
ate Canfe of the Sound, yet they influence thefe
infenfible Motions that immediately produce it;
and, for any Reafon we have to doubt of it, are
always proportional to them; and therefore
we may meafure Sounds as juftly in thefe, as
we could do in the other if they fell under our
Meafures* But even thefe feniible Vibrations
of the whole Chord cannot be immediately
mealured, they are toofmalland quick for that;
and therefore we muft feek another Way of
ttieafuring them, by . finding what Proportion
they have with fome other Thing: And, this
can be done by the different Ten/ions, or Groff-
7?g/j*,or Lengths of Chords that are in all other
refpefts, except any one of thefe mentioned^
equal and alike; the Chords in all Cafes being
fuppofed evenly and of equal Dimenfions through-
out : And of all Kind of Chords Metal or Wire-
ftrings are beft to make the following Experi-
ments with.
N o w, in general, we know by Experi-
ence that in two Chords, all Things being equal
and alike except the Tenfion or the Thichiefs
or the Length , the Tones are different ; there
muft therefore be a Difference in the Vibrations,
owing to thefe different Tenfion.s,c$r.which Diffe-
rence can only be in the Velocity of theCourfes
and Recourfes of the Chords, thro' the Spaces
in which they move to and again beyond the
ftraight Line : We are therefore to examine the
Proportion between that Velocity and the
Things mentioned on which it depends. And
mind that to prevent faying fo oft ceteris pari-
bus,-
§:*. efMUSICK. 47
bus, you are always to fuppofe it when I fpeak
of Two Chords of di^eventTenftons^Lengths, or
Groffnefs.
Proposition I. If the elaflick Chord
AB.(Plate i.fflg.i.)be drawn by any Point ofm
the Direction of the Line o D, every Vibration
it makes will be in a leffer Space as o d, till it
be at perfebl Reft in its natural Pofition A oB;
and the el a flick or reftituent Force at each
Point d of the Line oD (i. e. at the Beginning
of each Vibration ) will be in a fimple direel
Proportion of the Lines oD, o d, o d.
Demonstration. That the Vibrations
become gradually lefs till the Chord be at Ren\
is plain ; and that this muft proceed from the
Decreafe of the elaftick Force is as plain $ laft-
ly that this Force decreafes in the Proportion
mentioned, is proven by this Experiment made
upon a Wire-firing, viz, that being ftretched
lengthwife by any Weight, if feveral Weights
are applied fucceifively to the Point o, draw-
ing the Chord in the fame Direction as o D,
they bend it fo that the Diftances o D, o d, to
which the feveral Weights draw it, are in
fimple direel Proportion of thefe Weights : But
^.ction and Reaction are equal and contrary,
therefore the Refiftance which the Chord by
its Efefticity makes to the Weight, is equal
to the Gravity or drawing Force of that
Weight, u e. the reftituent Forces in the Points
D, d, are as the Lines o D, o d ; now it is the lame
Cafe whether the Chord be ftretcht by Weight
or any other Force ; for when we fuppofe it
ftretcht
,>*
48 A Treatise^ Cha?. It
ftretcht toD, or d,the elaftick Force is the fame
Thing, and in the fame Proportion at thefe
Points, whatever the bending Force is ; there-
fore the Proportion is true.
Corollary, The Vibrations of the fame
Chord are all performed in equal Time; becaafe
in the Beginning of each Vibration, the refti-
tuent or moving Force, is as the Space to be
gone thro' ; for it is as the half Space o D^ but
Halfs are as the Wholes.
Scholium. In the preceeding Experiment
( which is Dr. Grave/andes ) the Vibrations
are taken very final], that is \> at the greateit
bending the Line o D is not above a Quarter
of an Inch, the Chord being Two Foot and a
Half long. And if the Proportion be but phy-
fically true with refpecl: to the very fmall Vi-
brations, it will fufficiently anfwer our Purpofe j
for indeed Chords while they found vibrate in
very fmall Spaces.
But again^ as to the Cor ollary, which is the
principal Thing we have ufe for, it will perhaps
be objecled^that I have only considered the Mo-
tion of the Point o or D, without proving that the
elaftick Force in the reft of the Points are alfo
proportional to the Diftances; but as the whol$
bending Force is immediately applied to one
Point, (tho' thereby it acls upon them all )> the
reftitutive Force may be referred all to the fame
Point ; or, we may confider the whole Area
A B D, which is the Effect of the bending, as
the Space to be run thro 1 by the whole Body
or Chord A B D, and thefe Areas are as the
Lines
§; 2. of MUSICK. 49
Lines o D, o d, viz. The Altitudes of different
Figures having the fame common gafe A B,
and a fimilar Curve ADB, and A d B ; for
ftficlly fpeaking the Chord is a Curve in its
Vibrations ; anci if we take A D, and D B for
ftraight Lines, as they are very nearly, and
without any fenfble Variation in fuch fmall Vi-
brations as we now iuppofe, then it will be
more plain that thefe Areas #re as the Lines
o D, odj and becaufe in this Way we confider
the Action upon, and Rea&ion of all the Points
of the Chord, therefore the Objection is remo-
ved.
But there remains one Thing more, viz.
That the Conclusion is drawn from the Forces
or Velocities in the feveral Points 13, d, as if
they were uniform thro' all the Space $ where-
as in the Nature of the Thing they are accele-
rated from D to o, and in the fame Proportion
retarded on the other Side of o : . The Anjher
to this is plainly, that iince the Acceleration is
of the fame Nature in all the Vibrations, it mud.
be the fame Cafe with refpe£t to the Time as
if the Motion were uniform.
Now from the Confideration of this Accele-
ration, there is another Demonfiratlon drawn
oi the preceeding Corollary ; and that I may
fhow it, let me fir ft prove that there muft be
an Acceleration, and then explain the Nature
of it. Firft. Suppofe any one Vibration from
D to o, in that the Point D muft move into d, d,
fucceifively, before it come to O ; and if there
were no Acceleration, but that the Point D, in
D every
jo A Treatise Chap. II.
every Pofition of the Chord, as A d B, had no
more elaftick Force than is equsltoa Force that
could keep it in that Pofition- 'tis plain it co i
never pafs the Point o; becaufe thefe Force 3
as the Diftances, and therefore it is npihjti n
the Point oj but it actually pafifes that Point,
and confequently the Motion is accelerated; and
the Law of the Acceleration is this, In every
Point of the fame Vibration, the Point D is
accelerated by a Force equal to what would be
fufficient to retain it in that Pofition •> but thefe
Points being as the Diftances od, od, the Moti-
on of the Point D agrees with that of a Body
moving in a Cycloid^ whofe Vibrations the
Mathematicians demoriftrate to be of equal Du-
ration (vid. Keil's Introdu&fio ad <v 'tram phy fl-
eam) and therefore the Times of the Vibrations
of the Chord are alfo equal(wW. Gravesande's
mathematical Elements of Phyfecks. Book I.
Chap. 26.)
Before we proceed farther, I fhall apply
this Propofition to a very remarkable Pheno-
menon -, that Experience and our Reafonings
may mutually fupport one another. It is a very
obvious Remark, That the Sound of any Body
arifing from one individual Stroke, tho' it grows
gradually weaker, yet continues in the fame
"Tone : We fhall be more fenfible of this by ma-
king the Experiment on Bodies that have a
great Refonance, as the larger Kind of Bells
and long Wire-firings.
Now fince the Tone of a Sound depends
upon the Nature of thefe Vibrations, whofe
Dif-
§ i. of MUSIC K. ji
Differences we can conceive no otherwife than
as having different Velocities ,• and fince we have
proven that the fmall Vibrations of the fame
O>ord are all performed in equal Time -, and
JafHj) fince" it is true in FaA that the Tone
of a Sound which continues for fome Time after
the Stroke, is from firft to laft the fame j it
follows, T think, that the Tone is neceffarily
connected with a certain Quantity of Time in
making every fingle Vibration -, or, that a cer-
tain Number of Vibrations, accomplifhed in a
given Time, conftitutes a certain and determi-
nate Tone ; for this being fuppofed we have a
good Reafon of that Phenomenon of the Unity
of Tone mentioned : And this mutually confirms
the Truth of the Proportion, that the Vibrati-
ons are all made in equal Time $ for this Unity
of Tone fuppofes an Unity in that on which
the Tone depends, or with which our Per-
ception of it is connected ,- and this cannot be
fiippofed any other Thing than the Equality of
the Vibrations, in the Time of their Courfes
and Recourfes : For the abfolute Velocity, or
elaftick Force, in the Beginning of each Vibra-
tion is unequal, being proportional to the
Power that could retain it in that Pofiti-
on.
Again, if we could abfolutely determine
how many Vibrations any Chord, of a given
Length, Thichiefs and Tenfion, makes in a gi-
ven Time, this we might call a fixd Sound or
rather afixd Tone, to which all others might
be compared, and their Numbers be alfo deter-
D 2 mined
ji -^Treatise Chap. II.
mined,- but this is a mere Curiofity, which nei-
ther promotes , the Knowledge or Practice of
Mufick ; it being enough to determine nd
meafure the Intervals in the Proportions aad
relative Degrees of Tone^ as in the follow-
ing Proportions,
Proposition II. Let there be T-zvo elc±
ftick Chords A andC ( Plate i. Fig. 2. ) diffe-
ring only in Tenlion, i. e. Let them be fretcht
Length-wife by different Weights zvhich are
the Meajures of the Teniion ; the Time of a
Vibration in the one is to that of the other in-
'verfely as the fquare Root of the Tenfions or
Weights that ft retch them. For Example , if
the Weights are as 4 : 9. the Times are
as 3 : 4.
Demonstration. If Two Chords C and
A ( Plate 1. Fig. 2. ) differ only in Ten/Ion,
they will be bended to the fame Diitance O D
by Weights (fimilarly applied to the Points o )
which are direclly proportional to their Ten-
fions ; this is found by Experiment '(did. Grave-
fandeV Elements?) Again^ thefe Two Chords
bended equally, may be compared to Two
Pendulums vibrating in the fame or like Cycloid
with different accelerating Forces; in which Cafe,
the Mathematicians know, it is demonftrated,
that the Times are inverfely as the fquare Roots
of the Tenfions ", which are as the accelerating,
/". e. the bending Forces, when they are drawn
to equal Diftances; but the Proportion is true
whether the Diftances O D be equal or not ;
be-
§. i. of MUSICK. 53
becaufe all the Vibrations of the fame Chord
are of equal Duration by Prop. i.
Corollary. The Numbers of the Vibra-
tions accomplifhed in the fame Time are di-
rectly as the fqu are Roots of their Tenfions. For
Example, If the Tenfions are as 9 to 4. the
Numbers of Vibrations in the fame Time zvill
be as 3 to 2.
, Proposition III. The Numbers of Vi-
brations made in the fame Time by Two Chords,
A and B (Plate 1. Fig. 3. ) that differ only in
Thicknefs, are inverfely as the fquarg Roots of
the Weights of the Chords, i. e. as the Diameter _
of their JRafes inverfely.
Demonstration. We know by com-
mon Experience that the thicker and gr offer
any Chord is, being bended by the fame
Weight, it gives the more grave Sound ; fo
that the Tone is as the Thicknefs in general :
But for the particular Proportion, we have this
Experiment, viz. Take Two Chords B and C
( Plate 1. Fig. 3. ) differing only in Thicknefs ,-
let the Weights they are ftretched with be as
the Weights of the phords themfelves, i.e. as
the Squares of their Diameters ; their Sounds
are unifbn, therefore the Number of Vibrations
in each will be equal in the fameTime: And con-
fequently if the thick Chord B be comparedto a-
nother of equal Length A( in the fame Figure)
ftretched with the fame Weight, but * whole
Thicknefs is only equal to that of the
fmaller Chord C laft compared to it ; the
Numbers of Vibrations of B and A will be
, D 3 as
14 ^Treatise Ghap. II,
as the fquare Roots of the Weights of the
Chords inverfely: That is, inverfeJy as the
Diameters of their Bafes, or the B.res thro*
which the Wire is drawn.
Proposition IV. If Two Chords A
and B, in Plate i. Fig, 2, differ only in their
Lengths, the Time of a Vibration of the one
is to that of the other as the Lengths direbllyi
andconfequently as the Number of Vibrations in
the fame Time inverfely. For Example^Let the
one be Three Foot and the other Tzvojhe Firffi
pill make Two Vibrations and the other Three
in the fame Time,
Demon. 'Tis Matter of common Obfervati-*
on, that if you take any Number of Chords
differing only in Lengthy their Sounds will be
gradually acuter as the Chords are fljorter }
and for the Proportion of the lengths and
Vibrations, it will be plain from what has been
already laid 5 for the fame Tone is conftitute
by the fame Number of Vibrations in a given
'Time -, and we know by ' Experience that if
Two Chords C and B ( Plate 1. Fig, 2, )
differing only mLength^ are* tended by Weights
which are as the Squares of their Ijengths^
their Sounds are unijon ; therefore they make
an equal Number of Vibrations in ' the fame
Time. But again^ by Propofition 2, the
N mber of Vibrations of the longeft of
thefe Two Chords C, is to the Number in the
fame Time, of an equal and like Chord A
(in the fame Figure) lefs tended, as the fquare
IRoots of the Tenfions dke&Jy -, therefore if
v. "' A Is
§. z. of MUSIC K. j 5
A is tended equally with the ftiorter Chord
B (wbofb Vibrations arc equal to thofe of the
lor.ger Chord D that's mofttended)tis plain the
Number of Vibrations of thefe two muft be
as their Lengths, becaufe thefe Lengths are
diredtlyas the fquare Hoots of the unequal Tenfi-
ons.
ObsuvEj that if we fuppofe this Proportion
of the Time and Lengths to be otherwife de-
moniirated, then what is here advanced as an
Experiment will follow as a Confequence from
this Propofition and the Second. But I think
this Way of demonftrating the Propofition
very plain and fktisfying. You may alfo fee
from what Confide rations Dr. Gravefande con-
cludes it. Or we may prove it independently
of the Second Propofition^ after the Manner
of the Firft by the following
Experiment. Viz. If the fame or equal Weight
is iimilarly applied to fimilar Points O o, of Two
elaftick Chords AandB {Plate \. Fig. 2.) that
differ only in Lengths ; the Points O, o will
be drawn to the Diitances OD, o d, that fhall
be as the Lengths of the Chords A, B ; fo
that the Figures (hall be fimilar, and the, whole
Areas proportional to the Lengths of the
Chords.
Now the bending Forces in D and d are
equal and equally applied, therefore the rc-
ftituent Forces are equal - y the Times confe-
quently are as the Spaces, i. e. as the Areas or
the Chords A, B, and this holds whatever the
Difference of o d and O D is D fince all the Vi-
brations
j6 A Treatise Chap, II.
rations of the fame Chord are made in equal
Time ; and therefore, laftlj\ the Numbers of
Vibrations in a given Time areas thefe Lengths
inverfely.
Observe. From this Demonftration and
the Experiment ufed in the former Demonftrati-
on, we fee the Truth of Proportion 2. in ano-
her View.
General Corollary to the preceeding Pro-
portions. The Numbers of Vibrations made
in the fame Time by any Two Chords of the
fame Matter ^differing in L.engthfT hi chiefs and
Tenfwn, are in the compound Ratio of the
Diameters and Lengths inverfely ^and thefquare
Roots of the Tenfwns directly.
Now let us fum up and apply what has been
explained, and, f.rfi^ We have concluded that
the Differences of 7 one or the Intervals of
harmonick Sound are neceflarily connected
wirh the Velocity of the Vibrations in their
Courfes and Recourfes, i. e. the Number of Vibra-
tions made in equal Time by the Parts of the figr
norons-Body : And becaufe thefe Numbers can-
not be meafured m themfelves immediately,
we have fornd how to do it in Chords, by
the Proportions betwixt them and the diffe-
rent Tenfwns or Thi chiefs or Lengths-, we have
not fought any abfolute and determinate l Num-
ber of Vibrations in any Chord, but only the
Ratio or Proportion betwixt the Numbers ae-
compliftied in the fame Time, by fe vera! Chords
differing in Tenfwn or Thicknefs or Lengthy or
in ail thefe 5 therefore we have difcovered the
true
§V of MUSIC K. 57
true and.juft Meafures of the relative Degrees
of Tone, not only in Chords, but in all other
Bodies - } for if it is reafonable to conclude, from
the Likenefs of Caufes and Effects, that the
fame Tone is conftitute in every Body,by the fame
Number of Vibrations in the fame Time, it fol-
lows, that whatever Numbers cxprefs the Ratio
of any Two Degrees in one kind of Body, they
exprefs the Ratio of thefe Two Degrees univer-
sally : But this would hold without that
Suppofition, becaufe we can find Two Chords,
whofe Tones fhall be unifon refpeclively to any
other Two Sounds ; and therefore all the Con-
clufions we can make from the various
Compositions and Divisions of thefe Ratio's will
be*tuie of all Sounds, whatever Differences there
be in the Caufe.
I t follows again, that in the Application of
Numbers to the different Tones of Sound,
whereby we exprefs the Relations of one De-
gree to another, the grave is to the acute as
the leffer Number to the greater, becaufe the
graver depends upon the leaft Number of Vibra-
tions : But if we app T y thefe Numbers to theTimes
of the Vibrations, then, the grave is reprefanted
by the greater Number,and the acute by the leffer.
I f we exprefs the fame Tones by the Quanti-
ty of the different Tenfions of Chords that are
otherwife equal and like, then the Ratio will
be different, becaufe the Tenfions are as the
Squares of the Vibrations, and the grave will be
to the acute as the leffer to the greater : But
the Reafon. why we ought not to ufe thefe
Num-
y8 ^Treatise Chap. II.
Numbers is, that tho' different Tenfions make
different Tones, yet we can only examine the '
Grounds of Concord and Difcord, in the Ratio's
of tlie Vibrations, which are immediately the
Caufe of Sound ; and this is a more accurate
Way, becaufe thefe reprefent fomething that's
common in all Sounds ; and befides, being al-
ways leffer Numbers (&iz. the fquare Roots of
the other) are more convenient for the eafy
Companion of Intervals. As to the Diameters
or Lengths of different Chords, becaufe they
are in a fimple Proportion of the Numbers of
Vibrations, therefore the fame Numbers repre-
fent either them or the Vibrations, but inverfe-
ly ; fo that the graver Tone is reprefented by
the longer or grqffer Chord : And becaufe Ex-
periments are more eafily made with Chords
differing only in Lengths ; and alfo becaufe thefe
Proportions are more eafily conceived, and
more fenfibly reprefented by right Lines ; there-
fore we alfo reprefent the Degrees of Tone by
thefe Lengths, tho' in examining the Grounds
of Concord, we muft confider the Vibrati-
ons, which are expreft by the fame Num-
bers.
This brings to Mind a Queftion which Vin-
cenzo Galilei makes in his Dialogues upon Mu-
fick i he asks. Whether the expreffmg of the
Interval which we call an Octave by the Ratio
of i: 2. be reafonably grounded upon this. That
if a Chord is divided into Two equal Parts,
the Tone of the Half is an clave to that of the
Whole? The Reafons of his Doubt he propofes
thus,
§. i. ofMUSICK. 59
thus, fays he, There are Three Ways we can
make the Sound of a Chord neuter y viz. by
jhortning it, by a greater Tenjion, and by ma-
king it (matter^ ceteris paribus. By jhortning
it the Ratio of an OUiave is i : 2. By Ten (ion
it is 1 : 4. and by leaning the Thicknsfs it is
aMb 1 : 4. He means in the laft Caf% when the
Tones are meafured by the Weights of the Chord,
Now he would know why it is not as well 1 : 4.
as 1 : 2. which is the ordinary ExpreiTion : I
think this Difficulty we have fitffi.cient.-y anfwe-^
red above ; for thefe Weights are not the im-
mediate Caufe of the Sound j it is true we may fay
that the acuteTermof th.e05fave is to the gram
as 4. to i'. meaning only that the acute is pro-
duced by Four Times the Weight which deter-
mines the other ; and if Intervals are compa-
red together by Ratio's taken this W T ay, we can
compound and refolve them, and find their mu-
tual Connections and Relations of Quantity, as
truly as by the other ExprefTions $ but the Ope-
rations are not fo eafy, becaufe they are great-
er Numbers : And then, if the Sounds are pro-
. duced any other Way than by Chords of diffe-
rent Ten/ions or Thichiefs^ the Tones are to
one another as thefe Numbers in a very remote
Senfe ; for they exprefs nothing in the Caufe of
thefe Sounds themfeives,but only tell us, that Two
Chords being made uniions to thefe Sounds,
their Tenfions or Thicknefs are as thefe Numbers:
But, all Sounds being produced by Motion, when
we exprefs the Tones by the Numbers of Vibra-
tions in the fame Time, we reprefent fomething
that's
A Treatise Chap. II.
that's proper to every Sound ,- this therefore is
the only Thing that can be confidered in exa-
mining the Grounds of Concord and Difcord: And
becaufe the fame Numbers exprefs the Vibrati-
ons and Lengths of Chords, we apply them fome-
times alfo to thefe Lengths, for Reafons already
faid.
We have alfo gained this further Definition
of Acutenefs and Gravity, viz. That Acutenefs
is a relative Property of Sound,* which with
refpecl: to fome other is the Effect of a greater-
Number of Vibrations accomplifhed in the fame
Time, or of Vibrations of a fhorter Duration ,-
and Gravity is the Effect of a leffer Number
of Vibrations, or of Vibrations of a fhorter
Duration. And by confidering that the Vibrations
proceeding from one individual Stroke are
gradually in leffer Spaces till the Motion ceafe,
and that the Sound is always louder in
the Beginning, and gradually weaker, therefore
we may define Loudnefs the Effect of a greater
abfolute Velocity of Motion or a greater Vibra-
tion made in the fame Tinie^andlyOzvnefs is the
Effect of a leffer.
Before I end this Chapter, let us confider
a Conckfion which Kircher makes, in his
Mujhrgia univerfalis. Having proven in his own
Way, the Equidiurnity of the Vibrations of the
fame Chord, he draws this Conclufion, That
the Sound of a Chord grows gradually more
grave as it ceafes ( tho' he owns the Diiference
is not fenfible) becaufe the abfolute Velocity of'
Motion becomes h% i. e. That Velocity where-
§. 2. of MUSIC K. 6t
by the Chord makes a Vibration of a certain
Space in a certain Time. By this Argument
he makes the Degrees and Differences of Tune
| proportional to the abfolute Velocity : But if
this is a good Hypothefis, I think it will
folloWjContrary toExperience,that twoChords of
unequal Length ( ceteris paribus) muft give
an equal Tune ; for to demonftrate the recipro-
cal Proportion of the Lengths and the Number
of Vibrations, he fuppofes the Tenjion or
elaflick Force, which is the immediate Caufe
of the abfolute Velocity, to be equal when
the Chords are drawn out to propor-
tional Diftance ; for by this Equality the
fhorter Chord finifhes its Vibrations in
fhorter Time, in Proportion as the Spaces
are leffer, which are as the Lengths. A~
gaiiij the Elafticity of the Chord diminiflies
gradually, fo that in any ailignable Time
there is at leaft an indefinite Number 'of
Degrees ; and fince the Elafticity has fuch a
gradual Decreafe, it feems odd "that the Dif-
ferences of Tune^ if they have a Dependence
on the abfolute Velocity, fhould not be fenfible.
But in the other Hypothefis, where I fuppofe
the Degrees of Tune are connected with and
' proportional to the Duration of a fingle Vibrati-
on, and confequcntly to the Number of Vibrati-
ons in a given Time, there can no abfurd
Confequence follow. I am indeed aware of a
Difficulty that may be ftarted, which is this,
That the Deration of a fingle Vibration is a
Thing the Mind has nothing whereby to judge
b£
6i ATttAtisi ChaK if.
of, whereas it can eafily judge of the Difference
of abfolute Velocity by the different Percuffions
upon the Ear - } and the Defenders of this
Hypothefis may further aUedge, that the
Vibrations that produce Sound are the fmall
and almoft infenfible Vibrations of the Body <
£o far infenfible at leaft that we can only difcern
a Tremor, but no dimncl Vibrations ; and we
cannot, fay they, be furprized if the Differences
of Tune are infenfible. But I fuppofe the Degrees
of Tune of the firft Vibrations are predominant,
and determine the particular Tune of the Sounds
and then it is no lefs un -ccountable how
Two Chords dtawn out to fimilar Figures, as in
Prop. 4. ftiould not give the fame Tune^ and
indeed it feems impoffible to be otherwife in
this Hypothefis, which yet is contrary to Experi-
ence; and for the Difficulty propofcd in the other
Hypothefis it is at leaft but a Difficulty and no
Contradic~tion,efpecially if we foppofe it depends
immediately on a certain Number of Vibrations
In a given Time, which is the Confequence
of a fhorter Duration of every fingJe Vibration 3
and this again^ 1 own, fuppofes there can be
no Sound heard till a certain Number of Vibra-
tions are accomplifhed, the contrary whereof I
believe will be difficult to prove. I -iba.il there-
fore leave it to the PhilofophersJoecaxSe I think
the chief Demand of this particular Part is
fufficiently anfwered, which was to know how
to take thejuft Meafures of the relative Degrees
of Tune, and their" Intervals or Differences,
You'll remember too 3 what Reafon I have
already
i. of MUS1CK. <$3
already alledged for exprefling the Degrees
of Tunehy the Numbers of Vibrations accompJuTi-
ed in the fame Time j for whether the Caufe
of our perceiving a different Tone lies here or
not, the only Way we have of accounting for
the Concord and Difcord of different Tomij,
is the' Confideration of thefe Proportions, and
whatever may be required in a more univerial
Enquiry into the Nature and Phenomena
of Sound, this will be fufficient to . fuck
a Theory, as by the Help of Experience
and Obfervation, may guide us to the true
Knowledge of the Science of Mufick.
Besides, in this Account of the Caufe of the
; Differences of Tune, I follow the Opinion not
only of the Ancients but of our more modem
Philofophers j Dr. Holders whole Theory of
the natural Grounds and Principles of Har-
mony, is founded on this Suppofition,- take his
own Words, Chap, 2. " The Firft and great
u Principle upon which the Nature of harmo-
** nical Sounds is to be found out and difco-
tc vered is this : That the Tune of a Note (to {peak
cc in our vulgar Phrafe ) is conftituted by the
c Meafure and Proportion of Vibrations of the
c fonorous Body ; I mean, of the Velocity of
c thefe Vibrations in their Recourfes, for the
c frequenter thefe Vibrations are, the more a-
B cute is the Tune ; the flower and fewer they
c are in the flime Space of Time, by fo muck
' the more grave is the Tune. So that any
£ given Note of a Tune is made by one cer-
ft tain Meafure of Velocity of Vibrations, viz.
" inch
#4 ^Treatise Chap. II,
Cc fuch a certain Number of Courfes and Re-
Cc courfes, e . g. of a Chord or String in fuch a
cc certain Space of Time, doth conftitute fuch
L cc a determinate Tune.
Doctor Wallis'm the Appendix to his
Edition of Ptolomey\ Books oiHarmoii)\ owns
this to be a very reafonable Suppoiition^ yet
he fays he would not poiitively affirm,* that
the Degrees of Acutenefs anfwer the Number
of Vibrations as their only true Caufe,. becaufe
lie doubted whether it had been fu flic ientlycon-
hrm'd by Experience. Now that Sound depends
upon the Vibrations oi Bodies, I think, needs
no further Proof than what we have; but
whether the different Numbers of Vibrations
in a given Time, is the tine Caufe, on the
Part of the Object, of our perceiving a Diffe-
rence of Tune, is a Thing I . don't conceive
how we can prove by Experiments ; and to
the prefent Purpofe 'tis enough that it is a
reafonable Hypothefis ; and let this be the
only true Caufe or not, we find by Expert
ence and Reafon both, that the Diff?rences
of Tune are infeparably connected with the
Number of Vibrations; and therefore thefe, or
the Lengths of Chords to which they are pro-
portional, may be taken for the true Meafure
of different Tunes. The Doclor owns that the
Degrees of Acutenefs are reciprocally as the
Lengths of Chords, and thinks it fufficiently
plain from Experience ; fince we find that the
morter Chord (ceteris paribus) gives the more
acute Sound, /'. e. that tlie Acutenefs increaieth
as
$ i. of MUSICK. 6 5
as the Length diminiflieth,- and therefore the
Ratios of thefe Lengths are juft Meafures
of the Intervals of Tune^ whatever be the
immediate Caufe of the Differences, or what-
ever Proportion be betwixt the Lengths of
the Chords and their Vibrations. So far he
owns' we are upon a good Foundation as to th@
arithmetical Part of this Science $ but then in.
Philofophjy we ought to come as near the
immediate Caufe of Things as poflfibly we
can ; ' and where we cannot have a pofitive
Certainty, we muft take the moft reafonable
Suppofition j and of that we judge by its contain-
ing no obvious Contradiction ; and then by its
Ufein explaining the Phenomena of nature ,• how
well the prefent Hypothecs has explained the
fenfible Unity of Tune in a given Sound we
have already heard, and the Succefs of it in
the Things that follow will further confirm
it.
I {hall end this Part with obferving, that
as the Lengths of Chords determine the
Meafure of the Velocity of their Vibrations,
and this determines the Meafure of their Gravity
and Acutenefs^ fo 'tis thus that Harmony is
brought under Mathematical Calculation -, the
True object of tlie.Matheniatical Part of Mufick
being the Quantity of the Intervals of Sounds;
which are capable of various Additions, Sub-
ftraftions, &c. as other Quantities are; tho' per-
formed in a Manner fuitable to the Nature of
£he Thing,
E CHAR
ii A Treatise Chat.
C H A P. Ill,
TaturcofGoKCOKT>and'Disco%x$
as contained in the Caufes thereof.
§ i. Wherein the Reafons and Charafte?
rifticks of the fever al Differences of Concords
and Difcords are enquired into f
WE have already confidered the Rea-
fon of the Differences of Tune, and
theMeafures of thefe Differences, or
Qf the Intervals of Sound arifing from them :
We now enquire into the Grounds and Rea-
fons of their different Etfeds. When Two
Sounds are heard in immediate Succeffion, the
]y[in4 not only perceives Two fimple Ideas,
but by a proper Activity of its own, comparing
thefe Ideas, forms another of their Difference-
of Tune, from which arife to us various De-
grees of Pleafure or Offence $ thefe are the E£«
fects we are now to confider the Reafons
of.
But it will be fit in theFirft Place to know
what is mean'd by the Queftion, pr what we
propofe and expecl: to find , in order to this
cpferve^ That there is a great Difference be-
twixt knowing what it is that pleafes us., ancj
$r$y wg are pleafed with fucji a Thing ; Plea-
\ --••■-' " fur§
§ J* of MUSIC K. 67
fiire and Pain are fimple Ideas we can never
make plainer than Experience makes them,
for they are to be got no other Way - y and
for that Queftion, Why certain Things pleafe
and others not, as I take it, it (ignifies this,
viz. How do thefe Things raife in us agreeable
or difagreeable Ideas ? Or, What Connection
is there betwixt thefe Ideas and Things? When,
we confider the World as the Product of infinite
Wifdom, we can fay, that nothing happens
without a fufficient Reafoa, I mean, that what-
ever is, its being rather than not being is more
agreeable to the infinite Perfection of God,
who knew from Eternity the whole Extent ot
Poffibilky, and in his perfeffi Wifdom chofe to
call to a real Exiftence fuch Beings, and
make fuch a World, as fhould anfwer the beft
and wifeft End. The Actions of the Supreme
BEING flow from eternal Reqfons known
and comprehenfible only to his infinite Wifdom;
and here lies the ultimate Reafon and Caufe of
every Thing. To know how perfect Wifdom
and Omnipotence exerted it ielf in the Producti-
on of the World 5 to find the original Reafon
and Grounds of the Relations and Connection
which we fee among Things, is altogether out
of the Power of any created Intelligence ; but
not to carry our Contemplation beyond what
the prefent Subject requires, I think the Reafon
:>f that Connection which we find by Expe-
ience betwixt our agreeable and difagreeable
I .deas, and what we call the Objects of Senfe^
>ur Fhilofophj will never reach,' and for an$
■ " E % 3&in§
A Treatise Chap. Ill,
Thing we {hall ever find ( at leaft in our mor-
tal State ) I believe it will remain a Qpeftion
whether that Connection flows from any Ne-
ceflity in the Nature of Things, or be altoge-
ther an arbitrary Difpofition ; for to folve this,
would require to know Things perfectly, and
underftand their whole Nature; which belongs
only to that Glorious BEING on wlfbm
all others depend. We (hall therefore, as to
this Queftion, be content to fay, in thegeneralj
that 'tis the Rule of our Conftitution, whereby
upon the Application of certain Objects to
the Organ of Senfe, considered in their pre-
fent Circumftances, an agreeable or difagree-
able Idea (hall be raifed in the Mind. We
have a confcious Perception of the Exiftenc©
of other Things befides our felves, by the ir-
refimble Impreffions they make upon us ; if the
Effect is Pleafure we purfue it farther ; if it is
Pain we far lefs doubt of the Reality: And fo
in our Enquiries into Nature, we muft be fa-
tisfied to examine Obfervations already made,'
or make new ones, that from Nature's con-
ftant and uniform Operations we may learn her
Laws. Things are connected in a regular Or-
der ; and when we can difcover the Lam or
Rule of that Order, then we may be faid to
have difcovered thefecondaiy Reafoii of Things,-
for Jkample, tho' we are forced to refolve the
Caufe of Gravitation into the arbitrary Will of
GOD ; yet having once difcovered this Rule
in Nature, that all the Bodies within the At-
ppfphere of the Earth have a Tendency down-
ward
§ i; of MUSIC K. 6 9
ward perpendicularly as to a common Centre
within the Earth,and will move towards it in a
Right Line, if no other Body interpofes $ upon
this Principle we can give a good Reafon why
Timber floats in Water, and why Smoke a-
fcends. I call it afecondary Reafon, becaufe is is
founded on a Principle of which we can give no
other Reafon but that we find it constantly fb.
Accordingly in Matters of Senfe we have found
all we can expect, when we know with what
Conditions of the Object and Organs of Senfe our
Pleafure is connected; fo in theffarmoiiy of Sounds
we know by Experience what Proportions and
Relations oiTune afford Pleafure, what not ; and
we have alfo found how to exprefs the Diffe-
rences of Time by the Proportion of Numbers ;
and if we could find any Thing in the Relation
of thefe Numbers, or the Things they immedi-
ately reprefent, with which Concord and its va-
rious Degrees are connected ,• by this Means we
fliould know where Nature has fet the Limits
of Concord and Difcord- 3 we fhould with Cer-
tainty determine what Proportions conftitute
Concord^ and the Order of Perfection in the va-
rious Degrees of it ; and all other Relations
would be left to theCiafs of Difcords. And this
I think is all we can propofe in this Matter j
fo that we don't enquire why we are pleafed,
but what it is that pleafesus; we don't enquire
why, for Example, the Ratio of i : 2 conftitutes
Concord, and 6 1 7 Difcord, i. e. upon what ori-
ginal Grounds agreeable or difagreeable Idea's
are connected with thefe Relations > and the
E 3 prog
70 ^Treatise Chap. III.
proper Influence of the one upon the other ; but
what common Property they agree in that
make Concord ; and what Variation of it makes
the Differences of Concord; by which we may
&ko know the Marks of JDifcord : In Jhort, 1
would find, if poffible, the diftinguifhirig Cha-
racter of Concord and Difcord ; or, to what
Condition of the Object thefe different Effects
are annexed, that we may have all the Certain-
ty we can, that there are no other Concords
than what we know already ,• or if there are
we may know how to find them; and have all
poiTible AfTiftance, both from Experience and
Reafon, for improving the moft innocent and
ravifhing of all our fenfiial Entertainments ; and
as far as we are baffled in this Search, we
muft lit down content with our bare experi-
mental Knowledge, and make the beft Ufe of
it we can. Now to the Queftion.
By Experience we know,that theCeRatiof
of the Lengths of Chords, are all Concord, tho'
In various Degrees, viz. 2 : i, 3 : 2, 4 : 3, 5:4,
6 ' $■> 5 : 3? '8 : 5, that is, Take any Chord for
a Fundamental, which {kail be reprefented by
1. and thefe Sections of it are Concord with the
Whole, 'viz. r, h h fs is *> si for * as 2 to *>
fo is r to r, and fo of the raft. The firft Five
you fee, are found in the natural Order of
Numbers 1, 2, 3, 4, 5, 6; but if you go on with
the fame Series, thus, 7 : 6, 8 : 7, we find no
more Agreement ; and for thefe Two 3 : 5, and
5 : 8, they depend ' pon the others, as we fhall
fee. There are alio other Intervals that are
Qonz
§ t; '* e f music K. n
Concord befides thefe, yet none lefs than 2 : 1,
(the Oclave ) or whofe acute Term is greater
than i } nor any greater thanO£fo w,or whofe acute
Term is lefs than r 3 hut what are compofed of
the clave and fome leffer Concord, wliich is all
the Judgment of Experience.
I fuppofe it agreed to that the vibratory Mo*
tion of a Chord is the Caufe, or at leaft propor-
tional to the Motion which is the immediate
Caufe of its Sound; we have heard already that
the Vibrations are quicker, u e. the Courfesand
Recourfes are more frequent, in a given Time,
as the Chord is fhorterj I have obferved alfo
that acute and grave are but Relations, tho"
there muft be lomething abfolute in the Caufe
of Sound, capable of lefs and more, to be the
Ground of this Relation which flows only from
the comparing of that lefs and more ; and whe-
ther this be the abfolute Velocity of Motion,or the
frequency of Vibrations, I have alfo confideredj
and do here alfume the laft as more probable.
We have alfo proven that the Lengths ofChords
are reciprocally as the Numbers of Vibrations in
the fame Time; and therefore their Ratios are
the true Meafures of the Intervals of Sound.'
But I (hall apply the Ratios immediately to the
Numbers of Vibrations, and examine the
Marks of Concord and Difcord upon this Hypo-
thecs.
Now then, the universal Character where-
by Concord and Difcord are diftinguiihed, is to
be fought in the Numbers which contain and
cxprefs the Intervals of Sound: But pot in. thefe
Nunv
jt .^Treatise. Chap.III.
hers abftra&ly ; we muft confider them as expref-
(ing tb« very Caufe and Difference ofSound witk
refpect to Tune^ viz. the Number of Vibrations
in the fame Time : I fliall therefore pafs all thefe
Confederations of Numbers in which nothing
has been found to the prefent Purpofe.
Unisons are in the Firft Degree of Concord^
'or have the mod perfect Likenefs andAgreement
in Time ; for having the fame Meafure of
'Tune they affect the Ear as one fimple Sound ;
yet I don't fay they produce always the beft
•Effect in Mufeck -, for the Mind is delighted
with Variety ; and here I confider (imply the
Agreement of Sounds and the Effect of this
in each Concord fingly by it felf. Unifonance
therefore being the moft perfect Agreement
of Sounds, there muft be fomething in this,
neceffary to that Agreement, which is to be
found lefs or more in every Concord. The
Equality of Zzm^expreft by a Ratio ofEquality
r in Numbers ) makes certainly the moft perfect
Agreement of Sound ; but yet 'tis not true that
the nearer any Two Sounds come to an Equali-
ty of Tune they have the more Agreement,
therefore 'tis not in the Equality or Inequality
of the Numbers fimply that we are to foek this
fecondary Reafcn of the Agreement orDifagree-
ment of Sounds, but in fome other Relation of
them, or rather of the Things they exprefs.
I f we confider the Numbers of Vibrati-
' ons made in any given Time, by Two
Chords of equal Tune^ they are equal upon
the Hypothecs laid down $ and fo the
Vi-
k i. of, MUSICK. 73
Vibrations of the Two Chords coincide or
begin together as frequently as poffible with
irefpecl: to both Chords, p$z. at the leaft
i Number poffible of the Vibrations of each; for
1 they coincide at every Vibration : And in this
i Frequency of Coincidence or united Mixture
of the Motions of the Two Chords, and of
the Undulations of the Air caufed thereby, not
in the Equality or Inequality of the Number
of Vibrations, muft we feek the Difference of
; Concord and Difcord ; and therefore the nearer
;the Vibrations of Two Strings accomplifhed
in the fame Time, come to the leaft Number
poflible, they feem to approach the nearer to
the Condition, and confequently to the A-
greement 0/ Unifons. Thus far we reafoii
with Probability, but let us fee how Experience!
approves of this Rule.
I f we take the natural Series r, 2, 3, 4, 5;
6, and compare every Number to the next, as
expreflfing the Vibrations (in the fame Time)
of Two Chords, whofe Lengths are reciprocally
as thefe Numbers ; we find the Rule holds
exactly; for 1 : 2 is beft than 2 : 3, dye. and the
■ Agreement diminiflies gradually ; fo that after
1 6 the Coiijonance is unfufferable, becaufe the
. Coincidences are too rare ; but there are other
Ratio's that are agreeable befides what are found
in that continued Order, whereof I have
already mentioned thefe Two, viz. 3 : 5, and
5 : 8 which With the preceeding Five are all
the concording Intervals within, or lefs than
clave 1 : 2. u e* whofe acute Term is greater
than
74 ^Treatise Chap. lit
than r 3 the Fundamental being i. Now to judge
of thefe by the Rule laid down, 3 15 will be pre-
ferred to 4 s 5, becaufe being equal in the Numbed
of Vibrations of the acuter Term, there is an
Advantage on the Side of the Fundamental in
the Ratio 3 : 5, where the Coincidence is made
&t every Third Vibration of the Fundamental,
and $th of the acute Term : Again as to the
Ratio 5 : 8 'tis lefs perfect than 5 i 6, becaufe
tho* the Vibrations of the fundamental Term
of each that go to one Coincidence are equal,
yet in the Ratio 5 : 6 the Coincidence is
at every 6 of the acute Term, and only at
every 8 in the other Cafe. Thus does our Rule
determine the Preference of the Concords
already mentioned ; nor doth the Ear con-
tradict it ; fo that thefe Concords ftand in the
Order of the following Table, where I annex
the Names that thefe Intervals have in Pra-
ctice, and which I (hall hereafter affurrie till we
come to the proper Place for explaining the
Original and Reafon of them.
Vibrations.
tTnifoii*
Oblave*
Fifth.
Fourth.
Sixth greater.
Third greater*
Third kjfer*
Sixth lejjer.
acute
, grave}
1
: 1
2
• 1
3
: 1
4
• **
5
•- J 3
5
• 4
6
: 5
8
: 5
gram
, acute.
Lengths, Now
$ t, ef MUSIC K. 7$
Now you mu ft obferve that this Frequency
of Coincidence does not refped: any abfolute
Space of Time j for 'tis ftill an ObJave, for Ex*
ample, whatever the Lengths of the Chords are,
if they be to one another as i : 2; and yet 'tis
certain that a longer Chord, cater is paribus?
takes longer Time to every Vibration 5 It
has a Refpe& to the Number of Vibrations of
both Chords accomplifhed in the fame Time:
lit does not refpecl the Vibrations of the Funda-
mental only, for then 1 : 2 and 1 : 3 would be
: equal in Concord, and fo would thefe 4 : 7 and
1 4 : 5 which they are not nor can be • for where
the Ratios differ there muft the Agreement differ
from the very Nature of the Thing, becaufe it
depends altogether on thefe Ratios ; fothat
equal Agreement muft proceed from an equal
(i. e. from the fame) Ratio ; nor can itre-
, fpeel: the acuter Term only, elfe 3 : 5 and 4 : £
< would be equal j therefore neceffarily a Confide-
' ration muft be made of the Number of Vibrati-
ons of both Chords accomplished in equal
Time* And if from the known Concords
within an clave, we would make a gene-
ral Rule, it is this^ viz* that when the
Coincidences are moft frequent with refpeel:
to both Chords ( i. e. with refpe& to the
Numbers of Vibrations of each that go to
every Coincidence) there is the neareft Ap-
proach to the Condition of JJnifons : So that
when in Two Cafes we compare the fimilar
Terms (i. e. the Number of Vibrations of the
Fundamental of the one to that of tjie other,
1 Afid
r A Treatise Chap. Ill,
and the acute Term of the one to the acute
Term of the other) if both {imilar Terms
of the one are lefs than thefe of the other,
that one is preferable ; and any one of the
(imilar Terms equal and the other unequal,
that which has the leaft is the preferable hv
termly as we find by the Judgment of the
Ear in all the Concords of the preceedingTable.
Now if this be the true Rule of Na-
ture, and an univerfal Chara&er for judging
of the comparative Perfection of Intervals, with
refpect to the Agreement of their Extremes in
Tune ; then it will be approven by Experience,
and anfwer every Cafe : But it is not fo, for
by this Rule 4:7 or 5 : 7, both Li/cords, are
preferable to 5 : 8 a Concordat? indeed in alow
Degree ; and 1:3, an Oblave and Fifth com-
pounded, will be preferable to 1 : 4 a double
ObJave) contrary to Experience. But fuppofe
the Rule were good as to fuch Cafes where both
fimilar Terms of the one Cafe compared are
lefs tha;i thefe of the other, or the one fimikr
Term equal and the other not ,• yet there are
other Cafes to which this Character will not
extend, viz. when there is an Advantage ( as
to the Smalnefs of the Number of Vibrations
to one Coincidence) on the Part of the Funda-
mental in one Cafe, and on the Part of the
acute Term in the other ; which Advantage
may be either equal or unequal, as here 5 : 6
and 4:7,- the Advantages are equal,the Coinci-
dence in the Firft being made fooner, by Two
[Vibrations of the Fundamental^ than in the
Second
$, %;• cfMUSicK. 7 j
Second, which again makes its Coincidences
fooner by 2 Vibrations of the acute Term. If we
were to draw a Rule from this Comparifon,where
the Ear prefers 5:6a 3d leffer, to 4 : 7 a
Difcordy then we fliould always prefer that one,'
of Two Cafes whofe mutual Advantages are
equal, which coincides at the leaft Number of
Vibrations of the acute Term, But Experience
contradi&sthis Rule, for 3 : 8,an Offiave and 4th
compounded, is better than 4 : 7 ; fo that we
have nothing tb judge by here but the Ear. ffi
lajily, the mntual Advantages are unequal, we find
generally that which has the greateft Advantage
in whatever Term is preferable, tho' 'tis un-
certain in many Cafes. Upon the Whole I
conclude that there is fomething befides the
(Frequency of Coincidence to be confidered in
judging of the comparative Perfection of Inter-
j vols 1 which lies probably in the Relation of
the Two Terms of the Interval, i. e. of their
Vibrations to every Coincidence ; fo that it is
not altogether leffer Numbers,but this joined with
! fomething elfe in the Form of the Ratio, which
\ how to exprefs fo as to make a complete Role,'
no Body, that I know, has yet found.
As to the Concords of the preceeding Table
Vfome have taken this Method of comparing
I them : They find the relative Number of Coinci-
] dences that each of them makes in a given
; Time, thus, Find the leaft common Dividend
; to all the Numbers that exprefs the Vibrations
pf the Fundamental to one Coincidence,- take
this for a Number of Vibrations made in any
I Swfle. by a common fundamental Chord > if
_..._ it
f% ■ ^Treatise Chap. III.
it is divided feverally by the Numbers whofe
common Dividend it is, viz. the Terms of
the feveral Ratios that exprefs the Vibrations of
the Fundamental to one Coincidence; the Quotes
are the relative Numbers of Coincidences made
in the fame Time by the feveral Concords ;
thus, the common Dividend mentioned is 60,
and it is plain while the common Fundamental
makes 60 Vibrations, there are 60 Coincidences
of it with the acute Oblave^ and 30 Coincidences
with the sth^and fo on as in the Table annexed,
f he Preference in this Ratios [Coin*
Method is according to 80*,'
greater Number of Co* 5 th,
incidences, and where qth,
that is equal the Prefe- 6th gr.
rence is to that Interval 3d gr,
whofe acutefl Term has 3d leff.
fewer Vibrations to one 6th leff.
Coincidence. And fo the
Order here is the fame as formerly determined ;
but we are left to the fame Difficulties and Un-
certainty as before ; for this Rule refers all to the
Confideration of the Vibrations of the Funda-
mental to one Coincidence $ and therefore of
Two Cafes that whofe Jeher Term is leaft will
be preferable, whatever Difference there be of
the other Term, which is contrary to FJxpe-*
rience.
Merfennus, in his Book I. of Harmony ', Art,
[1. of Harmonick Numbers, has a Propofitioi&
which promifes an univerfal Character, for
g^ingttifhjng the Perfection of Intervals as to
ifef
2 : xi
6%
3 : 2
30
4 : 3
?Q
5 '• 3
20
5 ■ 4
15
fti.:s
12
8 ? 5
13
I i: _ of MUSI CR. ?f
[he Agreement of their Extremes in Tune:Th%
mbftance of the whole Jrt. I {hall give you
iriefly in the feveral Propofitions of it, became
t may help to explain or confirm what I have
delivered -, and then I fliall examine that particular
?ropofition which refpects the Thing directly
before us; he tells us, That, imo. Every Sound
las as many Degrees of Acutenefs as it confifts
>f Motions of the Air, i.e. as oft as theTympan
f the Ear is ftruck by the Air in Motion. 'Tis
.lain he means that the Degree of Acutenefs
lepends on the Number of Vibrations of the
\ir, and confequently of the fonorous Body,
xcompliftied in a dven Time, agreeable to
nrhat I have faid of it above, eife I do not un-
ierftand the Senfe of the Proportion. ido.
fhe Perception of Concord is nothing but the
ompanng of Two or more different Motions,
tfiich in the fameTime affecl: the auditory Nerve.
tio. We cannot make a certain judgment
if any Confonance until the Air be as eft ftruck
ti the fame Time, by Two Chords, or other
inftruments, as there are Unites in each Num-
ber, exprefling the Ratio of that Concord : For
Example, We cannot perceive a 5th, till 2 Vibra-
ionsof the one Chord, and 3 of the ether areae-
ompliflied together, which Chords are in Length
s 3 to 2. qto. The greater Agreement and
'leafure of Confonance arifes from the more
requent Union (or Coincidence) of Vibrations*
|ut, abferve, this is faid without determining
yhat this Frequency has refpeft to^ and
m fecomplete § guje it is 3 1 tkink we have
£lread$
$0 ^Treatise Chap. iff.
already feen. %to. That Number of Motions
(or Vibrations ) is the Caufe that the arithmeti-
cal Divifion of Confonancies (or Intervals) has
more agreeable Effects than the harmonic ah, but
this cannot be undreftood till afterwards. Now
follows the Propofition which is the qth in
] MerfennuS) but placed laft here, becaufe 'tis
what I am particularly to examine. 6to. The
more fimple and agreeable Confonancies are
generated before the more compound and harfo.
Example. Let t,~2, 3, be the Lengths of Three
Chords, 1 : 2. is an ddiave, 2:3a $th; and
it is plain 1: 3 is an Offiave and 5th compounded^
or a Twelfth, But the Vibrations of Chords
are reciprocally as their Lengths, therefore
the Chord 2 vibrates once while the Chord 1
vibrates twice, and then exifts an 06fave+ but
the 1 2th does not yet exift, becaufe the Chord
3 has not vibrated once, nor the Chord 1 vibrat*
ed thrice (which isuieceffary to a 12th;) again J
for generating a sth^ the Chord 2 muft vibrate
thrice, and the Chord 3 twice, which cannot
be unleis the Chord 1 in the fame Time vibrate
6 Times, and then the 12th will be twice pro-
duced, and the Offiave thrice, as is manifenY
for the Chord 2 unites its Vibrations foojier
with the Chord 1 than with the Chord 3,- and »-
they are fooner confonant than the Chord 1 or
'2 with 3. Whence many of the Myfteries of
Harmony, £&, concerning the Preference of
Concords and their Succeffion may be deduced,
by the fagacious Pra&ifer, Thus far Merfennus 1
$nd Kircher repeats his very Wordsj
- ~ r — — *—;-; g ufe
§ i. of MUSIC K. 8 1
Rut when we examine this Proportion by
other Examples, it will not anfwer j and we are
as far as ever from the univerfal CharaJcr
fought. Take this Example, 2:3: 6, the very
fame Intervals with Merfennus's Example^ only
here the 05iave is betwixt the TV r f) greater!:
Numbers,which was formerly betwix^'thc Two
letter ; now here the Chord 2 unites every
Third Vibration with every Second Vibration
of the Chord 3, and then the $th exifts ; but
alfo at every Third Vibration of the fame Chord
2 there is a Coincidence of every fingle Vibra-
tion of the Chord 6 ( becaufe as 2 to 6 fo 1 to
3 ) and then doth the 1 2th exift, and alfo the
Offiave, becaufe at every fecond Vibration of the
Chord 3, and every fingle Vibration of the
Chord 6, there is an Stave; fo that in 3
Chords whofe Lengths are as it 3: 6, containing
the Qdfave : 5th : 1 2/^,all the Three are generated
in the fame Time, viz. while the Chord 2
makes Three Vibrations,- for when the Chord
3 has made Two, precifely then the $th exifts ;
at the fame Time alfo the Chord 6 has made
; 1 Vibration, and then doth the 1 2th firft exift :
But while the Chord 3 vibrates twice ( i. e.
1 while the Chord 2 vibrates thrice ) the Chord
! 6 vibrates once, and not till then doth the
Offiave exift. From this Example 'tis plain the
[Propofition is not true in the Senfe in which
Merfennus explains it, or at leaft, that I can
underftand it in : It is true that taking the Series
h, 2 > 3? 4> Si 6, 8, and comparing every Three
of them immediately next other in the Mann/r
F of
■
8i A Treatise Chap.III.
of the preceeding Example, the Preference will
be determined the fame way as has been already
done, 'viz. Qffave : $th : qth ': 6th, greater; 3^
freater, 3d leffer, 6th leffer : But yet it will not
old of the v$ry fameConcords taken another way,
as is ma^^rfumciently plain in the laft Example.
Take this other, 6:4: 3, containing a $th,4th y
and Offiave ; while the .Chord 4 makes 3 Vibrati-
ons, the Chord 3 makes 4 Vibrations ; and then
there is a qth : Alfo while the Chord 4 makes
3 Vibrations, the Chord 6 makes 2 Vibrations^
and then there is a 5 th: So that we have here
a $th and qth generated in the fame Time;
tho' if you take the fame Concords in another
Order, thus, 2 13 14; then the Rule will hold.
Take laftly this Example : Suppofe Three
Chords a : b : r, tvhere a : b, is as 4 : 7, and
h : c as 5 : 6, while b vibrates 4, Times, a vi-
brates 7 Times, and then that Difcord 4:7
exifts; but the 3$ leffer, 5 : 6,is not generated till
£ has vibrated 6 Times, fo that the Difcord
4: 7 is generated before the Concord 5 : 6. It will
be fo alfo if you take them thus ; fuppofe a ; b
as 8 : 5, and # : c as 7 : 4, here xheDifcord exifts
whenever a has made 4 Vibrations, and the Con- i
cord not till a has made 5 Vibrations. Now if
this were a juft Rule, it would certainly anfwer
in all Pofitions of the Intervals with refpecl: to
one another, which it does not;- or there muft
be a certain Order wherein we ought to take
them; but no one Rule with refpect to the
Order will make this Character anfwer to Ex-^ \
perience in every Cafe.
Now
I
§ t; ef MUSIC K. S 5
Now after all our Enquiry for an univerfal
Chara&er, whereby the Degrees of Concord
may be determined, we are left to our Experi-
ence, and the Judgment of the Ear. We find
indeed that where the radical Numbers which
exprefs any Interval are great, it is always
grofs Difcord; and that all the Concords we
know are expreft by fmall Numbers : And of all
the Concords within an 051 aye, thefe are beft
which are contained in fmalleft Numbers ,• fa
that we may eafily conclude that the frequent
Coincidences of Vibrations is a neceffary Condi-
tion in the Production of Harmony ; but ftiJl
we have no certain general Rules that afford
an univerfal Character for judging of the Agree-
ment of any Two Souads, and of the Degree
of their Approach to the Perfection of Vmjons;
which was the Thing we wanted in all this
Enquiry : However, as to the Ufe of what we
have already done, I think I may fay, that in
a Philofophical Enquiry, all our Pains is not
loft, if we can fecure our felves from falfe and
incomplete Notions, and taking fuch for juft
and true ,- not that I fay 'tis a wrong Notion of
the Degrees of Concord, to think they depend
upon the more and lefs frequent uniting the Vi-
brations, and the Ear s being confequently more
or lefs uniformly moved - 3 for that this Mixture
and Union of Motions is the true Principle, or
at leaft a chief Ingredient of Concord, is fuffici-
ently plain from Experience ; but I fpeak thus,
becaufe there feems to be fomething in the Pro-
portion of the Two Motions that we have not
F 2 yet
A Treatise Chap. III.
yet found, which ought to be' known, in order
to our having an univerfal Rule, that will infal-
libly determine the Degrees oiConcord, agreeable
to Senfe and Experience. And if any Body can
be fatisfied with the general Reafon and Princi-
ple of Concord and Difcord already found, they ■
may take this Definition, viz. That Concord
is the Refult of a frequentUnion and Coinci-
dence of the Vibrations of Twofonorous Bodies*,
and confequently of the undulating Motions of
the Air*, which being caufed by thefe Vibratir
onsy are like and proportional to them ; which
Coincidence the more frequent it is with re'
fpedf to the Number of Vibrations of both Bo-
dies performed in the fame Time, ceteris pari-
bus, the more perfect is that Concord, till the
Rarity of the Coincidence in refpebl of one or
both the Motions become Difcord.
I can find no better or more particular Ac-
count of this Matter among our modern En-
quirers j you have already heard Merfennus, and
1 (hall give you Dr. Holders Definition in his
own Words, who has written chiefly on this One
Point, as the Title of his Book bears : Says he,
" Confonancy (the fame I call Concord) is the
<c Paffage of feveral tunable Sounds through the
<c Medium, frequently mixing and uniting in
" their undulated Motions caufed by the well
" proportioned commenfurate Vibrations of the
<c fonorous Bodies, and confequently arriving
" fmooth and fweet and pleafant to the Ear.
On the conti'aryjDiJifbnanfy is from difpropor-
tionate Motions of Sounds, not mixing, but
£ jarring
a
§ r. ofMUSICK. 85
fcC Jarring and cJafliing as they pafs, and arriving
<c to the Ear harfh and grating and orTcn-
<c five. ' If the Dr. means by our Pleafures be-
ing a Confequence of the frequent Mixture of
Motions, any other Thing than that we find
thefe Things fo connected, I do not conceive
it; but however he underftood this, he has ap-
plied his Definition to the Preference of Con-
cord no further than thefe Five, 1 : 2, 2 : 3,
3 : 4, 5 : 4, 5:6. Yet after all I hope we ftiall,
in what follows, find other Confiderations
to fatisfie us, that we have difcovered all the
true natural Principles of mufical Pleafure, with
refpecl to the Harmony of the different Tunes
of Sound ; and I ihould have done with this
Part, but that there are fome remarkable Pheno-
mena, depending on the Things already ex-
plained, which are worth our Obfervation.
§ 2. Explaining fome remarkable appearances
relating to this Subject, upon the preceeding
Grounds of Concord.
I. TF a Sound is raifed with any confide-
•* rable Intenfenefs y either by the human
Voice, or from any fonorous Body; and if there
is another fonorous Body near, whofe Time is
unifon or obJave above that Sound, this
Body will alfo found its proper Note unifon or
oftave to the given Note, tho' nothing viiibly
has touched it. The Experiment can be made
moft fenfibly with the Strings of a mufical In-
F 3 feu-
8tf [A Treatise Chap/ III,
ftrumentj for if a Sound is raifed unifon or
c5fa<ve below the Tune of any open String of
the Inftrument, it will give its Sound diftin&ly.
And we might make a pleafant Experiment
with a ftrong Voice Ringing near-^i well tuned
Harpfichord. We find the fame Phenomenon
by raifing Sound near a Bell, or any large Plate
offuch Metal as has a clear and free Sound $
or a large chryftal drinking Glafs. Now our
JPhilqfophers make Ufe of the Hypothefis al^
ready laid down to explain this furprizing Ap-
pearance i they tell us, That, for Example, when
one String is (truck, and the Air put in Motion,
every other String within the Reach of that Mo-
tion receives fome Imprelfion from it; but each
String can move only with a certain determi-
nate Velocity of Recourfes in vibrating, becaufe
all the Vibrations from the greateft to the leaft
are equidiurnal ; again, all Unifons proceed
from equal or equidiurnal Vibrations, and o-
ther Concords from other Proportions, which as
they are the Caufe of a more perfect Mixture
and Agreement of Motion, that is, of the un-
dulated Air, fo much better is that Concord and
nearer to Unifon : Now the unifon String keep-
ing an exa£t equal Courfe with the founded
String, becaufe it has the fame Meafure of Vi-
brations, has its Motion continued and impro-
ven till it become fenfible and give a diftind
Sound; and other concording Strings have their
Motions propagated in different Degrees, accor-
ding to the Commenfuratenefs of their Vibra-
tions with thefe of the founded String ; the
ObJave
$ i. of MUSICK. % 7
Oftave moft fenfibly, then the 5th,- but after
this the crbifing of the Motions hinders any
fuch Eflfefl: : And they illuftrate it to us in this
Manner ; fuppofe a Pendulum fet a moving, the
Motion may be continued and augmented, by
making frequent light Impulfes, as by blowing
upon it, when the Vibration is juft fmifhed and
the Pendulum ready to return $ but if it is
touched before that, or by any crofs Motion,
and this done frequently, the Motion will be
fo interrupted as to ceafe altogether ; fo of Two
unifon Strings, if the one is forcibly ftruck it
communicates Motion by the Air, to the o-
ther ; and being equidiurnal in their Vibrati-
ons, they finifh them precifely together,- and
the Motion of that other is improven by the
frequent Impulfes received from the Vibrations
of the Firft, becaufe they are given precifely
when that other Chord has finiflied its Vibrations,
and is ready to return - 3 but if the Two Chords
are unequal in Duration, there will be a crofnng
of Motions lefs or more, according to the Pro-
portion of that Inequality ; and in fome Cafes
the Motion of the untouched String is fo check-
ed as never to be fenfible, or at leaft to give
any Sound ; and in Fad we know, that in no
Cafe is this Phenomenon to be found but the
Vnifotty Odlave and Fifth; moft fenfibly in the
Firft, and gradually lefs in the other Two,
which are alfo limited to this Condition, that
the graver will make the acuter Sound, but not
contrarily. And as this is a tolerable Explica-
tion of the Matter, it confirms in a greatDegree the
F 4 Truth
88 ^Treatise Chap. III.
Truth of the Equidiurnity of the ] Vibrations of the
fame Chord, and the Proportion of the Lengths
and Duration of the Vibrations j for we know
that the Sound of the untouched Chord is wea-
ker than that of the other, and its Vibrations
confequently Jefs ; now if .they were not equidi-
urnal, and if the Proportion mentioned were not
alfo true, we fhould not have fo good a Reafon
of the Phenomenon) which joyned with the fen-
fible Identity of the Tune, is fuffrcient without
other Demonftrations to make it highly pro-
bable that the Vibrations are all performed in
equal Time, and that the Duration of a (ingle
Vibration of the one is to that of the other di-
rectly, or the Number of Vibrations in a given
Time reciprocally as the Lengths of the Chords
{ceteris paribus?)
If. I cannot omit to mention in this Place, how
the Gentlemen of the Academy of Sciences in
France apply this Hypothecs of harmonick
Mot i on, for explaining the ftrange Recovery of
one who has been bitten by the -Tarantella, the
Effect of which is a Lethargy and Stupifying
of the Senfes ; I {ball not here repeat the whole
Story, but in fliort, the Recovery is by Means
of Muficki 'tis not every Kind that will recover
the fame Perfon, nor the fame Kind every Per-
fon; but having tried a great many various
Meafures and Combinations of Tune and Time,
they hit at random on the Cure, which ex-
cites Motion in the Patient by Degrees, till he
is recovered. To account for this, thefe Philo-
foghers tell us^ that there is a certain Aptnefs
in
§ i. of MUSIC K. Z 9
in thefe particular Motions, to give Motion to
the Nerves of that Perfon (for they fuppofe
the Difeafe lies all there ) in their prefent Cir-
cumftances, as one String communicates Motion
to another, which neither a greater nor leffer,
nor any other Combination can do - 3 being ex-
cited to Motion the Senfes return gradually,
III. There are other Inftances of this wonderful
Power, and, if I may call it fo, fympathetick
Virtue in fonorous Motions; I have felt a very
fenfible tremulous Motion in fome Parts of my
Body when near a bafs Violin, upon the found-
ing of certain Notes ftrongly ftruck^ho* the Sound
of a Cannon would not produce fuch an Effect.
And from all our Obfervations we are allured that
it is not a great or ftrong Motion in the Parts of
one Body that is capable to produce Motion by
this Kind of Communication in the Parts of
another, but it depends on a certain inexpres-
sible Likenefs and Congruity of Motions,* where-
of take this one Example more, which is not
lefs furprifing than the reft : If a Man raifes his
Voice unifon to the Tune of a drinking Glafs,
and continue to blow for fome time in it with
a very intenfe or ftrong Voice, he ftiall not only
make the Glafs found, b .it at laft break it;
whereas a Motion much ftronger, if it is out of
Tune to the Glafs, will nevermake it found and
far lefs break it(I have known perfons to whom
this Experiment fucceeded,)The Reafon of this
feems very probably to be, that when the Glafs
founds, its Parts are put into a vibratory or
tremulous Motion, which being continued long
by
90 ^Treatise Chap. III.
by a ftrongVoice,theirCohefion is quite broken;
but fuppofe another Voice much ftronger, yet
if 'tis out of Tune, there will be fuch a eroding
of Motions that prevents both the Sound of
the Glafs and the breaking of it. It is a noted
Experiment, that by prefling one's Finger upon
the Brim of a Glafs, and fo moving it quickly
round, it will found ; and to demonftrate that
this is not effected without a very fwift Motion
of the infenfible Parts of the Glafs, we need
but fill fome Liquor into it, and then repeating
the Experiment, we fhall have the Liquor put
gradually into a greater Motion, till the Glafs
found very diftin&ly, and continuing it with a
brisk Motion, the Liquor will be put into a
very Ferment. The Confideration of this may
perhaps make the Explication of the laft Cafe
more reafonable.
IV. Doctor Holder ', to confirm his |
general Reafon of ' Confonancy alledges fome Ex-
periments that happened to himfelf, particularly,
<c fays he, w Being in an arched founding Room
ic near a ftirill Bell of a Houfe-elock,when the A- 1
cc larm ftruck I whittled to it, which I did with:;
<c Eafe in the fame Tune with the Bell ; but
endeavouring to whittle aNote higher or lower.
cc the Sound of the Bell and its crofs Motions
cc were fo predominant, that my Breath and
" Lips were checked, fo that I comld not
<c whiftle at all, , nor make any Sound of it in
." that difcording Tune. After I founded a
<c fhrill whiftling Pipe, which was out of Tune
£ to the Bell, and their Motions fo clafhed
that
§ i. of MUSIC K. 91
1 f that they feemed to found Jike Twitching one
cc another in the Air." To confirm this of the
I Doctor's, there is a common Experiment, that
I if Two Sounds, fuppofe the Notes of a
mufical Inftrument, are brought to unifon
Offiave or $th, and then one of them railed
or jAeprefled a very little, there wijl be a
; Clafhing of the Two Sounds, like a Beating,
as if they ftrove together ; and this will continue
till they are reftored to exact Concord, or carried
a little further from it, for then alfotliis Beating
will ceafe, tho' the Difcord will perhaps in-
i creafe. Now if we confider that Concords are
fuch a Mixture and Agreement of Sounds that
the compound feems not to partake more of
the one Simple than of the other, but they are
I fo evenly united that the one does not prevail
over the other fo as to be more obfervable;
We fee that this driving, in which we find an
alternate prevailing of either Sound, ought natu-
rally to happen when they are neareft to their
1 moft perfect Agreement,- but when they are
•; farther removed, the one has gained too much
j upon the other not to make that one nioft
i obfervable. All thefe Things ferve to fliow
us how neceffary an Ingredient in the Caufe
of Concord the Union and Coincidence of the
1 1< Motions is,and I fhall beg a little more of your
H Patience to confider the following Illuftration.
[ It is not an unpleafant Entertainment to con-
template the beautiful Uniformity of Nature in
her feveral Productions ; the Refemblance dis-
covered among Things., if it doivt let us farther
into
92 A Treatise Chap., Ill
into the Knowledge of the Effence and original J
Reafen of them, it does at leaft increafe our]
Knowledge of the common Laws of Nature J
and we are helped to explain and illuftrate one
Thing by another. To the Matter in Hand, i
we may compare Sight and Hearing, and to l
manage the Comparifon to greateft Advantage,
let us confider, Senfation is the fame Thing
with refpecl: to the Mind that perceives, what-
ever be the Instrument of Senfe, i. e. without
diftinguiftiing the external Senfe ( as Philofo-
phers fpeak) the internal is the fame, which
is properly Senfation, as this implies a certain
Mode of the Mind caufed by the Admittance
(or, with Mr. Lock> the a£tual Entrance) of an
Idea into , the Understanding by the Senfes ;
which is a Definition plainly unconfin'd to one
or other of the Five Ways whereby Ideas en-
ter, when the Mind is faid to perceive by the
Senfes ; hence we have good Reafon to think,
that it is not improper to compare one Senfe
with another, as Seeing and Hearing ; for tho\
their Objects are different, and the Means where-
by they make their Imprefiion on the Mind be:
fuited to them, by which Senfations very diftincl
are produced ; yet they may be equally agree-
able in their Kind, and have fome common Prin
ciple in both Cafes neceffary to that Agreeable-.!
nefs. We believe that Nature works by the!
mod fimple and uniform Ways; accordingly we
find by Experience that fimple Ideas have a much
eafier Accefs than compound ; and the more
Difficulty the lefs Pleafure ,• yet the more eaf)
arc
ft
§ i. of MUSIC K. 93
I are not always the moft agreeable ; for as we
! have no Pleafure in what falls confufedly on the
; Senfes, and wearies the Mind with the mani-
i fold and perplex'd Relations of its Parts >• nei-
j ther does that afford much Pleafure that is too
i- eafily perceived, at leaft we are foon cloyed with
j it $ but a middle betwixt thefe Extremes isbeft.
j Again^ we know that Variety entertains, both
of hmple Ideas and thefe varioufly connected
! and joyned together : And becaufe the Mind is
■ beft pleafed with Order, Uniformity, and the di-
f ftincl: Relation of its Ideas, the compound Idea
it ought to have its Parts uniform and regularly
j connected, and their Relations fo diftinct that
f the Mind may perceive them without Per-
plexity : In fhort, when the Caufe is moft uni-
\ form, and involves not too great Multiplicity in
■ the Senfation, the Idea will be entertained with
\ the more Pleafure - 3 hence it is that a very in-
; tricate Figure, perplex'd with many Lines, and
i thefe not very regular, nor their Ratios di-
i ftincl:,doesnot pleafe the Eye fo well as a Figure
■ of fewer Lines and in a more diftinft Rela-
% tion.
I But the Comparifon muft run between the
■ Eye and Ear in Perceptions that have fome-
■! thing common : Motion is the Object of Sight
■' very properly ; and tho' it be notfo of Hearing
i immediately ,• yet Sound being the immediate
i Producl: of Motion, we may conclude that if
the Eye is gratify'd with the Uniformity of
Motion, for the lame Reafon ( whatever it be
in its felf) will the Ear be with Uniformity in
Sounds ,-
94 4 Treatise Chap. IIL
Sounds, which ow themfeJves to Motion, and
are in a Manner nothing elfe but Motion for-
cing on us a Perception of its Exiftence by o-
ther Organs than the Eye, and therefore makes
that different Idea we call Sound. In Seeing the
Thing is plain ,- for if Two Motions are at
once in our View, where the Senfe attends to
nothing but the Motion, then, as the Relation
of the Velocities is more diftin£t, we compare
.the Motions, and view them with the greater
Plealbre ,• but were the Relation lefs fenfibky
there ccnld but little Pleafure arife from thefe
Ideas : Thus, were it obvious that the one Mo-
tion were to the other as 2 : 1 or 3 : 2 uniform-
ly and conftantly, we could look on them with
Delight y but were the Ratio lefs perceiveable
as 13 : 7; or the one being uniform Motion,
and the other irregularly accelerate $ the Mind
would weary in the Comparifon, and perhaps
never reach it, therefore find no Pleafure : I
do not fay that in many Cafes, which might be
viewed with Satisfaction, we could be determi-
nately fure what were the Ratio of Velocity ;
but from Experience we know, that the more
commenfurable the Extremes are to one another, ;
it is the more agreeable, becaufe diftincl ;
therefore it is certain we perceive the one
more than the other : And in many Cafes there
would be a Pain in viewing fuch Objects, the
Irregularity of the Motion creating a Giddinefs
in the Brain, while we endeavour to entertain
.both the Motions j and by Experience we
know \
1
$i. of MU SICK. 9 f
know, that to follow very quick Motions with
the Eye, efpecially if circular, this is conftantly
the Effect. It is the fame Way in Hearing,
fome fimple Sounds are painful and harfh, be-
caufe the Quicknefs of the Vibrations bears no
Proportion to the Organs of Senfe, which is ne-
ceffary to all agreeable Senfation. But we have
a particular Example that comes nearer the
Purpofe.
Let us view the Motion of Two Pendu-
lums ; if they are of equal Length, and let fall
from equal Height they defcribe equal Arches;
their Motions continue equal Time, and their
Vibrations begin always together: The Motions
of thefe Two Pendulums are like and equal,
fo that if we fuppofe the Eye to follow the one,
and defcribe an equal Arch with it ( which
would be if the vifual Ray in every Point of the
Arch were perpendicular to the pendulum
Chord ) then that one would always eclipfe the
other, and the Eye perceive but one Motion;
and fuppofe the Eye at a confiderable Diftance,
it would not perceive Two different Motions,
tho' itfelf moved not; confequently there could
I be no jarring of thefe Ideas : This is exactly the
I Cafe of Two Chords every way the fame, and
equally impelled to Motion; for their Vibrations
j give the Parts of the Air alike and equal Motion,
1 fo that the Ear is always ftruck equally and at
the fame Time, hence we perceive but one
fimple Sound ; and with refpecl: to the Effed;
it is no more a compound Idea than Two
Bottles of Water from the fame Fountain make
$6 y? Treatise Chap. III.
a compound Liquor, which only increafe the
Quantity,- as thdforefaid Unifons only fill theEar
with a greater Sound increafing the Intenfenefs.
I f in the fame Cafe we fuppofe the
JSve fo {ituated as to fee diftin&ly the Motion
of both Pendulums - 3 or fuppofe the Pendulums
fall from different Heights, then this Variety
would afford a greater Pleafure ; for the Mind
perceives a Difference, but a very diftinct Relati-
on } becaufe we fee the Vibrations begin always
at the fame Times and this explains the greater
Pleafure we have in Unifons which proceed
from Chords differing in fome Circumftances,
as if the one were more intenfe or of a di£
ferent Species ; in which we perceive the Unity |
of Acutenefs, but thefe other different Circum-
ftances make them perceiveably diftincl: Ample
Sounds, which heightens the Pleafure* If we
carry this Companion further, we'll find, that
if Two Pendulums of unequal Length be let
fall together from fimilar Points of their Arch>
they begin not every Vibration together,but"they
will coincide more or lefs frequently, according
to a certain Proportion of their Lengths, which
is always reciprocally fubduplicate; and tho*
this is quite another Proportion than that of
(imple Chords which are in reciprocal fimple
Proportion of their Number of Vibrations to
every Coincidence, yet the Illuftration drawn
from this Comparifon ftan ds good, becaufe we
confider only the Ratios uf the Number of
Vibrations to each Coincidence in both Cafes ,•
and in this we firjd it true in general, that the
more
§ i. of MUSIC K. pr
more frequently the Vibrations coincide, the Pro-
fpect is the more agreeable,- but it is alfo according
to the Number of Vibrations of both Pendu-
lums in the fame Time^ in fo much that the
fameNumbers which make Iefs or more Concord
in Sound, will alfo give a greater or lefs pleafant
Profpecl, if the Pendulums are fo proportioned^
according to the known Laws of their Motion ;
and if the Pendulums feldom or never coincide, of
begin their Vibrations together, there will b©
fuch a thwarting of the Images as cannot
mifs to offend the Sight.
CHAP. IV,
Containing the Harmonical Afithtiietick*
EllE t propofe to explain as mucti
of the Theory of Numbers as is neeef-
fary to be known, for making and un-
derftanding the Comparifons of mufical Inter-
ew/r,which are expreft by Numbers^ in order to our
i finding their mutual Relations,Compofitions and
' Refolutions.But I muft premife T woThings*i^f /?«,
That I fuppofe the Reader acquainted with tne
more general and commonProperties and Opera-
tions of Numbers ; fo that I fliall but barely propofe
what of thefe I have Ufe for, without any De-
imonilrationj and demonftrate Things that are
g id?
j>3 ^Treatise Chap. IV.
lefs common. Second, That I confine my felf
to the principal and more neceffary Things ;
leaving a Thoufand Speculations that may be
madej as lefs ufeful to my Defign, and alfo be-
caufe thefe will be eafily underftood when you
meet with them, if the fundamental Things
here explained be well underftood.
§ i* Definitions.
I.HTHERE is a twofold Comparifon of
■*- Numbers, in both of which we diftin-
guifh an Antecedent or Number compared, and
Confequent or Number to which the other is
compared* By the.fflrft we find how much
they differ, or by how many Units the Ante-
cedent exceeds or comes fhortof the Confequent;
which Difference is called the arithmetical
Ratio ( or Exponent of the arithmetical Rela-
tion or Habitude) of thefe Two Numbers :
So if 5 and 7 are compared, their arithmetical
Ratio is 2 ; and all Numbers that have the
fame Difference, whatever they are them-
felves, are in the fame arithmetical Habi-
tude to one another. By the Second Compari-
fon we find how oft or how many Times the
'^Antecedent contains (if greater!:) or is contained
iif leaft) in the other ; and this Number is cal-
ed the geometrical Ratio (or Exponent of the
geometrical Relation) of the Numbers compa-
red;
§ k: of music fc. <>£
red ; fo compare 12 to 4, the Ratio is $,
fignifiying that 1 2 contains 4, or that 4 is con-
tained in 12, thrice.
The geometrical Ratio thus conceived is
always the Quote of the greater divided by the
leffer : But obferve when the leffer is antece-
dent to the greater, the Senfe of the Compare
fon is alfo this, viz. To find what Part or Parts
of the greater that leffer is equal to ,• and ac-
cording to this Senfe the geometrical Ratio of
Two Numbers is made univerfally the Quote
of the Antecedent divided by the Confequent ±
and is expreft by fetting the Antecedent over
the Confequent Fraction-wife > fo that if th&
Antecedent is greateft, the Ratio is an imprcn
per Fraction, equal to fome whole or mix'd
Number, and fignifies that the Antecedent Con-
tains the Confequent as many Times, and Parts
of a Time, as that Quote contains Units and
Parts of an Unit. Example, The Ratio o£
is" to 4 is {* equal to 3 ( for 1 2 contains a
thrice. ) The Ratio of 18 to 7 is - equal
j to 2*-, fignifying that 18 contains 7 Two
Times and f- Parts of a Time, u e* f Parts of
7; which is plainly this, that 18 contains 2
Times 7, and 4 over. But if the Antecedent
is leaft, the Ratio is a proper Fraction, figni*
I fying that the Antecedent is fuch a Part of the
i Confequent So the Ratio of 7 top Is^i. e^that 7
' is I Parts of 9.
In what follows I fhall take the geometrical
i Ratio of Numbers both ways, as it happens to
fce mod convenient*
* Gi II.
%iS ^Treatise Chap. IV.
I L. An Equality of Ratios conilitutes
Proportion, which is arithmetical or geo-
metrical as the Ratio is. A Ratio exifts
betwixt Two Terms, but Proportion requires
at leaft Three ; fo thefe i, 2, 3, are in arith-
metical Proportion, or thefe, 2, 5, 8, becaufe
there is the fame Difference betwixt the Num-
bers compared, which are 1 to 2, and 2 to
3, or 2 to 5, and 5 to 8. Again thefe are in
feometrical Proportion, 2, 4, 8, or 9, 3, 1,
ecaufe as 2 is a Ha f of 4, fo is 4 of 8, alfo as
9 is triple, of 3 fo is 3 of 1.
Observe, imo. In all Proportion, as there
are at leaftTwo Couple of Terms,fo the Compa-
rtfon muft run alike in both, i. e. if it is from
the leffer to the greater, or contrary, in the one
Couple, it muft be fo in the other ajfoj thus in
2, 6, 9 the Proportion runs, as 2 to 6 fo is 6
to 9, or as 9 to 6 fo 6 to 2.
2 do. I f three proportional Numbers are right
clifpofed, it will always be, as the lft to the 2 d y
fo the 2d to 3 J, as above ; but 4 Numbers
are in Proportion when the ift is to the id as
the 3d to the 4^,without confidering the Ratio
of the id and id; as here 2 : 4: 3 : 6; for in
a proper Senfe Proportion is the Equality
of the Ratios of Two or more Couples of
Numbers, whether they have any common
Term or not; and fo, ftri&ly, there muft be
Four Terms to make Proportion, tho' there
need be but Three different Numbers.
III. From the laft Thing explained we have
a Diftin&ion of continued and interrupted Pro-
"T V for-
§i. ofMUSlCK. io i
portion Continue ^Proportion is when in a Serfe>
of Numbers there is the fame Ratio of every
Term to the next, as of the ift to the id ; as
here 1:2:3:4:5, which is arithmetical
and i, 2, 4, 8, 16, which is geometrical.
Interrupted is when betwixt any Two Terms
of the Series there is a different Ratio from
that of the reft; as 2 : 5 : 6 . : 9, arithmetical,
where 2 is to 5 as 6 : 9 (-*'. e. differing by 3,)
but not fo 5" and 6, or 2, 4, 3, 6 '; geometrical,
where 2 is to 4 as 3 to 6 (i- e. a Halfy but not
! fo 4 to 3; and obCerve that of 4 Terms, if there
i is any Interruption of theRatio it muft be betwixt
1 the id and 3^, elfe thefe 4 are not proportional,
IV. Out of thefe Two Proportions arifes a
Third Kind, which we call harmonical Pro-
portion^ thus constituted j of Three Numbers, if
the ift be to the 3 J in geometrical Proportion,
as the Difference of the ift and 2d to the
Difference of the id and 3 J, thefe Three Num-
bers are in harmonical Proportion. Example,
2:3: 6 are harmonical, becaufe 2 : 6 : :
1 : 3 are geometrical. And Four Numbers are
harmonical, when the ij? is to the 4th, as the
Difference of the ift and id to the Difference
j of the 3d and 4?^, as here 24 : 16 1 12 : p
are harmonical, becaufe 24 : 9 : : 8 : 3 are
geometrical.
Again, of 4 or more Numbers,if every Three
immediate Terms are harmonical, the Whoie
isaSeries of continual harmonical proportionals
as 30 : 20 : 15* : 12 : 10. or if evefy 4 immediate-
ly next are karmonical, 'tis alfo a continued
Q 3 Series:
ici A Treatise Chap. IV.
Series^ but of another Species, as 3, 4, 6, 9
18, 36.
How this came by the Name of harmonica!
Proportion ftiall be fliewn afterwards ; and here
I {hall explain the fundamental Properties of
this Kind, having firft propofed as much of
the Doctrine of arithmetical and geometrical
Proportion as is neceffary for the Explanation
of the other.
jj 2, Of Arithmetical tf/zJ Geometrical Pro-
portion.
rHEOR EM I. If any Number is given as
the Firft of a Series of Proportionals, and
aHb the common Ratio^ the Series may be
continued thus : imo. In arithmetical Pro-'
portionby adding the Ratio ( or' common Biffed
rence) to the ift -Term given, and then to the
Sum - y and fo on to every fucceeding Sum ; thefe !
feveral Sums are the Terms fought in an in-
creasing Series ^ which may be. continued in infi-
nitum. But to make a 'decreasing Series , fub-
fira£t the Ratio from the Firft Term, and from
every fucceeding Remainder ,• the feveral Re-
mainders are the Terms fought, But 'tis plain
this Series has Limits, and cannot defcend in
infinitum* Example > Given 3 for the ift Term
of an increasing Series, and 2 the arithmetical
flatio, or common Difference $ the Series is
h % 7* #? &c. Qr 3 given § the ift Terr%
§ 2. of MUSIC K. 103
and 3 the common Difference in a decreafing
Series, it is 8, $, 2, and can go no further in
pofitive Numbers, ido. In geometrical Propor-
tion, by multiplying the given Term into the
Ratio (which I take here for the Quote of
the greater Term divided by the leffer)and
that Product again by the Ratio, and fo on
every fucceeding Product by the Ratio; the
feveral Products make the Series fought increas-
ing, but for a decreafing Series divide. Ex-
ample. Given 2 the ift Term, and 3 the Rath
for an increafing Series it is 2 : 6:18, 54, 162.
(re. Or, given 24 the ift Term and the Ratio 2,
the decreafing Series is 24 : 1,2 : 6 : 3. ir,&c It
is plain a geometrical Series may increafe or
decreafe in infinitum in pofitive Numbers,
TheoremII. If Three Numbers are in
arithmetical or geometrical Proportion, the
Sum of the Extremes in the firft, and the Pro-
duel in the fecond Cafe, is equal to double the
middle Term in the ift, and to the Square of
the middle Term in the fecond Cafe. Example.
3:7: ii* arithmetical, the Sum of the Ex-
tremes 3- and 1 1 is equal to twice 7, viz. 1.4.
And in thefe, 4:6:5? geometrical, the Product
of 4 and 9, viz, 36, is equal to the Square of
6, rvr 6 Times 6.
Corollary. Hence the Rule for finding
a Mean proportional, either arithmetical or
geometrical, betwixt Two given Numbers is
very obvious, viz. Half the Sum of the Two
given Numbers is an arithmetical Mean, and
the Square Root of their Product i$ a geomtrical
Hean e The or,
104 -^Treatise Chap. IV,
Theorem III. If Four Numbers are in
Proportion ari thmetical ovgeometrical,whether
continued or interrupted, the Sum of the Ex^
tremes in the firft Cafe, and Product in the id y
\s equal to the Sum of the middle Terms in
the lft and the Product in the id Cafe, Ex*-
Ample, In thefe, 2:3:4:5 arithmetical, the
Sum of 2 and 5 is equal to the Sum of 3 and
4 j and thefe geometrical 2:5:4:10. the Pro-?
duel of 2 and 1 o is equal to that of % and 4,
viz* 20,
Corollary. If Four Numbers reprefented
thus, a ' b :: c : d, are proportional 'either arith-
metically ox geometrically, comparing a to £ an 4
ftod} tne y will alfo be proportional taken
inyerfely, t\ms,d :c :: b : a, or .alternately thus,
a : c *. : b : d, or inverfely and alternately thus,
d : b : : c : a. The reafon is obvious, becaufe
in all thefe Forms the Extremes and the middle
Terms are the fame, whofe Sums, if they
are arithmetical, orProdutls if geometrical, be-
ing equal, is a Sign of their Proportionality by
. this Theorem,
Theorem IV. In a Series of continued-
Proportionals, arithmetical or geometrical, the
Two Extremes with the middle Term, or
the Extremes with any Two middle Terms
at equal Diflanee from them a are alfo. propor-
tional. Example, 2, 3^ 4, 5, 6, 7, 8 arith-
metical,hexe 2 a 5, 8 3 are arithmetically pro-
portional, alfo 2 3 4, 6 % 8, or 2 ; 3 : 7 : 8. Again
in this geometrical .Series^ 2:4: 8 : 16 ; ; 32 ;
im
§ i. of MUSICK. ioj
64 : 128, thefe are geometrically proportional
2 : i6 9 128, or 2 : 8, 32 : 128.
Theorem V. If Two Numbers in any
geometrical Ratio are added to_, or fubftra&ed
rfom other Two in the fame Ratio (the lefs
with the lefs and greater with the greater) the
Sums oxDifferences are in the fame Ratio. Ex-
ample, 6 : 3 :: 10 : 5 are proportional^the com-
mon Ratio being 2, and 6 added to 10 makes
1 6 j as 3 to 5 makes 8, and 16 to 8 are in
the fame Zfo/70 as 6 to 3 or 1 o to ft and again
16 being to 8 as 6 to 3, their Differences ,10
and 5 are in the fame Ratio.
The Reverfe of this Propofition is true, w'js.
That if to or from any Two Numbers be added
or fubftrafted other Two, then, if the Sums or
Pifferences are in the fame geometrical Ratio
of the Firft Two, the Numbers added or fub-
ftra&ed are in the fame Ratio*
Corollary, If any Two given Numbers
are equally multiplied or divided, i. e. mul-
tiplied or divided by the fame Number, the
Two Products or Quotes are in the fame Ratio
with the given Numbers, /. e\ are proportional
with them, Example. 3 and 5 multiplied each
fry 7 produce 21 apd 35*, and thefe are propor-
tional 3:5', 21 : 35". Again 24 and i<5, divi-
ded each by 8 quote 3 and 2 and thefe arepro^
portional 24 : 16, 3 : 2,.
I t follows alfo that if every Term of any
continued Series is equally multiplied or divided
it is ftilj a continued Series in the fame Ra->
tio.
Theorem.
io6 A Treatise Chap. IVI
Theorem VI. If Two Numbers in any
arithmetical Ratio be added to other Two in
the fame Ratio (the lefs to the lefs and greater
to the greater) th.e Sums are in a doub e Ratio,
i.e. their Difference is double that of the refpe-
6tive Parts added ,- fo, if to thefe 3 : 5", you add
thefe 7 : 9 the Sums are 10, 14 whofe Difference
4 is double the Difference of 3 : 5 or 7 : 9. And
if to this Sum you add other Two in the fame
Ratio, the Difference of the laft Sum will be
triple the Difference of the Firft Two, and fo
on.
Observe. If Two Numbers in any arith-
metical Ratio are fubftrafted from other Two
in the fame Ratio ( the lefs from the lefs, (3t.)
the arithmetical Ratio of the Remainders is o,
fo from 7 : $ take 3 : 5 the Remainders are
* 4 * ■ *
Corollary. If Two Numbers in any arithme-
tical Ratio be both multiplied by the fameNum-
ber, the Difference of the RroduUfs fhall contain
the Firft Difference, as oft as the Multiplier
contains Unity; fo 3, 5* multiplied by 4 produce
12, 20, whofe Difference 8 is equal to 4 Times
2 ( the Difference of 3 and 5 ) and fo if any
continued arithmetical Series has each Term,
multiplied by the fame Number, tlie Produces
will make a continued Series with a Difference
containing the former Difference as oft as the
Multiplier contains Unity. But if divided, the
Difference of the Quotes will be fuch a Part
of the Firft Difference as the Divifor denomi-*
nates.
Theorem
§*; of MUSIC K. 107
Theorem VII. If Two Numbers in any
"Ratio arithmetical or geometrical^ be added
to, or multiplied by other Two in any other
Ratio of the fame Kind (the lefier by the leffer,
and. the greater by the greater ) the Sums in
the one Cafe and Products in the other are in
a Ratio which is the Sum or Product of the
Ratios of the Numbers added, or multiplied :
An Example will explain it, Let 2 : 4 and
3 : 9 be added in the Manner mentioned, the
Sums are 5, 13, whofe arithmetical Ratio or
Difference is 8 the Sum of 2 and 6 the Diffe-
rences of the Numbers giveiij or if they are
multiplied, piz t 2 by 3, and 4 by 9, the Pro-
duces 6 and 36 are in the geometrical Ratio of
6, equal to the Product of 2 and 3 the Ratios
of the given Numbers,
Theorem VIII. If any Two Numbers are
multiplied by fame Number, and the Produces
taken for the Extremes of a Series, they will
admit of as many middle Terms as the Multi-
plier contains Units lefs one ,• and the whole
Series will be in the arithmetical Ratio of the
I Firft Numbers ; fo let 3 and 7 be multiplied:
by 4 the Prodn£ts are 12 and 28 ( in the fame
•geometrical Ratio as 3 "and 7 by Corollary to
\ Theorem $t}o) and their arithmetical Ratio or
! Difference 1 6, is 4 Times as great as that of 3
and 7, which is 4 (by C'orol. to Theor. 6.) and
therefore they are capable of 3 fuch middle
Terms as that the common Difference of the
Svhole Series (hall be 4,- the Series is 12 : 16,
20 i
108 A Treatise Chap.IV.
10 : 2,4 : 28. Corollary. Hence we have a So-
lution to this Problem.
Problem J. To find an arithmetical
Series^ of a given Number of Terms, whofe
Extremes fhail be in the geometrical Ratio^ and
the intermediate Terms in the arithmetical
Ratio of Two given Numbers,- the Rule is,
Multiply the given Numbers by the Number of
Terms lefs i, and then fill up the middle
Terms by the given Ratio. Example. Let 5
to 5 be given for the Ratio of the Extremes,
and 10 for the Number of Terms ,- I multiply
3 and 5 by 9, which produces 27 and 45, ana
the Series is 27, 29, 31, 33, 35, 31, 39, 4 1 *
43, 45- •■
L e t us now compare the arithmetical and
geometrical Proportions together.
' Theorem IX. If there is a Series of Num-
bers in continued arithmetical Proportion^
then the geometrical Ratios of each Term to
the next muft neceflfarily differ,- and from the
leaft Extreme to the greateft, thefe Ratios ftill
increafe ,- but from the greateft they decreafe,
comparing always the lefler to the greater ; but
contrarily if we compare the greater to thelefler.
Example. In this arithmetical Series 1, 2, 3,
4,5, 6. the geometrical Ratios are 7, f-, J, % i y
increasing from ~, and confequently decreafing
from §•. Jlgain 9 if we take a continued geome-
trical Series^ the arithmetical Ratios or Diffe-
rencesinere afe from the leaft Extreme to the
greateft, and contrarily from the greateft to the
leaft. Example. 1, 2, 4, 8, 16^ the arithmetical
Ratios are 1, 2, 4 3, Cqrq^
§ 3 . of MU SICK. io 9
Corollary. It is plain, that if an arith-
metical Mean is put betwixt Two Numbers,
the geometrical Ratios betwixt that middle
Term and the Extremes are unequal ; and that
of the leffer Extreme to the middle Term is
lefs than that of the fame middle Term to
the other Extreme. Example. 2, 4, 6 the two
geometrical Ratios are - and - comparing
the leffer Number to the greater ; but it is con-
trary if we compare the greater to the lef-
fer.
§ 3. Of Harmonical Proportion,
THEOREM X. If Three or Four Num-
bers in harmonical Proportion are multi-
i plied or divided by any the fame Number,
I the Produces or Quotes will alfo be in harmo-
| nical Proportion ; becaufe as the Products or
1 Quotes made of the Extremes are in the fame
1 Ratio of the Extremes, fo . the Differences of
t the Produces of the intermediate Terms, tho'
they are greater or leffer than the Differences
1 of thefe Terms, yet they are proportionally fo,
being equally multiplied or divided.Z'xample. If
6, 8, 12, which are harmonical, be divided by
2, the Quotes are 3, 4, 6, which are alfo har-
monical ; and reciprocally, firice 3, 4, 6, are har-
monical, their Produ&s by 2, 273. 6. 9 8, 1 2 are
harmonical
Theorem,
tio yf Treatise Chap.IW
Theorem XI. If double the Product of
any Two Numbers be divided by their- Sum 3
the Quote is an harmonical Mean betwixt
them* Example. Let 3 and 6 be given for
the Extremes to find an harmonical Mean,
their Product is 18, which doubled is 36 ; this
divided by 9 (the Sum of 3 and 6 ) quotes 4, and
thefe Three are in harmonical Proportion, viz..
3:4: 6.
To them that have the leaft Knowledge of
^Algebra, the following Demonftration will be
plain ; fuppofe any Two Numbers a and b t and
a the greater, let the harmonical Mean fought
be x ; from the Definition of harmonical
Proportion, we have this true in geometrical
Proportion^ Wkl a : b : : a-x : x-b. And by
Theorem 3d, ax-ab^ab-xb: Then> ax^bx^Zab t
find kftly, x^m W. D.
T heorem XII. Take any Two Num-
bers in Order, and call the one the Firft
Term, and the other the Second ; if you mul->
tiply them together^ and divide the Product by
the Number that remains^, after the Second li
fubftractecl from double the Firft, the Quote is
a Third in harmonical Proportion, to be taken in
the fame Order. Example, Take 3 : 4 their
Product is 1 2, which being divided by 2 ( the
Remainder after 4 is taken from 6 the double
of the Firft) the Quote is 6, the Third harmoni-
cal Term fought : Or reverfely, take 6, 43
their Product is 24, which divided by ,8 ( the
Difference of 4 and 1.2) quotes $ 3 the Third
Term fought,
Dei
§ 3 . of MUSIC K. in
Demonstration. Take a and b known
Numbers, and a the greateft; let* be the Third
Term fought, lefs than b ; then, fince thefe
are harmonic al^ <viz. #, b, #, thefe are geome*
trical^ viz* a : x : : a-b : b-x (by Definition 4.
§ 1. of this Chapter ) then, taking the Products
of the Extremes and Means, we have ab-ax=ax-
xb; and ab=2ax~xb. Andh&ly k—J& W. W. 2X
The Demonftration proceeds the fame way when
a is fuppofed lefs than Z>, and x greater.
Observe. When a is greater than £, then
x can always be found becaufe in the Di-
vifor (ia-b) ia is neceffarily greater than
b. But if a is lefs than Z>, it may happen that
ia fhall be equal to or lefs than k and in that
1 Cafe x is impoflible. Example. Take 3 and 6 9
if a 3d greater than 6 be required it cannot be
found ; for 2^, $te. twice 3, or 6, is equal to
b or 6 -and fo theDivifor is o; or if 2 a be great-
I er than Z>, as here 3, 5, where twice 3 or 6 is
; greater than ?, then it is more impoffible.
Hence again ohferve^ that from any given
1 Number a Series of continued harmonical Pro-
portionals (of the ifi Species,/, e. where every
3 immediate Terms are harmonical ) may be
i found decreasing in infinitum but not increa-
! fmg.
Lastly, observe this remarkable Difference
i of the Three Kinds of Proportionals, <viz. That
! from any given Number we can raife by Theorem
1. a continued arithmetical Series increafing in
infinitum ; but not decreaiing. The harmonical
is decreafable but not increafable in infinitum
H
U2 ./f Treatise Chap.IV*
i by the prefent Obferve ; the geometrical is both
(by Theorem i. )
Theorem XIII. Take any Three Numbers
in Order, multiply the ift into the 3d, and
divide the Producl by the Number that remains
after the middle or id is fubftra&ed from double
the lji ; and that Quote fhall be a /\th Term
In harmonical Proportion to the Three given.
Example, Take thefe Three, 9, 12, W6 3 a.'qtfc
will be found by the Rule to be 24.
Demonstration. Let any Three given
Numbers be a, b, c, and a lefs than b, let the
Number fought be # greater than c, then by Uefi-*
nition 4th, it is a : x :: b-a : x-c^ and ax-ac=bx*
ax, laftly x=^$. The Demonstration is the fame
when a is greater than b, and x lefs then c.
Obferm here alfo that i£b is equal to or greater
than 1a, then there can be no 4th found, fo
that x is impoifible. But this can only happen
when the Terms increafe, i. e. when a is lefs
than£,and c lefs than x. See this Example, j , 2,'
3, to which a 4th harmonical is impoffible.
Theorem XIV. Take any Series of continued
arithmetical Proportionals, and. out ofthefe may
be made a Series of continued harmonical Pro*
portionals of the firft Species, where every Two
Terms (hall be in a reciprocalgeo/tfemcj /Proporti-
on of the correfpondent Terms of the arithmetical
Series. TheRuk is, Take the Two firft Couplets
of the arithmetick Series, fet them down in a
reverfe Order, ( as in the Operation below )
multiply each of the ifl- Couple by the greater
p£ the id, and the leffer of the one by the
leffeV
§ 3 . of MU SICK, 113
lcflTcr of the other; and fet down the Products,*
then, take the next Couplet, and multiply each
of the la ft Products by the greater of this Couplet, 1
and alfo the leaft of thefe Produces by the leaft
of this Couplet, and fet down thefe new Pro-
duels : Repeat this Operation with every Couplet,
and the Jail Line of Produces is the Series fought.
The following Example and Operation will
make it plain.
Arithmetical Series, NOTE, After
2 :2:a: 5 : 6 &c. this Operation
•^— ■ — ^ « is:finiflied,.the-
Series found
may be redu^
eect.by, equal .
Div!iicn,upofc
fible „• To the i
Series found in
this Example*
4 :
3
12
: 8
j
6
5 •
4
6o
40 :
:
24
6
'• 5
360 : 240 : 180 : 144 : 120, (jc, is, reduced to
Harmonic al Series. this, 30, 2 a, 1 5, '
; 12, 10.
T he TJemonfiration of this Rule is calily made, 1 1
11110. If we take any Three Numbers in. arith-
metical Proportion^ and multiply them according
to the Rulej 'tis manifeft the Products will be
harmonica!; for the Two Extremes of, the Three
arithmetical being multiplied by the fame middle
Term, their Produces ( which are the Extremes
of: the Three harmonic al) are in the {amc#geo-
metrical Ratio ; and then the Two Extremes
being multiplied together, and the Product made
the middle Term, it mull: be an harmonical
■ H Mcc.n^
"n4 -^Treatise Chap. IV.
'Mean, becaufe the arithmetical Ratio of the
Two Couplets being equal, and the ift Couplet
being multiplied by the greater Extreme, and
the other by the leffer Extreme, the Differences
of the Products are increafed in Proportion of
thefe Multipliers (0/2. the Extremes) confequent-
ly the Three Products are in harmonica! Pro-
portion, according to the Theorem, But the
fame being true of every Three Terms im-
mediately next in the arithmetical Series
thus multiplied - 3 and it being alfo true by The-
orem 1 o. that the Terms of any harmomcal
Series being equally multiplied the Produces
are alfo harmonic al, and in the fame geo-
metrical Ratio, it will be evident that working
according to the Rule we mult have an harmoni-
cal Series.
The Reverfe of this Theorem is alfo true,'
4>iz. that if you take a Series of continued Har-
monicals of the ift Species, and multiply them
in the Manner prefcribed in the Rule, there
will come out a Series of Arithmeticals, whofe
every Two Terms fhall be reciprocally in the
geometrical Ratio of their correspondent Har-
monicals. Example. Take 3, 4, 6, the Products
according to the Rule are 24 :. 18 : 12, or by
Reduction 4:3: 2, which are arithmetic 'afa fee
the Operation The Reafon is plain, for the
" Difference of the Two Couplets
4 : 3 and 6 : 4 being geometri-
cally as the Extremes 3 : 6>
when the ift Couplet is multi-
plied by the greater Extreme,
and the other by the Jeaft, the
Dife
3 •
4 :
6
4
4 :
3
6 :
24
: 18
:
12
§ jV cf MUSIC K. ji;
Differences of the Products muft be equals every
Thing elfe is plain. ^
Corollary. From the Demonjiration of
this Theorem it follows, that taking any Series
of whatever Nature, another may be made out
of it, whofe every Two Terms lhall be refle-
ctively in a reciprocal geometrical Proporti-
on ot their Correspondents in the given Series^
Theorem XV. In a Series of continued
Harmonic ah of the ifi Species, any Term with
iany Two at equal Diftance from it are in har-
monica! Proportion. Example, io, 12^ 15, 20^
30, 60 ; becauie every Three immediate Terms
are harmonic al, therefore thefe are fo, io, i$±
30 ; and thefe 5 12, 20, 60. The Reafon is eafily
deduced from the laft. But of Harmonic als
of the 2d Species^ ( See Definition 4. ) it will
not always hold that any Two with any other
Two at equal Diftance are alfo harmonical ; an
Example Will demonftrate this : See here 3, 4,
6, % i&; 365 tho' every Four next other are
harmonic a\ yet thefe are not fo, 3 : 6 : 9 i
Theorem XVI. If there are Four Numbers
idifpofed in Order, whereof one Extreme and
the Two middle Terms are in arithmetical
Proportion., and the fame middle Terms with
the other Extreme are in harmonical Propor-
tion., the Four are in geometrical Proportion*
as here, 2:3:4 : 6, which are geometrical,
and whereof 2:3:4 are arithmetical, and
3>
4, d harmonic ah
H i PEM'
Xi6 A Treatise* Chap. IV.
Demonstration. This Theorem con-
tains 4 Cafes, ffiid. If the Firft Three Terms
are arithmetical increafmg, and the laft Three
harmonica!) the Four together are geometrical*
Demonftr ation. Let a : b : c : d be Three Num-
bers, whereof a, b, c are arithmetical incre^r
fingfronxtf, and Z>, r, d harmonical^ then are #, b 9
r, d, geometrical; for fince out of the Jfarmo^.
nicals we have this geometrical Proportion^iz.
1) : d : ': c-b : d-c and alfo -b-a=c-b (fince a> b, c
are arithmetical) therefore b. : d : b-a : d-c ; and
confequently (- by 7heor. .5. ) b : d : .'' a,: c y or
a : b ■: : c : d- W. W. fi> Example. 2, 3, 4, 6.
ado. If the Firft Three are harmonic al decrea-
fing, 'and' the laft Three arithmetical, the Four
are geometrical ; this is but the Reverfe of the
Lift Cafe, and heeds no other Proof, pio. If
the Firft Three are arithmetical decreasing,
and the other Three harmonical) the Four are
geometrical) fuppofe a$ b) c are arithmetical
decreaiing, and b, c\ d ? harmonical) then //, b,
r, i, are geometrical, for out of the Harmonic als
we have this geometrical Proportion, viz. b : d
; ; h-c (=a-b) : c-d) therefore b j d : :~a : c, and
a :b : : c : d.i Example. 8 * 6 :: 4 : 3. qto,.
If the firft Three are harmonic al increafjng, and
the other Three -arithmetical) the Four are
geometridal; this is ■ the Reverfe of the- laft,
Ob serv'e. ".It muft hold reciprocally that if
Four Numbers are geometrical) and the- firft Three
arithmetical or harmonic al) the other Three
muft be contrariiy harmonical or arithmetical;
for to the fame Three Numbers there can be but
* ■' on©-
§ J. of MUSICK. n 7
one individual Fourth geometrical, and to the
Two.laft of them but one individual Third
arithmetical or harmonical, therefore the Ob-
ferve is true.
Theorem XVII. If betwixt any Two Num-
bers you put an arithmetical Mean, and aJfo an
'harmonical one,thc Fcur will be in geometrical
Proportion. Example. Betwixt 2 and 6 an arith-
metical Mean is 4, and an harmpnical one is 3,
and the Four are 2 : 3 : : 4 : 6 geometrical; the
Demonftration you'il find here.: Let ^ and £ be
•Two given Numbers, , an arithmetical Mean
by Theor. 2, is ~ and an harmonical Mean by
jtbeor. 11. ,|~ , and thefe Four are geometrical
a i- re : : ^ : b, which is proven by the equal
Products of the Extremes and Means.
§ 4. 57.^ Arkhmetick of Ratios geometrical, or
of the Compofition and Refolution of Ra-
' tios., -
BY the proceeding Definitions, the Exponent
of the geometrical Relation of TwoNum-
'bers is a proper Fraction, when we compare
the leffer to the greater, fignifying that the
• leffer is fuch a Part or Parts of the greater ,- fo
the Ratio of 2 to 3 is ~\ Signifying that ^2 is
:Two thirds of 3. Or, if we compare. the grea-
ter to the leffer, it is an improper Fraction,
which being reduced to its equivalent Whole
. . ' H 3 or
Ii8 A Treatise Chap.
or rhix'd Number, exprefles how many Times
and Parts of a Time the greater contains the
lefler ;• fo the Ratio of 1 3 to 5 is p m 2}, for
13 is equal to 2 Times 5, and 3 oyer : Or being
kept in the fractional Form fignifies that the
greater is equal to fo many Times fuch a Part
of the lefler as that lefler denominates ; and
this Difference of comparing the greater as An-,
tecedent to the lefler, or the lefler to the grea-?
ter,conftitutes Two different Species of Ratios \
One Number is faid to be compofed of o-
thers a when it is equal to the Sum of thefe o^
thers ; the Compound therefore rnuft be greater
than any of thefe of which it is compofed ;
and this is the proper Senfe of Compofition of
Numbers, fo 9 is compofed of 4 and 5, or 6
and 3, &c. alfo \ is compofed of, or equal to the
Sum off and J,. But thd"Ratios are Fractions
proper or improper, as they exprefs what Part
or Parts, or how many Times fuch a Part of
one Number another Number is equal to ,• yet
in the Arithmetick propofed they are taken in
a Notion very different from that of mere
Numbers ; for if we take the Exponents of
Two Relations as Numbers, and add them
together, the Sum is a Number compounded
of the Numbers added, but it is not a Ratio or
the Exponent of a Relation compounded of
the other Two Ratios $ fo that Compofition
and Refohtion of Ratios is not adding and
fubftrafting them as Numbers. What it is fee
|n the following Definition^ wherein I take the
Math pr ffxpomnt pf the Relation of Twq
§ 3 . of MUSIC K. iij>
Numbers to be the Quote of the Antecedent di-
yide4 by the Confequent.
Definition. One Ratio is faid to be
i compounded of others, when it is equal to the
Ratio betwixt the continual Product of the An-
tecedents of thefe others,and the continual Product
oft heir Confequent s multiplied as Numbers(V. e. by
the Rules of common Arithmetic^) or thus,one
Ratio is compounded of others, when, as a
Number, it is equal to the continual Product
of thefe others considered alfo as Numbers.
Example. The Ratio of i to 2 is compound-
ed of the Ratios of 2 to 3, and 3 to 4, be-
: caufe f is equal to | multiplied by J, alfo 40 to
> 147 is in the compound Ratio of thefe, viz. 2:
3, 5 : 7 and 4 : 7.
Theorem XVIII. Take any Series whate-
ver, the Ratio of the Firft Term to the laft
conlidered as a Number, is equal to the conti-
nual Product of all the intermediate Ratios mul-
tiplied as Numbers, taking every Term in Order
from the Firft as an Antecedent to the next.
I For Example. In this Series 3, 4, 5, 6, the Ra-
tio of 3 and 6 is ~, equal to the continual Pro-
1 duct of thefe J-, f-, r, {or when all the Numera-
t tors are multiplied together, and all the Deno-
1 imitators, it is plain the Products are as 3 to 6 y
becaufe all the other Multipliers are common to
both Products ; and it muff be true in every Se-
ries for the fame Reafon.
Corollary. If the Series is in continued
geometrical Proportion, the Ratio of the Ex-
tremes is equal to the common Ratio taken and
H 4 . null-
no ' ^Treatise Chap. IV,
multiplied into it fclf, as a Number, as oft as
there are Terms in the Series lefs one.
, Pr.oble M II. To find a Series of Numbers
which fliall be to one another (comparing them
.in Order each to the next) in any given Ra-
tios, taken in any Order ailigned, Rule. Mul-r
tiply both Terms of the ifl Ratio by the An-
tecedent of the 2d, and the Confequent of this
by the Confequent of the ift ; and • thus you
Jiave the ift Two Ratios reduced to Three
Terms, which multiply by the Antecedent of
the %d Ratio, and the Confequent of this by
the laft of thefe Throe, and ycu have the ift
r £\iTQQ Ratjos reduced to 4 Terms :'■ Go on
thus, multiplying the laft Series by the Antece-
dent of the next Ratio, and the Confequent of
this by the laft Term of that laft Series, v The
Juftnefs of the Rule appears from this, That
the Terms of eacli Ratio are equally multiplied.
JSxampk. The Ratios of 2 : 3, of 4 : $ and 6 :
s] are reduced to this -Series 48 : 72 : .go : 1-05*.
See the Operation.
- '..-., Observ.e. From the O-
2 :\3 peration .of this Rule it is
4:5 plain, , that the Extremes of
""""• the Series found are, the
8. 1,2. 15:
Product of all the Antece
One equal to the continual
6 : 7
48 : 72 : 90 : 105 dents, and the other to the
— — continual Product of all the
Cbnfequents of the. given
'Ratios ' y fo that- thefe Extremes are in thecoma
found Ratio of the given Ones \ which is other*
* -'"'■ - ' ' wife.
$4- of MUSIC K. hi
wife plain from the laft Proportion, fince all the
intermediate Terms of this Scries are in the
'Ratios given refpeftively. .And it- follows alfo,
that where any Number of Ratios are reduced
to a Series, tho' the Number of the Series will
differ according to the different Orders, yet be-
eaufe the intermediate Ratios are -the fame
in every' Order, the Extremes muft {till be in
the fame Ratio*
Theorem XIX. Every Ratio is compofed
of an indefinite Number of other Ratios -, for,
by Coral, to Theor. 5. if - any Two Numbers
are equally multiplied, the Produces are in the
fame geometrical Ratio, and by CoroL to Theor*
6. their Difference contains the Firft Difference,
as oft as the Multiplier contains Unity; therefore
it is plain that thefe Produces are the Extremes
of a Series, which can have as many middle
Terms as their Difference has Units JeTs one ; and
confequentJy by taking the Multiplier greater
you make the Difference of the Products greater,
which admitting flill a greater Number of mid-
dle Terms, reduces the Ratio given into more
intermediate Ones : So take the Ratio of 2 : 3,
multiply both Terms by 4, the Products are
8:12, and the Series is 8 : 9 : 10 : 11 : 12, but
multiply by 7, the Series is 14 : 15 : 16 : 17 :
18 : 15? : 20 : 21,
Observe. We may fill up the middle
Terms very differently, fo as to make many
different Series betwixt the fame Extremes : And
hereby we learn how to take a View of all the
mean
122 A Treatise Chap. IV.
mean Ratios, of which any other k compo-
fed.
Theorem XX. The geometrical Ratio of
any Two Numbers taken as a proper Fraction^
( /. e . making the leffer Number the Antecedent}
is lefs than that of any other Two Numbers
which are themfelves refpectively greater, and
yet have the fame arithmetical Ratio or Diffe-
rence. Example. The Ratio 2 : 3 taken as a
Fraction is \ lefs than that of 3 : 4, viz* |, or
than 5 : 6, viz. \.
Demonstration. Let a and a>^b repre-
fent any Two Numbers, let afyc and a>fac>fab
reprefent other Two which are refpeclively
greater than the firft Two, but have the fame
Difference b; take them Fraction-wife thus >
~ b and a 45r w if we reduce them to one com-
mon Denominator, the new Numerators will
be found aa^aofcab, and aa^ac^ab^bc %
which is greater than the other by be-, therefore
the Firft Fraction, to which the Numerator
aa^ac^ab correfponds, is leaft.
Problem HI. To reduce any Number of
Ratios to one common Antecedent or Confe-
quent. Rule. Multiply all their Antecedents
continually into one another, that Product is
the common Antecedent fought : Then multi-
ply each Confeqnent into all the Antecedents
(except its own) continually, and the laft Pro-
duct is the Confeqnent correfpondent to the
Confeqnent that was now multiplied. Or, mul-
tiply all the Confequents for a common Confe-
quentj and each Antecedent into all the Confe^
quent s (except its own) for a new Antecedent. So
thefe
§ 4 . of MUSIC K. 123
thcic Ratios •, 2 : 3, 3 : 4, 4 : 5 reduced to one
Antecedents are 24 : 36, 24 : 32, 24: 30, which
in one Scries are 24 : 36 : 32 : 30.
The Reafon of the Rule is plain from this,
that the Terms of each Ratio are ecjually mul-
tipli^ • (
ADDITION of RATIOS.
Problem IV. To add one or more Ratios
together, or to find the Compound of mdhllatigf*
Rule. Multiply all the Antecedents continually
into one another, and all the Confequents • the
Two Produces contain the Ratio fought ; which
is plainly this; Take the Ratios Fra&ion-wife,
(the Antecedent or each, whether 'tis greater
or lelfer than the Confequent^ being the Nume-
rator, and the Consequent the Denominator )
and as fractional Numbers multiply them con-
tinually into another, the laft Product is the
, Exponent of the Relation fought. Example.
Add the Ratios of 2 : 3, 5 : 7 and 8 :p, the Sum
ox: compound Ratio fought is 80:189. The
1 Reafon of the Rule is plain from the Definition
: of a compound Ratio in § 4. of this Chap-
1 ter.
Observe imo. To underftand in what
1 Senfe this Operation 'is called Addition of Ratios,
\ we muft confider that to compound Two or
more Ratios is in effect this, vip* to find the
Extremes of a Series whofe intermediate
Terms are refpectively in the Ratios given ; fo
to compound or add the Ratios^ 2 : 3 and 4 : 5V
is
H4 ^Treatise Ghap. IV.
is to rind the Extremes of Three Numbers,
whereof the ift fhall be to the 2 d as 2 to 3,
and the id to the 3d as 4 to 5. Such a Series
may in any Cafe be found by ProbL 2. and
in this Example it is8 : 12 : 15, for 8 igto 12
as 2 to 3, and 12 : 15 as 4 : 5, and 8 5SV is
the compound Ratio fought, which is called the
Sum of the given Ratios, becaufe it is the Effedfc
of taking to the Confequent o£ the ift Rati®,
conf dered now as an Antecedent, a new Con-
fequent in the id Ratio ; and fo of more Ratios
added.
■ ido. There is no Difference, as to this Rule,
whether all the Ratios to be added are of one
Species or not, u e. whether all the Antecedents
are greater than their Confequent s, or all lefs, or
feme greater fome lefs. For in this Rank 3 :
4:5:2 the Ratio of 3 to 2 is .compounded of
the intermediate Ratios 3 : 4, 4 : 5, and 5:2:
tbo' the laft is of a different Species from the
other Two; what Difference there is in the
'Application to nnijical Intervals fhall be explained
in its Place.
SUBS TRACTION of RATIOS;
Problem V. To fubftrac~t one Ratio from
'another. Rule. Multiply the Antecedent of
the Subftrahend into the Confequent of the Sub-
ftra&or, that Product is Antecedent of the
Remainder fought; then multiply the Antecedent
of the Subftra&or into the Confequent of the
■Subftrahend, and that Product is the Confequent
of
§ 4- of MUSIC K. ny
of the Remainder fought \ which is plainly
this ; Take the Two Ratios Fraction-wife, and
divide the one by the other according to the
Rules of Fractions. .Example. To fubftracl:
the Ratio of 2 : 3 from that of 3 : 5 ; the Re^
mainder is 9 : 1 o, for \ divided by \ quotes
?.
,0 "
The Re of on of this Rule is plain ; for, as-
the Senfe of Subftra&ion is oppofite to Addition,
lb mult the Operation be; and to fubftracl; one
Ratio from another fignifies the finding a Ratio,
which being added (in the fenfe of Probl. 4.)
to the Subftracl:er,or Ratio to be fubftracl:ed, the
Compound or Sum fliall be equal to the Sub-
ftrahend ,• and therefore, as Addition is done by
multiplying the Ratios asFraclions, fomuft Sub-
ftrattion be done by dividing them as Fraclions^
and fo in this Series 6:9 : 10, the Ratio 6:
1 o ( or 3 : 5 ) is compofed of 6 : 9 ( or 2 : 3)
and 9 : ioj which Compofition is done by
multiplying ~ Q into f whofe Product is jj or | :
So to fubftracl: 6 : 9 or 2 : 3 from 6:10 or 3 :
1 5, it muftbe done by a reverfe Operation divid-
ing ~ by \ whofe Quotient is * .
Ob s e r v e. As in Addition, the Ratios added
may be of the fame or different Species, fo
it may be in Subftra&ion; but it is to' be obfrrv-
ed here that the Two given Ratios to be fub-
ftracted, being confidered as Fractions, and both
proper Fractions, then, the lead being fubftra-
cl:ed from the greater, the Remainder is a Ra-
tio of a different Species, as in. this Series, 5* :
2 : 7, for take 7 from ft he Remainder is h But take
the
I ■
ii6 ^Treatise Chap. IV,
\he greater from the lefler, and the Remainder
U of the fame Species •, fo f from | there re-
mains ~, as in this Series 2:5: 7. Again fup-
pofe both the given Ratios are improper Fra-
ctions (/. ?> the Antecedents greater than the
Confequents ) if the leaft is fubftra&ed from the
greater^ the Remainder is of the fame Species j
but the greater from the leffer and the Remain-
der is of a different Species. Example. } from
£• remains r, as in this Series 7 15:2. But 7 - from
I remains 7, as here 7 : 2 i 5 ; thefe Obfervatt*
pns are all plain from the Rule*
MULTIPLICATION of RATIOS*
Problem VI. To multiply any Ratio bjj
a Number. This Problem has Two Cafes*
Case I. To multiply any Ratio by a whole
Number. Rule. Take the given Ratio as oft
as the Multiplier contains Unity, and add them
all by Probl. 4th. Example. 2 : 3 multiplied by
4, produces 16 : 8 1 ; or thus, Take the Ratio
as a Fraction, and raife it to fuch a Power as
the Multiplier expones, that is, to the Square if
'tis 2, to the Cube if 3, and fo oh.
For the Reafon of the Ride confider, That
as the multiplying any Number fignifies the
adding it to it felf, or taking it fo many Times
as the Multiplier contains Unity, fo to multiply
any Ratio fignifies the adding or compounding
it with it felf, fo many Times as the Multiplier
contains Unity, i. e. to find a new Ratio that
frail be equal to the given one fo oft compound-
ed
§ 4. of MUSIC K. n 7
ed,thus,to multiply the Ratio of 2 : 3 by the Num-
ber 4 fignifies the finding a 2fo/io equal to the
compound Ratio of 2 .- 3 taken 4 Times, which
is ]<5 : 81; for 2 : 3, 2 : 3, 2 : 3) 2 : 3, being
added by Prohh 4. amount to 16 : 81, and
to fill up the Series apply Probl 2.
Observe. The Product is always a ite/o
of the fame Species with the given Ratio's as is
plain from the jR»fc And if you'll complete
the Series by Probl 2. fc * turn the given
Matwio oft taken as the Multiplier exprcfles
into a Series it will be a re^/W £^
mr/i/ one. Thus, 2, 3 multiplied by £ pro-
duces i5, 8 1, and the Series hit: 24 : L l
54 : 81 ; and this Series fliows clearly the Im-
port of this Multiplication, that it is the finding
the Extremes of a Series, whofe intermediate
J-erms have a common Ratio equal to the gi-
ven Ratw.and which contains that Ration oft
repeated as the Multiplier contains 1.
Case II. To multiply any Ratio ;by a
. Iia&on that is, to take any Part of a piven
tor of the Fraction, according to the M Cafe
and divide that Product which is alfo a $$&
[by the Denominator, after the Method! of
i/tLVV r f f owin g iWrt the Quote
Sv rS: Rc T • fou S ht ^f#. To myi-
tis R T s i 27 > h A- **ft I*"*
;tipJy 8 . 27 by 2, the Product is 6a : 720 and
:this divided by 3 , according to the'nexr >/ S
? quotes the «** 4 i P, fo'that the Ratio,. *
8 ? Parts of the Ratio $ : 27. ' * *
1 The
ii% ^Treatise Chap. IV.
The Re of on of the Operation is this,
fince j Parts of i ( i. e. of once the Ratio to
be multiplied ) is equal to ~ Part of z
( or of twice the Ratio to be multipli-
ed ) therefore having taken that Ratio twice,
I muft take a Third of that- Product, to
Lave the true Product fought : And fo of other
Cafes* The Senfe of this Cafe will appear plain in
this Series 8:12:18:27 which is in continued,
geometrical '•' Proportion^ the common Ratio be-'
jflg that of 2 : 3 j confequently 8:27: contains 2 : 3
Three Times j or 2:3 multiplied by 3 produces
8 : 27 : Alfo 8 : 18 (equal to 4 : 9) contains
2 : 3 twice, and confequently is equal to \, Parts
of 8 : 27.
O b s e r v e. It produces the fame Thing to
divide the given Ratio by the Denominator of
the given Fraction, and multiply the Quote
( which is a Ratio) by the Numerator ; becaufe,
for Example, 2 Times f of a Thing is equal
to J of twice that Thing.
Corollary.. To multiply a Ratio by a
rnix'd Number, we mud: multiply it feparately,
Fir ft ^ Ey the integral Part (by Cafe 1. ) and'
then by the fra&ional Part ( by Cafe 2. ) and
fum thefe Produces (by ProbL 4.) or reduce the
mix'd Number to an improper Fraction,, and.
apply the Rule of the Iaft Cafe. Example. To .
multiply 4:9 by ir or "r," the Producl: is 8 ; 27,.
for in this Series 8 : 12 : 18 : 27, it is plain 6 :
27 is 3 -Times 2:3. And this is ^ of 4 19
(equal to 8:18) confequently 8 : 27 is equal to
3 Halfs or 1 and £ of 4 : 9,
Dt-
§ 4- vf MUSIC £ tfy
DIVISION of RATIOS.
Problem Vll. To divide any Ratio by $
Jvfumber. This Probl. has Three Cafes-,
Case L To divide any Ratio by a whole
Number, that is, to find fuch a Ratio as being
multiplied (or compounded into it felf) as oft
as the Divifor contains Unity^ ftiall produce
the given Ratio, Rule* Out of the Ratify
taken as a Fraction^ extract fuch a Root ias
the Divifor is the Index of, /. e-. thefquare Root
if the Divifor is 2 i the cube Root if the Divifor
is 3 5 &c and that Root is the Exponent oi the-
Relation fought. Example. To divide the
Ratio of 9 : 16 by 2, the fquare Root
of % is | which is the /foft'o fought.
The itejf/ora of this i£«/e? is obvious, iromt
its being oppofite to the like Cafe in Multiplicati-
on i and is plain in this Series^ 9 : 12: i6, which
is in the continued Ratio of 3 t 4* and imce the
multiplying 3 : 4 by 2, to produce 9 * 16', is
performed by multiplying \ by }, or fquar-
ing J, the Division of 9 : id by 2 to find 3 : 4*
can be done no other ways than by extracting
\ the fquare Root of ~ $ which is \ • and fo of
other Cafes; which will be all very plain to thenl
who underftand any Thing of the Nature of
Powers and Roots. Or folve the Probl. thus <
Find the firft of as many geometrical 'Means be-
twixt the Terms of the given Ratio as the
Divifor Contains of Units Jefs one,that .compared
with the leffer Term of the given Ratio con~
I tains
130 ^Treatise Chap, IV.
tains the Ratio fought ; thus 9:12 is the
Anfwer of the proceeding Example.
Case II. To divide a Ratio by a Fraction,
that is^ to find a Ratio of which fuch a Part
or Parts as the given Fraction expreffes fhall
be equal to the given Ratio. Rule. Multiply
it by the Denominator (by Probl. 6. 1 Cafe) and
divide the Producl: by the Numerator ( by Cafe
-1 of this Probl.) the Quote is the Ratio fought.
Or divide the Ratio by the Numerator, and
multiply the Quote by the Denominator. Ex-
ample. To divide 4:9 by f or to find -Parts
of 4 : o I take the Cube of f, it is f-^, whefe
fquare Root is f 7 the Ratio fought. The Reqfon
or the Operation is contained in this, that it is
oppofite to Cafe 2. of Multiplication. And
becaufe 8 : 27 multiplied by ~, produces
4 : 9, fo 4 : 9 divided by \ ought to quote
B : 27.
. Corollary. To divide a Ratio by a mix'd
Number ; reduce the mix'd Number to an
improper Fraction, and divide as in the laft
Cafe. - -
Case III. To divide one Ratio by another,,
both being of one Species; that is , to find how
oft the one is contained in the other ; or how
oft the one : t ought to be added to it felf to
make a Ratio equal to the other. Rule*
$ub(tra£fc the Divifor from. the Dividend (by
Probl. k. ) and the fame Divifor again from
the laft Remainder ,- and fo on continually, 1
till the Remainder be a Ratio of Equality i
and then the Number. of Subtractions is the
Number
§ 4- , of MUSIC K ^ 13 1
Number fought ; or, till the Species of the
Ratio change, and then the Number of Subtra-
ctions Ms one is the Number of Times the
w h of e Divifor is found in the Dividend, and the
Ikil Remainder except one is what the Dividend
contains over fo many Times the Divifor. Ex-
ample. To divide the Ratio 16:81 by 2 : 3,
1 fubftract 2 : 3 from 16 : 8i, the Remainder is
48 : 162 equal to 8 : 27 • from this I iubftradfc
2 gp, the id Remainder is 24 : 54, equal to
4 # j from this I fubftra£t 2 : 3, the 3^ Remain-
der is 1 2 : 1 8 or 2 : 3,- from this I fubflracl: 2 : 5,
the qth Remainder is 6 : 6 or 1 : i, a Ratio of
Equality; therefore the Quote fought is the
Number 4, iignifying that the Ratio 2 : 3 taken
4 Times, is equal to 16 : 54 5 as you fee it
all in this Series 16: 24: 36: 54: 81. '.£#-
fl//z/>/s 2. To divide 1 2 : 81 by 2 : 3, proceed ill
the .feme Manner as before, and you II find the
Remainders to be 2 : o, 1 * 3, 1 • 2, 3 : 4, 9: 8>
and becaufe the kft changes the Species, I juftly
conclude that the Ratio 12 : 81 does not con-
tain 2 : 2 five Times, but it contains it 4
Times and 3 : 4 over ; for 2 : 3 multiplied by
4 produces 16 : 81, which added to 3 : 4 makes
exactly 12:81, as in this Series 16 : 24 : 36 :
5*4 : 81 :ib8 whofe Extremes 16 : 108, (equal
to 12 : 81 ) is in a Ratio compounded of
16 : 81 and 81 : 108 (equal to 3. 4. )
, Observe, i The Two Ratios given muft
be of one Species - 3 becaufe the Senfe of it is,
to find how oft the Divifor muft be added to
It felf to make a Ratio equal to the Dividend;
I 2 and
t$z A Treatise Chap, IV.
and in multiplying, any Ratio by a whole
Number, that Ratio and the Product are al-
ways of one Species, as was obferved in ProhL
6. therefore 'tis plain that the Ratio of the
Dividend, taken as a Fra&ion, muft be leffer
than the Divifor fo taken, the Antecedent being
leaft, i. e . thefe Fractions being proper, and con-
trarily if they are improper; the Reqfon is plain,
becaufe in an increafing Series, i. e. where all
the Antecedents are leffer than their ConfeqilHits^
the Ratio of the Firft to the leaft Extreme is
lefs than the Ratio of any Two of the inter-
mediate Terms, and yet, according to the
Nature of Ratios^ contains them all in it; but
in a decreasing Series* u e. where all the
^Antecedents are greater than the Confequents y
the iffl to the leaft, or the greateft Antecedent
to the leaft Confequent, is in a greater Ratio
than any of the intermediate, and alfo contains
them all : So in this Series 2:3:4:5, the
Ratio 2 : $ contains all the intermediate Ratios^
and yet } is lefs than £ or | or \ ; but take
the Series reyerfely, then J is greater than } or
a
§ j. of MUSIC K. i 33
(j$. Containing an Application of the preceed-
ing Theory of Proportion to the Intervals
of Sound*
IT has been already fliewn that the Degrees
of Tune are proportional to the Numbers of
Vibrations of the fonorous Body in a given
Time, or their Velocity of Courfes andRecour-
fes j which being proportional, in Chords, to
their Lengths (ceteris paribus) we have the juft
Meafures of the relative Degrees of Tune in the
Ratios of thefe Lengths -, the grace Sound be-
ing to the acute as the greater Length to the
lener.
The Differences of Tune make Difiance or
Intervals in Mufick> which are greater and lef-
fer as thefe Differences are, whofe Quantity is
the true Object of the mathematical Part of
Mufich Now thefe Intervals are meafured,
not in the fimple Differences, or arithmetick
Ratios of the Numbers expreifmg the Lengths
or Vibrations of Chords, but in their geometrical
Ratios ; fo that the fame Difference of Tune %
k e, the fame Interval depends upon the lame
geometrical Ratio ; and different Quantities or
Intervals arife from a Difference of the geome-
trical Ratios of the Numbers exprefling the Ex-
tremes, as has been already fliewn • that is,
I 3 eoual
134 -^Treatise Chap. IV.
equal geometrical Ratios betwixt whatever
Numbers, conftitute equal Intervals Jo\xt unequal
Ratios make unequal Intervals.
But now ob/erve, that in comparing the
Quantity of Intervals, the Ratios expreffing
them muft be all of one Species 3 otherwife
this Abfurdity will follow, that the fame Two
Sounds will make different Intervals ;. for JE%>
ample, Suppofe Two Qhords in Length, as 4 and
5, 'tis certainly the fame Interval of Sound, whe-
ther you compare 4 to 5, or 5 to 4, yet the
Ratios of 4 : 5 and 5 » 4 taken as Numbers, and
expreft Fraction-wife would differ mQuantity,and
therefore differentRatios cannot without thisQua-
Hflcation make in every Cafe different Intervals.-
I n what Manner the Inequality of Intervals
are mcafured, (hall be explained immediately ,
and 'here take this general Character from the
Things explained, to know which of Two or
more Intervals propofed are greateft. If all the
Ratios are taken as f roper Fratlions, theleaft
Fraction is the greateft Interval. But to fee
the Reafon of this, take it thus,- The Ratios
that exprefs feveral Intervals being all of one
Species, reduce them (by Probl. *». of this Chap.):
to one common Antecedent, which being lefler
than the Con/eqiientSyth^t Ratio which has the
greateft Consequent is the greateft Interval.
The Reafon is obvious, for the longeft Chord
gives the graved Sound, and therefore muft be
at greateft Diftance from the common acute
Sound, Or contrarily, reduce them to one
common Confequent greater than the Antece-
dents 2
§ f. of MUSIC K. 135:
dents, and the leffer Antecedent cKpreffes the
ac liter Sound, and eonfequcntly makes with that
common fundamental or graveft Sound, the
greater Interval.
I t follows that if any Series of Numbers are
in continual arithmetical Proportion^ comparing
each Term to the next, they cxprefs a Series
of Intervals differing in Quantity from mil: to
laft; the greateft Interval being betwixt the
Two leaft Numbers, and fo gradually to the
greateft, as here 1 : 2 : 3 : 4. 1 : 2 is a grea-
ter Interval than 2:3, as this is greater than
3 : 4. The Reafon why it muft hold fo in eve-
ry Cafe is contained, in Theor. 20. where it
was demonftrated that the geometrical Ratio
of any Two Numbers taken as a proper Fraction
( i. e. making the leffer the Antecedent ) is lefs
than that of any other Two Numbers, which are
themfelves refpectivejy greater, and yet have the
fame arithmetical Ratio or Difference : And
by what has been explained we fee that the
leffer proper Fraction makes the greater Interval.
Thus we can judge which of any Intervals
propofed is greateft, and which leaft, in gene-
ral ; but hew to meafure their feveral Differences
or Inequalities is another Queftion ; that whofe
Extremes make the leaft Fraction is the great-
eft Interval^ and fo, in genera], the Quantities of
feveral Intervals are reciprocally as thefe Fracti-
ons; but this is not always in a fimple Propor-
tion. For Ex 'am fie , The Interval 1:2, is to
the Interval 1 : 4 cxactiy as \ to \ (or as 1 to 2)
the Quantity of the laft being double the other.
",': 1 4 But:
l^G A Treatise Chap. IV.
But 2 i 3 to 4 : 9 is not as f to *-, but as i to 2,
as fliall be explained., Sounds themfelves are
expreffed by Numbers, and their Intervals are
reprefented by the Ratios of thefe Numbers,
fo thefe Intervals are compared together by
comparing thefe Ratios^ not as Numbers, but as
Ratios i and I fuppofe every given Interval is I
expreffed by expreffing diftin&Iy the Two Ex-
tremes, i r ii their relative Nurhbers.
I fliall now explain the Compofition and Re~
folutimi of Intervals^ which is the Application
of the preceeding Arithmetick of Ratios ,• and
tl^is I fhall do, Firft'm general, without Regard
tp the Difference of Concord and IJi/cord,
which (hall imploy the reft of this Chapter ;
and in the next make Application to the
various Relations and Compofit ions ok Concords,
and after that of Di fiords in their Place.
In what Senfe Ratios are faid to be added
and fubftracled, &c. has been explained, but
in the Compofition of Intervals we have a
more proper Application of the true Senfe of
adding and fubftra&ing, (jc. The Notions of
Addition and Subftraclion,Cjf. belong to Quan-
tity ; concerning which it is an Axiom, that
the Sum, or what is the Refult of Addition,
muft be a Quantity greater than any of the
Quantities added, becaufe it is equal to them
all ; And in fubftra&ing we take a lelfer
Quantity from a greater, and the Remainder is
Jefs than that greater, which is equal to the
Sum of the Thing taken away andtheRemainr
der, A mere Relationcannot properly be called
guan-
§ y. of MUSIC K. 137
Quantity, and therefore the geometrical Ratio
of Numbers can be no otherwife called Quan-
tity than as by taking the Antecedent and Con-'
fequent Fraction-wife, they exprefs what Part
or how many Times fuch a Part of the Con-
fequent the Antecedent is equal to ; and then
the greater Fraction is always the greater Ratio,
But the Compofition of Ratios is a Thing of a
quite different Senfe from the Compofition of
mere Numbers or Quantity \ for in Quantities;
Two or more added make a Total greater
than any of them that are added ; but in the
Compofition of Ratios, the Compound confider-
cd as a Number in the Senfe abovementioned,
may be lefs than any of the component Parts.
Now we apply the Idea of Diftance to
the Difference of Sound in Acutensfs and
Gravity in a very plain and intelligible Manner,
fo that we have one univerfal Character to de-
termine the greater or leffer of any Intervals
propofetl i according to which Notion of Great-
nefs and Littlenefs all Intervals are added and
; fubftracted, &c, and the Sum is the true and
proper Compound of feveral leffer Quantities,-
f and in Subtraction we actually take a leffer
Quantity from a greater; but the Intervals
I themfeives being expreffed by the geometrical
Jiatio of Numbers applied to the Lengths of
Chords ( or their proportional Vibrations) the
1 Addition and Subtraction, eye. of the Quantities
of Intervals is performed by Application of
|he preceeding Arithmetick of Ratios.
on.
138 ^Treatise Chap. IV<
Note. In the following Problems I conftant-
ly apply the Numbers to the Lengths of Chords,
and fo the leffer of Two Numbers that expre£
any Interval I call the acute Term a nd the
other the grave.
ADDITION <£ INTERVALS.
Problem VIII. To add Two or more In-
tervals together. Rule. Mutiply all the acute
Terms continually, the Produd is the acute
Term fought - 3 and the Product of the grave
Terms continually multiplied, is the grave Term
fought ; that is, Take the Ratios as proper
Fractions,- and add them by Probl. 4. -Efc-fe
ample. Add a %th 2 : 3 and, a qth 3 : 4, and a |
3d g. 4 : 5, the Sum is 24 : 60 equal to 2 : 5.I;
a %d g. above an clave.
Observe. This is a plain Application of J
the Rule for adding of Ratios, and to make |
it better underftood, fuppofe any given Sound
represented by a, and another Sound, acuter
or graver in any Ratio, reprefented by b ; if
again we take a Third Sound fall acuter or
graver than b, and call it c, then the Sound
of c being at greater Diftance from a, towards
Acutenefs or Gravity, than b is, the Interval
betwixt a and c is equal to the other Two
betwixt a b and b cl And fo let any Number
of Intervals be propofed to be added, we are
to conceive fonie Sound a as one Extreme of
the Interval fought ,• to this we take another
Sound b acuter ox graver in any given Ratio ; then
a Third Sound c acuter or graver than b in
'at. an-
§ f, of MUSIC K. i 3?
another given Ratio, and a /\th Sound d acuter
or graver than c, and fo on > every Sound always
exceeding another in Acutenejs ox Gravity, and
all of them taken the fame way, i, e. all
acuter, or all graver than the proceeding, and
confequentiy. than the firft Sound a ; and then
the fiiit and laft are at a Distance equal to the
Sum of the intermediate Distances. For
Example. If 5 Sounds are reprefented by a, b,
c, d, e exceeding each other by certain Ratios
I oiAcutenefs or Gravity from a to £,the Interval
1 a : e is equal to the Sum of the Intervals a :
1 b, b : c, c : d, d : e.
■ Now that the Rule for finding the true Di-
J fiance of a : e is juft,you'll eafily perceive by con-
.iidering that Intervals are reprefented by Ratios;
therefore feveral Intervals are added by com-
) pounding theRatio s that exprefs them; for if the
(given Intervals or Ratios are reduced, by
jProbl. 2. to a Scries continually increasing or
decreasing, wherein every Number being
'antecedent to the next, they Shall contain in
Order the Ratios given, i. e. exprefs the
'given Intervals, 'tis plain the Ratio of the
•Extremes of this Series Shall ' be compoled of
'all the intermediate (which are the given)
o\RatioSj and therefore be the Sum of them
k according to the true Senfe in which Intervals
are added, , as it has been explained ; fo in the
proceeding Example, in which we have added
a 5/"/? 2 : 3, a qth 3 : 4 and sl-$4 g- 4 : S-> the
Compound of thefe Ratios is 24 : 60 or 2 :
5 i for take ttiem in tae Order propofed they
are
140 -^Treatise Chap. IV.
are contained in this fimple Series, 2:3 .'4 :
5, which reprefents a Series of Sounds gradually
exceeding each other in Gravity from 2 to $
by the intermediate Degrees or Ratios pro-
pofed ; fo that 2 : 5 being the true Sum of
thefe Intervals, and the true Compound of the
given Ratios, fliews the Rule to be juft.
Again take Notice, that tho' in the Com-
pofition of Ratios it is the fame Thing whether
they are all of one Species or not, yet in their
Application to Intervals they muft be of oner
Kind. I have already (hewn what Abfurdity
would follow if it were otherwife, but you may
fee more of it here ; fuppofe Three Sounds re-
prefented by 4 : 5 : 3, tho 4: 3 is the true Com-
pound of thefe Ratios 4 : 5 and 5 : 3, yet it
cannot exprefs the Sum of the Intervals re-
prefent ed by thefe j for if 4 reprefent one
Extreme and 5 the middle Sound {graver than
the former) 3 cannot pofiibly represent another
Sound at a greater Diftance towards Gravity,
becaufe 'tis acuter than 5, and therefore in^
ftead of adding to the Diftance from 4, it
diminiflics it; but it is the fame Interval (tho*
in fome Senfe not thp fame Ratio) whether
the Jefler or greater-re antecedent \ and the Sum
of thefe Two Intervals cannot be reprefented
but by the Extremes of a Series continually
increasing or decreafing from the leaft or
greateft of the Numbers propofed, becaufe they
cannot otherwife reprefent a Series of Sounds
continually riling or falling, the Ratio of the
Extremes of which gind of Series can only he
§ tf ef MUSIC K. 141
called the Sum of the intermediate Diftances
or ntervalof Sound; and fo the preceeding Ex-
ample muft be taken thus* 3:4:5, where 3 :
5 is not only the compound Ratio of 3 : 4 and
4 : 5, but expreffes the true Sum of the Inter-
nals reprefented by thefe Ratios.
It is plain then from this Explication, that
in Addition of Intervals the Sum is a greater
Quantity than any of the Parts added, as it
ought to be, according to the )uft Notion of
the Quantity of Intervals . but it would be
otherwife and abfurd if the Ratios expreffing
Intervals were not taken all one way ; fo in the
preceeding Example tho' 4 : 3 is the Compound
of 4 : 5 and 5 : 3, yet eonfidered as a Fracti-
on I it is greater than \, and confequently a
letter Interval, by the Character already efta-
bliftied.
Problem IX. To add Two or mora
Intervals, and find all the intermediate Terms ,-
a certain Order of their Succeflion being affigned,
from the graveft or the acutefi Extreme.
Rule. If the given Intervals are to pro-
ceed in Order from the acutefi Term, make
the letter Numbers Antecedents; if from the
\gravefi, make . the greater Antecedents, and
ithen apply the Rule of ProbL 2.
Example. To find a Series of Sounds, that
from the acutefi to the graveft fhall be in Or-
der (comparing the ifl to the 2d, and the 2 d
to the 3d,and fo on ) a %d g : qth : 3d I : $th:
(Working by xhe Rule I fad this Series 120: 150:
200
14% /^Treatise Chap»IV,
200 .240: 360, or reduced to lower Terms by Di-
vifion they are 12 : 15 : 20 : 24 : 36. See the O
peration here. But if the fame Intervals arc
to proceed in.
4:5 - - - - - • 3d gr.
3:4---- qth.
12 : 15
60 : 75
20
5
100 : 120
2
3 J Jeff.
I - 5^
120 : 150 : 200 : 240 : 360
that Order from
the grave/} Bx-
tremes, the Se-
ries is 90 : 72
54 : 45" : 30.
Observe, Id
adding feveral
Intervals in a
coritinuedSeries,
the Sum or Ra-
tio of the Extremes muft always be the fame
whatever Order they are taken in ; becaufe in
any Order the Ratio of the Extremes is the
true Compound of all the intermediate Ratios.
or the Ratios added, which being individually
the fame, only in a different Order, the Sum
muft be the fame ; but then according to the
different Orders the Series of Numbers will be
different, fo if we add a /\th 3:4, %d gr.4 : 5
and a %d lejj'. 5 : 6, thev
can be taken in Six dif-
ferent Orders, which are
contained in thefe Six
different Series, whicl
contain all the different
Orders both from Gravy
tj and Acutmefs*
SUB-
3 : 4 :
4 : 5 :
5:6:
c :
6
8
6
8
I 10
16 : 8 :
10
20
12 : 15
: 20
: 24
l$ 1 20
: .24
! 3°
§ 5 . of MUSICK. 143
SUBSTRACTION of INTERVALS.
Problem X. To fubftrad a leffer Interval
from a greater. Rule. Multiply the acute
Terms of each of the given Intervals by the grave
Term of the other, and the Two Products are
in the Ratio of the Difference fought, that is^
take the Ratios given as proper Fractions, and
fubftract them by Prohl 5.
Example. Subftracl a $th 2 : 3 from an
ObJave 1 : 2, the Remainder or Difference is a
$tk 3 : 4. See the Intervals in this Series (made
■ by reducing both the Intervals given to a com-
js; mon Fundamental by Probl. 3 ) 6 : 4:3 the
I Extremes 6 : 3 are OVtave^ the intermediate
) Ratios are 6:4a 5th , and 4:3a 4^, therefore
\ any one of them taken from OtJave leaves the
j other.
II The Reafon and Senfe of the Rule is
if obvious ; for as Subflraclion is oppofite to Additi-
i on, fomuft the Operation 'be; and this is a plain
! !i Application of the Subtraction of Ratios^ with
I [» the fame Limitation as in Addition, viz. that
t! \ the Ratios mnft be taken both one way, fo
1 11 that we take always a leffer Quantity from
i i a greater, and the Remainder is lefs than that
[ greater, according to the true Chara&er where-
by the greater and teMntervals arediftinguiflied.
Observe. The Difference of any Two
Intervals expreffes the mutual Relation betwixt
any Two of their fimilar Terms, i, e. Suppofe
• any Two Intervals reduced to a common acute
or
144 A TiUATiSE Chap. IV.
or grave Term, their Difference ts the Interval
contained betwixt the other Two Terms,- and the
Ratio expreflingit is called the mutual Relation
of the Two given Intervals -> fo the Difference or
mutual Relation of an Otlave and $th is a qth
MULTIPLICATION otlNTERVALS.
Because it is the fame Interval whether
the greater or lefTer Number be Antecedent oi
the Ratio^ and in- all Multiplication the Multipli-
er muft be an abfolute Number^ therefore
Multiplication of Intervals is an Application
of Probl, 6. without any Variation or Limi-*
tation. I need therefore only make Examples^
and refer to that Problem for the Rule.
Problem XL Cafe i .To multiply an Interval
by a whole Number. Example. To multiply
a fib 2 : 3 .by 4. the Produtt is 16 : 81 the
qth Power of 2 and 3 ; and the Series of in-
termediate Terms being filled up is 16 : 24 :
36 : 54 : 8 1, expreifmg 4 Intervals in the-
continued Ratio of 2 : 3.
Case II. To multiply an Interval by a
Fraction. Example. Multiply the Interval 8 2
27 by |, the Product, 7. e. \ Parts of the given
Interval is 4 : 9, for £ is the Square of the cube
Root of | 7 . See this Series, 8 : 12 : 18 : 27, in
the continued Ratio of 2 : 3, where 8:18
(or 4 : 9) is plainly 2 Thirds of 8 : 27.
Note. If thefe Two Cafes are joyned we can
multiply any Interval by any mixt Number : Or
we may turn the mixt Number to an improper.
Fraction, and apply the id Cafe % (Zo* ,
§ y. tf MUSICK. _ i 4 y
Corollary. From the Nature of Multi-
plication it is plain, that we have in thefe Cafes
a Rule for finding an Interval, which ftia U be
to any given One, as any given Number to any
other ; for 'tis plain if we take thefe given Num-
bers in form of a Fraction, and by that Fraction
multiply the given Interval, we flail have the
Interval fought, which is to that given as the
Numerator to the Denominator •> fo in the
preceeding Example, the Interval 4 : 9 is to
i8 : 27 as 2 to 3. Eut ob/erve, if the Rcot to
be extracted cannot be found, then the Problem^
ftricUy fpeak'ing, is impo(fible,and we can exprefs
the Interval fought only by irrational Num-
bers; Example, To multiply a 4th 3 : 4 by f-,
i Le. to take f- Parts of it,it can only be expreifed
by the Ratio of the Cube Root of 9 to the Cube
Root of 1 6, or the Square of the Cube Root of
5, to the Square of the Cube Root of 4. And
thebeft We can do with fuchCaies,if they are to
be reduced to Pra<5tice,is to bring the Extraction
of the Root as near the Truth as may ferve
Our Purpofe without a very grofs Error,
But if 'tis propofed to find Two Intervals
:hat are as Two given Numbers* this can eahly
)e done by multiplying any Interval^ taken at
Pleafure, by the Two given Numbers feverally ,•
tis plain the Produ&s are in the Ratio of thefe
lumbers*
K DIVI-
i4<£ A Treatise Chap.
DIVISION of INTERVALS.
Here alfo there is nothing but the Applica-
tion of ProbL 7. to whicn I refer for the
Rules j and only make Examples.
Problem XII. Cafe 1. To divide an Inter-
val by a whole Number, i. e. to find fuch an
aliquot Part of that Interval as the given Num-
ber denominates.
Example. Divide the Interval 4 : 9 by %'
that is, find the Half of it ; the Anfwer is a
%th 1 : 5, for Two $hts make 4 : 9 y as in thisi
Series, 4:6:9.
Case II. To divide an Interval by aFra&i
on, that is, to find an Interval that fhall be to,
the given one,as the Denominator of the Fracti-
on to the Numerator.
Example. Divide the Interval 1 : 4 by \\
the Quote is 1 : 8, which is to 1 : 4, as 3 to 2.,
See this Series, i, 2, 4, 8.
NO TE. To divide by a mixt Number, wd
can turn it to an improper Fraction, and do as
in Cafe 3.
Observe. As Multiplication and Divifior/
are direcUy oppofite, fo we have by Di virion a:
well as by Multiplication, a Rule to find ar
Interval, which fhall be to a given .one, as an\
given Number to another : Thus, if the Inter
val fought rauft be greater than the given one
make the leaft of the given Numbers the Nu
merator, and the other the Denominator of \
Fraction, by which divide the given Interval
bu
% f t of MUSIC K. t 4 r
but if the fought Interval muft be leffer than
the given, make the greater Number the Nu-
merator j which is all directly oppofite to the
Rule of Multiplication : And, as I have already
obferved in Multiplication, if the Roots to be
extracted by the Rule cannot be found, then
there is nolnterml that is accurately to the gi-
ven one as the Two given Numbers.
Case III. To divide one Interval by ano-
ther, that is i to find how oft the leffer is con-
tain'd in the greater. Rule. SubftracT: ( by
ProbL 10.) the leifer from the greater, and
the fame Divifor from the laft Remainder
: continually till the Remainder be a Ratio
of Equality, or change the Species ; the
} Number of Subtractions, if you come to a
\ Ratio of Equality, is the Number of Times
the whole Divifor is to be found in the Divi-
f dend : But if the Species change, the Number
f of Subftra&ions preceeding that in which the
Remainder changed,is the Number fought: But
then, there is a Remainder which belongs alfo
'to the Quote, and it is the Remainder of the
Operation preceeding that which changed ,• fo
that the Dividend contains the Divifor fo oft as
that Number of Subftraclions denotes and con*
tains that Remainder over, which is properly
the Remainder of the Divifion.
Example I. To find how oft the Inter*
fval 64 : 125 contains 4 : 5* By the Rule I find
ThreeTimes.
, Example IL To find how oft an Sve 1 : 2
contains a 3d g. 4 : jr. you'll find Three Times,
W>- K 1 and
148 ^Treatise Chap. IV.
snd thk ht&@<0 over, viz. 125 '• 128. For, Fir ft ,
I iiibftra^: 4^S" om l • ' 2 » tne ^rft Remain-
der is 5" : 8 Hfem this I fubftracl 4 : 5, the zd
Remainder is 2.5: 32 ; from this I fubftrad 4 :
5, the 3^ Remainder is 125 : 128 ; from this I
lubftrajtS T4 ?;.? 5 5 the 4^ Remainder is 62?: 51 2,
which-- ili"©f a different Species, the Antecedent
being here greateft, which in the other Ratio
is lead-; therefore the Quote is 3, and the Ra-
tio or Interval 125 : 128 over. See the Proof
in this Series, 64 : 80 : 100 .'125 : 128. which
is in the continued Ratio of 4 : 5. 64 : 125
is equal to Three times 4 : 5, and 64 : 128 is
equal to 1:2.
Thus far only I proceeded with the Anfwer
in Cafe 3. of ProbL 7. for dividing of one Ratio
by another. Now I add, that if we would
make the Quote complete and perfect, fo that
it may accurately fhew how many Times and
Parts of a Time the Dividend contains the Di«
vifor, ( if 'tis poflible ) then proceed thus, viz*
Take the Remainder preceeding that which:
changed, by it divide the given Divifor, until
you come to a Ratio of Equality, or till the j
Species change, and then take the Remainder
(preceeding that which changed of thiiDivifon);
and by it divide the laft Divifor; and fo on
continually till you find a Divifion.tllat ends in
& Ratio of Equality ; then take the given Di-
vidend and Divifor, and the Remainders of each;
Divifion, and place them all in order from Left
to Right, as in the following Example. Now,
each of thete Ratios having been divided by the
n^v§
§ jr. of MU SICK. 149
next towards the right Hand, they have aJl
been Dividends except the lea ft ( or that next
the right) therefore over each I write the Quote
or whole Number of Times the next leffer was
found in it ; then nnmbring thefe Dividends
and Quotes from the Right, I fct the firft Quote
under the firft Dividend, and multiplying the
firft Quote by the fecond, and to that Produd
adding i, I fet the Sum under the id Dividend:
Again, I multiply that laftSum by the 3 J Quote,
and to the Product add the Quote fet under the
firft Dividend ; and this Sum I let under the 3 d
Dividend; again, I multiply the laft Sum by the
qth Quote, and to the Product add the Number
fet under the id Dividend, and I fet this Sura
under the qth Dividend ; and fo on continually,
multiplying the Number fet under every Divi-
dend by the Quote fet over the next Dividend
(on the Left), to the Product I add the Num-
ber fet under the laft Dividend (on the Right) :
When all this is done, the Numbers that ftand
under each Dividend, exprefs how oft the
laft Divifor (which is the firft Number en the
Right of the Series of Dividends) is contained in
each of thefe Dividends \ and confequently
thefe Dividends arc to one another as the Num-
ber fet under them : Therefore, in the iaft
Place, if the Numbers under the given Divi-
dend and Divifor are divided, the greater of
them by the teller, the Quote iignines how oft
the Interval given to be divided contains the
other given one.
K 3 Example.
♦
2
1
3
2048,
i:
**>
1 :
8,
II
4
3
ijo A Treatise Chap. IV,
' Example. Divide the Internal 1 : 2048 by
1 : 16". According to the Rule I fubftradt 1 : 16
from 1 : 20485 and have two Subftraclions, with
a Remainder 1:8 (for the qth Subftraclion
changes the Species ) then I fubftraft 1 : 8 from
1 : 1 (£5 and after one Subtraction there remains
1 : 2 ( the 2^ Subftra&i on changing.) Again Ifuh*
{tract 1 : 2 from 1 : 8, and after Three Subftra-*
£tions there remains a Ratio of Equality. Now
place thefe according to the Rule, as in the fol-
lowing Scheme, and divide 11 by 4, the Quote
fliews, tha*t the
given Dividend
2 1 : 20485 contains
the Divifor 1 :
165 2 and J-
Parts of a Time, L e, that it contains 11 16
twice ,- and moreover. 3 qth Parts of 1 : icT,
which you may view all in this Series 1 : 2:4:
8 ; 161 32: 64? 128: 256: 512: 1024: 20485
in the continual Ratio of 1 : 2 $ in which we
fee 1 : 16 contained two Times, as in thefe
three Terms 1 : 16: 2565 then remains 25*6':
20485 equal to 1 : 8. which you fee is equal to
3 4^ h Parts of 1 : 1 6 9 viz. three Times 1 : 2,
which is a qth of 1 : 1 6, as you fee in the Se*
ries.
For a more general Demon ftration, fup->
pefe any Quantity , Number or Interval, repre-
sented by a and a leffer by b ; let a contain b
Two Times ( which Two is fet over a ) and
c the Remainder. Again let b contain c Three
times ( which Three is fet over b ) and d the
Rej
<§ r . of MUSIC K. i 5 t
Remainder. Then let c contain d Five times
( which Five is fet over c ) and e the Remain-
der. Laftly^ Let d contain e Four times
( fet over d ) and no Remainder ( i. e. a Ratio
of Equality.- ) Now becaufe d contains e Four
times, I fet 4 under J,
2354 then c containing d Five
ax b: c: d: e times, and ^containing
155: 67:21:4 e Four times, therefore c
; • muft contain e as many
times as the Product of Five into Four, viz.
Twenty times > y but becaufe c is equal to Five
times d and to e over> and e is contained in the
Remainder, viz, it felf once, therefore e is con-
tained in c Twenty one times. Again b con-
tains c Three times and d over, and c contains
e Twenty one times precifely, therefore b muft
contain e as oft . as the Sum of Three times
21, viz. 63 and 4 which is 6j ; then a con-
tains b Two times and c over, aifo b contains
e Sixty feven times, therefore a contains e as
oft as the Sum of Two times Sixty feven, viz.
134 and 21, which is 155. The other Inferen-
ces are plain, viz. 11110. That each of thofe In-
tervals a : b : c, &c. are to one another, as the
Numbers fet under them 3 - for thefe are the Num-
bers of Times they contain a common Meafure
e , And confequently, ido. If any of thefe Num-
bers be divided by another, the Qiiote will
fiiew how oft the Interval under which the
Dividend ftands, contains the other.
Corollary. Thus we have found a Way
to difcover the Ratio betwixt any Two Inter-
K 4 vals 9
iji ^Treatise Chap. V,
mls^ if they are commenfurable \ fo in the pre* I
ceedirig Example^ the Interval i 12048 is to ]
1 ; 1 6 y as the Number 11 to 4. But obferve^ if
the Divifions never came to a Ratio of Equa-i
lity, the given Intervals are not commenfurable,
tor as Number to Number ; yet we may come
n^ar.the Truth in Numbers, by carrying pn the
PiViilon a considerable Length,
CHAP. V.
Containing a more particular Confide ra^
tion of the Nature, Variety and Com^
pofition of Concords, in Applka-*
tion of the preceedmg Theory.
J J E have already diftinguifhed and de-
fined fimple and compound Intervals^
which we fliaJl now particularly apply
to that Species of Intervals which is callee}
CoNCORP.
Definition. A fimple C o N c o R D is fuch 5
whofe Extremes "are at a" Diftsnce Jefs than
the Sum of .any Two other Concords, A com-
pound Concord is equal to Two or more
Concords. This in general is agreeable to the
common Notion of fimple and compound , but
1 he Definition is alfo taken another Way a-?
jnong the Writers on Mufick 1 thus an Q&ave
l \
§ r, of MUSIC K. iyj
i : 2, and all the lefler Concords ( which have
been already mentioned ) are called fimple and
original C 6 n c o r d s j and all greater than an
Octave are called compound Concords, * be-
caufe all Concords above an Octave are compc-
fed of, or equal to the Sum of one or more
Otlaves y and fome {ingle Concord left than an
Octave-, and are ordinarily in Practice called by
the Name of that Jimpk Concord; of vvhicn
afterwards.
Jj i. Of the original Concords, their Rife
and Dependence on each other? Sec,
Ee thefe original Concords again in the fol-
lowing Table^ where I have placed them in
Order, according to their Quantity,
'Table offimple
Concords.
!
5 : 6 a 3d /, L e t us now firft examine
4 : 5 a 3d g. the Compojition and Relations
3:4a 4th. of thefe original Concords a-
£ : 3 a 5th, mong themfelves.
5:8a 6th /. I f we apply the preceeding
3:5a 6th g» Rules of the Addition and Sub-
1:2 a 8ve. ftra&ion of Intervals to thefe
Concords^ we fliall find them
divided into fimple and compound^ according to
the
iy4 ^Treatise Chap. V.
the firft and more general Notion, in the Man-
ner expreffed in the following Table*
Simple.
6. a 3d I.
5
a idg.
a 4th.
ith.
6th /.
6th g.
8ve.
Compound.
3d g . & 3d I
3d /.
$>:
4th. or
3d I. or
Sdg. or
3d /.4th.
o
u.
E
o
4th,
4th,
fjfe
] <5:h I.
Proof
in
Numb
4-
5-
6.
5.
6-
8.
h
4-
5-
■
1.
3-
4-
3-
S-
6.
+•
5-
8.
*•
5-
6.
8.
The 3 d I. 3d g. and qth, are equal to the
Sum of no other Concords ; for the 3d /. is it
felf the leaft Interval of all Concords. The 3d
g. is the next, which is equal to the 3d /. and
a Remainder which is Difcord. The qth is
equal to either of the ^ds and a Difcord Re-
mainder ^ and thefe Three are therefore the
leaft Principles of Concord, into which all other
Intervals are divihble : For the Competition of
the 5th, 6th and 8c<?, you fee it proven in the
Numbers annexed ,• and that they can be com-
pounded of no other Concords, you'll prove by
applying the Rules of Addition and Subtra-
ction.
A s to the Proofs in Numbers which are an-
nex'd, they demonftrate the Thing,, taking the
component Parts in one particular Order ; but
it is alfo true in whatever Order they are taken,
as is proven in Probl. 2. Chap. 4. Or fee all
the Variety in this Table ; in the laft Column
of which you fee the Names of all the compo-
nent Parts fet down in the feveral Orders of
which
1 1. 0/MUSICK. ijj
which they are capable, either from the acutefi
Term or the graveft.
TAB LE of the 'various Orders of the har-
monkal Parts of the greater Concords.
ph 7 2 : 3 \ 4
6th J. $ ; S<
IO
IS
5
12
6
20
6
IS
8
24
/ 3 • 4
«".,£• j • 3 ^12 . 15 , 20
3J £. 3^ /•
3^ 7. 3^g.
3 J /. 4?^
qth. 3d I.
qth, 3d g.
3dg, qth.
5 th, qthl
4th, 5th.
6th g. 3d L
3d I 6th g.
3^g. 6th L
6th L 3d g,
4*h idg. 3d I
3d g. 3d I qth.
3d I, qth. 3d g.
3dl. 3d g. yh.
3d g. 4th. 3dL
24 . 30 4^3 3 d I. 3d g.
Here you may obferve, that the Varieties of
the Compofition of Octave by Three Parts, viz,
3d g. 3d /. 4th, include the other Three Ways
by Two Parts ; and alfo all the Varieties
of the Compofition of the 5 th and 6th.
We
Bye 1
<
I
<
i
1
1 c
u
2< {
3
5
4
5
3
4
5
10
12
U5
3
4
5
6
5
8
- 4
• 5
. 6
,12
15
20
• 4
, 6
. 6
. 10
s 8
. 10
5 ; 6*
6. 8
8 . 10
15 . 20
20 . 24
iy<5 ^Treatise Chap. V.
We have already, by Addition of the vari-
ous Concords within an ObJave, found and pro-)
ven that the $th^ 6th s and $ve, are equal to*
the Sum of lefler Concords, as in the preceeding
Table : Now we (hail confider, by what Laws.
of Proportion thefe Intervals are refolvable back
into their component Parts ; or, how to put i
fucli middle Numbers betwixt the Extremes of j
thefe Intervals, that the intermediate ifo/mrl
(hall make harmonic al Intervals ; by which we
(hall have a nearer View of the Dependence of j
thefe original Concords upon one another.
O f the Seven original Concords we examine I
their Compqfition among themfclves, i, e, what]
lefler ones the greater are equal to ; therefore
the Oblave being the greater!, its Re/blutionsl
muft include the Refolntions of all the reft.
Proposition I. If betwixt the Extremes I
Qifih Octave we place au arithmetical Mean I
(by Cor oh to Theor.2. Chap. 4.) it (hall refolve I
it into Two Ratios, which are the.Concords of I
Kth and ^h ; and the $$h (hall be next the Je£- 1
fer Extreme : So betwixt 1 and 2 an arithme- I
tic al Mean is \\ ; or becaufe 1 and 2 can have I
no middle Term in whole Numbers ; therefore I
if we multiply them by 2, the Products 2 and 4 I
being in the fame Ratio x can receive one arith- I
metical Mean (by Theor. %tfi) which Mean is |
3, and the Series 2:3:4, $&, a $th and a
4Z&.
Proposition II. If betwixt the Extremes
of an Oblave we take an harmonic al Me an Joy
Theor, 1 1 thy the intermediate Ratios (hall be
a
§ i. of MUSIC K 15?
•i a 4th and a 5 thy and the qth next the leffer
■ii Extreme ;fo betwixt 1 : 2 an harmonic al Mean
31 is 1 J ; or multiplying all by 3, to bring them to
g wh oh Numbers, the Series is 3, 4, & which
if is harmomcaL
Corollary. 'Tis plain, that if betwixt the
I Extremes of the Ofytave we put Two Msans %
\ one arithmetical and one harmonical, the Four
1 Numbers fhall be in geometrical Proportion, as
gi here, 6, 8, 9, 1 2. The Reafon is, that the
I qth and 5^ are the Complements of each other
i to an Qclave ; and therefore a qth to the lower
te Extreme leaves a $th to the upper, and contra-
il rily : And in this Divifion of the OUlave, we
y have the Three Kinds of Proportion, Arith-
1 metical, Harmonical and Geom e-
j TRiCAL,mixt, for 6 :g : 12. viz. the 5 th, qth,
\i and 8#?, are arithmetical ; 6 : 8 : 1 2, the 4^,
k 1 5"^, and 8^, are harmonical; and 6:8:9 : 12,
I geometrical,
j Observ E.The 5*^/^ and qth are the Refult of
K the immediate and moft iimple Diviiion of the
p Offiave into Two Parts : The 4^/? is not refol-
p vable into other Concords, fince the only leffer
p Concords are the 3 # 7 g, and 3d Land either of thefe
i ! taken from a 4? h, leaves a Di/bord ; and there-
fore 'tis in vain to feek any mean Terms that
will refolve it into Concords. ' Tis natural there-
fore next to enquire into the Refolutions of the
$th, which by a remarkable Uniformity, we
find reducible into its confcituent leiler Concords
by the fame Laws of Proportion,
I p 3 02
ij8 ^? Treatise Chap. V.
Proposition III. An arithmetical Mean
put betwixt the Extremes of a $th$ refolves it
it into a 3d g. and a 3 d L with the id g. ; next
the leffer Extreme, as here, 2:2^: 3. which
multiplied by 3 are reduced to thefe whole Num*
bers 4:5:6.
P r o p o s 1 t 1 o n IV. An harmonica! Mean put
betwixt the Extremes of a 5th, refolves it into
a 3d g. aad 3d L with the 3d L next the leffer
Extreme ; as 2 : 2* : . 3, which multiplied by 5
are reduced to thefe, 10 ; 12 : 15,
Corollary. The fame Thing follows
here as from the two hrft Proportions, viz.That
' taking both an arithmetical and harmonica}
Mean betwixt the Extremes of a $th y the Fouir
Numbers are in geometrical Proportion, as in
thefe, 20, 24, 25, 30.
N o w out of the various Mixtures of thefe
fimple Divifions of the $<ve and 5th, we can
bring not only all the Refolutions of the 6th,
and the other Refolutions of the 8c^, but afl
the Varieties with refpe£t to the Order iii
which the Parts can be taken, as follows, viz,
into. If with the arithmetical Diviiion of
the 0.5ia<ve, we mix the arithmetical Divi-
fion of the $th, L e. if we put an arithmetical
Mean betwixt the Extremes of theOGfave, and
then another arithmetical Mean betwixt the
leffer Extreme and the laft mean Term founds
and reduce all the 4 to whole Numbers, then
we have this Series 4, 5, 6, 8, in which we
have the Q&iave refclved into its three ®eonfti->
tuent Concords, 3d greater,3 d leffer, and qth ; and
within
§ i. of MUSIC K. ij 9
within that Series the $th refolved into its two
conftituent Concords, 3d greater, and 3d lef-
fer : And if we confider the Extremes of the
hfflave with the leaft of the two middle Terms
f, then thefe 4, 5, 8 (hew us the Offiave re-
folved into a 3d g. and a 6th L Laftly. It
ftiews us the 6th L refolved into a 3d I. and
a 4?#> 0/3. 5) ^, 8.
I 2^0. 1 f we mix the harmonic al Divifion of
Octave, with the. arithmetical Divifion of
the $th, i. e, if we put an harmonic al Mean
betwixt the Extremes of Slave, and then an
arithmetical Mean betwixt the greateft Ex-
treme and middle Term laft found, as in this
Series, 3, 4, 5, 6, then we have the Refoluti-
on of the Offiave into a 6th g. and 3d I, as
in thefe 3, 5, 6 ; alfo the 6th g. refolved into
a 4^ and 3d?g. in thefe, 3, 4, 55 and taking
the whole Series, we have a 2d Order of the
Three Parts of the Offiave.
W e have feen all the harmonical Parts of
the Offiave and 5th, and both the 6ths ; and as
to the Variety of Order in which thefe may be
placed betwixt the Extremes, it may all be
found by other Mixtures of the Parts of the
Offiave, and $th or 6th -, as you'll eafily find by
comparing the 6 Orders of the Competition of
Offiave by 3 Concords, in the preceeding Table,
O r, you may find them all in one Series, if
you'll divide the Ocfave thus, viz. Put both an
arithmetical and harmonical Mean betwixt its
Extremes,and you'll have a qth and $th to each
of .the Extremes ; both of which $ths divide
arith-
itfo ^TRfiATISE CHAf.
arithmetically and alfo harmonically , and at I
every Divifion reduce all to a Series of whole
Numbers $ and 'tis plain you'll have a Series of 1
8 Terms, among which you'Jl have Examples l
Of the 7 original Concords with their Compo*
JitiOnS) and all the different Orders in which
their Parts caft be taken. Or, you may make
the Series by taking the 7 Concords , and redu-
cing them to a common Fundamental^ by Pro*
bkfn 3. the Series is 360 : 300 t 288 : 270 :
240 : 225 : 216 : 180. See Plate 1. Fig* 4*
wherein I have connected the Numbers fo as
all the Compofition may be eafily traced.
There is this remarkable in that Series*
that you have all the Concords in a Series, both
afcending toward Acutenefs from a common
Fundamental^ or greateft Number 360^ and de-
fending towards Gravity r , from a common a-
cute Term 180. and for that Reafon the Se-
ries has this Property, that taking the Two
Extremes, and any other Two at equal Di-
itance, thefe 4 are in geometrical Propor*
Hon.
Nota. 1 f betwixt the Extremes of any In*
terval you take Two middle Terms, which (hall
be to the Extremes in the Ratios of any Two
component Parts of that Interval^ i. e. if the
two middle Terms divide the Interval into thg
fame Parts only m a different Order, the Four
Numbers are always geometrical.
N o w, from the Things laft explained, wd
ihall make feme more particular Ohfervations
concerning
§ i. rfMUSlCK. 161
concerning the Dependence of the original Con-
cords one upon another.
The clave is not only the greateft Interval
of the Seven original Concords, but the firft in
Degree of Perfections the Agreement of whofe
Extremes is greateft, and in that refpecl: moll
like to Unifons : As it is the greateft Interval,
fo all the lefter are contained in it ; but the
Thing moft remarkable is, the Manner how
thefe lefter Concords are found in the Oblave,
which fhews their mutual Dependences ; by tak-
ing both an harmonical and arithmetical Mean
betwixt tke Extremes of the Oblave, and then
both an arithmetical and harmonical Mean be-
twixt each Extreme, and the moft diftant of
the Two Means laft found, viz, betwixt the
i Ieffer Extreme, and the firft arithmetical Mean,
alfo betwixt the greater Extreme and the firft
i 'harmonical Mean we have all the. lefter Con*
\ cords t Thus if betwixt 360 and 180 the Ex^
1 tremes of Oblave, we take an arithmetical
Mean, it is 270, and an harmonical Mean is
. 140 ; then betwixt 360, the greateft Extreme,
and 240, the harmonical Mean, take an arith-
metical Mean, it is 300, and an harmonical
Mem is 288; again, betwixt 188 the lefter
(Extreme of the Oblave, and 270 the firft arith-
metical Mean, take an arithmetical Mean, it is
1225, and an harmonical it is 216, and the
whole Numbers make this Series^ 360 : 300 :
288 : 270 : 240 : 216 : 180.
Observe. The immediate Divifion of the
Oblave refolves it into a qth and $0o\ the a-
L rithmetical
\6% ^Treatise Chap. V.
rithmetical Divifion puts the 5th next the lei 1
fer Extreme, as here 2, 3, 4, and the har-
monical puts it next the greater Extreme, as
here 3:4:6; and you may fee both in thefe
four Numbers 6, 8, p, 12. Again the im-
mediate Divifion of the $th produces the Two
3 ds ; the arithmetical Divifion puts the leffer
3 d> and the harmonic al the greater 3d next the
leffer Extreme ; as in thefe 4, 5, 6, and 1 o,
12, 15 ; or fee both in one Series, 20, 24, 25,
30. The two 6th s are therefore found by Divi-
fion of the Oclave^ tho' not by any immediate
Divifion. The fame is true alio of the two sds;
fo that all the other fimple Concords are found
by Divifion of the Otlave. The 5th and qth
arife immediately and direc~f ]y out of it, and the
3 ds and 6th s proceed from an accidental Di-
vifion of the Otlave ; for the %ds arife imme-
diately out of the 5th, which having one Ex-
treme common with the Otlave, the mean
Term which divides it dire&Iy, divides the
Octave in a Manner accidentally.
Now, if we confider how perfectly the Ex-
tremes of an clave agree, that when they are
founded together, 'tis impoflible to perceive two
different Sounds ; fo great is their Likenefs, and
the Mixture fo evenly, that it is impoffible to
conceive a greater Agreement ; we fee plainly
there is no Heafon to expecl: that there fhould
be any other Concord within the Order of Na-
ture that comes nearer, or fo near to the Perr
fe&ion of Unifbns : And if we confider again,
how thefe Seven original Concords gradually
decreafe
§ i. of MUSIC K. \6i
decreafe from the Offiave to the leffer 6th>
which has but a fmall Degree of Concord ; and
with that Confederation joyn this of the mutual
Dependence of thefe Seven Concords upon one
another, and efpecially how they all rife out of
the Divifion of the Odlave, according to a moll
fimple Law, <viz. The taking an arithmetical
and harmonical Mean betwixt its Extremes
which gives the Two Concords next in Per-
fection to the QbJave, whereof the $th is beft •
and the fame Law being applied to this, difco-
vers all the reft of the Concords ; for out of the
$th arife immediately the two ^ds, whofe Com-
plements to clave are the two 6th s ; and for
that Reafon thefe 6ths and ids are faid to rife
accidentally out of the Oblave ; (and afterwards
we ftiall fee how by the fame Law, fome other
principal Intervals belonging to the Syftem of
Mufick are found.) Upon all thefe Confedera-
tions we may be fatisfied, that we have difco-
vered the true natural Syftem of Concords with-
in the OEiave > and that we have no reafonable
Ground to believe there are any more, nor even
a Poffibility of it, according to the prefent State
and Order of Things.
Now as to the Order of their Perfection, We
have already ftated them according to the Ear
tkusficlave, 5th, qth, 6th gr. 3d gr. 3d lejf. 6th
kjfi In which Order we find this Law, That
the beft Concords are expreft by leaft Numbers*
Yet, as I obferved, this is not an univerfal Cha-
racter ,• and we are only certain of this from
Experience,that the frequent Coincidence ofVi-
L 2 brations*
i<$4 -^Treatise Chap. V.
brations, is a neceffary Part of the Caufe of
Harmony ; Senfe and Obfervation muft fupply
the reft, in determining the Preference of Con-
cords ; and fo we take thefe 7 original Concords
in the Order mentioned ,♦ and upon what Con-
fiderations they are otherways ranked by
practical Muficians, (hall be explain'd in its
proper Place.
Yet before I go further, let us notice this
one Thing concerning the Difference of the
arithmetical and harmonical Divifion. An a-
rithmetical or harmonical Mean put betwixt
the Extremes of any Interval, divides it into
two unequal Parts 5 the arithmetical puts the
greatefi Interval next the lefTer Extreme, the
harmonical contrarily, as in thefe, 2:3:4, and
3:4:6, where the Q&ave is divided into its
constituent 5th and qth ; or the Refolutions
of the $th, as here 4:5: 6, and 10 : 12 :
1 5. Now let us apply thefe Numbers either
to the Lengths of Chords or their Vibra-
tions, and we find this Difference, that applied
to the Vibrations, the arithmetical Divifion
puts the beft Concord next the fundamental, or
grave Extreme, and the harmonical puts it next
the acute Extreme ; but contrarily in both when
applied to the Lengths of Chords. As thefe two
Divifions- refolve the Oclave or $th into the
fame Parts, they are in that refpecl: equal ;
but if we" fuppofe the Extremes of the Oftave
or 5th, with their arithmetical or harmonical
Means, to be founded all together, there will be
a considerable Difference ; and that Divifion
which
§ i. of MUSIC K. t 1 6 j
which puts the beft Concord loweft is beft,
which is the arithmetical if the Numbers are
applied to the Vibrations, but the harmonic al
it applied to the Lengths of Chords. The ob-
ferving this ftiali be enough here; I fhall mors
fully explain it when I treat of compound Sounds,
under the Name of Harmony. This however
we find true. That geometrical Proportion af-
fords no fimfle Concords (how it comes among
the compound fhall be feen prefently ) and it
has no Place in the Relation and Dependence.
of the original Concords, but fo far. as a Mix-
ture of the arithmetical and harmonic al produ-
ces it, as in thefe, 6, 8 5 9, 1 2. And here I fhall
obferve. That the harmonical Proportion re*
ceived that Denomination from its being
found among the Numbers, applied to the
Length of Chords, that exprefs the chief Con-
cords in Mufick, viz. the OcJave, $th, and^th^
as here, 3, 4, 6, But this Proportion does not
always conftitute Concords, nor can poffibly do,'
becaufe betwixt the Extremes of any Interval
we can put an harmonical Mean, yet every In-
terval is not refolvable into Parts that are Con-
cords ; therefore this Definition has been reject-
ed, particularly by Kepler > and for this he in-
ftitutes another Definition of harmonical Pro-
portion, viz. When betwixt the Extremes of
any Ratio or Interval, one or more middle
Terms are taken, which are all Concord among
themielves, and each with the Extremes, then
that is an harmonical Divifion of fuch an Inter-
val j fo that Qt~tave, 6th and 5th are capable
L 3 o*
166 ^Treatise Chap. V.
of being harmonically divided in this Senfe ; all
the Variety whereof you fee in a Table at
the Beginning of this Chapter: And thefe middle
Terms will be in fome Cafes arithmetical Means,
as i ; 2 : 3 j in fome geometric al, as i, 2, 4; in
fome harmonic al (in the firft Senfe ) as 3 : 4 ; 6 •
and in others they will depend on no certain
Proportion, as 5, 6, 8.
Hitherto we have considered the Refolu-
tion and Compofition of Intervals, as they are
expreft by Ratios of Numbers ; but there are
other Ways of deducing the. Relation and De-
pendence of the Concords, not from the Divi-
sion or Refolution of a Ratio,hut the Divifion of
a fimple Number, or rather of a Line expreft by
that Number, which may be cali'd the geome-
trical Part of this Theory. But it will be bet-
ter if I flrft confider and explain the remaining
Concords belonging to the Sjftem of Mujick y
which are particularly called compound Con-*
Cords,
1 1&m+amm—mi an
S z.Qf
§ i. Of MUSIC K. 167
§ 2. 0/Compound Concords; and of
the Harmoniek Series ; with fever al Obfer-
vations relating to both fimple and compound
Concords,
HITHERTO we have taken it upon Ex-
perience, That there are no concording
Intervals greater than OcJave, but what are
compofed of the 7 original Concords within an
Oblave-y the Hea/bn of which is deduced from
the Perfection of the OdJave. We have feen
already how all the other fimple and original
Concords are contained in, and depend upon the
I OcJave, and derive their Sweetnefs from it, as
1 they arife more or lefs directly out of it : We
I have obfervcd, that it has in ail Refpects the
I greater!: Perfection of any Interval^ and comes
i neareft to Unifons ; and tho' there feems to be
I fomething {kill wanting, to make a general Cha-
racter, by which we may judge of the Ap-
| proach of any Interval to the perfect Agree-
ment of Unifons ", yet 'tis plain the 061 ave 1 : 2
comes neareft to it; for 'tis contained not only
in the leaft of all Numbers, but that Proporti-
on is of the moil perfect Kind, mz. Multiple ;
and of all fuch it is the moft fimple, which
makes the greater]: Degree of Commenfurate-
nets or Agreement in the Motions of the Air
that produce thefe Sounds. Let me add this
L 4 other
i68 A Treatise Chap. V.
other Remark, That if Wind-inftruments are
overblown, the Sound will rife firft to an OBave^
and to no other Concord \ why it fliould not as
well rife to a qth, &c. is owing probably to
the Perfe&ion of Otlave, and its being next to
Wriifon, Again, take into the Confideration
that lurprifng Phenomenon of Sound being raif-
ed from a Body which is touched by nothing
but the Air, moved by the fonorous Motion of
another Body> particularly that if the Tune of
the untouched Body be' Octave above the
given Sound, it will be moft diftin&ly heard $
and fcarcely will any other but the O clave be
Jieard?
From this flmple and perfect Form of the
Octave, arifes this remarkable Property of it,
that it may be doubled, tripled, (?c. and ftill be
Concord, u e. the Sum of Two or more
Octaves are Concord, tho' the more compound
will be gradually lefs agreeable ; but it is not fo
with any other Concord lefs than Octave, the
Double, &c. of thefe being all Difcords ; and
as continued geometrical Proportion conftitutes
a Series of equal Intervals^ fo we fee that fuch
a Series has no Place in Mufick but among
OtJaves,t\ie Continuation of other Concords pro?-
ducing Difcord, Thefe Things remarkably con-?
firm to us the Perfection of the Octave : There
is fuch a Likenefs and Agreement betwixt its
Extremes, that it feems to make a Demon-
ftration a priori, that whatever Sound is Con-?
cord to one Extreme of the Octave, will be fo
%q the other alfo jj and in Experience it is fo*
§ i. of MUSIC K. i6>
We have feen already, that whatever Sound
betwixt the Extremes of an Q0ave 9 is Concord
; to the one, is in another Degree Concord to
i the other alfo ; for we found that the Octave
j is refolvable into Concords, Again* if we add
any other fimple Concord to an Octave, we find
by Experience that it agrees to both its Ex-
tremes,- to the neareft Extreme it is afmple
i Concor J,and to the fartheft it is a compound Con-
i cord : Now, take this for a Principle, That
whatever agrees to one Extreme of Octave, a-
grees alfo to the other, and we eafily conclude,
, That there cannot be any concor ding 'Interval
j greater than an Octave, but the Compounds of
; an Octave and fome leffer Concord : For if we
.1 fuppofe the Extremes of any Interval greater
i than an OcJave to be Concord, 'tis plain we can
|i put in a middle Term, which fliall be Octave
i to one Extreme of that Interval* confequently
I the other Extreme fhall be alfo Concord with
this middle Term, and be diftant from if*
by an Interval lefs than an Octave ; and there-
; fore if we add a Difcord to one Extreme of an
\ Octave, it will be alfo Difcord to the ether ,-
the fame will apply alfo to the Compounds or
Two or more - Octaves $ but the Agreement
will ftill be lefs a§ the Compofition is grea-
I ter,
I cannot but mention here how If Cartes
concludes this Principle to be true; he ob-
serves, what I have done, That the Sound of
a Whiftle or Organ-pipe will rife to an
2^§I e ? if Vw forcibly Horn $ which proceeds?
fays
\yo ^Treatise Chap. V.
fays he, from this, That it differs leafi from
Unifon. Hence again, fays he, I judge that no
Sound is heard, but its acute Octave feems
fome way to eccho or refound in the Ear \ for
which Reafon it is that with the grojfer Chords
(or thofe which give the graver Sound) of fome
firinged Inftruments (he mentions the Teftudo)
others are joyned an Octave acuter, which are
always touched together, whereby the graver
Sound is im proven, fo as tp be more diftinctly
heard. From this he concludes it plain, That
no Sound which is Concord to one Extreme of an
O&ave, can be Difcord to the other. From
all this we fee how the Octave comprehends the
whole Syflem of Concords, (excepting the Uni*
fon) becaufe they are all contained in it, or
compofcd of it and thefe that are cotained in it.
The Author already mentioned of the E-
lucidationes Phjficce upon Z)' Curies 's Compend
of Mufick, advances an Hypothefis to explain
how this happens, which IS Cartes affirms,
viz. That the Fundamental never founds but
the acute Octave feems to' do fo too. He fup-
pofes that the Air contains in it feveral Parts of
different Constitution, capable, like different
Chords, of different Meafures of Vibrations,
which may be the Reafon, fays he, that the
liuman Voice or Inftruments, and chiefly thefe
of Metal never found, but fome other acuter
Sounds are heard to refound in the Air.
In the Beginning of this Chapter 1 obferved
two different Senfes in which Concords were
called fimpk and compound; The Octave and
5 1. of MUSICK. i 7 1
all within it are called fimple and original Con-
cords ; and all greater than an clave, are com-
pound, becaufe all fuch are compofed of an
Octave, and fome leffer Concord. Now, the
t$b± 6th s and Oliave are alfo compofed of the
ids and /$hs which are the mod: fimple Con-
wds; but then all the 7 Concords within an
Iffiave have different Effects in Mufick, where-
[s the compound Concords above an Octave have
ill in Praaice the fame Name and Effect with
hefe fimple ones, lefs than an Octave, of which
vith the Octave they are compofed ; fo a $th
md an Octave added make 1 : 3, and is called
1. compound 5th. Now as there are 7 original
Concords, fo thefe 7 added to Octave, make 7
ompound C oncor ds ; and added to two Octaves,
aake other 7 more compound, and fo on.
W have feen already, in Prob. 8. how to add
ntervahy and according to that Rule I have
iade the following Table of Concords, which
place in Order, according to the Quantity of
lie Interval, beginning with the leaft. I fup-
>ofe 1 to be a common fundamental Chord,
nd exprefs the acute Term of each Concord by
liat Fraction or Part of the Fundamental that
rakes fuch Concord with it, and have reduced
ach to its radical Form, i. e. to the loweft
dumber; fo an Oclave and 5th added, is in the
latio 2 : 6, equal by Redu&ion to 1 : 3 j and
Ithers.
, follows the general Table of Cox cords,
Octaves
t?t
A Treatise
Chap. V.
Otfaves
I;
i:
o
i:
O
I .
*~*
*—
o
►—
O
2
4 :
3
8:
B
6th g.
6th I
$]
7
C/5
3
y
J
IO
T3
O
PS
: 3
if
PS
a
4/3
(ft.
^:
cT
16
I
P
* 2 i
P
cr
5th
, 2
o
o
<
, i
<
1
o
3
CD
\-6
Q
qth
3
4
p
o
o
l-S
1
o
p
CD
3'
16 l
O
3d g.
I
: 4
:7
4(0
x
2
;7
o
Si
* i
O
1
%d L
tf
1*2*
5?
24:
These Compounds are ordinarly called by the
Name of the fimple Concord of which they are
compofed, tho' they have alfo other Names, of
which in another Place,
I f this Table were continued infinitely, 'tis
plain we ftiould have all the poflible harmonic al
Ratios , and in their radical Forms ; 'tis alfo
certain, that there fhould be no other Numbers
found in it than thefe, 1, 3,5, and their Mul-
tiples by 2, i. e. their Produ&s by 2, which are
2, 6, 10, and the Produ&'s of thefe by 2,ws|
4, 1 2, 2 o,and fo on in infinitum^rrmltliplym^ the laft
Three Produces by 2. The Reafon of which is,that
in this Series i, 2, 9, 4, 5, <5, 8, we have no
other Numbers but i, 2, 5, and their Products
by 2 j and we have here alfo all the Numbers
that belong to the fimple original Concords ; and
if we conlider how the Compounds are raifed by
adding an Qjftave continually, we fee plainly that
n@
:§ »; of MUSIC K. 173
no new Number can be produced, but the Pro-
,du£t of thefe that belong to the fimpk Concords
multipiied by 2 continually. All which Num-
bers make up this Series, viz. 1, 2, 3, 4, 5,
4 8, 10, 12, i£ 20, 24, 32, 40, 48, 64,
$o,(5r. which is continued after the Number 5,
by multiplying the laft Three by 2, and their
Products in infinitum by 2 ; whereby 'tis plain,
'we fhall have all the Multiples of thefe original
Numbers 1, 3, 5, arifing from the continual
Multiplication of them by 2. And this I call the
Harmonical Series, becaufe it contains
all the poflible Ratios that make Concord^
: either finish or compound : And not only fo,
.but every Number of it is Concord with every
I other, which I {hall eafily prove: That it con-
tains allpornble Concords is plain from the Way
\ of raifing it, fince it has no other Numbers than
J what belong to the preceeding general Table of
^Concords ; and that every Number is Concord
( iwith every other is thus proven : After the Num-
j ber 5 every Three Terms of the Series are the
1 Doubles or the laft Three ; but the Numbers 1,
1 2, 3, 4, 5, are Concord each with another, and
\ confequently each of thefe muft be Concord
I with every other Number in the Series, fince all
j the reft are but Multiples of thefe ; for whatever
Concord any leffer Number of thefe 5 makes
with another of them that is greater, it will with
the Double of that greater make an OStave more,
j and with the Double of the laft another Ottave
more, arid fo on : Thus, 2 to 3, is a 5th, and
1 1° 6 i§ %l Sth arjjd §ve j but 3 comparing any
greater
174 ^Treatise Chap. V.
greater Number of thefe Five with a lerTer,what~
ever Concord that is, it wiJJ with the Double of
that leffer be an $<ve lefs, providing that Double
be ftill lefs than the Number compared to it, (fa
5 to 2 is a %d g. and %<ue, and 5 to 4 is only a
3^?g.)But if 'tis greater, then it will be the Com-
plement of the firft Concord to 8#?, i.e. theDif'
ference of it and S<ve, (fo 5 to 6 is a 3d /. the
Complement of a 6/-/? £.5:3 to an 8ce) and
taking another Double it will be an %<ve more
than the laft, and fo on. Now the Thing be-
ing true of thefe Five Numbers compared toge-
ther, and with all the other Numbers in the Se-
ries, it muft hold true of all thefe others compa-*
red together, becaufe they are only Multiples
of the firft. The Ufe of this harmonick Series
you'll find in the next Chapter. I fhall end this
with fome further Obfervations relating to the
harmonic al Numbers, and the whole Syjletn of
Concords both Jimple and compound.
In the preceeding Chapter I have endeavou-
red to difcover fome Character, in the Propor-
tion of mufical Intervals, whereby their various
Perfections may be ftated, tho' not with all the
Succefs to be wiflicd - y fo that we are in a great
Meafure left to Senfe and Experience. We have
feen that the principal and chief Concords, are
contain'd within the firft and leaft of the natu-
ral Series of Numbers ; the Ocfave, $th, qth,
and^ds, in the natural Progreilion 1, 2, 3, 4, 5,
6i and the Two 6th s arife out of the Divilion
of the Octave, and are contain'd in thefe Num-
hers 3, $,% Considering what a neceffary Con-
ditioa
§2. of MUSICK. 175
"dition of Concord, frequent Union and Coinci-
dence of Motion is, we have concluded, that
±e fmaller Numbers any Proportion confifts of,
mteris paribus, the more perfecl is the Inter-
nal cxpreffed by fuch a Proportion of Numbers.
But then I obfcrved, that befides this Smalnefs
of the Numbers on which the Coincidence de-
fends, there is fomething ftiJl a Secret in the
Proportion or Relation of the Numbers that re-
present the Extremes of an Interval, that we
DUght to know for making a general Character,
whereby the Degrees of Concord may be deter-
mined i fo 4 : 7 is Difcord, and yet 5 : 6 is
Concord, and 5 : 8. Now again we fee in this
Table of Concords, that the Smalnefs of the
Numbers does not abfolutely determine the Pre-
ference, elfe 1 : 3 an ObJave and $tb, would be
ibetter than 1:4a double Octave, which it is
aot, and fo would all the other compound $ths
in infinitum. Again, the compound id 1 : 5
would be better than either the compound Oblave
1 : 8, or the compound 5th 1 : 6, which is all
contrary to Experience ; and this demonftrates,
ithat there muft be fomething elfe in it than
barely the Smalnefs of the Numbers. Z)' Cartes
obferves here, that the 3d 1 : 6, compos'd of
Two Slaves, is better than either the fempfa
%d, 4 : 5, or the firft Compound 2:5; and
gives this Heafon, viz, that 1 : 5 is a multiple
Proportion, which the others are not; and o ,t
of multiple Proportion, he fays, the belt Concords
proceed, becaufe it is the moft iimplc Form,
and eafily perceived : By the fame Heafon all
the
\?6 ^Treatise Chap. V.
the compound $ths are better than the Jimple
$th$ and 7)' Cartes himfelf makes the firft
compound $th i : 3 the moft perfect, becaufe it
is Multiple, and in fmaller Numbers than the
jimple 5th. But we mnft obferve, that every
multiple Proportion will not coniiitute Concord,
fo 1 : 9 is grofs Difcord, being equal to Three
05la<ves> and this Difcord 8 : p. Now confi-
der either the Numbers or their multiple Pre*-
portion, and this of 1 : 9 fhould be better
than 3 : 8, or than 3 : 16; yet it is otherwife,
for thefe are compound qths, which are Concord 1
we muft therefore refer this to fome other things
in the Relation of the Numbers, that we can-
not exprefs.
Observe next how Z)' Cartes ftates thefe
Concords -, he puts them in this Orderj OffiaveJ
5th> 3d g. 4th, 6th g. 3d I. 6th 1; and gives
this Reafon, 0jz. That the Perfe&iou of any
Concord is not to be taken from its Jimple Form
only, but from a joynt Confideration of all its
Compounds \ becaufe, fays he, it can never be
heard alone fo {imply, but there will be heard
theRefonance of its Compound ; as in the TJhi*
fori, or a (ingle given Sound, the Relonance of
the acute OSra-ve is contained j and therefore he
places the 3d g. before the qth becaufe being
contain'd in lefter Numbers, it is more perfect.
But we muft obferve again, that as Concord
does not depend altogether upon multiple Pro-
portion, neither does it upon the SmaJnefs of
the Numbers ,- for then & Cartes fhould have
put the $th before the O&aves, becaufe all its
Com-
§ i. df MUSIC K. tyy
Compounds are contained in letter Numbers thari
the OcJaves. We fee then how difficult it is to
deduce the Perfection of the Concords from the
Numbers that exprefs them.
\ L e t us confider this other Remark of If
Cartes, he obferves that only the Numbers 2,
jl 5, are ftriclly mufical Numbers, all the other
Numbers of the Table being only Compounds
or Multiples of thefe Three, which belong in
the firft Pkce to the Oclave, 5th, and ^d g*
which he calls Concords properly, and per fe, as
he calls all others accidental, for Reafons I
ifliall fliow you immediately*
Now, tho' the compound Kths are contain'd
in leffer Numbers than the Oftaves^ perhaps
the Preference of the Octaves is due to the ra^
dical Number 2, which belongs originally and
ij$i the firft Place to that' Concord > whereas the
compound $ths depend on the Number 3 which
is more complex : But we fhall leave this Way
!of Reafoning as uncertain and chimerical $ yet
this we have very remarkable, that the firft fix of
the natural Series ofNumbers,^.!, & } 3, 4, 5, 6 i
>are Concords comparing every One with every
other, which is true of no other Series of Num^
ibers, except the Equimultiples of thefe 6,
which, in refpecl of Concord, are the fame with
Ithefe. Again, 'if each of thefe Numbers be
imultiplied by it felf, and by each of the reft,
and thefe Produces be difpofed in a Series^ each
Number of that Series with the next conftitutes
fome Interval that belongs to the Syftem of
Mufick, tho' tliey are not at all Concord^ as
M , will
iy% A Treatise Chap. V.
will appear afterwards : ' That Series is i. 2. 3.
4. 5. 6. 8. 9. 10. 12. 15. 16. 18. 20. 24.
2/?. 30. 36. It would be of no great Ufe to re-
pete what wonderful Properties fome Authors
have found in the Number 6, particularly Kir-
Cher, who tells us, that it is the only Number
that is abfolutely harmonic al, and clearly repre-
fents the divine Idea in? the Creation, about
which he imploys a great deal of Writing.
But thefe are fine imaginary Difcoveries, that "I
fnall leave every one" to fatisfy himfelf about,
by confulting their Authors or Propagators.
Another Thing remarkable in this Syftem
of Concords is, that the greateft Number of Vi-
brations of the Fundamental cannot be above
5, or 5 there is no 'Concord where the Funda-
mental makes more than 5 Vibrations to one
Coincidence with the acute Term : For fince it
is fo in the fimple Concords, it cannot be other-
wife in the Compounds, the Octave being *,
which by the Rule of Addition can never alter
the leffer Number of any fimple Concord to which
• it is added. It is again to be remarked, that
this Progrefs of the Concords may be carried on
to greater Degrees of Compofitioh in infinitum*
but the more compound (till the lefs agreeable,
if you'll except the Two Cafes abovementioned
of the $th 1:3, and $d 1 : 5 ; fo a fingle 0-
Bave is better than a double C Stave, and this
better than the Sum of 3 Oc~iaves, &c. and fo of
$ths- and other Concords. And mind, tho' a
compound OcJave is the Sum of 2 or more O-
ifaves, yet by a compound $th or other Con-
cord ?
§i.' of MUSIC K. t7 9
cord, Is not meant the Sum of Two or more
Sths but the Sum of an Ociavc and fth 9 or of
Two Ocf.'Wes and a g$#j &c. Now, tho' this
Competition of Concords may be carried on in-
finitely, yet 3 or 4 O^g^j" is the grcat^ft Length
we go in ordinary Practice; the old Scales
O.f Mujlck were carried no further than 2, or at
moft 3 Odfaoejy which is fully the Compafs of
any ordinary Voice : And tho- the 05?a<ve is the
moix perfect Concord^ yet after the Third 0-
Ba-ve the Agreement diminishes very fail > nor
do we go even fo far at one Movement, as
from the one Extreme to the other of a triple
or double Odfave, and feldorn beyond a fmgle
Qffiavei yet a Piece of Mufick may be ■ carried
agreeably thro' all the intermediate Sounds,
Within the Extremes of 3 or 4 QVtaves ; which .
wid afford all the Variety of Pleafnre the Har-
mony of Sounds is capable to afford, or at leaft
We to receive : For we can hardly raife Sounds
beyond that Compafs n either by Voice or In (frag-
ments, that fliail not offend the. Ear. Chords
are fitted for railing a great Variety of Degrees
of Sound j and if we fuppofe any Chord c Foot
long, which is but a fmall Length to give a good
Sound, the Fourth Offave below muff be Eight
Foot, which is fo long, that to give a. clear
Sound, it muft have a °;ocd Decree of Tendon \
and this will require a very great Tenfion in the
i Foot Chord: Now if we go beyond- the
Fourth OdiwDe, cither the acute Term will be
too fhort, pr the grave Term too long,- and
if in this the Length be {implied by the Groii-
2 neis
1S0 A Treatise Chap. VI.
nefs of the Chord, or in the other the Shortnefs
be exchanged with the Smalnefs, yet the Sound
will by that means become fo blunt in the one,
or fo (lender in the other, as to be ufelefs.
IS Cartes fuppofes we can go no further than
Three ObJaves, but he muft mean only, that
the Extremes of any greater Interval heard
without any of the intermediate Terms, have
little Concord to our Ears; but it will not follow,
that a Piece of Mufick may not go thro' a grea-
ter Compafs, e/pecially with many Parts.
CHAP. VI.
Of the Geometrical Part 0/ Mufick $ or y
how to divide right Lines, fo as their
Sections or Parts one zvith another •, or
with the Whole, Jhall contain any given
Interval of Sound.
1~*HE Degrees of Sound with refpect to
Tune, are juftly expreft by the Lengths
of Chords or right Lines ; and the Pro-
portions which we have hitherto explained be-
ing found, firfl by Experiments upon Chords,
and
§ i. of MU SICK. 181
and again confirmed by Reafoning ; the Divi-
(ion of a right Line into fuch Parts as fhall con-
ftitute one with another, or with the Whole, a~
ny Interval of Sound is a very eafy Matter :
For in the preceeding Parts we have all along
fuppofed the Numbers to reprefent the Lengths
of Chords; and therefore they may again be
eafily applied to them, which I fhall explain in
a few Problems.
§ i. Of the more general JDivifwn of Chords.
Problem ' I s O aflign fuch a Part of any
I. X right Line, as fhall confti-
tute any Concord ( or other Interval ) with
tlie Whole.
Rule. Divide the given Line into* as many
Parts, as the greateft Number of the Interval
has Units ; and of thefe take as many as the
leffer Number ; this with the Whole contains the
Interval fought. Example. To find fuch a Part
of the Line A B, as fhall be a $th to the
Whole. The $th is 2 : 3, therefore I divide the
Line into Three Parts, whereof 2, viz. A C,
is the Part fought $ that is, Two Lines, whofe
Lengths are as A B to A C, cater is paribus^
make a $th.
C
A - 1 1 -B
M 3 Corol-
i8x ^Treatise Chap. -VI.
Corollary. Let it be propofed to find
Two or more different Sections of the Line A
B. that {hall be to the Whole in any given Pro-
portion, 'Tis plain, we muft take the given
Ratios, and reduce them to one Fundamental
( if they are not fo ) by Prohl. 3. Chap, 4.
end then divide the Line into as may Parts as
that Fundamental has Units -, fo, to find the
Sections of the Line A. B. that fhall be Qdl'ave-,
$th and 3d g. I take the .Ratios 1:2, 2:3,
sfiiid 4 : 5, and reduce them to One Fundamen-
tal, the Series is 30 : 24, 20 : 15. the Funda-
mental ? is 30, and the Sccticns fought are 24
the id g. 20 the nth, and 15 the Octave.
Problem II. To find feveral Sections of
a Line, that, from the leaft gradually to the
Whole, (hall contain a given Series of Intervals^
in a given Order, i. e. fo as the leaft Section to
the next greater fliall contain a certain Inter-val y
from that to the next fhall be another; and fo
on. Rule. Reduce ail the Ratios to a conti-
nued Series, by Probl. 2. Chap. 4. Then di-
vide the Line into as many Parts as the greateft
Extreme of that Series ; and number the Parts
from the one End to the other, and you have
the Sections fought, at the Points of Division
anfvvcring the feveral Numbers . of the Series.
Fxaniple. To find feveral Sections of the Line
A R, io that the leaft to the next greater
fhall contain a ^d g. that to the next greater a
$th % and that to the Whole an Qffiave., The
"Three Ratios 4:5, 2:3, .1 : 2, reduced to
One Scries, make 8 : 10 : 15 : 30. ' So the Line
§ i. of mUSICK. 183
A B being divided into Thirty equal Parts,
we have the Sections fought at the Points C
Druid E, foas AC to A D is a 3d g. AD
to A E & S^h and A E to A B 052ave.
8 10 15 30
A ■ 1 — 1 1 ~. B
CLE
Problem III. To divide a Line into Two
Parts, which Dial! be any given Internal, Rule,
Add together the Numbers that contain- the
Ratio of that Interval, and divide the Line in-
to as many Parts as that Sum ; the Point of
Divifion anfwering to any of the given Num-
bers is the Point which feparates on either Hand
the Parts fought. Example. To divide the Line
A B -into Two Parts which {hall contain be-
twixt them a *qth y I add 3 and 4, .and. divide
the Line into 7 Parts, and the Point 4 or C
gives the Thing foughtj for A C is 4, and C
B is 3. A- — 1 — 1 — 1 — \—\ — 1 — r B.
NO TA. The Difference of this and the laft
Problem is, that there we found feveral Sccuons
of the Line which were not considered as alto-
gether precifely equal to the Whole ; but here
the Point fought muft be fuch as their Sum fhall
be exactly equal to the Whole.
Corollary. If it is propofeel to divide a Line
into more than Two Parts,which fhall be to one
another as any given Intervals from the lead to
the greatcfl; ^ we muft take the given Ratios,
and reduce them to one continued Series, as in
the laft Probl. and add them all together,- then
divide theJLine into as many Parts as that Sum.
M 4 '-/'■''■ " Mz
-/f Treatise . Chap. VI.
Example. To divide the Line AB into 4 Parts,
which fhall contain among thenv from the leaft
to the greateft, a 3 J g, qth and $th, I take the
Ratios 4:5, 3:4 and 2 : 3, which reduced to
one Series, it is 12 : 15 : 20 : 30, whofeSum is
77 ,• let the Line be divided into 77 Parts j and
if you firft take off 1 35 then 1 5, then 20, and
kitty 30 Parts, you, have the Parts fought e^
quai to the Whole.
The preceeding Problems are of a more ge-
neral Nature, I fhall now particularly treat of the
harmonical Divifion of Chords,
§ 2. Of -the harmonical Divifion of Chords,
I Explained already T wo different Senfes in
which any Interval is faid to be harmoni-
cally divided ; the Firft, When the Two Ex-
tremes with their Differences from, the middle
Term are in geometrical Proportion ; the 2d,
when an Interval is fo divided, as the Ex-
tremes and all the middle Terms are Concord
each with another, Now, we are to eonfider,
not the harmonical Divifion of an Interval Or
Ratio, but the Divifion of a fingle Number or
Line, into fuch Sections or Parts as, compared
together and with th*? Whole, fhall be harmoni-
cal in either of the Two Senfes mentioned, L e.
either with refpeCt to the Proportion of their
Quantity, which is the firft Senfe^ or of their
§ 2. of MUSIC K. i8j
Quality or Tune, wliich is the fecond Senfe of
harmonica!- Divifion.
Problem IV. To find Two Sections of a
Line which with the whole ihall be in harmoni-
ca! Proportion of their Quantity. To anfwer
this Demand, we may take any Three Numbers
in harmonica! Proportion^ 3,4,6,and divide the
whole Line into as many Parts as the greateft
of thefe Three Numbers (as here into 6),and at
the Points of Divifion anfwering the" other two
Numbers (as at 3 and 4) you have the Sections
fought. And an infinite Number of Examples
of this Kind may be found, becaufe betwixt any
Two Numbers given, we can put an harmonical
Mean, by Theor. 11. Chap. 4.
Note. The harmonica! Sections of this Pro-
blem added together, will ever be greater than
the Whole, as is plain from the Nature of that
Kind ; and this is therefore not fo properly a
Divifion of the Line as finding feveral Sections,
or the Quotes of feveral diftincl: Divifions.
Thefe Sections with the Whole,will alfo con-
ftitute an. harmonica! Series of the id Kind, but
not in every Cafe ; for Example, 2, 4, 6, is har-
monical in both Senfes ,- alfo 2:3:6; but2i 5
24, 28 is harmonical only in the Firft Senfe be-
caufe there is no Concord amongft them but
betwixt 2i, 28, (equal to 3 : 4.)
T o know how many Ways a Line may be
divided harmonically in both Senfes, fliall be pre-
fently explained.
Problem V. To find Two Sections of a
Line 3 that together and with the Whole fhall bs
har~
i'8<J ^Treatise .'Chap. VI.
harmonical in the Second Senfe ; that is, in re-
fpeft of Quality or Tune, Rule* Take any
Three Numbers that are Concord each with a-
■nother, and divide the Line by the greater!:, the
Points of Divifion anfwering the other Two
give the Sections fought : Take, for Example,
the' Numbers 2, 3, 8, or 2, 5, 8, and apply them
according to the Rule,
I obferved in the former Problem, That the
Two Sections together are always greater than
the whole Line ,- but here they may be either
greater, as in this Example, 2, 3, 4, or lefs, as
in this Example, 1, 2, 5, or equal, as here, 2,
3, 5, which laft is moft properly Divifion of
the Line, for here we find the true constituent
Parts of the Line : They may alfo be harmoni-
cal in the firft Senfe, as 2:3: 6, or otherwifeas
2:3:4. ■
N o w, to know all the Variety of Combina-
tions of Three Numbers that will folve this
Problem, we muft confider the preceeding gene-
ral Table of Concords, Pag. 172. and the harmc-
nical Series made out. of it, which contains the
Numbers of the Table and no other. I have
fliewn that all the Numbers of the Table of
Concords, are Concords one with .another, as well 1
as thefe that are particularly connected : We
have alfo feen that, tho' the Table were carried
on in infinitum, the lefler Number ci every Ra-
tio is one of thefe, 1. 2. 3. 4. 5; and the greater
Number of each Ratio one of thefe, 2. 3. 5. or
their Produces by 2. in infinitum, 'Tis plain
therefore, that if we fuppofe this Table of Con- \
cords
§ i. of MUSIC K. 187
cords carried on in infinitum*, we can find in it
infinite Combinations of Three Numbers that
fliall be all Concord. For Example^ Take any
Two that have no common Divifor, as 2 : 3,
you'll find an Infinity of other Numbers greater
to loyn with thefe jfor we may take any of the
Multiples in infinitum of either of thefe Two
"Numbers themfelves, or the Number 5, or its
Multiples : But if we fuppofe the Table of r Con-
cords Iimitecl(as with refpecfc to Practice it. is) fo
will the Variety of Numbers fought be: Suppoie
it limited to Three O&aves, then the harmoni-
ca! Series goes no farther than the Number 64,
as here, 1. 2. 3. 4. 5. 6. 8. 10. 12. 16. 20. 24.
32. 40. 48. 64, (jr. and as many Combinati-
ons of Three Numbers as we can find in that
Series, which have not a common Divifor, fo
many Ways may the Problem be folved. But
befides thefe we muft corifider again, that as
many of the preceeding Combinations as are
arithmetically proportional (fuch as 2. 3. 4, and
2. 5. 8) there are fo many Combinations of cor-
j refpondent Harmonicals ( in the firft Senfe )
which will folve this Problem. Thefe joyned to
i the preceeding, will exhauft all the Variety with
which this Problem can be folved, fuppofing 3
I Oblaves to be the greateft Concord. Again y
\ we are to take Notice, that of that Varie-
! ty there arc fome, of which the Two leffer
Numbers will be exactly equal to the greateft,
as 1. 2. 3. tlio' the greater Numbers .are other-
wife,
I fliall
188 ^Treatise Chap. VI.
I fhall now in Two diftincl: Problems ftiow
you, Firft, The Variety of Ways that a Line
may be cut, fo as the Sections compared toge-
ther and with the Whole fhall be harmonical in
both the Senfes explained ; and ido. How ma-
ny Ways it may be divided into Two Parts e-
qual to the Whole, and be harmonical in the
Second Senfe j for thefe can never be harmoni-
cal in the Firft Senfe, as fhall be alfo fhewn.
Problem VI. To find how many Ways
'tis poffible to take Two Sections of a Line, that
with the Whole fhall conftitute Three Terms
harmonical both in Quantity and Quality.
From the harmonical Series we can eafily
find an Anfwer to this Demand : In order to
which confider, Fir ft ^ That every Three Num-
bers in harmonical Proportion (of Quantity)
have other Three in arithmetical Proportion
correfponding to them, which contain the fame
Intervals or geometrical Ratios^ , tho' in a diffe-
rent Order ; and reciprocally every arithmetical
Series has a correfpondent Harmonical^ as has j
been explained in Theor. 14. Chap* 4. Let us I
next confider. That there can no Three Num- I
bers in arithmetical Proportion be taken, which I
fhall be all Concord one with another, unlefs they \
be found in the harmonical Series : Therefore it I
is impofiible that any Three Numbers which are j
in harmonical Proportion (of Quantity) can be I
all Concord unlefs their correfpondent Arithme- 1
ticals be contain'd in the harmonical Series, j
Hence 'tis plain, that as many Combinations of j
Three Numbers in arithmetical Proportion as I
cag I
I
§ 2. of MUSICK. 189
can be found in that Series, fo many Combina-
tions of Three Numbers in harmonical Propor-
tion are to be found, which fhall be Concord
each with another i and fo many Ways only
can a Line be divided harmonically in both
Senfes.
And in all that Series 'tis impoflible to find
any other Combination of Numbers in arith-
metical Proportion, than thofe in the following
Table-, with which I have joyned their correfpon-
dent ffarmonicals.
Arithmet.
3
4
1
3
12
10
3
5
Harmon.
.3.6
4 . 6
15 • 2 °
12 . 1?
5 • 15
8 . 20
N o w,to fbow
that there are
no other Com-
binations to be
found in the
Series to an-
fwer the pre-
fent Purpofe,
obfer've, the Three arithmetical Terms muft be
in radical Numbers, elfe tho' it may be a diffe-
rent arithmetical Series, yet it cannot contain
different Concords, fo 4 : 6 : 8 is a different
Series from 2:3:4, yet the geometrical Ratios,
or the Concords that the Numbers of the one
Series contain, being the fame with thefe in the
other, the correfpondent harmonical Series gives
the fame Divifion of the Line. Now by a fliort
and eafy Induction, I fhall fliow the Truth of
what's advanced : Look on the harmonical Se-
ries, and you fee, 177/0. That if we take the
Number 1, to make an arithmetical Series of
Three'
icjoc A Treatise Chap. VI.
Three Terms, it can only be join'd with 2:3.
or 3 : 5, for if you make 4 the middle
Term, 'the other Extreme muft be 7, which is
not in the Series ; or if you make 5 the Middle,
the other Extreme is % which is not in the Se-
ries : Now all after 5 are even Numbers, fo that
if you take any of thefe for the middle Term,
the other Extreme in arithmetical Proportion
with them, muft be an odd Number greater
than 5, and no foch is to be found in the Sc-
ries : Therefore there can be no other Combi-
nation in which 1 is the lelfer Extreme, but
thefe in the Tahle.
ido. Take Two for the leaft Extreme, and
the other Two Terms can only be 3 : 4, or 5 1
8 j for there is no other odd Number to take
as a middle Term, but 3 or 5 ; and if we take
4 or any even Number, the other Extreme muft
be an even Number, and thefe Three will ne-
ceffarily reduce to fome of the Forms wherein
1 is concerned, becaufe every even Number is
divifible by 2, and 2 divided by 2 quotes 1,
■%tio. Take 3 for the letter Extreme, the other
Two Terms can only be 4, 5; for if 5 is the
middle Term, the other Extreme muft be 7,
which is not in the Series : But there are no o*
ther Numbers in the Series to be made middle
Terras, 3 being the lefter Extreme, except even
Numbers • aiid 3 being an odd Number, the o-
ther Extreme muft be an odd Number too, but
no fuch is to be found in the Series greater than
5. /\to. The Number 4 can only joyn with m
6 9 for all the reft are even Numbers, and where
§ x. of MUSICK. i 9 i
the Three Terms are all even Numbers, they
are reducible. 5^0. There can be no Combina-
tion where 5 is the leaft Extreme, becaufe all
greater Numbers in the Scries are even; for
where one Extreme is odd, the other muft be
odd too, the middle Term being even. Laft^
lj. All the Numbers above 5 being even, are
reducible to fomc of the former Cafes : There-
fore we have found all the poffible Ways. any
Line can be divided, that the Ssclions compared
together and with the Whole, maybe harmo-
nical both in Quantity and Quality, as thefe are
. explain'd.
Problem VII. To di vide a Line into
Two Parts, equal to the Whole, fo as the Parts
among themfelves, and each with the Whole
fhall be Concord* and to difcover all the poifible
! Ways that this can be done. For the firft Part '
1 of the Problem, 'tis plain, that if we take
Three Numbers which are all Concord among
! themfelves, and whereof the Two leaft are e-
• qual to the greateft, then divide the given Line
into as ' many Parts as that greateft Number
contains Units, the Point of Divifion anfwering
; any of the lelfer Numbers folves the Problem :
So if we divide a Line A B into Three Parts,
one Third AC, and Two Thirds C B, or A
7) and D B are the Parts fought, for all thefe
are Concord 1 : 2, 2 : 3, 1 : 2. A — \ — \ — ~B
I fhall next (hew how many different Ways
this Problem can be folvcd ; and I affirm, that
there can be but Seven Solutions contained in
the
191 A Treatise Chap. VI.
the following Table., in which I have diftingui-*
fbed the Parts and the Whole.
That thefe are har-
monical Sections is plain,
becaufe there are no other
Numbers here but what
I ►{< I =
2
* >b 2 =
3
1^3 =
4
i * 4 =
5
i Hh 5 "
6
2 Hh 3 =
5
3*5-
| Parts.
8
Whole.
belong to the harmonical
Series; and 'tis remarkable
too, that there are no o-
ther here b llt what belong
to the Jimple Concords.
But then to prove, that
there can. be no other har-
monical Sections, •confider
that no other Number can poflfibly be any radi-
cal Term of a Concord, belides thefe ofthepre-
ceeding harmonical Series. Indeed we may
take any Ratio in many different Numbers, but
every Ratio can have but one radical Form,
and only thefe Numbers are harmonical ; fo 5 :
15 is a compound $th 3 yet 15 is no harmonical
Number, becaufe 5 : 15 is reducible to 1 : 3;
alfo 7 : 14 is an OcJave, yet neither 7 nor 14
are harmonical, face they are reducible to 1 : 2.
Now fince ail the poflible harmonical Ratios,
in their radical Forms, are contained in the Se-
ries, 'tis plain, that all the polfible harmonical
Sections of any Line or Number are to be found,
by adding every Number of the Series to it felf,
or every Two together, and taking thefe Num-
bers for the Two Parts, arid their Sum for the
whole Line. Now let us coniider how many
of fuch Additions will produce harmonical Se-
ctions,
§.z of MUSIC K. 193
jifions, and what will not : It is certain, that if
the Sum of any Two Numbers of the Series be
a Number which is not contained in it, then
the Divifioh of a Line in Two Parts, which are
in Proportion as thefe Two Numbers, can ne-
ver be harmonical j for Example the Sum
of 3 and 4 is 7, which is not an har-
monical Seffiioii) becaufe 7 is no harmoni-
cal Number,* or is not the radical Num-
ber of any harmonical Ratio, Again 'tis cer-
tain, That if any Two Numbers, with their
Sum, are to be found all in the Series, thefe
-Numbers conftitute an harmonical Sett ion. But
obferve, if the Numbers taken for the Parts are
reducible, they muft be brought to their radical
Form > for the Concords made of fuch Parts as
are reducible, muft neceffarily be the fame with
thefe made of their radical Numbers ; fo if we
take 4 and 6 their Sum is 1 o, and 4 : 6 are
harmonical Parts of 10 ; but then the Cafe is
not different from 2. 3. 5. Next^ We fee that
all the Numbers in that Series after the Num-
ber 5 3 are Compounds of the preceeding Num-
bers, by the continual multiplying of them by 2 ;
therefore we can take no Two Numbers in that
Series greater than 5, (for Parts) but what are
reducible to 5, and fome Number lefs, or both
lefs ; and if we take 5 or any odd Number lefs,
and a Number greater than 5, they can never
be harmonical Parts, becaufe their Sum will be
an odd Number,and all the Numbers in the Series
greater than 5, are even Numbers $ therefore
that Sum is not in the Series ; and if we take
N an
194 A Treatise Chap. VI.
an even Number Ids than ?, and a Number grea-
ter, the Sum is even and reducible ; therefore
all the Numbers that can poifiblymake the Two
Parts of different harmonic al Sections, are thefe^
i. 2. 3. 4. 5 j and if we add every Two of
thefe together, we find no other different
'harmonic al Sections but thefe of the pro-
ceeding Table, becaufe their Sum is either
odd or reducible ; and when the Parts are
equal, 'tis plain there can be But one fuch
Section, which is 1 : 1 : 2, becaufe all other
equal Sections are reducible to this.
§ 3. Containing farther Reflections upon the
Divifwn of Chord s.
"E have feen, in the Jaft Table ^ that the
harmonicalDivi&ons of a Line depend
upon the Numbers 2. 3, 4. 5* 6. 7. 8 ; and if
we reflect upon what has been already obferved
of thefe 1. 2. ?. 4. 5. 6. viz. That they are Con-
ecuh comparing every one with every other,
we craw this Conclufion, That if a Line is di-
vided into 2 or P, 4, 5 or 6 Parts, every Section
or Number of fiich Parts with the Whole, or
one with another, is Concord '; becaufe they are
all to one another as thefe Numbers I. 2. 3. 4.
£. 6. I fhall add now, that, taking in the Num-
ber 8, it will flill be true of the Series, 1. 2. 3*
4*
§ 3 . cfMUSICK. i 95
4. 5% 6. 8. that every Number with every other
is Concord ; and here we have the whole origi-
nal Concords. And as to the Conclnfion laft
drawn, it will hold of the Parts of a Line divi-
ded into 8 Parts, except the Number 7, which is
Concord with none of the reft. So that We
have here a Method of exhibiting in one Line
all the fiinple and original Concords, viz. by di^
vidingit into 8 equal Parts 5 and of thefe, tak^
ing 1. 2. 3, 4. 5. 6. and comparing them
together, and with the whole 8.
But if it be required to (how how a Line
may be divided in the moft llmple Manner to
exhibite all thefe Concords-, here it is : Divide
the Line A B into Two equal Parts at C\ then
divide the Part C B into Two equal Parts at
JD ; and again the Part G & into Tvvo equal
Parts at E 'Tis plain that AC or C^ are
each a Half of A ' B; and C D or B D are
each equal to a qth Part of the Line A B ; and
C E ox ED are A — — f--4--?— -£,
each an Sth Part of A B ; therefore A E is
equal to Five %th Parts of A B • and A D is
Six %th Parts, or Three qth Parts of it ; and
A E is therefore Five 6th Parts q£A7J. A-
gain, fince A B is Three qth Parts of A J$$
and A C is a Half^or Two j\ths of yf i?, there-
fore ^ C is Two 3 d Parts o£ A D ; then, be-
caufe A E\s Five 8*& Parts of ^f i? 5 and A
C Four 8?/'j ( or a Half) therefore ^f C is Four
Sti&sr of A E* Lafilys E Bis Three %ths of
^f .Z?. Conjequemlj AG to A B is znOcJa-zvi
ACto AD asthi AD to si B y i\ 4th; A C
N a to
io6* ^Treatise Chap. VI.
toii'a 3d g. AEto A D a 3d I A E to •
EB & 6th g. A E to A B a 6th I which- is
all agreeable to what has been already explain-
ed ; for AC and A B containing the Octave^
we have A D an arithmetical Mean, which
therefore gives us the g^with the acute Term
A C, and a /\th with the lower Term A B
of the Oclace. Again, A E is an arithmeti-
cal Mean betwixt the Extremes of the $th
AC and AD, and gives us all the reft of the
Concords.
I t will be worth our Pains to confider what
7)' Cartes obferves upon this Divifion of a Line :
But in order to tlie underftanding what he fays
here, I rauft give yon a fliort Account of fome
general Premifles he lays down in the Beginning
of his Work. Says he, c Every Senfe is capable
c of fome Pleafure, to which is required a cer-
c tain Proportion of the Objed to the Organ :
c Which Object muft fall regularly, and not very
c difficultly on the Senfes, that we may be able
c to perceive every Part diftin&ly : Hence.*
c thefe Objects are moft eafily perceived, whofe
c Difference of Parts is leaft, i, e. in which there
c is leaft Difference to be obferved ; and there-
c fore the Proportion of the Parts oucht to be
c arithmetical not geometrical j becaufe there
c are fewer Things to be noticed in the qrith*
€ metical Proportion^ fince the Differences are
c every where equal, and fo does not weary the
c Mind fo much in apprehending diftin&ly e-
c veiy Thing that is in it. He gives us this
I Example : Says he, The Proportion of thefe
t Lines
§ 3. ofMUSICK. 197
c Lines 3=£=i~ is cafier diftinguifhcd by the
c Eye, than the Proportion ofthefe ^~ r^
c becaufe in the firft we have nothing to notice
c but that the common Difference of the Lines
c is 1. ' He makes not the Application of this
exprefly to the Ear, by f ohfidering the Number
of Strokes or Imptilfe^made upon it at the fame
Time, by Motions of various Velocities ; and
what Similitude that has to perceiving the Dif-
ference of Parts by the Eye : He certainly
thought the Application plain \ and takes it alfo
for granted. That one Sound is to another in
Tuners the Lengths of Two Chords,r^mV pa-
ribus. From thcfe Premiffes he proceeds to
find the Concords in the Divifion of a Line, and
obferves, That if it be divided into 2, 3, 4, 5,
or 6 equal Parts, all the Sections are Concord ;
the firft and beft Concord OcJacc proceeds from
dividing the Line by the firft of all Num-
bers 2, and the next beft by the next Num-
ber 3, and fo on to the Number 6. But
then, fays he, we can proceed no further,
becaufe the Weaknefs of our Senfes cannot eaii-
]y diftinguhli greater Differences of Sounds : But
he forgot the 6th leffer, which requires a Divi-
fion by 8, tho' he elfewhere owns it as Concord.
We fhall next coniider what he fays upon the
preceeding Divifion of the Line A B y from
which he propofcs to (how how all the other
• Concords are contained in the OcJave^ and pro-
ceed from the Divifion of it, that their Nature
may be more diftin&ly known. Take it in his
N 3 own
ro8 /^Treatise Chap. VI.
own Words, as near as I can tranflate them.
■ ■ Fir ft then, from the Thing premifed it is
cc certain,this Divifion ought to be arithmetical^
" or into equal Parts, and what that is which
u ought to be divided is plain in the Chord AB\
" which is diftant from A C by the Part C B i
Cl but the Sound of A B, is diffant .from the
rc Sound of A C by an Qffave - 3 therefore the
" Part C B fiiall be the -Space or Interval of an
^ OcJave .; This is it therefore which ought to
" be divided into Two equal Parts to have the
cc whole Otlave divided, which is done in the
" Point D i and that we may know what Con-
" cord is generated properly and by it felf (pro-
cc prie & perje 9 as he calls it) by this Divifion,
* c we muft confider, that the Line A B> which
cc is the lozver or rraver Term of the Otlave*
< c is divided in Z), not in order to it felf (non
* c in ordine adfeipfum^ I fuppofe he means not
cc in order to a Comparifon of AB with A B)
* c for then it would be divided in C, as is al-
* c ready done (for AC compared to AB makes
* c the Octave) neither do we now divide the
" Unifon (viz. AB) but the Otlave, -{viz. the
" Interval of 8c^, which is CB^is he faid alrea-
fC dy) which confifts of Two Terms 3 - therefore
" while the graver Term is divided 5 that's done
* c in order to the acuter Term, not in order to
cc it felf. Hence the Concord which is properly
cc generated by that Divifion, is betwixt
" the Terms A C and A Z), which is a $th y
not betwixt A Z), Ji B^ which is a 4? h -, for
£he Part D B is only a Remainder, and
" generates
cc
§ 3- efMUSlCK. i 99
" generates a Concord by Accident, becaufe
<c that whatever Sound is Concord with one
cc Term o£Offave, ought aifo to be Concord
" with the other." In the fame Manner he
argues, that, the yd g. proceeds properly, & per
fe out of the Divifion of the nth, at the Point
E-, whereby we have A E a 3 d g, to the acute
Term of the $th, viz. to A C (for A C to A 7)
is 5th) and a]] the reft of the Concords are ac-
cidental ,*' and thus aifo he makes the tonus ma-
jor (of which afterwards) to proceed directly
from the 3d g, and the tonus minor and Semi-
tones to be all. accidental : And to fliow. that
this is not an imaginary Tiling, when he fays,
the 5th and 3d g. proceed properl)' from the Di-
vifion of Otlave, and the reft by Accident, he
lays, He found it by Experience in ftringed In-
ftruments, that if one String is ftruck, the Mo-
tion of it {hakes all tke Strings that are acuter
by any Species of 5th or 3 d g.but not thefe that
are qth or other Concord ; which can oniy pro-
ceed, fays he, from the Perfection of thefe Con-
cords, or the Imperfection of the" other, viz.
that the firft arc Concords per fe, and the others
per accidens, becaufe they flow neceffarily from
them. H Cartes feems to think it a Demon-
stration a priori from his Premiftes, that if there
is fuch a Thing as Concord among Sounds, it
muft proceed from the arithmetical Divifion of
a Line into 2. 3, eye. Parts, and that the more
fimple produce the better Concords. 'Tis true,
that Men muft have known by Experience,
that there was fuch a Thing as Concord before
N 4 they
20o >? Treatise Chap. VII.
tliey reafoned about it ) but whether the gene-
ral Refle&ion which he makes upon Nature, be
fufficient to conclude that fuch Division muft
infallibly produce fuch Concords, I don't fo
clearly fee; yet I. muft own his Reafoning is
very ingenious, excepting the fubt.il Diftin£tion
of Concords per fe (j per accidens, which I
don't very well underftand; but let every one
take them as they can,
CHAP. VII.
Of Harmo n y, explaining the Nature .
and Variety of it y as it depends upon the
various Combinations of concording
Sounds*
IN Chap. II. § i. I (hewed you tlieDiftincli-
on that is made betwixt the Word Con-
cord, which is the Agreement of Two
Sounds confidered either in Confonance or Sue-
ceffion, and Harmony, which is the Agreement
of more, confidered always in Confonance, and
requires at leaft Three Sounds. In order to pro-
duce a perfect Harmonj, there muft be no Dif-
cord
of MUSI CK. 201
cord found between any Two of the fimpleSounds;
but each n>uft be in fome Degree of Concord to
all the re& Hence Harmony is very well de-
fined, The Sum o/Concords arifing from
the Combination of Two or more Concords, i.
e. of Three or more fimple Sounds ftriking the
Ear all together ; and different Compofitions of
Concords make different Harmony*
To underftand the Nature, and determine
the Number and Preference of Harmonies, we
muft confider, that in every compound Sound,
where there are more than Two Simples,
we have Three Things obfervable, ift. The
primary Relation of every fimple Sound to the
Fundamental (or graveft) whereby they make
. different Degrees of Concord with it. idly. The
mutual Relations of the acuter Sounds each
with another, whereby they mix either Concord
or Difcord into the Compound, ^dly. The fecon-
dary Relation of the Whole, whereby all the
Terms unite their Vibrations, or coincide more
or lefs frequently.
The Two firft of thefe depend upon one a-
nother, and upon them depends the laft. Let
us fuppofe Four Sounds A* B. C D. whereof
A is the gravefi, B next acuter, then C, and
D the acuteft \ A is called the Fundamental,
and the Relations of B, C, and D, to- A, are
primary Relations : So if B is a 3d g. above A,
. that primary Relation is 4 to 5 ; and if C is
%th to A, that primary Relation is 2 to 3 ;
and if D is %ve to A, that is 1 to 2. Again,to
£nd the mutual Relations of all the acute Terms
$C 9
%oi ^Treatise Chap. VII.
Mi Q A we.muft take their primary Relations
to the Fundamental^ and fubftrac't each Jeffer j
from each greater, by the Rule of Subjlra&ion
of Intervals ; fo in the preceeding Example, B J
to. C is g to 6, a 3 J L B to D is 5 to 8, a <fr&
/, and C to 7) 3 to 4, a 4/7?. Or, if we take
all the primary Relations, and reduce them to
one common Fundamental, by Probl. 3. C/;^.
4* we ilia 11 fee all the mutual Relations in one
Series ,* fo the preceeding Example is 30. 24.
20* 15.
AG A IN, having the mutual Relations of
each Sound to the next in any Series, we may
find the primary Relations, by Addition of 7/z-
tercals ; and then by thefe all the reft of the
mutual Relations ; or reduce the given Relati-
ons to a continued Series by Probl. 2. Chap. 4.
and then all will appear at once. Laftly, to
find the fecondary Relation of the Whole, find
the leaft common Dividend to all the letter
Terms or Numbers of the primary Relations,
u e. the lea ft Number that will be divided by
each of them exactly without a Remainder,-
that is the Thing fought, and fliows that all the
iimple Sounds coincide after every fo many Vi-
brations of the Fundamental as that Number
found cxpreffes ' So in the preceeding Example,
the leffer Terms of the Three primary Relations
are 4. 2* 1. whofe leaft common Dividend is 4,
therefore at every Fourth Vibration of the Fun-
damental the Whole will coincide 3 - and this is
what I call the fe condar y Relation of the Whole.
I fhail firft (Sow how in every. Cafe you may
find
§ i. of MUSIC K. 203
find this leaft Dividend, and then explain how
it cxpreftes the Coincidence of the Whole.
Problem. To find the leaft common Divi-
idend to any given Numbers. Rule. imo. If
each greater of the given Numbers is a Multiple
of each IeiTei^ then the greateft of them is tha
(Thing fought ; as in the preceeding Example,
, zdo, If 'tis not fo, but fome of them are com-
menfurable together, others not ; take the grea~
,teft of all that are commenfurable, and, paffing
their aliquot Parts, multiply them together, and
vwith the reft of the Numbers continually, the
ilaft Product is the Number fought. Example,
2. 3. 4. 6. 8. Here 2. 4. 8, are commenfurable,
and 8 their leaft Dividend ; alfo 3. 6 commenfu-
rable and 6 their leaft Dividend : Then 8. 6,
.multiplied together produce 48, the Number
ifought. Take another Example. 2. 3. 5, 4*
'Here 2 . 4 are commenfurable and all the reft
■incommcnfurable, therefore I multiply 3. 4, 5,
continually, the Product is 60 the Number
fought. 3#z'o. If all the Numbers are incommen-
surable, multiply them all continually, and the
iaft Product is the Anfwer. Example. 2. P, $,
7. the Product is 210. The Reafon of this
■Rule is obvious from the Nature of Multiplica-
tion and Divifion.
N o w I {hall fnow that the leaft common Di-
vidend to the leffer Terms of any Number of
■wimary Relations*, expreffes the Vibrations or
\ the Fundamental to every Coincidence. ■ Thus,
.^f the Numbers that exprefs the Ratio of any
Jnterval ? the leffer is the Length of the ac titer
1 Chord,
xo4 A Treatise Chap. "VII.
Chord, and the greater the Length of the gra-
ver : Or reciprocally, the leffer is the Number
of Vibrations of the longer, and the greater the
Vibrations of the fhorter Chord, that are per-*
formed in the fame Time ; consequently the lef-
fer Numbers of all the primary Relations of any
compound Sound, are the Numbers of the Vibra-
tions of the common Fundamental which go to
each Coincidence thereof with the feveral a-
cute Terms ; but 'tis plain if the Fundamental
coincide with any acute Term after every 3 (for
Example) of its own Vibrations, it will alfo co-
incide with it after every 6 or p, or other Mul-
tiple, or Number of Vibrations which is di-
visible by 3, and fo of any other Number ; con-
fequently the leaft Number which can be ex-
actly divided by every one of the Numbers of
Vibrations of the Fundamental^ which go to a
Coincidence with the feveral acute Terms, muft
be the Vibrations of that Fundamental at which
every total Coincidence is performed. For_Z£tf-
ample, fuppofe a common Fundamental coin-
cide with any acute Term after 2 of its own Vi-
brations, and with another at 3 ; then what-
ever the mutual Relation of thefe Two acute
Terms is^ it is plain they cannot both together
coincide with that Fundamental^ till Six Vibra-*
tions of it be finifhed ; and at that Number pre-
cifely they muft ; for the Fundamental coin-
ciding with the one at 2, and with the other
at 3, muft coincide with each of them at Six .;
and no fooner can they all coincide, becaufe
6 is the leait Multiple to both 2 and 3 : Or thus,
of MUSIC K. io T
the Fundamental coinciding with the one after
2 5 muft coincide with that one alfo after 4. 6.
)8. &c. ftill adding 2 more ; and coinciding .
•with the other after 3. muft coincide with it
.alfo after 6. 9. 12. &c. ftill adding 3 more - 9 fo
. that they cannot all coincide till after 6. be-
.! caufe that is the leaft Number which is com-
J mon to both the preceeding Series of Coinci-
J dences. Next for the Application of this to
\ Harmony.
Harmony is a compound Sound confifting
(as we take it here) of Three or move Jimpk
; Sounds - s the proper Ingredients of it *are Con-
I cords ; and therefore all Di/cords in the prima-
ry Relations efpecially, and alfo in the mutual
Relations of the feveral acute Terms are
j abfolutely forbidden.
'T 1 s true that Difiords are ufed in Mufick^
\ but not for themfelves (imply • they are ufed
\ as Means to make the Concords appear more
agreeable by the Oppofition ; but more of this
in another Place.
Now any Number of Concords being pro-
pofed to ftand mprimary Relation with a com-
! mon Fundamental ; we difcover whether or
no -they conftitute a perfect Harmony, by find-
ing their mutual Relations. Example. Suppofe
thefe primary Intervals, which are Concords,
viz. 3d g. Kth, %ve, their mutual Relations qxq
all Concord, and therefore can ftand in Harmo-
ny ; for the 3^ g. and 5th, are to one another
as 5 : 6 a %d. I. The -$d g. and OSiave as 5 :
8, a 6th L the $th and OBave are as 3 : 4, a
qh
ig(S ^Treatise Chap. VIL
qth ; as appears in this Series to which the
given Relations are reduced, viz. 30 : 24 : 20 :
15. Again, take qth, 5th, and Octave, they
cannot ftand together, becaufe betwixt the /\th
and $th is a Difcord, the Ratio being 8 : 9*
Or fuppofing any Number of Sounds, which are
Concord each to the next, from the loweft to
tile higheft j to know if they can ftand in Har+
mony we mnft find their primary Relations^
and all the other mutual Relations, which mnft
be all Concord; fo let any Number of Sounds bd
&§ 4 : 5 : 6 : 8 they can ftand in Harmony, be-
caufe each to each is Concord ; but thefe can-
not 4. 6. 9, becaufe 4 : 9 is Difcord.
We have confidered the neceflary Conditi-
ons for making Harmony, from which it will
be eafy to enumerate or give a general Table
of all the poilible Variety ; but let us firft exa-
mine hocv the Preference of Harmonies is to
be determined ; and here comes in the Consi-
deration of the fecondary Relations. Now up-
on all the Three Things mentioned, ©/«. the
primary, fecondary, and mutual Relations, does
the Perfection of Harmonies depend $ fo that
Hegard mnft be had to them all in mak-
ing a right Judgment : It is not the beft fri*
mary Relation that makes beft Harmony ; for
then aqth and nth muft be better than a qth and
6th j yet the firft Two cannot ftand together,
becaufe of the Difcord in their mutual Relati*
m : "Nor does L the heft fecondary Relation
Carry it ,• for then alfo would a qtft and §t%
V^YsKq fecondary Relation with a common Fun*
da mental
V
§ i. of MUSIC K. zor
damental is 6, be better than %d I. and 5/7^
whofe fecondary Relation is 10; but here alio
the Preference is due to the better mutual Re-
lation of the id I. and 5//.?, which is a 3^ g,
and a qthandOSfave would be equal to a 6th g.
and Obi a^e,th.Q fecondary Relation of both being
3, which cannot polfiblybe, the Ingredients being;
different. As to the mutual Relations, thuy
depend altogether upon the primary ', yet not io
as that the bed: primary Relation fhall always
produce the beft mutual Relation 5 for 'tis con-
trary when two Terms are joyned to a Funda~.
mental ; fo a <$th and Odiave contain betwixt
'them a qth„ ; and a qth and 05fa-ve contain a
Jr&. But the primary Relations are by fai*
[more confide rable, and, with the Bcm^ry, af<
lord us the following Rule for determining the
[Preference of Harmony, 'in which that muft al-
ways be taken for a neceifary Condition, that
there be no DiJ'cord among any of the Terms ;
therefore this is the Rule, ' that comparing Two
[Harmonies (which have an equal Number of
Terms) that which has both the beft primary
land fecondary Relation, is mofr perfect \ but in
iTwo Cafes, where the Advantage is in the
primary Relations of the one, and in the fecon-
\dQ,ry of the other, we have no certain Rule;
'the primary Relations are the principal and
mod considerable Things ; but how the Ad-
vantage here 'ougnt to be proportioned to the
Difadvantage in the fecondary, or contranly,
I in order to judge of the comparative Perfection,
, is a Thing we know not how to determine 5
and
io8 ^/Treatise Chap. Vll.
and therefore a well tuned Ear muft be the laft
Refort in thefe Cafes.
L e t us next take a View of the poffible
Combinations of Concords that conftitute Har-
mony ; in order to which confider, That as we
diftinguifhed Concords into fimple and compound,
fo is Harmony diftinguifhable : That is fimple
Ffarmony, where there is no Comoro] to the
Fundamental above an clave, and it is com"
pound, which to the fimple Harmony of one
Offiave, adds that of another OcJave. The In-
gredients of fimple Harmony are the 7 fimple 0-
riginal Concords, of which there- can be but 18
different Combinations that are Harmony,
which I have placed in the following Table.
TABLE of Harmonies.
»
2 dry
ReL
2
3
3
4
5
5
2 dry
ReL
4
10
3
12
5
1 15
$th $ve
qth five
\6th g. %ve
,%d g. %ve
\^dl. %ve
■6th I. %ve
idg.^th
idl. $th
4th, 6th g.
■Zdg.6thg.
3d I. 6th l.
4?h, 6th I.
3dg.$th, %*&€
3d I. $th, %ve
qth, 6th g. %<ve
3dg.6thg.Sye
id l.6th%Sve
/\th,6thl. %ve.
I f we reflect on what has been explained of
thefe original Concords, we fee plainly that
here are ail the poffible Combinations that make
Harmony ; for the Offiave is compofed of a
%th and qth, or a 6th and 3 d, which have a Va-
riety of greater and leffer : Out of thefe are
* --- ': - - : ■ the
*/ MUSIC K. 109
'the firft Six Harmonies compofed ,• then the $th
being compofed of 3d g. and 3d /. and the 6th of
qth and 3d, from thefe proceed the next Six of
the Table • then an OcJave 5°y ne d to each of
thefe Six, make the laft Six.
Now the firft 1 2 Combinations have each 2
Terms added to the Fundamental, and their
Perfection is according to the Order of the
Table: Of the firft 6 each has an O clave >
and their Preference is according to the Per-
fection of the other leffer Concord jOvned to that
OcJave, as that has been already determined $
and with this alfo agrees the Perfection of their
fecondary Relations. For the next 6, the Pre-
ference is given to the Two Combinations with
the $th, whereof that which hath the 3d g*
is beft ; then to the Two Combinations with
the 6th g. of which that which has the 4th is
beft : Then follows the Combinations with the
6th I. where the id /. is preferred to the qth 5
for the great Advantage of the fecondary Rela-
tion, which does more than balance the Advan-
tage of the qth above the 3d /. So that in thefe
Six we have not followed the Order of the fe-
condary Relations*, nor altogether the Order of
the primary, as in the laft Cafe. ° Then come
in the laft Place the Six Combinations ariiing
; from the Divifion of the Odfave, into 3 Con-
: cords, which I Jiave placed kit, not becaufe
c they are leaft perfect but becaufe they are moft
l complex, and are the Mixtures- of the other 1 2
one with another 5 and for their Perfection^
they are plainly preferable to the immediately
O pre^
aio A Treatise Chap. VIL
proceeding Six, becaufe they have the very fame
Ingredients, and an O&ave more, which does not
alter the fecondary Relation? and fo are equal
to them in that Refped 4 but as they have an
Octave? they are much preferable ; and being
compared with the firft Six, they have the fame
Ingredients, with the Addition of one Concord
more, which does indeed alter the fecondary
Relations? and make the Compofition more fen-
fible, but ye adds an agreeable Sweetnefs, for
which in fome Refpecl: they are preferable.
For compound Harmonj? I fliall leave yon
to find the Variety for your felves out of the
Combinations of the jimple Harmonies of feve-
ral Qffaves. And obferve? That we may have
Harmony when none of the primary Intervals
are within an Oclave? as if to a Fundamental
be joyned a $th above OSiave? and a double
Oclave. Of fuch Harmonies the fecondary Rela-
tions are ever equal to thofe o£ the Jimple Har-
monies? whofe primary Intervals have the fame
Denomination } - and in Practice they are rec-
koned the fame, tho'. feldom are any fuch ufed,
I have brought all the Combinations of Con-
cords into the Table of Harmony which anfwer
to that general Chara&er^/^.That there muft be
no Difcord among any of the Terms,- yet thefe
few Things jmift be obferved. imo: That in
Practice Dif cords are in fome Circumftances ad-
mitted, not for themlelves, fimply confidered,
but to prepare the Mind for a greater Relifh of
the fucceeding more perfect Harmony,. 2M
,That tho' the qtk 7 taken by it felf 3 is Concord,
% and
of MUSIC K> in
nnd in the next Degree to the Kth ; yet in Pra-
ctice 'tis reckoned a Difiord when it (rands next
to the Fundamental ; and therefore thefe Com-
binations of the prececding Table, where it
poifeffes that Place, are not to be admitted as
Harmonies ; but 'tis admitted in every other
Part of the Harmony ,fo that the qth is Concordat
Di/cord> according to the Situation $ for Ex-
ample, if betwixt the Extremes of an Qffiave is
placed an arithmetical Mean, we have it divi^
ded into a qth and a $th 2. 3. 4. which Num-
bers., if we apply to the Vibrations of Chords,
then tjie $th is next the Fundamental, and the
fecondary Relation is in this Cafe, 2. But take
an harmonical Mean, as here 3. 4. 6, and the
qth is next the Fundamental, and the fecon-
dary Relation is 3, Now in thefe Two Cafes,
I the component Parts being the fame, viz* a qth y
$th, 8cv?, differing only in the Position of the
qth and $th, which occafions the Difference o£
I I the fecondary Relation, the different Effects
j can only be laid on the different Pofitions of
jl the /\th and 5th j which Effect can only bemea-
j fured by the fecondary Relation ; and by Ex-
:j perience we find that the heft, fecondary Rela-
tion makes the beft Compofition, fo % 3. 4. is
'better than 3:4:6: And thus in all Cafes,
? where the fame Interval is divided into the fame
Parts differently fituated, the Preference will an-
cfwer to the fecondary Relation, the leffer mak-
ing the beft Compofition, which plainly depends
upon the primary Relation ; but the 4th next
the Fundamental is not on'y worfe than the
O 2 iih
211 ^Treatise Chap. VII.
5th, b ut is reckoned Difcord in that Pofition ;
and therefore all the other Combinations of the
Table are preferr'd to it, or rather it is quite re-
jected ; the Reafon affigned for this is, that the
graver Sounds are the molt powerful, and raife
our Attention moft; fo that the qth being next
the Fundamental, its Imperfection compared
with the OVtave and $th is made more remark-
able, and confequently it muft be lefs agreeable
than when it is heard alone > whereas when it
ftands next the acute Term of the Odfave, that
Imperfection is drowned by its being between
the $th and clave, both in primary Relation
to the Fundamental. But this does not hold
in the 6th and id, becaufe they differ not in
their Perfe&ion fo much as the 5th and 4th. But
we fhall hear If Cartes reafoning upon this. Says
lie, H<ec infalicijfimafiuz. The 4th is the mqft un-
happy of all the Concords, and never admitted
in Songs, but by Accident (he means not next
the Fundamental, but as it falls accidentally a-
mong the mutual Relations) not that it is more
imperfeffi than the 3d or 6th, but becaufe it is
too near the 5th, and lofes its Szveetnefs by this
Neighbourhood ; for underftanding which we
muft notice, That a 5th is never heard, but the
acuter 4th feems fome way to refound y zvhich is
a Confequent of what was J 'aid before, that the
Fundamental never founds but. the acuter Octave
feems to dofq too.
LET?; the Tines A C andD B be a 5th, ana
the Line .|£ F 5 an- acuter Octave to A C> it zvil
he a 4th to D B ; and if it refound to the Fun-
daments
of MUSIC K. 2tj
damental, then, when the 5 th is founded zvith
a c the Fundamental, this
E f Relbnance is a 4th a-
bo<ve the 5 th that always follows it, -which is
the Reafon it is not admitted next the Bafs \ for
fince all the reft of the Concords in Mufick are
only nfefulfor 'varying of the 5 th, certainly the
4th zvhich does not Jo is iifelefs, which - is plain
from this, That if we put it next the Bafs, the
acuter 5th will re/bund, and there the Ear will
obferve it out of its Place, therefore the 4th
would be very difpleafing^ as if we had the Sha-
dow for the Sub/lance, an Image for the, real
1 Thing, Elfewhere he fays it ferves in Com-
; petition where the fame Reafon occurs not,
which hinders its (landing next the Bafs. It is
5 well obferved, that the reft of the fimple Con-
I cords ferye only for varying the 5th ; Variety
: is certainly the Life of all feniual Ple-afbre, with-
fi out which the more exquiiite but cloy the foon-
; er; and in Mufick, were there no more Con-
1 cords but clave and $th, it would prove a very
; poor Fund of Pleafurej but we have more, and
agreeable to Li' Cartes's Notion, we may fay,
They are all defigned to vary the $th, for they
■ all proceed from it, as we faw in the Divificns
| of the upper and lower $t'h of the Oc~fave\ : in
Chap. 5. and that all the Variety in Mufick
' proceeds from thefe $ds and 6th s arifing from
the Divifion of the %th directly or accidentally,
1 as we Oiall fee more particularly afterwards :
Mean time obferve, that as the qth rifes na-
1 turally from the* Divifion of the Slave, fo it
1 O 3 ferves
H4 ^Treatise Chap. VII.
ferves to vary it, and accordingly is admitted
in Composition in every Part but next the Fun-
damental or Bafs; for the 5th being more per-
feci: and capable of Variety (which the qtb is
not, fince no leffer Concord agrees to both its.
Extremes) by Means of the 3^, ought to Itand
next the Fundamental. Now if the qtp muft
not fland with the Fundamental, then this 4^,
with the Oblave, muft not be reckoned among
fimple Harmonies. To prove that the qth con**
fidered by it felf is a Concord, Kircher makes a
very odd Argument, Says he, A qth added to a
$th makes an OtJave, which is Concord; but
nothing gives what it has not, therefore, the
4\th is a Concord : But by the fame Argument
you may prove that any Interval lefs than
OtJave is a Concord.
I have obferved of the Series 1. 2. 3,. 4. 5. 6«
8. that they are Concords each with other. They
contain all the original Concords, and the chief
of the compound ; and they ftand in fuch Order
that Seven Sounds in the Proportions and Order
of this Series pyned in onelfarmony is the moft
complete and perfect that can be heard : For
here we have the chief and principal of all the
Harmonies of the preceeding Table, as you'll
fee by comparing thefe Numbers with that
Table ; fo that in this fhort and fimple Series
we have the whole eflential Principles and Inr.
gredients of Mufick ; and ail at once the moft
agreeable Eife6t that Sounds in Confonance can
have.
of MUSI CK. 21 f
Let us now confider how thefe Sounds may
be raifcd ,• this will be eafily i'ound from th 3
Things already explained; but we muft hrft ob-
ferve, that there will be a great Difference be-
twixt applying thefe Numbers to the Lengths of
Chords,and to their Vibrations : If they are ap-
plied to the Chords, then 'tis eafy to find Seven
Chords which fliall be as thefe Seven Numbers ;
but 8 being the Jongeft Chord, the lefs perfect
Concords ftand in primary Relation to the Fun-
damental ,• and the fecondary Relation is 15:
But if we have Seven Sounds whofe Vibrations
are as thefe Numbers, then 1 is the Vibration
of the Fundamental, and fo on in Order to 8
the Vibration of the acntefi performed in the
fame Time : And thus the beft Concords ftand
in primary Relation to the Fundamental, and
•1 is the fecondary Relation : Therefore to
afford this moft perfect Harmony,- we muft find
Seven Sounds which from the loweft to the
higheft (hall be as 1 : 2 : 3 : 4 : 5 : 6 : 8, the
leaft Number reprefenting the graveft Sound.
Now, to do this, let us mind that the Lengths
.of Chords are in fimple reciprocal Proportion of
theirVibrations accomplifhed in the fame Time,
out of which I fliall draw the Two following
Problems, whereof the firft ftiaH folve the Que-
ftion in hand.
Problem I. To find the Lengths of feve- **
ral Chords, whofe Vibrations performed in the
fame Time, {hall be as a given Rank of Num-
bers. Rule. Take the given Series, and out of
it find another reciprocal to it, by Th;or. 14.
p 4 Obf.
tx6 ^Treatise Cha#; VII,
Chap. 4. which, according to the Demonftra-
tion there given, and what I have premifed
here, is the Series of Lengths fought, fo the
preceeding Series 1.2. 3. 4. 5. 6. 8, being given
as a Series of Vibrations performed in the fame
Time, the Lengths of Seven Chords, to which
that Series of Vibrations agrees, are 120,60.40,.
30. 24. 20. 15. And thefc Seven Chords being
in every other Rcfpecl: cqral and alike, and alt
founded together, fhall produce the Harmony
required.
Problem II. The Lengths of feveral Chords
being given, to find the Number of Vibrations
of each performed in the fame Time. This is
done the fame Way as the former : And fo if
the Series 1. 2. 3. 4, y. 6. 8, &c. be the Length
of Seven Chords, their Vibrations fought are
120. 60. 40. 30. 24. 20. 15.
Note. From what has been explained in
Theor. 14. Chap. 4. we fee that if one of thefe,
viz. the Lengths of feveral Chords, or their
Vibrations accomplifhed in the fame Time, make
a continued arithmetical or harmonical Series,
the other will be reciprocally an harmonical or
arithmetical Series,fo the preceeding Series 1. 2.'
3. 4, 5. 6\ being continuedly arithmetical, its
correfpondent Series 120. 60. 40. 30. 24. 20. is
continuedly harmonical ; but the Number 8 in
the firft Series interrupts the arithmetical Pro-
portion^ and fo is the harmonical 'Proportion
interrupted by its Correfpondent 15. But as in
the firft, 2. 4. 6. 8. are continuedly arithmetical^
fo are thefe correfpondent to them in the other
harmo^
§ r. of MU SICK. ii7
harmonica I \ viz. 60 : 30 : 20 : 15, Alfo it will
hold univerfally, that taking any Numbers out
of the one Series in continued arithmetical or
fcarmonical Proportion, their Correspondents in
the other will be reciprocally harmonical or a-
fithmetical.
"
CHAP. VIII.
(yconcinnous Intervals, and the Scale of
Mufick.
§ 1. Of the Necejfity andUfe of concinnous Dif-
cords, and of their Original and Dependence
-on the Concords.
1
TT E have, in the prcceeding Chapters,
f coniidered the firft and moft eltenti-
al Principles [as far as concerns the
firft Part of the Definition ] of Mufick, viz.
thcfe Relations of Sound in Acutenefs and Gra-
vity whofe Extremes are Concord ; for without
tliefe there can be no Mufick : The indefinite
Number of other Ratios being all Difcord, be-
long not effentially to Mufick, becaufe of them-
felvos
2i8 A Treatise Chap. VIII.
felves they produce no Pleafure ; yet fome of
them are admitted into the Syflem as necefTary
to the better being of it, both with refpeel: to
Confonance arid SucceJJfion> but moft remarkably i
~m this } and fuch are called concinnous Inter-
vah\, as being apt or fit for the Improvement of
MuficfcAM otherDifcords are called inconcinnous.
To explain what thefe concinnous Intervals are,
their Number, Nature and Office, {hall employ
this Chapter.
I n order to which, I fhall firft offer the fol-
lowing Considerations, to prove that fome o-
thcr than the harmonic al Intervals of Sound
( h e. inch whofe Extremes are Concord) are
necelfary for the Improvement, or better Being
of Mufich
W e know by Experience how much the
Mind of Man is delighted with Variety : It can
fraud no Difpute, whether we confider intellecl:u-
al or fenfible Pleafures ; every one will be con-
felons of it to himfelf : If you ask the Reafon,
I can only anfwer, That we are made fb : And
if we apply this Rule to Mrtjick, then it is plain
the more Variety there, is in it, it will be the
more entertaining, unlefs it proceed to an Ex-
cefs \ for fo limited are our Capacities, that
too much or too little are equally fatal to our
Pleafures. Let us then confider what muft be
the EfTed of having no other but harmonic al
Intervals in the S?/iem oi Mu/ick, and,
Firft) With refpeel: to a fin de Voice, if that
fhduld move always from one Degree of Time
to another^ fo as every Note or Sound to : the
next
§ I, $f MUSIC K. u 9
.next were in the Ratio of fome Concord, the
, Variety which we happily know to be the Life
,iof Mufeck would foon be exhaufted. For ta
,imove by no other than harmonica! Intervals,
: would not only want Variety, and fo weary us
( with a tedious Repetition of the fame Things;
^but the very Perfection of fuch Relations of
.Sounds would cloy the Ear, in the fame Man-
|rier as fweet and lufcious Things do the Tafte,
which are therefore artfully feafoned with the
.Mixture offowr and bitter: And fo mMufick the
j Perfection of the harmonic al Intervals are fet
| ofr^ and as 'twere feafoned with other Kinds of
J Intervals that are never agreeable by themfelves,
,but only in order to make the Agreement of
< the other more various and remarkable. D*
) Cartes has a Notion here that's worth our con-
) fidering. He obferves, that an acute Sound re-
quires a greater Force to produce it either in
I ,the Motion of the vocal Organs of an Animal,
i ©r in ftriking a String; which we know by Ex-
| perience, fays he, in Strings, for the more they
, are {tretched they become the acuter, and re-
| quire the greater Force to move them : And
I hence he concludes, that acute Sounds, or the
Motion of the Air that produce them immedi-
i ately, ftrike the Ear with more Force : From
which Obfervations he thinks may be drawn the
true and primary Reafon why Degrees ( which
are Intervals lefs than any Concord) were in-
vented j which Reafon he judges to be this, Left
if the Voice did always proceed by harmonical
Pittances^ there iliould be too great Difpropor-
tion
220 ^Treatise Chap. Vlli;
tion or Inequality in the Intenfenefs of it (by*
which Intenfenefs he plainly means that Force
with which it is produced, and with which al-
fo it ftrikes the Ear) which would weary both
Singer and Hearer. For Example. Let A and
JB be at the Diftance of a greater 3d, if one
would afcend from A to i?, then becaufe
JB being, acuter ftrikes the Ear with more
Force than A ; left that Difproportion fliould
prove uneafy, . another Sound C is put between
them, by which as by a Step we may afcend
more eafily, and with lefs unequal Force in rai-
fing the Voice. Hence it appears, fays he, that
the Degrees are nothing but a certain Medium
contrived to be put betwixt the Extremes of
the Concords ", for moderating their Inequality,
but of themfelves they have not Sweetnefs e-
nough to fatisfy the Ear, and are of Ufe only
with regard to the Concords ; 10 that when the
Voice has moved one Degree,the Ear is not yet
iatisfied till we come to another, which there-
fore muft be Concord with the firft Sound. Thus
far H Cartes reafens on this Matter - 3 the Sub-
itance of what he fays being plainly this, viz.
That by a fit Divifion of the concording Inter-
vals into lefferOnes, the Voice will pals fmooth-
ly from one $%)te to another, and the Hearer
be prepared for a more exquifite Reiiili of the
perfecter Intervals, whofe Extremes are the
proper Points in which the Ear finds the ex-
pected Reft and Pleafare. ' Yet moving hy bar-
monical Diftances is alfo neceflary, but not fo
frequently: The Thing therefore required as
to
§ i. of MUSIC K. in
'to this Part is, fuch Intervals lefs than any har-
mimical one, which iliall divide thefe, in order
(that the Movement of a Sound from their one
Extreme to another, by thefe Degrees, may be
ifmooth and agreeable ; and by the Variety im-
prove the more effential Principles of Mupck to
i a Capacity of affording greater Pleafure, and
all together make a more perfect Syftem.
idly. Letus confider Mufick in Parts, i. e.
when Two or more Voices joyn mConfonance;
' the general Rule is. That the fucceffive Sounds
i of each be fo ordered, that the feveral Voices
I {hall always be Concord. Now there ought to
I be a Variety in the Choice of thefe fucceffive
i Concords, and alfo in the Method of their Sue-
ceffions ; but all this depends upon the Move-
ments of the fingle Parts. And if thefe could
move in an agreeable Manner only by harmo-
nical Diftances, there are but a few different
Ways in which they could remove from Con-
cord to Concord, and hereby we fliould lofevery
much of the Ravifhment of Sounds in Confo-
nance. As to this Part then, the Thing de-
manded is, a Variety of Ways, whereby each
Jingle Voice of more in Confonance may move
agreeably in their fucceffive Sounds, fo as to
pafs from Concord to Concord, and meet at e-
very Note in the fame or a different Concord
from what they flood at in the laft Note. In
what Cafes and for what Reafons Di/cords are
allowed, the Rules of Compofition muft teach :
But joyn thefe Two Confederations, and you fee
marufeftly how imperfect Mufick would be with-
out
in -^Treatise Chap. VlIL
out any other Intervals than Concords ; tho*
thefe are the principal and moft effential, and
the others we now enquire into but fubfervient
to them, for varying and illuftrating the*Plea-
fure that arifes immediately out of the harmo-
nical Kind.
But, laftly^ eonfider, that tho* the Melody
of a fingle Voice is very agreeable, yet no Con-
fonance of Parts can have a good Effect fepa-
rately from the other ; therefore the Degrees
which anfwer the firft Demand, muft ferve the
other too, elfe, however perfect the Sj'ftem be
as to the firft Cafe, it will be ftill imperfect as
to the laft.
W h e n a Qiieftion is about the Agreeable-
nefs of any Thing to the Senfes,thc laft Appeal
muft be to Experience, the only infallible Judge
in thefe Cafes ; and fo in Mufick the Ear muft
inform us of what is good and bad ; and no-
thing ought to be received without its Ap-
probation. We have feen to what Purpofes
other Intervals than the harmonic al are necef*
fary , now we {haJ fee what they are ; and a-
greeable to what has beenfaid, we fhaJl make
JTxperience the Judge, which approves of thofe,
and thofe only, with their Dependents (befides
the harmonica} Intervals) as Parts of the true
natural Syfiem of Mufick, viz, whofe Ratios
are 8 : 9. called a greater Tone, 9:10 called
a kjfer 2.bne 9 and 15 : 16 called a Semitone:
And thefe are the lefier Intervals, particularly
called Degrees, by which a Sound can move up-
wards or downwards fucceOively, from one Ex-
treme
§ i. ofMUSICK. 2*3
treme of any harmonic al Interval to another,
and produce true Melody ; and by Means where-
of alfo feveral Voices arc capable of the ne-
ceffary Variety in palling from Concord to Con-
iCord. By the Dependents of thefe Degrees, I
mean their Compounds with OcJa®e> (which
are underftood to be the fame Thing in Practice,
as we obferved in another Place of compound
poncords) and their Complements to an OBave
i (or Differences from it) viz, 9 : 16, 5 1% 8 :
15, which are alfo a Part of the Syftem^ tho r
.more imperfect, but of thefe afterwards : As to
1 the Semitone^ 'tis fo called, not that it is geo-
smetrically the Half ©f either of thefe which wo
.call Tones (for 'tis greater) but became it comes
|near to it -> and 'tis called the greater Semi-
itonej being greater than what it wants of a
\Tone.
NOTE, Hitherto we haveufed the Words,
\Tone and Tune indifferently, to fignify a cer-
tain Quality of a fingle Sound ; but here Tone
j is a certain Interval^ and (hall hereafter be con-
stantly fo ufed, and the Word Tune always ap-
plied to the other.
Our next Work fliall be to explain the 0-
riginal of thefe Degree t, and their different Pcr-
fe£Uons- and then fnew how they anfwer the
Purpofes for which they were required ; and, in
doing this, Ifhall' make fuch Reflections upon the
; ! Connection: and Dependence of the feveral Parts
of the Syjtejit, that we may be confirmed both
by Senfeand Reafon in the true Ppnciphs of
Mufick % .
As
224 ^f Treatise Chap. Vllt
A s to the Original of thefe Degrees ", they
arife out of the fimple Concords, and are equal]
to their Differences, which we take by Probl.
icf, Chap. 4. Thus 8 : 9 is the Difference
of a gth and 4^. 9 : 1 o is the Difference of a
3^ /. and qth, or of 5/7? and 6th g. 15 :'' 16, the
Difference of 3d g. e^A qth, or of 5^ and
6th L
We fhall prefently fee the Reafon why no o^,
ther Degrees than fuch as are the Differences of
Concords could be admitted ; but there are o-
ther Differences among the fimple Concords, be-,
{ides thefe ( which you may obferve do all a-
rife from a Comparifon of the 5/7:? with the Ga-
ther Concords) ^et none elfe could anfwer the
Dehgn, which I fhall fliew immediately, and
give you in the mean Time a Table of all thefe
Differences of Jimple Concords^ which are not
Concords themfelves.
Differences of Ratios*
I 3dg- ■■
3d L and < qth
[6thg.
3 ^ and i&i. :
4th and $th
5 th and £**'* '
&& /. apdi 6/£ g # = 24 : 25
2 4
• 25
9
: 10
18
•* 25
15
: 16
25
! 32
8
: P
15
: 16
9
: 10
I fhall
now ex^
plain how
thefe De-
grees con-
tribute to
the Inv
provement
of the Sy
fiem oi
Mufick
In doing
which
M
§ i. of MUSIC K. 115
{hall endeavour to give the Reafon why thefe
only are proper and natural to that End.
Degree's were required both for improving
the Melody of a [ingle Voice considered by it
»felf ; and that feveral Voices, while they move
mel^dioully each by it felf, might alfo joyn to-
gether in an agreeable Variety of Harmony ;
and therefore I obferved, that the Degrees re-
quired muft an&rer both thefe Ends, if polfible ;
^accordingly, mture has bounteoufly afforded
"us. thefe neceffary Materials of our Pleafure, and
[made the preceeding Degrees anfwer all our
fWifh, as I fhall now explain*
I fhall firft Confider it with refpe&to thcCon-*
%nance of Two or more Voices. SuppofeTwo Vol-
ses^fand B, containing between them any Con-
:ordi they can change into another Concord only
Two Ways, una. If the one Voice as A keeps
Its Place, and the other B moves upward or
downward ■■(/„ e, becomes either acuter or gra-
ver than it was before* ) Now if the Movement
Jgf B can only be agreeable by harmonic al In*
''ervals, they can change only in thefe Cafes,
nz. if the firft Concord be Qblave, then by B's
noving nearer the Pitch of A$ either by the
'Diftance of a 6th, $th, $th or 3d, the Two
J/oices will concord in a 3d, j\th, $th or 6th,
'vhich is plain from the Compofition of an 0-
\fave : And confequently by B's moving far-
mer from A, the Voices can again change from
'\ ny of thefe leffer Concords to an clave? Or
fippofe them at firft at a 6th, by B's moving
ither a qth or 34 they will meet in a 3d or
[■i : K ¥h
n6 A Treatise Chap. VIII.
4th, or being at a qth or id, they may meet in
a 6th, becaufe a 6th is compofedof 4^ and 3d,
And laftly, being at a 5th, they may meet in a
3^3 and contrarily. But by the Ufe of thefe De-
grees the Variety is increafed j for now fup-
pofe A&vA B diftant by any fimple Concord, if
B moves up or down one of thefe Degrees 8 : p,
or 9 : 10, or 15 : 16, there fhall always be a
Change into fome other .Concord* becaufe thefe
Degrees are the very Difference of Concords,
Then, ido. If we iuppofe both the Voices to
move, they may move either the fame Way
( i. e. both become neuter Or graver than they
were ) or move contrary to one another; and in
both Cafes they may increafe their firft Diftance,
or contract it, lb as to meet in a different Con-
cord i but then if the Movements be by har mo-
nk al Intervals, the Variety will be far lefs here
than in the firft Suppofition ; but this is abun-
dantly fupplied by the Ufe of the Degrees,
Youmuft obferve again, that befides the Want
of Variety in mod: of the Changes that can be
made,- from Concord to Concord, by the (ingle
Voices moving in harmonical Diftances, there
will be too great a Disproportion or Inequality
of the Concord you pafs from, and that you
meet in, which muft have an ill Effect : For
by Experience we are taught, that Nature is
beft pleafed., where the Variety and Changes of
our Pleafure ( arifing from the fame Objects )
are gradual and by fmooth Steps ; and there- 1
fore moving from one Extreme to another is
to be feldom pra^is'd ; for this Heafon alfo
the
§ i. ofMUSlClC H7
the Degrees are of neceffary Ufe for making
the Paflage of the Concords eafy and fmooth,
which generally Ought to be from one Concord
into the next, which is confident with the Mo-
tion of one or both Voices. But ]et me make
this laft Remark, which we have alfo con-
firmed from Experience, mz% That of Two
Sounds in Confonance, 'tis required not only
that every Note they make together be Con-
cord ( I have faid already that there are fome
Exceptions to this Rule ) but that, as much as
poillble, the prefent Note of the one Voice be
] Concord to the immediately preceeding Note of
the other; which can be done by no Sleans fo
i well as by fuch Degrees as are the Differences
i|pf Concords ( where thefe happen to be Dtp-
4 cord) Muficians call it particularly Relation
VmharmonicaL ) And indeed upon this Principle
li it can eaiily be {hewn-, that 'tis impoffible there
)! can be any other Degrees admitted^ than what
5 are equal to the Differences okfemple Concords:
jlf only one Voice move, the Thing is plain ; if
jbofh move, let Us fuppofe A B at any Concord^
biand to move into another, and there let the
/.Two new Notes be expreffed by a b> Then
jlince a B muft be Concord, it follows^ that the
IjDiftance of a and A is equal to the Difference
e of the Two Concords A B, and a B -, the fame
Way 'tis proven that b B is the Difference of
the Concords A B^ and b A.
'Tts a very obvious Queftion here^ why the
ucceflive Notes of Two different Voices may
lot as well admit of Difcords, as thefe of the
P a fame
228 ^Treatise "Chap. VIII.
fame Voice j to which the Anfwer feems plain-
ly to be this, that in the fame Voice,' the De-
grees^ which are the only Difcords admitted,
are regulated by the harmonical Intervals to
which they are but fubfervient ; and the Melo-
dy is conducted altogether with refpe£t to thefe ;
for the Degrees of themfelves without their
Subferviency to the Concords could make no
Mufick) as {hall be further explained afterwards :
But in the other Gafe, the facceifive Motions
can be brought under no fuch Regulation, and
therefore mufl be harmonical as much as pof-
jtible, left it diminiih the Pleafurc ofthefucceed-
ing Concord -, befides, coniider the Difcords that
are mod ready to occur here, are greater than
the Degrees, and would be intolerable in any
Cafe.
But now, fuppofmg that only thefe Difcords
belong to the Syftem of Mufick, which are the
Differences of Concords, you'll ask why the o-
t her Differences marked in the preceeding Table
are excluded, viz. 24 : 25 the Difference of the
Two 3^j, or the Two 6ths\ 18 .-25 the Diffe-
rence of the 3d /. and 6th g. 25 : 32 the Diffe-
rence of 3d g. and, 6th I. To fatisfie this, we
are to confider, Firft, that the Paffage of feve-
ral Voices from Concord to Concord does not
need them, there being a fnnicient Variety from
the other Differences ; but chiefly the Reafon
feems to be, that they don't anfwer the De-
mands of a fingle Voice, which I {hall explain
in the next §, and dehre you here only to ob-
fervc
§ i. of MU SICK. 119
ferve that they arifc out of the imperfect Con"
cords> viz, ids and 6ths.
J 2. Of the life of Degrees in the Coiiftrubliou
of the Scale of Mutick.
WE have already obferved, that the Cor^
cords arc the effential Principles of
Mufick as, they afford Pleafure immediately and
of themfelves : Other Relations. belong to Mu-
fick only as they are fhbfervieiit to thefe. We
have alfo explained what that Subferviency re-
quired is, viz. That by a fit Divifion of the
harmonica! Intervals a fingle Voice may pafs
fmoothly from one Extreme to another, wnere-
i>y the Pleafure of thefe perfect Relations may
be heightned, and we may have a Variety
neceffary to our more agreeable Entertain-
ment: It follows, that to anfwer this End, the
Intervals fought, or fome of them at leafl, muft
be lefs than any harmonical one, i. e. lefs than
& id I. 5 : 6 ; and that they ought all to be lefs,
will prefently appear from the Nature of the
Thing. For the Degrees fought we have al-
ready affigncd thefe, viz. 8 : 9 called a greater
Tone^ 9:10 called a leffer Tone, and 15 : 16
called a greater Semitone : Now that every har-
monical Interval is compofed of, and confe-
quently refolvable into a certain Number of
thefe Degrees , will appear from the following
P 3 Table,
230 ^Treatise Chap. VIII.
Table, wherein I give you the Number and
Kinds of thefe Degrees that each Concord is e-
qual to, which you can prove by the addition
of Intervals^ Chap. 4, Or you'll find it more ea-
fily afterwards, when you fee them all {land in
order in the Sea Is j we fhall afterwards confider
in what Order thefe Degrees ought to b,e taken
in the Di virion of any Interval.
NOT£ %
That as in
this Table fo
afterwards I
(hall for Bre-
vity mark a
greaterTone
thus t g, a
leffer thus tl x
a Semitone
thus /;
But now, obferve y that fince we can con- '
ceive a Variety of other Intervals that will di-
vide the Concords befides thefe, we are there-
fore to confider for what Ileafon they are pre-
ferable to any other : To do this, I fhall firft
fliew you, that no other but fuch as are equal
to the Differences of Concords are fit for the
Purpofe, and then for what Reafon only thefe
Three are cliofen.
For the Fir ft ^ confider, that every greater
Concord contains all the leffer within it, in fuch
a Manner, that betwixt the Extremes of any
greater Concord, as many middle Terms may
TABLE of th
>e. component \
Pat-
ts of Cone
ords.
id I.
1 igy
&.i/
idgi
'• * *S>
1 tk
Mh
S 1 *&
1 tl,
if
5th
i. 2 *&>
I tfy
if
6th I.
# 2 *&
I tl y
*/
6th g.
2 tg,
2 tl,
if
%ve
3 tg,
2 tly
*/
§ i. of MUSIC K. 231
be placed as there are leffer Concords ; which
middle Terms fliall be to any one Extreme of
that greater Concord in the Ratio of thefe leffer
Concords ; fo betwixt the Extremes of the 8<ve
maybe placed 6 Terms, which (hall make all
the leffer Concords with any one of the Ex-
tremes, as in this Series,
T . * • i • 1 • i ' • L • L • i
where comparing each Term with i, you have
all the fimple Concords in their gradual Order*
3 J /. 3^^. 4^, 5th j 6th L 6th g. %ve\ and the
mutual Relations of the Terms immediately
next other in the Series arc plainly the Differences
of the Concords which thefeTerms make with the
Extreme. Now it is natural and reafonable that if
we would pafs by Degrees from one Extreme to
another of any greater harmonical Inter val^ntho
moft agreeable Manner, we ought to choofe fuch
middle Terms as have an harmonical Relation
to the Extremes of that greater, rather than
fuch as areDifcordi for the fimple Concords be-
ing different in Perfection, vary the Plcafure in
this Progreflion very agreeably ; but we could
not bear to hear a great many Sounds fucceed-
ing one another, among which there were no
Concord^ or where only the laft is concord to
the Firft : And therefore it is plain that the
Degrees required ought to be equal to the Dif-
ferences of Concords^ as you fee evidently they
muff \>q where the middle Terms are Concord
P /j with
2,32 . ^Treatise Chap. VIII.
with one or both the Extremes, But of all the
difcord Differences of Concords, only thefe are
agreeable, viz. 8 : 9, 5? : 10, 15 : 16 ; the o-
ther Three are rejected, viz. 24 : 25, 18 : 25,
25 : 3 2 j the Reafon of which fecms to be, that
the Two laft are too great, and the firft too
fmalTj but particularly 25 : 32 is an Interval
greater than a qth, as 18 : 25 is greater than a
'%d g. and therefore would make fucb a di£
proportioned and unequal Mixture with the o-
ther Degrees, that would be infuffcrable. Then,
for 24 : 25 it is too fmall, and would alfomake
too much Inequality among the Degrees. But
at kft we (hall take Experience for the infallible
Proof that we have chofen the only proper De-
grees : Our Reafpn in Cafes like this can go no
"further than the making fuch Obfervations upon
the Dependence and Connection of Thing?,
that from the Order and Analogy of" Nature
we may draw a probable Conclufion that we'
have difcovered the true natural Rule. And of
this Kind we (hall immediately have further Dct
monfrrations that the only true natural Degrees
are thefe already affign'd.
W e come now to confider the Order in which
the Degrees ought to be taken, in this Divifion
of the harmonical Intervalsfov confliluting the
Scale oiMufick •> for tho' we have the tme De-
grees, yet it is not every Order and Progreflion
of them that will produce true Melody. For
Example, Tho' the greater Tone 8 : 9 be a true
Degree, yet there could be no Mufick made of
any Number of fuch Degrees, becaufe no Num.-*
§ i. of MUSIC K. 233
bcr of them is equal to any Concord ; the fame
is true of the other Two Degrees ; which you
)may prove-by adding Two or Three, (jc. of
lany one Kind of them together, till you find the
Sum exceed an OcJave, which it will do in 6
greater Tones ; or 7 leffer Tones, or ir
Semitones -, and compare the Sum of 2, 3, 4,
fc\ of them, till you come to that Number,
you'll find them equal to no Concord. There-
fore there is a Neceifity that thefe Degrees be
mixt together to make right Mufick ; and 'tis
plain they mult be fo mixt, that there ought
never to be Two of one Kind next other. But
this we {hall have alfo confirmed in examining
the Order they ought to be taken in.
The OcJave containing in it all the other
r imple Concords, and the Degrees being the Dif-
ferences of thefe Concords, 'tis plain that the
Diviiion of the Oblave will comprehend the
Divifions of all the reft ; Let us therefore joyn
ill the fimple Concords to a common Funda-
mental, and we have this Series,
. 5 : 4 . 3 .
l-\6 ' 7 • 4 •
7 • S • 7 • SL-
Fund. 3d I, 3dg. 4th,
$th 6th I. 6th g. %Q?.
Now if we fhould afcend to an Od?a<ve by
:hefe Steps, 'tis evident we have all the pofli-
^>le harmonical Relations to the Fundamental ;
md if we examine what Degrees are in this A-
feent
234 -^Treatise Chap. VIII.
fcent, or the mutual Relations of each Term to
the next, they are thefe.
But this we know is far from being a melodi-
ous Afcent; there is too great Inequality among
thefe Degrees ; the firft and laft are each a 3J
/. which ought alfo to be divided,- it is e-1
qual to a £g. and f. and fo inftead of { we (hall 1
have thefe Two Degrees 8 : 9 and 15 : 16. But
when'. this is done, yet the Diviflqn of the 0-
ciave will not be perfect 5 for we have too ma-
ny. Degrees* and an Excefs is as much a Fault
as a Defect : So many fmall Degrees would nei-
ther be eafily raifed, nor heard with Pleafure :
The Two 3 as and Two 6th s have fo fmall a 1
Difference, 24 : 25, that the Division of the 0-
if am does not require nor admit them both to-
gether, the Progrefs being fmoother where we
have but one of the %ds and one of the 6th rl
If this Degree 24 : 25- be expelled, then w|
9 : 10 have Place in the Series, which is not only
a better Relation of it felf, as it confifts of leffer
Numbers, but it has a nearer Affinity with the
other Two 8 : 9 and 15 :iS 9 all thefe Three
proceeding from the 5^, as I have already
noted.
Now then if we take only one of the ^ds
and one 6th in theDivilion of the %ve we have
thefe Two different Series,
The
4 x i i i
■-• 5* *4* T* 5* 2
Fund. 3d g. Ath, $th,6th g.%ve
of MUSIC K. % 5j
The 3^ /.
and 6th I. are
|> taken togeth-
er, as the 3d g,
and 6th g. be-
caufe their Re-
lation is the
)Concord of a 4/^ ; whereas the 3d L and 6th g,
jjaifo the id g. and 5^ /. are one to the other a
grofs Difcord j and 'tis better how many s Con-
cords are among the middle Terms j but if in
fome particular Cafes of Practice this Order is
changed, 'tis done for the fake of fome other
Advantage to the Melody, of which I have an
Occafion to fpeak afterwards. But the 3 ds next
each Extreme are yet undivided, which ought
to be done to complete the Division of the
'DtJave.
I n the firft of the preceeding Series we have
the 3 d L next the Fundamental, and the 3d g.
next the other Extreme: In the Second we
Jhave the 3d g. next iheFundamental,and the 3d
% next the acute Extreme. Now it is plain
what Degrees will divide thefe 3ds, becaufe we
fee them divided in the Divifions already made ;
Tor in the firft Series, betwixt the 3d /. and the
\th we have a 3 d g. (which is their Difference)
Jdivided into thefe Degrees, and in this Order
iafcending, mz. tl. and tg. and betwixt the
Ath and 6th L we haye a 3d I. (which is their
Di£
t$6 ^Treatise Chap. VIII.
Difference ) divided into t g. and /. We have
the fame Intervals divided in the other Se-
ries betwixt the 3d g. and $th, and betwixt
the 4th and 6th g. but the Order of the De-
grees here is reverfe of what it is in the
other Series : And the Queftion now is, what
is the moft natural Order for the Divifion of
thefe ids that ly next the Extremes in the 0-
ctaves ? It may at firft feem that we have got
a fair and natural Hint from thefe Places men-
tioned, and that the %ds ought to be ordered
the fame Way towards the Extremes of each;
Series, as they are in thefe Places of it. In the:
ids next the Fundamental I have followed that,
Order, but not for that Reafon ; and in the up-
per 3 ds I have taken the contrary Order, which
fee in the Two following Series, where I have
marked the Degrees from every Term to the
next ; and you fee I have divided
; witha 3 ^A- I . j. ? . -. J. g-. ~. 7
tg' ./•' tL tg.f. tg, tl.
.1 ' : j , . 8 . 4 . 3 . 2 . 3 . 8 . I
with a 3d g, - I . — . — i — . — . — . — . *
*g- */• ,/: *g. * /. fg. f.
the %d g. ( which is in the upper Place of the
one and lower of the other Series ) in this Or-
der afcending, viz. t g. and 1 1. And the 3d 7.
(which is alfo in the upper Place of theone
and
I
I i. of MUSIC K. z 37
and lower of the other ) in this Order afccrr
iling, viz. t g. and/ The Reafon of this
Choice I (hall thus account for. Flrjl } As to
-he id next the Fundamental^ I place the t g.
oweft, becaufe it is the Degree which a natu-
ral Voice can molt eailly raife, being the moft
perfect of the Three, and we find it fo by Ex-
perience -j and if you confider, that it is the Dif-
ference of a qth and 5^,' which two Concords
rthe Ear is perfectly Judge of, by pra&ifingthefe
one learns very eaiily how to raife a t g. with
Exaclnefs : But for the t I. ( the other Part of
the id g. ) it is not fo eafily learned, for the
(Difference betwixt the Two Tones being but
ifmall, one cannot be fure of it, but will readily
'fall into the more perfect It is true, that in
riling from any Fundamental to a 3 d g. we take'
ia t L at the fecond Step; but then. I believe,
our taking it exactly here, is owing to the Idea
of the Fundamental^ to which the Ear fceks
jthe harmonic al Relation of id g. where it refts.
uvith Pleafure; and whenever a Reafon like this
occurs, the Voice will eaiily take a t I. even
'at the firft Step; for iExampk^ Subpofe Two
;' Voices concording in a 6th g, if one of them
keeps its Tune^ and the other moves to meet
it in a $ih 3 then muft that Movement be a t L'
which is the Difference ok 6th g. and^th : As
) to the Parts of the id I. obferve, that the t g.'
' and/; being remarkably different, there would
be no Hazard of taking the one for the other $
therefore as to that, any of them might ftand
: next the Fundamental^ yet the t g, being a
more
i$% -^Treatise Chap. VlIL
more perfed Relation, it is eafier taken, and
makes a more agreeable Afcent, the' I know
that in fome Circumftances the/.' is placed next
the Fundamental ( as I (hall mark in its pro-
pet Place. ) Nbzu for the Degrees of the upper
Third, the t g. is fet in the loweft Place in both
the Series j the Erfe£t of which is, that the
middle Term proceeding from that Order, is
in an harmonical Relation to more, and the
more principal of the other Terms in the Seri-
es* Kepler upon harmonical Proportions pla-
ces the t g. next both the Extremes in the 0-
f&dwf 9 and gives this Reafon for it, left the fe~
cend and feventh Term of the one Series differ
from thefe in the other ( for it feems he would
have them differ as little as. poffible, viz. only
in the $d$ and 6ths ) and this he concludes
with a Kind of Triumph againft the Authorities
of Ptolomjy G alliens and Zarline^ whom he
mentions as contrary to him in this Point. But
indeed I cannot fee the Sufficiency of this Rea-
fon, there is nothing in it drawn from the Na-
ture of the Thing r And as to 3d in the upper
Fkce 5 the Order in which I've placed its /)<?-
gfeeS) is approven by Experience, and is I think
the conftant Practice.
THUs-we have the O&ave completely di-
vided into all its concinnous Degree j", and in it
the Divifion of all the Jefler Concords^ with the
moil natural and agreeable Order in which
thefe Degrees can follow^ in moving from any
given Sound through any harmonical Inter-
val, There are only thefe Three different
Dqt
§ i. of MUSI CK. t 39
Degrees, viz. t g. 8 ; p y t L 5 : 6 9 and f.
15 : 16. And how many of each Kind every
harmonic al Interval contains, is to be feen in the
proceeding Series, which eafily confirms and
proves the Table of Degrees given a little a-
bove, where you fee alfo the natural Order,
viz. in afcending, it is t g. t 1, f. t g. t I. t g.
/.' Or this, t g.f. 1 1, t g. f. t g. t I. ac-
cording as you chofe the 3d I. or %d g. to as-
cend by ; and in [deicending we take that Order
jufbreverfe, by taking the fame individual middle
Terms.
Now the Syftem of 05fave containing all the
original Concords, and the compound Concords
being the Sum of Odfave and fome leffer Con-
cord-, therefore 'tis plain, that if we would
have a Series of Degrees to reach beyond an
Octave, we ought to continue them in the fame
Order thro' a fecond Oflave as in the firft, and
fo on thro* ft third and fourth Odfave, &c. . and
fuch a Series is called The Scale of Mufick,
which as I have already defin'd, exprelfes a Se-
..* ries of Sounds, rifing or falling towards Acute-
nefs or Gravity, from any given Pitch of
Time, to the greatelt Diftance that is fit or
practicable, thro' fuch intermediate Degrees as
makes the Succeflion moft agreeable and per-
fect ; and in which we have all the harmonical
Intervals moft concinnoujly divided. And of
this we have Two different Species according
as the id I. or 3d g. and 6th I. or 6th g, are ta-
ken in, which cannot both ftand together in
r ?^°n to one Fundamental, and make an har-
monical
240 A Treatise .Chap. VIIL
monical Scale* But if either of thefe Ways we
afcend from a Fundamental or -given Sound to
an Oftave, the Succeifion is very melodious, tho'
they make different Species of Melody, It is
true, that every Note to the next is Difcord,
but each of them is Concord with the Funda-
mental, except the id and jth, and many of
them among themfelves, which is the Ground
of that Agreeablenefs in the SuccefTion ,• for we
mud; reEeft upon what I have elfewhere obfer*
ved, that the graver Sounds are the more
powerful, and are capable of exciting Motion
and Sound in Bodies whofe Tune is acuter in
a Relation of Cone or d, particularly 8^ and $tb,
which an acute Sound will not effect with re-
IpeiQ to a -grave. And this accounts for that
Maxim in -Practice, That all Miifick is counted
upwards $ the Meaning is, that in the Conduct
of a fucceflive Series of Sounds, the lower or
gr&ver Notes influence and regulate the acuter,
in fuch a Manner that all thefe are chofenwith
refpeel: to fome fundamental Note which is
cafled theKej ; but of this only in general here,
in another Place it fhall be more particularly
confiderecl.
W e have expreft the feveral Terms of the
Scale by the proportional Sections of a Line re-
prefented by 1, which is the Fundamental of
the Series ; but if we would exprefs it in whole
Numbers, it is to be done by the Rules of Ch.
4. by which we have the Two following Se-
ries, in each of which the greateft Number
§ j 4 of MUSIC K. t 4 t
expreiles the longeft Chord&nd the other Num-
bers the reft in Order*
540 : 480 : 432 : 4<d£ : 360 : 324 : 288 : 270
tg. tl. f. tg. tl. tg. f.
216 : 192 : 180 : 162 : 144 : 135 : 120 : 108
tg. f. tl. tg. f. tg. tl.
The firft Series proceeds by &3d g. and the"
other by a 3 d L and if any Number of Chords
are in thefe Proportions of Length, ceteris pari-
huS) they will exprefs the true Degrees and In*
tervals of the Syftem of Mufich, as 'tis contain'd
in an %ve concinnou/ly divided in the Two dif-
ferent Species mentioned*
$ 3. Containing further Reflections upon the
Conftitution of the Scale of Mufick ; and ex-
1 ' plaining the Names of 8ve, 5th, &c. which
have been hitherto ti/ed without knowing all
theif Meaning ; Jhewing alfo the proper
Office of the Scale*
E confidered in Chapter 5. th e Divifiort
of the Concords, in order only to find
what Intervals they were immmediately divi-
fible into: We find that either an harmonic al or
arithmetical Mean divides the %ve into a nth
Q. and
242 ^Treatise Chap. VIII.
and qth<> with this Difference, that the harmo-
nic al puts the 5 thy and the arithmetical the qth
next the Fundamental : And from this the In-
vention of the tg (which is the Difference of /\th
and $th) was very obvious; Thefe Divifions of
the Sjvc we fuppofe indeed made only for dis-
covering the immediate harmonical Parts of itj
but taking in both thefe middle Terms, then
we fee the %ve refolved into thefe Three Parts,
and in this Order, viz, a 4^, a tg and a /\th>
as in thefe Numbers 6:8:9:12. where 6 and
1 2 are Bve ; 8 is an harmonical Mean, and 9
an arithmetical Mean - y 6 : 8 is a 4/^ $ 8 ,* 9 a
£g. and 9:12a tfh ; that thefe Two middle
Terms are at a Diftance proper for making
Melody ', and confequently that their Relation
8 : 9 is a continuous Interval, we have infallible
Aflurance of from Experience.
But I propofed to make fome Obfervations
on the Connection and Dependence of the feve-
ral Parts of the Syfiem ofMufak; and Mrft^we
are to remark, that this Degree 8 : 9 proceeds
from the Two Concords that are of the next ;
perfect Form to 8ve y viz. 4th and 5th, which
are the harmonical Parts of it,* and ftands foin
the middle betwixt the upper and lower qth>
that added to cither of them it makes up the $th^
and fo joyns the harmonical and arithmetical
Divifion of %ve in one Series : and this tg be-
ing the Difference of Two Concords of which
the Ear is perfectly Judge, we very eafily learn
to raife it ; and in Fad we know it is the De-
gree which a natural Voice can with moft Eafe
and
§ f. of MUSIC K. 145
and Certainty raife from a Fundamental or gi-
ven Sound. Again, we found that the fame
Law of an harmonic al and arithmetical Mean
refolved the $th into 3d L and 3d g. By the
harmonic al the %d g. being next the greater
Number, as here 10 : 12 : 15, and by the arith-
metical the 3d 1. lowed, as here 4:5 : 6; and
applying this to the upper and lower $th pro-
ceeding from the immediate Divifion of the Sve,
We have 4 more middle Ternis within the %<oe$
whereof the lower Two are ids to the Funda-
mental and 6th s to the other Extreme, and the
upper Two are 6ths to the Fundamental, and
%ds to the other Extreme, as you fee in the
preceeding Series : And this produces Two new
Degrees, viz.- 24 : 25^ the Difference of 3d Land
3d g* or of 6th L and 6th g. and 15 : 16, the
Difference of 3d g. and 4th, or ofc^th and 6th L
but this Degree 24 : 25 is too fmall, and upon
that Account rejected, as I have already faid.
Now we are to find why this Degree 24 : 25 is
inconcinnous, and 15:16 continuous, fromfome
^ .-fettled Conftitution and Rule in Nature, which
we fliall have from this Obfervation, bi'Z\ That
if we apply the fame Law which refolved the
mte and $th into their harmonica I Farts, to the
3d g. we have it divided into a tg. and a tL as
in this arithmetical Series 8:9: 10; or this
harmonical,36 : 40 : 45 j and if we confiderthis
Analogy, it feems to determine thefe Two De-
grees of tg. 8 : 9 and 1 1. 9 : 10, to be the true
continuous Parts of 3d g. and thereby excludes
i4 ; 2 5> and confequently the Two 3$ and
Q. 2 Two
244 A Treatise Chap. VIII.
two 6th s from {landing both together in oneScale.
And nozVy fince the $th does not admit of both
thcfe middle Terms together which proceed
from its harmonkal and arithmetic '<3/Divifion,it
feems to be but the following of Nature, i£ we
apply the fame Kind of Di virion to the upper and
lower 5th of the See; the Effect of which is,
that as by the harmonicalDivifion of the lower
%th we have a %d g. next the Fundamental; fo
by the harmonicalt)ivifion of the upper $th we
have a 6th g. to the 'Fundamental $ and by the
arithmetical Divifions we have contrarily the
%d I. and 6th /.next the Fundamental^ you fee
in the proceeding Series : And this is a Kind of
natural Proof that the id I. and 6th I. alfo the
3d g. and 6th g. belong to one Series ; and here
we have the Difcovery of the tl. which lies na-
rally betwixt the 3 d I. and qth, or betwixt the
5th and 6th g. But tho' the Two 3 ds and Two
6ths cannot ftand together, yet there muft
none of them be loft, and therefore they con-
ftitute Two different Scales. But the Diviflon
of the %ve is not finiihed, for the ^dr that ly-,
next the Extremes are undivided ,- as to the %d
g. we fee how naturally 'tis refolved into a
tg. and tl. which is another Way of difcover-
ing thefe Degrees ; and 'tis worth remarking,
that the fame general Rule which by a gradual
Application refolved the %w immediately into a
$th and qth* and then the $th immediately in-
to ^d g. and 3 d I. (by which Divifions the Two
6th s were alfo found indirectly) being applied
to the %dg* produces immediately the Two prin-
cipal
§ 3 . ofMUSICK. 145
cipal continuous Intervals ; and for the Original
of the/J 15 : 1 6. we fee 'tis the Difference of id g.
and 4th, and rifes not from the immediate Di-
vifion of any other Interval, but falls here by-
Accident, upon the Application of the proceed-
ing general Rule to the Sve and $ih* But we
have yet the 3d /. which is next the Extremes
to confider ; of what continuous Parts it confifts
was eafy to fee betwixt the 3d g. and $th. viz*
af. and tg\ but next the Extremes of the 80s
they muft be in this Order attending, viz. tg.
and/.' Of the Reafon of this I have faid e~
nough already: And now the Divifion of the
Octave being completed, we have the whole
original Concords concinnoufly divided, and
thefe Intervals added to the Syftem, viz. 8 : <?>
9 : 1 o, and 15 : 16. which have all this in com-
mon, that they are the Differences of the $th
and fome other Concords.
Of the particular Names of Intervals, as Sve, 1
5 th, &c.
W e have confidered the continuous Divifion
of every harmonic al Interval, and we find the
Sve contains 7 Degrees ; ,the 6th, whether lef-
fer or greater, has 5; the $th has 4; the /\th
has 3 j- the 3d, leffer or greater, has 2 : And if
we number the Terms ©r Sounds contained with-
in the Extremes (including both) of each har-
monical Interval, there will be one more than
there are of Degrees, viz. in the %ve there are
Q. 3 8. in
246 ^Treatise Chap. VIII.
$, in the 6th 6, in the 5th 5. in the 4th 4. and
in the 3J 3. And now at laft we underftand
from whence the Names of %ve, 6th, $th, &c,
come,- the Relations to which thefe Names
are annexed are fo called, becaufe in the natu-
ral Scale of Mufick the Terms that are in thefe
Relations to the Fundamental are the Third,
Fourth, &c in order from that Fundamental
inclufively. Or thus, becaufe thefe harmonical
Intervals being concinnoujjy divided, contain
betwixt their Extremes (including both) fo
many Terms or Notes as the Names Site, 6th,
&c. bear. For the fame Reafon alfo, the Tone
orf. (whichever of them (lands next the Funda-
mental) is called a 2d, particularly the Tone
(whofe Difference of greater and leffer is not
(tricky regarded in common Practice) is called
the id g, and j\ the id\. Alfo that Term
which is betwixt the 6th and 8ve, is called the
r/th, which is alfo the greater 8:15, or the lef-
fer $ : 9. Concerning this Interval we muft
here remark, that as it ftands in primary Rekh
tion to the Fundamental in the Divifion of the .
8ve, it does in this refpect belong to the Syfiem
of Mufick : But it is alfo ufed as a Degree with-
out Divifion, that z'jyin Practice we move fome-
times the Diftance of a jth at once ; but it is
in fuch Circumftances as removes the Offence
that fo great a Difcord would of it felf create ;.
pf which we fhall hear more in the next Chap-
ter; and here obferve, that it is the Difference
of %ve and the Degrees of Tone and Semitone*
§ 3 . of MUSIC K. z 4 r
A s to the Order in which the Degrees of this
Scale follow, we have this to remark, that if ei-
ther Series, (viz. that with the id I. or with
the id g. ) be continued in infinitum, the Two
Semitones that fall naturally in the Divifion of,
the 8<ve, are always afunder 2 Tones and 3 Tones
alternately, i. e. after a Semitone come 2 Tones,
then a Semitone, and then 3 Tones ; and of the
Two Tones one is a greater and the other a
leiTer ; of the Three, one is leiTer in the
middle betwixt Two greater. If you continue
either Series to a double Offiave, and mark the
Degrees, all this will be evident. Obferve alfo,
that this is the Scale which the Ancients called
the Diatonick Scale, becaufe it proceeds by
thefe Degrees called Tones (whereof there are
Five in an $<ve) and Semitones (whereof there
are Two in an Offave) But we call it alfo the
Natural Scale, becaufe its Degrees and their
Order are the moft agreeable and continuous,
and preferable, by the Approbation both of Senfe
and Reafon, to all other Divifions that have e-
ver been inftituted. What thefe other are, you
{hall know when I explain the ancient Theory
of Mufick i but I fliall always call this, The Scale
of Mufick, without Diftinaion, as 'tis the only
true natural Syfiem.
We have already obferved, that if the Scale
of Mufick is to be carried beyond an OEfave, it
muit be by the fame Degrees, and in the fame
Order thro' every fucceflive Offiave as thro' the
firft. How to continue the Series of Numbers
by a continual Addition, is fufficiently explained
ft 4 $k
248 'A Treatise Chap, VIII,
already j and for the Names there are Two
Ways, either to compound the Names of the
fimple Interval with 'the Stave thus, viz, tg*
ox J. or 3 J, &c, above an Oblave> or above Two
OdfaveSj &c. or name them by the Number
of Degrees from the Fundamental^ as gth y i oth y
&c. but the firft Way is more intelligible, as
it gives a more .diftincl: and fimple Idea of the
Pittance, juft as we conceive a certain Quantity
of Time more eafily, by calling it, for J?x~.
ample ) 9 Weeks, than 63 Days. But that you
may readily know how far any Note is remo-.
ved from the Fundamental^ if you know how-
far it is above any Number of Oblaves. See the
following Table^ wherein the firft Line con-,
tains the Names of the Notes within one 0-
ffiave i the fecond Line the Names ( tvith re--
fpe£t to the firft Fundamental ) of thefe Terms
that are as far above one Oclave y as thefe ftan-r.
ding over them in the firft are above the Fun-
damental \ and the Third Line the Names o|
$hefe at>ove Two Octaves.
JFund.
id. \ id \ qth\ $th\ 6th\ Jth [ 2 th'
gth\ ioth\ \\th\ i2tb\ fc%th\ i^th\ i$tb>.
1 1 6th | i Jth | i%th\ 19th \ 2 cth J 2 iji j 2 2d
And this Table may be continued as far as you
pleafe $ or if you take the Columns of Figures
downward, then each Column gives the Names
of the ffotes or Terms that are equally remo-
ye$ from the Fundamental^ from the firft 0-
£fave ?
§ 3 , of MUSICK. 249
Uave, the fecond Octave, &c. Thus the firft
Column on the left fticws the Names of fuch
as arc a 2d above the Fundamental, above the
firft Stave, &c. if we confider what is practi-
cal then the Scale is limited to Three or
Four OBaves, otherwife 'tis infinite. Again
obferve, that let the Scale be continued to any
Extent, every Otlave is but a Repetition of the
firft; and therefore an Otlave is faid to be a
perfect Scale or Sjflem, which comprehends
Eight Notes with the Extremes ; but the Eighth
being fo like the firft, that in Practice it has
the fame Name, and is the fame Way funda-
mental to the Degrees of a fecond Octave, and
fo on from one Octave to another, gave Occafiort
to fay there are but feven different Notes in the
Scale of Mufick; or that all Mufick is compre-
hended in feven Notes j becaufe if we take 0-
ther feven Notes higher, they are but Repetitions
of the firft feven in Otlave, and have the fame
Names.
Of the Office of the Scal e.
The Conftitution of the Scale being already
explained, the Office and Ufe of it fhall be next
treated of^ which you have expreft in general
in the proceeding Definition of it ; but that, you
may have a diftincl: and clear Notion, I {hall
be a little more particular. The Defign then
of the Scale of Mufick is to fliow how a Voice
may rife or fall, lefs than any harmonical In-
tervaly and thereby move from the one Ex-
treme
2o A Treatise Chap. VIII.
treme of any of thefe to the other, in the moft
agreeable Succeflion of Sounds : It is a Syftem
which ought to exhibite tousthe whole Principles
diMufick, which are either Concords or contin-
uous Intervals : The Concords or harmonical In-
tervals are the effential Principles, the other
are fubfervient to them, for making their Ap-
plication more various. Accordingly we have
in this Scale the whole Concords, with all their
concinnous Degrees, placed in fuch Order as
makes the moft perfect Succeflion of Sounds
from any given Fundamental, which I fuppofe
reprefented in the preceeding Series by i ; fo
that the true Order of Degrees thro' any har-
inonical Interval is, that in which they ly from
"i upwards, to the acute Term of the given Con-
cord, as to I for the Octave, \ for the jth, &c.
or downwards from thefe Terms to the Fun-
damental i. The Di virions of the Oclave, $th
and tfh are different, according to the Difference
of the 3 ds, and thefe Intervals are to be found
in primary Relation to the Fundamental, in
both the preceeding Scales-, but the ^dl. and
6th I. belong to the one, and -$dg. and $th g.
to the other Scale.
This Scale not only fhews us, by what De-
grees a Voice can move agreeably, but gives us
alfo this general Rule, that Two Degrees of
one Kind ought never to follow other immedi-
ately in a progreifive Motion upwards or down-
wards j and that no more than Three Tones
( whereof the middle is a leffer Tone, and the
other Two greater Tones ) can follow other,
but
§ 3- of MUSIC K. iji
but &f. or iome harmonic al Interval muft come
next j and every Song or Compofition within this
Rule is particularly called diatonick Mufick^
from the Scale whence this Rule arifes j and
from the Effect we may alfo call it the only
natural Mufich : If in fomc Inftances there are
Exceptions from this Rule, as I fhall hereafter
have more particular Occafion to obferve, 'tis
but for Variety, and very feldom pra&is'd : But
this general Rule may be obferved, and yet no
good Melody follow ; and therefore fome more
particular Rules muft be fought from the Art
of Compofition, While we are only upon the
Theory ', you can expect but Theory and gene-
ral Notions^ yet I fhall have Occafion after-
wards to be more particular on the Limitations,
which are neceflary for the Conducl: of the true
mufical Intervals in making good Melody^ as
thefe Limitations are contained in the Nature
of the Scale of Mufich But don't miftake the
Defign of this Scale of Degrees , as if a Voice
ought never to move up or down by any other
immediate Diftances, but by Degrees > for tho'
that is the moft frequent Movement, yet to
move by harmonic al Diftances at once is not ex-
cluded, and 'tis abfolutely neceflary : For the
Agrceablenefs of it, you may confider the De-
grees were invented only for Variety, that we
might not always move up and down by har-
monical Intervals •, which of themfelves are the
moft perfect, the others deriving their Agreeable-
nefs from their Subferviency to them. Qbferve y
thefe Tones and Semitones are the Diajiems
or
2jt ^Treatise Ghap. VIII.
oxfimple Intervals of the natural or diatonick
Scale. In Ch, 2. J) 1. I have defined a JD/tf-
^/fcf/z, fuch an Interval as in Practice is never di-
vided, tho' there may be of thefe fome greater •
fome lelfer. To underftand the Definition per-
fectly, take now an Example in the diatonick
Scale : A Semitone is, lefs than a Tone^ and both
are Diaftems;we may raife a Tone by Degrees^
firft railing a Semitone^ and then fuch a Diftance
as a jTb/z? exceeds a Semitone^ which we may
call another Semitone ', f. ^ . from <3 to b a &/«/-
fo#f 9 and then from b to c the Remainder of a
7"b/^^ which is fuppofed betwixt a c. But this is
never done if we would preferve the Character
of 'diatonick Mufick> becaufe in that Scale Two
Semitones are not to be found together ,- and if
we rife to the Diftance of a Tons^ it muft be
done at once ,- all greater Intervals are divifible
. in Practice of this Kind of Melody •> but in other
Kinds pra£tis'd by the Ancients^ we find that
the Tone was a Syftem 9 and fome greater In-
tervals were prachs'd as Diaftems^ which fiiall
be cxplain'd in another Place.
W e dial! ftill want fomething toward a com-
plete and fin idled Notion of the Ufe and Office
of the Scale of Mufiek-> till we underftand di-
flincUy what a Song truly and naturally concin-
nous is, and particularly what that is which we
call the Key of a Song • and the true Notion of !
thefe we {hall eaiily deduce from the Things al-
ready expkin'd concerning the Principles of
Mujichi but I find it convenient firft to difpatch
-fome remaining Considerations of the Intervals
of.
§ 4 . of MUSIC K. xyj
of Mufick, particularly as they regard the
Scale.
§ 4. Of the accidental Difcords- in the Syftem
</Mufick.
WE have considered thefe Intervals and
Relations of Time that are the imme-
diate Principles of Mufick, and which are direct-
ly applied in the Practice ; I mean thefe Inter-
vals or Relations otTune^ which, to make true
Melody^ ought to be betwixt every No re or
Sound and the immediately next ; thefe we
have considered under the Distinction of Con-
cords and continuous Intervals. But there are ci-
ther difcord Relations that happen unavoidably
in Mufick) in a kind of accidental and indirecl
Manner -, thus, in the Succession of feveral Notes
there are to be considered not only the Re-
lations of thefe that fucceed other imme-
diately, but alfo of thefe betwixt which other
Notes intervene. New the immediate Succe£
fion may be conducted fo as to produce good
Melody ', yet among the distant Notes there may
be very grofs Difcords^ that would not be tole-
rated in immediate Succejfion^ and far lefs in
Confonance. But particularly let us consider
how fuch Difcords are actually contained in the
Scale of Mifick : Let us take any one Species,
fuppofe
ij4 ^Treatise Chap. VIIL
fuppofe that with the 3d g* as here, in which 1
mark the Degrees betwixt each Term, and
the next.
Names ] Fund, idg. idg. qth&hfithg. jthg. %<ve
Ratios, p T.~°*~i~.~i~4 — .-.*•
Degr. J ?g : tl :/•: fg : */: <g : /
Now tho' the Progreflion is melodious, as
the Terms refer to one common Fundamental,
yet there are feveral Difcords among the mu-
tual Relations of the Terms, for JSxample i
from qth to jth g. is 32 : 45, alfo from 2J g.
to 6*/? g. is 27 : 40, and from 2<s?g. to qth is
27 : 32, all Difcords. And if we continue the
Series to another Slave, then 'tis plain we
(hall find all the Difcords, lefs than Offiave, that
can poffibly be in fiich a ^raZz, by comparing
every Term, from 1 in order upwards, to every
other, that's diftant from it within an 05fa<tfe ,-
and tho' there be Difference of the Two Scales
of Afcent, the one ufing the id I. and 6th I. and
the other the 3d g. and 6th g. yet all the Re-
lations that can poffibly happen in the one, will
alfo happen in the other, as I fliall immediately;
{how you.
Let us therefore take any one of thefe Se-
ries, as that with the %d g. and 6th g. and
continue it to a double Offiave, and then exa-
mine the Relations of each Term to each. In
order to this> I fhall anticipate a little upon that
§ 4 . of MUSIC K. zjy
Part where I am to explain the Art of writing
Mufick i and here fuppofe feveral Sounds in the
Order of the preceeding Scale to be reprefented
by fo many Letters ; and becaufe every Odfave
is but the Repetition of the ift, fo that from
every Term to the %th inclufwe^ is always a
juft Octave in the Relation of I : 2; therefore
to reprefent fuch a Scale by Letters, we need
but 7 different ones, a, b, c, d, e, f, g, which
will anfwer the firft 7 Terms of the OtJave,
and the Sth will be reprefented by the firft
Letter,- andfo in order again to another OEfave.
And that all Things may be as diftincl; as pof-
fible, we {hall make every 7 Letters in order
from the Beginning of a different Character,-
but for a Reafon that will appear afterwards,
inftead of beginning with A, I ftiall k begin with
C, and proceed in this Order,
c : d : e : F : g : a : b : c : d : e :f:'g : a :h:: cc,
where Creprefents die Fundamental and loweft
Note of the Scale ; and the reft are in order
acuter. And now when any Interval is expreffed
by Two Letters, it will be eafy to know in
which Offiave ( i. e . whether in the firft or fe-
cond in order from the Fundamental) each Ex-
treme is ; for if they be both one Kind of Cha-
racter, then they are both in one Qftave^ as
C-F i otherwife they are in different Oclaves^&s
A -f. And it will be eafily known whether the
Interval be equal to, or greater or lefs than an
Offiave j for from any Letter to the like Letter
k
ijS ^Treatise Chap. VllL
is an Odfave, or Two Odlaves, as c-c is an
Offiave, or C-cc Two Ottaves^ confequently
Jl-b is known at Sight to be greater than an
OEfave^ even as far as b is above a • and _Zfc
J9 to be lefs. Again^ by this Means we eatily
know whether the Example is taken afcend-
ing or defcending, fo 'tis plain, that from D to
a is afcending, or from d to g ; but from f to
d is defcending, or from d to J^: The Order of
the feveral Letters, and their different Cha-
racters determine all thefe Things with great
Eafe. \ t
According to this Suppofition, then, I have
exprefs'd the Scale by thefe Letters, in a
Table calculated for the Purpofe of this Sediion^
(See Plate i. Fig. 5.) In the firft Column on
the left you have the Names of the Intervals^
as they proceed in Order from a common Wim*
damental > in the id you have the Progreflioit
of Degrees from every Term to the next,- in the
id you have the feveral Terms ex^prefTed by
Letters ; in the p h Column you have the Num-
bers that exprefs the Relations of every Term
to the Fundamental C (which is 1 ) as far as
Two Odfaces, taken in the natural Order of
the concinnous Parts of the Offiave^ as above
divided and explained, thefe being fuppofed to
be fixed Eolations , then in the other Columns
you have expreffed. the Relations of every Term,,
in order upwards from C, to all thefe above
them, as far as an Otlave ; reduced to a com-
mon Fundamental 1, which is the firft Number
in every Column^ and fignifics that the Letter
or
§ 4 . cf MUSIC K, ij7
or Note againft which it {hinds, is fuppofed to
be a common Relative to the 7 Terms that
fraud next above it,, i. e. 1 hat the other Num-
bers of that Column compared to 1, cxpre-fs the
Relations which the Notes, or Letters aeainft
which they ftand, bear to that againft which
the 1 of that Column (rands, according to the
iixt Relations fuppofed in the Fourth Co umn
of Numbers. The nth Column is the flime
with the ifti and. if we would carry on that
Table in infinitum^ it would be brt a Repetiti-
on of the proceeding 7 Columns of Numbers ;
which (hews us that Two Octaves was funi-
cient to difcovcr all the fimple Difcords that
could polTibiy be in the. Scale. I have carried
thefe Columns no further than one Octave^ ex-
cept the firft,becaufe all above are but an Scr, and
fomeleffercompoundedjand therefore we needed
only to find ail the fimple Difcords lefs than an
Sve: Eut the ift Column is carried to Two SveSy
becaufe the reft are made out of it ; for thefe
other exprefs the mutual Relations of each
Term of the ifi Column to all above it within
an OcJave^Tcduced to a common Fundamental 1 .
I'll next (how you that there are no ' other
Relations in the other Series, which afcends
by a 3d I. and 6th I. than what are here. The
two Species differ only in the jthSj 6tbs and sds y
and if you'll look but a little back, you'll fee the
true Relation of the Terms of that other Series
to the Fundamental, which if you compare
with that Column in this Table, which begins
againft E^ you'll find them the fame in every
R Term
258 ^Treatise Chap. VIII.
Term but one ; for here the id Term is 15 :
16 which there is 8 : 9 ; but if you compare
the Column which begins againft A, you'll find
that agree with the Scale we are fpeaking of in
every Term but the qth, which is here 20: 27,
and there 3 : 4, the one wants the true id, and
the other the true qth ; but both thefe are in
the firft Column which begins at C ; therefore
'tis plain that .if thefe Columns are continued,we
muft find in them all the Relations that can
polfiblybe in that Scale ; which a little Exami-
nation will foon difcover.
■N o w befides the harmonic at Intervals and
Degrees already explained, we have in this Ta-
ble the following difcord Relations, which pro-
ceed from the Differences of the Degrees, and
the particular Order in which they follow other
in the Scale ; for we
'£xa. Ratios
B F * 27 : 32
F B * - 32 J 45
may conceive a great
Variety of other Bif-
cords from different
\#&A B s 20 : 27 I Combinations of thefe
o B £A E 27 : 40 Y Degrees, but the Spe-
M B° F = 45 : 64 culation would be of
F B * 16:27 noUfej 'tis enough
D C K 9 1 i6\ to confider what are
_ . ; j inavoidable in the Or-
v " der of the Scale of
Mufick, which are thefe mentioned. Again,
from the Table We find plainly that from any
Note or Letter of the Scale, to the 2d, 3d, qtm
$th, &c. inclufive, either above or below, is not
always the fame Internal ; becaufe tho' there is
I 4. of MUSIC K. ij 9
an equal Number of Degrees in every fucli Cafe,
Vet there is not always an equal Number of the
fame Degrees ; fo, from <?to F, there are three
Degrees, whereof 1 is a tg. 1 is t /. and 1 a/;
tut from F to B there are Three Degrees,
whereof 2 are ?g. and 1 is a £ /.
W e have already fettled the Definitions of
a id,qth, &c. as they are harmorical Intervals,
they are either to be taken from the true Ra-
tios of their Extremes >• or, refpecting the Scale
of Mufick, from the Number and particular
Kinds of Degrees ; yet we may make a general
Definition that will ferve any Part of the Scale i
and call that Interval, which is from any Let-
ter of the Scale to the 2d, 3d, Ofh\ &c. inclufive^
a id, a 3 d, a J\th, &c. But then ive muft make
a Diftin&ion, according as they are harmoni-
cal or not -, under which Diftinclion the Otlaves
will not come, becaufe every Eight Letter
inclufive is not only the fame, but is a true
Oclave in the Ratio of 1 : 2, • which is plain
from this, That every Oclave in order from the
Fundamental or loweft Note of the Scale,
is divided the fame Way, into the fame
Number of the fame Kind of Degrees^ and
in the fame Order : And for other Intervals
lefs than an OBave, we have Three of each
Kind, differing in Quantity • which Differences
arife from the Three different Degrees, as I
have expreffed them in the following Tabh<,
wherein the greateft (lands uppermolt, and fo in
i6o ^Treatise Chap. VIH.
ids. \ %ds. \ /\ths. \ $ths. I 6th s. \ jths.\
8:9)4
5 3 2 : 45 1 2
3
16 : 27!
8:i5|
9 • io| 5
6
20 : 27 1 27
: 40
3:5
5-.p r
15 : i6\ 27 :
32
3:44*
: 64
5:8 J
9 : if]
The Three 2^/j- or Degrees are all concin-
mous Intervals ; of the 3 ds one is Difcord, viz.
27 : 32, and therefore called o.fal/'e 3d 5 the o-
ther Two are particularly known by the Names
of -$dg. and ^dh of the qths and 5^ Two
&xq J)if cords s and called /jI^ 4^' and 5/7tf; 'and
therefore when we fpeak of a 4th or $th) with-
out calling itfal/bj 'tis underflood to be of the
true harmonical Kind ; of the 6th s one isfalfe,
and the other Two which are harmonical, are
called 6/7? g. and 6/7?/. the jths are neither
Ijarmonical nor concinncus Intervals, yet of Ufe
in Mufick, as I have already mentioned ; the
Two greater are particularly known by the
Name of greater or lelfer nth, tho' fome I know
make the leaft 9 : 16 the jth lelfer; I mean
they make that Ratio a Term in the Divifion
of the Otfave by 3 d L and 6th I. but I (hall have
Occaflon to confider this more particularly in a-
nother Place. Motv, as to the Compoiition of
the Ofifa-ve out of the Intervals of this Jafl
Fable, we have, this to remark, that if we
comparer" the 2 ds with the yths, or the %ds
with the 6th s ) or qth's with sths, the greater
of the one added to the lefier of the other, or
the Middle of the one added to the Middle of
the
§ 4 . of MUSIC K. itfi
the other, is cxaclly equal to O&tave ] and ge-
nerally add the greater!: of any Species of In-
tervals ( for Example $th$ ) to the leffcr of
any other ( as ^ds ) and the leaft of that to
the greater of this; alfo the Middle of the one
to the Middle of the other, the Three Sums or
Intervals proceeding from that Addition are e-
quaJ.
We. fhall next confidcr what the Errors of
thefe falfe Intervals are. The Variety, as to
the Quantity, of 'Intervals that have the fame
Number of Degrees in the Scale, arifes, as I
have already faid, from the Differences of the
Three Degrees ; and therefore the Differences
among Intervals of the fame Species and De-
nomination, i. e. the Exceiles or Defects of the
falfe fromthe true, are no other than the Dif-
ferences of thefe Degrees, viz. 80 : 81, the Dif-
ference of atg. and tl. which is particularly
called a Comma among Muficians ; 24 : 2$, the
Difference of a /' /. and f. which is fometimes
called a leffer Semitone, becaufe it is lefs than
15 : %6 ; then 128 : 135, the Difference of a
tg. andy^ which is a greater Difference than the
laft, and is alfo called a leffer Semitone, and is
a Middle betwixt 15: : 16, and 24 : 25. Be-
twixt which of the greater Intervals thefe Dif-
ferences do particularly ex? ft, will be eafily
found, by looking into the former Table, and
applying Problem 1 o. of Chap, 4. that is, mul-
tiplying the Two Ratios compared crofs-ways,
the greater Number of the one by the leffer of
the other, the Produces contain the Ratio or
R 3 Dif-
t6t x /^Treatise Chap. VIII.
Difference fought. Obferve alfo, that the. great-*
eft of the /{ths, viz. 32 : 45 is particularly cal-
led a Tritone^ for 'tis equal to 2 ?g. and 1 t k
and its Complement to an Qtfave^ viz. 45 : 64,
which is the leaft of the sths, is particularly
called a leffer $th or Semidiapente ( the Origi-.
na! of the Jaft Name you'll hear afterwards. )
Thefe Two are the fa (fe qth and $th^ which
are ufed as Df cords in the Brffmefs of Uarmo-
nj\ and they are the Two Intervals which di-
vide the Qffiave into Two Parts neareft to E-
quality, for their Difference is only this very
final] Interval 2025 : 2048. And becaufe in
common Practice the Difference of tg. and tk
is neglected, tho' it has its Influence, as wo
{hall hear of, therefore tbefe Intervals are only
called falfe, which exceed or come fhort by a
Semitone ; and upon this Supposition therefore
there is no falfe 3d or 6th ', nor any falfe qth
or 5th, except the Tritone and Semidiapente
mentioned, which with the jths and ids are
slU the D if cords reckoned in the Syftem\ how-
ever when we would know the Nature of Things
accurately, we muft neglect no Differences.
The Diftinclions already made of the Inter-
vals of the Scale of Mufic% regard their Con-
tents as to the Number and Kind of I)egrecs ;
but in the Scale we find Intervals of the fame
J3xtent, differing in the Order of their Degrees.
We iliall eafily find the whole Variety, by exa-
mining the Scales of Mufick ; for the Variety is
increafed by the Two different Series or Scales
&bove explained, there being fome in the one
that
§ 4 . of MUSIC K. i6$
that are not to be found in the other. I fliall
leave it to your felves to examine and find out
the Examples, and only mention here the 0-
fifaves, whereof there are in this refpeft feven
different Species in each Scale, proceeding from
the feven different Letters $ for it is plain at
fight, that the Order of Degrees from each of
thefe Letters upward to an clave is different ;
and that there can be no more Variety if the
Scale were continued in infinitum, becaufe from
the fame Letter taken in any Part of the Scale,
there is always the fame Order. What Ufe
lias been made of this Diftincfcion of Intervals,
and particularly Octaves, falls to be confidered
in another Place ; I fliall only obferve here,
that tho' all this Variety happens actually with-
in the Compafs of Two Offiaves, yet if you
ask, what is the moft natural and agreeable Or-
der in the Divifion of the Otlave, it is that
which belongs to the 061 ave from C in the pro-
ceeding Scale i or change the 3 d 9 6th and jth
from greater to leffer, and that makes another
concinnous Order ; the Degrees of each as they
follow other, you have already fet down. Now
if you begin and cany on the Series in any of
thefe Two Orders to a double clave, none of
the accidental Difcords will give any Offence
to the Ear, becaufe their Extremes are not
heard in immediate Succeffion ,- and the Difcord
is rendred altogether infeniible by the immediate
Notes j cfpecially by the harmonious Relation
of each Term to the common Fundamental,
and the manifold Concords that are to be found
K 4 among
2^4 4 Treatise Chap. VIII.
among the feveral middle Terms. For the Po-
fitions of the Degrees, which occaflon thefe
Difcords, if we confider them with refped: to
the Fundamental C, they are truly continuous,
but with rofpeft to the lowed of Two Notes,
betwixt which they make the Difcord, they
follow inconcinnouPj from it, becaufe they
were not defigned to follow it as a Fandamen-
tad, and fo are not to be referred to it : There-
fore in all the Scale, only C can be nvide funda-
mental, hecaiife from none of the other Six Let-
ters do the Degrees follow in a right continuous
Orde-, unlefs, as I faid before, we neglect the Dif-
ference gl tg. and th and then the Octave from
y/wi!l be a right continuous Series, proceeding by
a %& /. when it proceeds by a ydg. from C, and
contrarily ; and hereby we fiiall have both the
Species in one Series j otiierwife there are Three
Terms that are variable, which are the id, 6th
and -]th from the Fundamental, i. e. E, A, B,
when the Fundamental is called C; and this
nuift be carefir'y minded when we fpeak of the
Scale of Mufick. How unavoidable thefe Kinds
of Bij 'cords arc among the Notes of the Scale^
we have fen ; but, as I have already obferved,
there are other Suocefficns that are melodious,
bolides a conftant Succefficn of Degrees ; j for
thefe are mixt in Pradice with harmonical In-
tervals : And here alio the immediate Succef-
iion.many be melodious, tho' there be many Dif-
cords among the diflant Notes, whofe Harfh-
nefs is rendred altogether infenllble from their
Situation, eipeciaily becaufe of the harmonical
Relation
§i. ofMUSICK. i6 5
Relation of the feveral Notes to fome funda-
mental or principal Note, which is called the
K )\ w th a particular Refpecl: to which the
reft of ilia Notes are chofen.
<&®&®<&!^®&®^®i&®<&>®<&%
CHAP. IX.
.Of the Mode or Key m Mufick ,♦ and a
further Account of the true End and Of-
fice 6 f the Scale ^Mufick.
§ i. Of the Mode cr Key.
"\1\7T E have already diyided the Applicati-
ve on of the Tune of Sounds inco thefe
Two, Melody and Harmony. When
feveralfimple Sounds fucceed other agreeably in
the Ear, that Effect is called Melody ; the pro-
per Materials of which are the Degrees and
I harmonious Intervals above explained. But 'tis
$ct every Succeilion of thefe that can produce
this Pleaiure $ Nature has marked out certain
Limits for a general Rule, and left the Applica-
tion to the Fancy and Imagination ; but alwiys
und^r the Direction of the Ear. Tnc other
chief Ingredient in Mufick is the Duration^ or
Difference of Notes with refpecl to their unin-
terrup-
*66 ^Treatise Chap. IX.
terrupted Continuance in one Tune y and the
Quicknefs or Slownefs of their Succeflion ,• tak-
ing in both thefe, a melodious Song may be
brought under this general Definition, viz, A
Collection of Sounds or Notes ( however pro-
duced) differing in Tune by the Degrees or har-
monious Intervals of the Scale o^Muiick, which
facceeding other in the Eai\ after equal or un-
equal Duration in their refpstlive Tunes, affect
the Mind with Pleafure, But the Defign of
this Chapter is only to confider the Nature and
general Limits of a Song, with refpect to Tune^
which is properly the Melody of it; andobferve,
That by a Song I mean every fingle Piece of
Mufick, whether contrived for a Voice or In-
(tniment.
A Song may be compared not abfurdly to an
Oration ; for as in this there is a SuhjeB^ viz.
fome Per/on or Thing the Difcourfc is referred
to, that ought always to be kept in View, thro'
the Whole, fo that nothing unnatural or foreign
to the SubjebJ may be brought in ; in like.Man-
ner, in every regular and truly melodious Song,
there is "one Note which regulates all the
reft ; the Song begins, and at leaft ends in this,
which is as it were the principal' Matter, or
mimical Subject that demands a fpecial Regard
to it in all the other Notes of the Song. And
as in an Oration, there may be feveral diftincl:
Parts, which refer to different Subjects, yet
fo as they muft all have an evident Connection
with the principal Subject which regulates and
influences the Whole j fo in Melodr, there may
§ i. of MU SICK. %6 7
be feveral fubprincipal Subjects, to which the
different Parts of that Song may belong, but
thefe are themfelves under the Influence of the
principal Subject, and muft have a fenftble Con-
nection with it. This principal Note is called
the Key of the Song, ov the principal Key with
refpecl; to thefe others which are the fubprin-
cipal Keys, But a Song may be fo fliort, and
{imply contrived, that all its Notes refer only to
one Key.
T h"a T we may underftand this Matter di£
tin&ly, let us reflect, on fome Things already ex-
plained : We have feen how the Octave con-
tains in it the whole Principles of Mufick, both
with refpect to Confonance (or Harmon}') as it
contains all the original Concords, and the har-
monical Divifion of fuch greater, as are equal to
the Sum of leffer Concords ; and with refpecl; to
Sticcejfion (or Melody) as in the concinnous Di-
vifion of the Oclave, we have ail the Degrees
fubfervient to the harmonical Intervals, and the
Order in which they ought to be taken to make
the moft agreeable Succeflion of Sounds, riling
ir falling gradually from any given Sound, i. e,
any Note of a given and determined Pitch of
Tune; for the Scale fuppofes no Pitch, and &£&
,y atfigns the juft Relations of Sound which
nake true mufical Intervals : But as the
ids and 6th s are each diftinguifiied into greater
md leffer, from this arifc Two different Species
m the Divifion of the Ot^avf. We have alfo
)bferved, That if ■ either Scale ( viz. That
vhich proceeds by the yl I. or by the 3 d g. )
is
16*8 ^Treatise Chap.- IX.
is continued to a double QBave, there {hall be
in that Cafe 7 different Orders of the Degrees
of an 8^, proceeding from the 7 different Letters
with which the Terms of the Scale are marked;
none of which Orders but the tuft, mz, from
C is the natural Order ,• and tho' in railing the
Series from C to the double Octave, we actually
go through the Degrees in each of thefe Orders,
yet C only being the Fundamental, to which all
the Notes of the Series are referred, there is no-
thing offenfive in thefe different Orders, which
' are but accidental ; fo that in every QbJave con^
cinnoufly divided, there are 7 different Intervals
relative to the Fundamental, whofe acute
Terms are the effential Notes of theOBa ve,&ti£
they are thc[e,vtz. the idg. 3d g. qfhy <th, 6th g.
yth g. Sve, or id g. 3 d /. /\th, zth, 6th I. 7th
I. %®e*
N o w, let us fuppofe any given Sound, u e«
a Sound of any determinate Pitch of Time, it
may be made the Key of a Song, by applying
to it the Seven effential or natural Notes that a-
rife from the concinnous Divificn of the §>ve, as
I have juft now fet them down, and repeating
the %<ve above or below as oft as you pleafe.
The given Sound is applied as the principal Note
or Key of the Song, by making frequent Clofes
or Cadences upon it ; and in the Courfe- or
Progrefs of .the Melody, none other than thefe
Seven natural Notes can be brought in, while
the Song continues in that Key, becaufe eveiy
other Note is foreign to that Fundamental or
Key.
To
§ i. of MUSICK. i6 9
T o underftan'd all this more diftin<5tly, let us
confider, That by a Clqfc en Cadence is meant a
l terminating or bringing the Melody to a Period
lor Reft, after which it begins and fets oat a-
mew, which is like the finifliing of fome diftinct
IPurpofe in an Oration j but you mutt get a per*
(fed: Notion of this from Experience. Let usfiip-
> pofe a Song begun in any Note, and carried on
[upwards or downwards by Degrees and haft
\ monical D{ftances, fo as never to touch any
Notes but what are referable to that firft Note
, as a Fundamental^ i. e. arc the true Notes of
I the natural Scale proceeding from that Funda-
i mental ; and let the Melody be conducted fo
I through thefe natural Notes, as to dole and
r terminate in that Fundamental, or any of its
%ves above or below ; that Note is called the
r Key of the Melody, becaufe it governs and re-
| gulates all the reft, putting this general Limita-
tion upon them, that they mull: be to it in the
j Relation of the Seven effcntial and natural Notes
i of an,8c\? 5 as abovementioned ; and when any
f other Note is brought in, then 'tis faid to go out
of- that Key : And by this Way of fpeaking of
a Song's continuing in or going out of a Kej\
we may obferve, that the whole 8^, with all
its natural and concinnous Notes, belong to the
Idea of a Key, tho' the Fundamental, being
I the principal Note which regulates the red:, is in
i a peculiar Sen-fe called the Key, and gives De-
nomination to it in a Syflem of fixt Sounds, and
in the Method of marking Sounds by Letters,
as we fliall hear of more particularly afterwards.
And
270 A Treatise Chap. IX*
And in this Application of the Word Key to one
fundamental Note, another Note is faid to be
out of the Key, when it has not the Relation
to that Fundamental of any of the .natural Notes
that belong to the continuous Divifion of the
%ve. And here too Xve muft add a neeeffary
Caution with refpecl: to the Two different Di-
vifions of the 2<ve, viz. That a Note may be-
long to the fame Key, i. e. have a juft mufical
Relation to the fame Fundamental in one Kind
of Divifion, and be out of the Key with refpeft
to the other : For Example, If the Melody has
nfed the 3 J g. to any Fundamental, it requires
alfo the 6th g. and therefore if the 6th I. is
brought in, the Melody is out of the firft Key.
Now a Song may be carried thro' feveral
Keys, i* e. it may begin in one Key, and be led
out of that to another, by introducing fome
Note that is foreign to the firft, and fo on to
another : But a regular Piece muft not only re-
turn to the Erik Key, thefe other Keys muft alfo
have a particular Connection and Relation with
the firft, which is the principal Key. The Rule
which determines the Connection of Keys, you'll
find diftinctly explained in Chap. 13. for we may
not change at random from one Key to another j
I fhall only obferve here, that thefe other Keys
muft be fome of the Seven natural Notes of the
principal Key b yet not any of them; for which
fee the Chapter referred to;
But that you may conceive all this yet more
clearly, we (hall make Examples. Suppofe the
following Scale of Notes expreft by Letters,
where-
§ i. cfMUSICK. i7i
wherein I mark the Degrees thus, &iz, a t g.
iwith a Colon (:) at I. with a Semicolon ($)
[^Semitone with a Point (.) And here I mark
I the Series that proceeds with the 3d g, &c.
C:B\E.F:G;A:B. C: d ;e.f:g ; a :b.C
■
' The firft Note reprefents any given Sound, and
! the reft are fixt ^according to their Relations
to it, expreft by the Degrees: Let the firft Note
of the Song, which is alfo the defigned Key, be
taken Unifcn to C. (which reprefents any given
Sound) all the reft of the Notes, while it keeps
within one Key, muft be in fuch Relation to the
firft, as if placed according to their Diftances
from it in a direel: Series, they fhall be unifon
each with fome Note of the preceeding Scale :
j The Example is of a Key with the 3d g, &c.
which is eaiily applied to the other Species. Let
j us now fuppofe the Conduct of the Melody fuch,
that after a Cadence in C the Song fhall
make the next Cadence in a 3 d g. above, &i%*
j ini?, and this is a new Key into which the jlftf-
I lody goes.
W e have obferved in the preceeding Chap*
I, that the Order of Degrees from each of the
Letters of the diatonick Scale, is different ;
and therefore while the Relation of thefc
Notes are fuppofed fixt, 'tis plain none of the
Notes of that Scale except C can be made a
Key,' became the Seven Notes within the %®e
are not in the true Relation of the eflential and
natural Notes of an %ve concinnoufly divided \
and
272 ^Treatise Chap. IX.
and therefore the natural Scale{i.e. the Order from
C) muft be applied anew from every new Key ;
as in the preceeding Fxampk, the id Key h_E,
which in that Scale has" a 3d l. at G, but it has
not all its Seven Notes in juft Relation to the
Fundamental, the firft Degree bring a f. which
ought to be tg -, and therefore if the Melody in
that Ky be fo managed as to have Ufe for all
the Seyen natural Notes, they cannot be all
fouiid in the Series that proceeds concimioufiy
from C, but requires the Application of the na-
tural Scab to that new Pitch, /. e. requires that
we make a Series of continuous Degrees from
that ngw Fundament al; which we may exprefs
either by calling itC,and applying the fameNamcs
to tfe whole %ve^ above or below it,as to the for-
mer Ky, or retaining ftill the Names E F, &c.
to -an hve y but fuppohng their Relations chan-
ged.
A Song may be fo ordered, that it flialJ not
require all the Seven natural Notes of the Keys
and if the Melody be fo contrived in the Jub-
pfmcipal Ktfji of the Song, that it fliall ufe none
of the effential Notes of thefe Keys^ but iiich as
coincide with thefe of the principal Key, then
is the whole of that Song more (trictly limited
to the principal Ky: So that in a good Senfe
it ^lay be fai never to go out of it -, but then
there will be Jefs Variety urfderfiich Limitations:
And if a Song may be fuppofed to go through
feveral Kys, the principal being always perfeel:
as from C, and the Subprincipals taken withfuch
Imperfections as they unavoidably have, when
wc
§ t. of MUSIC K. £73
we are confined to one individual Scries of de-
terminate Somids,thcMiifick may be faid alfo in
this Cafe never to depart from the principal
Key; but 'tis plain, that the ufmg fuch Inter*
vols with refpecl to the fuhprincipal Keys, will
make the Melody imperfect, and alfo occafion
Errors of worfe Gonfequence in the Harmony
of Parts fo conducted.
T i s Time now to Confder the JbiftinBipns
of Keys. We have feen that to conflitute any
Note or given Sound a Key or fundamental
Note, it muft have thefe Seven effential or na-
tural Notes added to it, viz. id g. 3d g. or 3^/,
4th, $th, 6th g. or 6th I. jth g. or jth I. %ve
out of which, or their %ves i all the Notes of
the Song muft be taken while it keeps within
that Key, u e. with in the Property of that Fun*
dameiital '% 'tis plain therefore, that there are
but Two different Species of Keys, according
as we joyn the greater or leffer 3d, which are
'always accompanied with the 6th and jth of
the fame Species, viz. the 3d g. with the 6th g.
and jth g ; and the ^dL with the 6th L and
jth I; and this Diftin£iion is marked with the
Names of A Sharp KEY,wkieh is that with the
$dg, &c. and A FlatKey with the 3d I, &c.
Now from this it is plain, that however many
different Glofcs may be in any Song, there can
be but Two Keys, if we conhder the effential
Difference of Kjys ; for every Key is either
JJjarp or flat, and all Jharp Keys are of the fame
Nature, as to the Melody, and fo are all fla
Keys ; for Example, Let the principal Key a
S
1/4 -^Treatise Chap. IXL
a Song be C ( with a 3^ g. ) in which the final
C ofe is made, let other Clofes be made in
E ( the 3 d or the principal Key ) with a 3^^.
and in yf ( the 6/i> of the principal Key ) with
a 3^//. yet in all this there are but Two diffe-
rent Keys^jharp andjlat: But obferve, in r om-
mon Practice the Keys are faid to be different
when nothing is confidercd, but the different
Tune or Pitch of the Note in which the diffe-
rent Clofes arc made ; and in this Senfe the
fame Song is faid to be in different Keys, ac-
cording as it is beaun in different Notes or De-
grees of Tune, But that we may fpeak accu-
rately, and have Names anfwering to the real
Differences of Things, which I think neeeffary
to prevent Confuiion, I would propofe the
Word Mode, to exprefs the melodious Confiitu-
tion of the Odiave, as it confifts of Seven effen-
tial or natural Notes, befides the Fundament al\
and becaufe there are Two Species, let us call
that with a ^dg. the greater Modt\ and that
with a 3 dU the kjfer Mode : And the Word
Key may be applied to every Note of a Song,
in which a Cadence is made, fo that all thefe
( comprehending the whole OtJave from each )
may be called different Keys, in refpeel: of their
different Degrees of Tunes, but with refpeel;
to the effential Difference in the Constitution
of the ObJaves, on which the Melody
depends, there are only Two different Modes,
the greater and the leffer. Thus the Latin Wri-
ters ufe the Word Modus, to fignify the parti-
cular Mode or Way of conftituting the Offiaye;
a»d
§ i. ofMUSICK. try
and hence they alio called it Conftitutio ; hut of
this in its own Place.
r Tis plain then, that a Mode (or Key in this
Senfe) is not any (ingle Note or Sound^ and can-
not be denominated by it, for it fignifies the par-
ticular Order or Manner of the continuous De-
grees of an %*ve> the fundamental Note of which
may in another Senfe be called the K.£?' 5 as it
figniiios that principal Note which regulates the
reft, and to which they refer : And even when
the Word Key, applied to different Notes, fig-
nifics no more than their different Degrees of
Tunc, thefe Notes are always confidered as
Fundamentals of an %<ue conclnnoufly divided,
tho' the Mode of the Divilion is not confidered
when we call them different Keys ; fo that the
whole %ve comes within the Idea of a Key in
this Senfe alfo : Therefore to diftinguifh proper-
ly betwixt Mode andfQ>', and to know the real
Difference, take this Definition, viz* an &ve
with all its natural and continuous Degrees is
called a Mode, with refpect- to tke Con'ftitution
or the Manner and Way of dividing it; and with
refpect to the Place of it in the Scale of Mufick,
i. c. the Degree or Pitch of Tune, it is called a
■Key, tho' this Name is peculiarly applied to the
Fundament ah Hence it is plain, that the fame
Mode may be with different K^eys, that's to fay,
an 052 ace of Sounds may be railed in the fame
Order and Kind of Degrees, which makes the
fame Mode-, and yet be begun higher or lower,
i. e. taken at different Degrees of Tune, with
refpecl: to the Whole, which makes different
S x Kjys>
Z7<$ ^Treatise Chap. IX.
Keys. It follows alio from thefe Definitions,
that the fame Key may be with different Modes>
that is, the Extremes of Two Offiaves may be
' in the fame Degree of Tune> and the Divifion
of them different. The Manner of dividing the
Offave, and the Degree of Tune at which it is
begun, are fo diftinct, that I think there is Rea-
fon to give them different Names ; yet I know,
that common Practice applies the Word Key to
both ; fo the fame Fundamental conftitutes
Two different Keys y according to the Divifion
of the Offave; and therefore a Note is faid to be
out of the Ke}\ with refpc6t to the fame Fun-
damental in one Divifion, which is not fo in a-
nother, as I have explained more particularly a
little above ; and the fame Song is faid to be in
different Keys, when there is no other Diffe-
rence, but that of being begun at different
Notes. Now, if the Word Key muft be ufed
both Ways, to keep up a common Practice,
we ought at Jcaft to prevent the Ambiguity,
which may be done by applying the Words
fljarp and flat. For Example. Let the fame
Song be taken up at different Notes, which we
call C and A y it may in that refpecl be faid to
be in different Keys, but the Denomination of
the Key is from the Clofe ; and Two Songs clo-
fing in the fame Note, as C, may be faid to be
in different Keys, according as they have a grea-
ter or leffer 3d; and to diftinguifh them, we fay
the one is in the (harp Key C, and the other in
the flat Key C; and therefore, when jharp or flat
is added to the Letter or Name by which any
funda*
§ i. of MUSI CK, 177
fundamental Note is marked, it expreiles both
the Mode and Key, as I have diftinguiflied them
above ; but without thefc Words it expreffes no-
thing but what I have called the Key in Diftin-
tfion from Mode. But of the Denominations of
Keys in the Scale of Mufick, we (hall hear par-
ticularly in Chap, n.
Obferve next, that of the natural Notes
of every Mode or Oclave, Three go under
the Name of the ejfential Notes, in a pecu-
liar Manner, viz. the Fundamental, the 3d, and
$th, their Oft ayes being reckoned the fame,
and marked with the fame Letters in the Scale ;
the reft are particularly called Dependents. But
again, the Fundamental is alfo called the final \
becaufe the Song commonly begins and always
ends there: The $th is called the Dominant e,\>^~
caufeit is the next principal Note to the finals
and moft frequently repeted in the Song $ and if 'tis
brought in as a new Key, it has the moft per-
fect Connection with the principal Key : The
3d is called the Mediante, becaufe it ftands be-
twixt the Final and Dominante as to its Ufe.
But the 3d and $th of any Mode or Key defcrvc
the Name of ejfential Notes, more peculiarly
with refpect to their Ufe in Harmony ', becaufe
the Harmony of a 3d, $ih and %ve, is the moft
perfect of all others $ fo that a 3d and a $fk 9
applied in Confonance to any Fundamental^
gives it the Denomination of the Key ; for chief-
ly by Means of thefe the Cadence in the Key
is performed. The Bafs being the governing Part
with reipeft to -the Harmony ' 3 ought finally to
S3 c!c-S
^7% A Treatise Chap.. IX.
clofe in the Key ; and the Relation or Harmo-
ny of the Parts at the final Clofe, ought to be
fo perfecl, that the Mind may find entire Sa-
tisfaction in it, and have nothing farther to ex-
pert Let us fuppofe Four Voices, making to
gether the Harmony of thefe Four Notes
G — c — e -- g y where G is the Fundamental, f
a qth, e a 6th g. and g an %m j fo that c - e
is a %dg. and e — g a 3d I. and c — g a $th.
The Ear would not reft in this Clofe, becaufe
there is a Tendency in it to fomcthing more
perfect - } for the true Key in thefe Four is r, to
which the id and $th is applied ; the Bafs
clofing 'mG puts the <th out of its proper Place,
for it ought to ftand next the Fundamental \
nor can the 3 d be fcparate from the 5/"/;, which
can ftand with no other. Now the Thing re?
quired is, to reftore the $th to its due Place,
and this is done, by removing the qth to the
upper Place of the Harmony; fo in the proceed-
ing Example, fuppofe the Bafs moves from G
to c, and the reft move accordingly till the
Fcur make thefe c — c — g — cc, in which
c — e is %dg. c—g a $th} then we have a
perfect Clofe?, and the Mufick is got into the;
true and principal K.ty, which is c.
W e have one Thing more to obferve as to
the jtby wki«k is natural to every Mode ; in
the greafer Modes or flarp Keys 'tis always
the 7th g. but flat Keys ufe both the "jthg. and
jthL in different Circnmftances : The ythL
moft naturally accompanies the 3 dl. and &#'/.
which conftitute 2. flat Key, and belongs to it
necef-
§ z. of MUSIC K. t? 9
neceffarily, when we confider the continuous
Divifion of the Qfifave, and the mod agreeable
' Succeifion of Degrees ; and it is ufed in every
I Place, except it is fometimes toward a Clofe,
i efpccially when we afccnd to the Key, for then
the Jthg. being within a f, of the Key, makes
a fmooth and cafy Paffage into it, and will fome-
times alfo occaiion the 6th g. to be brought in.
Again, 'tis by Means of this jthg. that the
Tranfition from one Key to another is chiefly
performed,- for when the Melody is to be trans-
ferred to a new Key, the Jthg, of it ( whether
'tis sijharp or flat Key ) is commonly introdu-
ced: But you {hall have more of this in Chap. 13,
I have laid, that the yth is ufed in Melody as
a iingle Degree, but in fucli Circumftances as
removes the Harfhnefs of fo great a Difcord t as
particularly in quick Movements ; and we may
here confider, that a qth. being the Comple-
ment of a true Degree to Ovlave, partakes of
the Nature of a Degree fo far, that to move up »
ward by a Degree, or downwards by its Cor-
refpondent yth, and contrarily downwards by a
Degree, or upwards by a jth, brings us into the
fame Note; and from this Connection of it with
the true Degrees, 'tis frequently ufeful.
§ 2. Of the Office of the' Scale of Mufick.
NO w from what has been explained, we ve-
ry eaiily fee the true and proper Office of
the Scale of Mufick, which, ftri&ly fpeaking, is
all comprehended in an ObJave, what is above or
S 4 below
i8o ^Treatise Chap. IX,
below being but a Repetition. The Scale fup^
pofes no determinate Pitch of Tv.ne^ but that
being affigncd to the Fundamental^ it marks
out the Tune of the Reft with relation to it.
We Jearn here how ' to pafs by Degrees moft
mclodioufJ)\ from any given Note to any har->
wonical Diftance. The Scale (hews us, what
Notes can be naturally joyned to any Fundament
fal % and thereby teaches us the juft and natural
Limitations of Melody. It cxhi bites to us all
the Intervals and Relations that are ehential
and ncceflary in Mujick, and contains virtually
all the Variety of Orders, in which thefe Re^
lations can be taken fuccctfively ; if a Song is
confined to one Ke}\ the Thing is plain, if 'tis
carried thro' feveral Kcjs^ it may feem to re*
quire fcveral diftincl: Series; yet the Mu/jck in
every Part being truly diatcnick, 'tis but the
fame natural Scale (with its Two different Spe-
cies ) applied to different fundamental Notes.
And this brings us to confider the Erfe£t of
having a Series of Sounds fixt to the Relations
of the Scale : If we fuppofe this, it will eafily
appear how infufiicient fuch a Scale is for all
the agreeable Variety of Melody: But then,
this Imperfection is not any Defect in the natu-
ral Syfiem^ but follows accidentally, upon its be-
in 3 confined to this Condition : For this is
not the Nature and Office of the Scale of Mu-
Jtck, that fappofing its Relations all expreffed
in a Series of determinate Sounds, that indivi-
dual Series fliould contain all the Variety of
Notes, that can melodioujjy &cceed other j un-
' ' " lefs
§ i. tf MUSIC K. tli
lefs you'll fuppofe every Song ought to be limi-
ted to one Key; but otherwife one individual
diatonick Series of fixt Sounds is not fufftcient.
Let us fuppofe the Scale of Mufeck thus defin'd,
viz. a Series of Sounds, whofe Relations to one
another are fuch, that in one individual Series,
determined in thefe Relations, all the Notes
may be found that can be taken fucceffrvely to
mate true Melody; fuch a Syftem would indeed
be of great Ufe, and be juftly reckoned a per-
fect Syftem ; but if the Nature of Things will
not admit of fuch a Series, then 'tis but a Chi-
mera ; and yet it is true, that the natural
Scale is a juft and perfect Syftem, when we
confider its proper OfUce as I have expreft it a-
bove, and as we fliall underftand further from the
next Chapter, in which I fliall confider more
particularly the DefccJ of Inftriiments having
fixt and determinate Sounds, and the Remedy
applied to it; and comparing this with the Ca-
pacity of the human J^oice^ we ftiall plainly
underftand, in what different Senfcs the Scale of
Mufick explained, ought to be called a perfedf
or imperfect Syftem,
CHAP.
SO -
\
itz A Treatise Chap. X.
C H A P. X.
Concerning the Scale of Mufick limbed to
fixed Sounds ; explaining the Defects of
Infrruments, and the Remedies thereof > r
wherein is taught the true Ufe and Ori-
ginal of the Notes we commonly call
fharp and flat.
§ i. Of the Defeats of Inftruments, and of the
Remedy thereof in general, hj the Means of
what we call Sharps and Flats.
H E Ufe of the Scale of Mufick has been
largely expkin'd, and the general Limi-
tations in Melody contained in it. Why
the Scale exhibited in the preceeding Chapters
is called the natural, and the diatonick Scale,
has been alfo faid, and how Mufick compofed
under the Limitations of that Scale is called
diatonick Mufick*
Let us now conceive a Series of Sounds de-,
termined and fixt in the Order and Proportions
of that Scale , and named by the fame Letters.
Suppofe, for ^Example, c,n Organ ov Harpfichord,
the loweft or graveft Note being taken at any
Pitch of Tune; it is plain 3 i;/zo. That we can pro-
ceed from any JS T oteonly by one particularOrder
Of
i. ! of MUSIC K. z8 3
of Degree s $ for we havefhewn before, that from
i every Letter of the Scale to its Octave, is con-
itain'd a different Order of the Tones and Semi-
tones, ido. We cannot for that Keafon rind a-
ny Interval required from any Note or Letter
upward or downward ; for the Intervals from
every Letter to all the red are alfo limited ; and
therefore, %tio. A Song ( which is truly dia-
tonick ) may- be fo' contrived, that beginning
at a particular Letter or Note of the Inftrument,
all the Intervals of the Song, that is, all the o-
thcr Notes, according to the juft Diftances and
Relations deiigned by the Compofer, fhall be
found exactly upon that Inftrument, or in that
fixt Series $ yet fhould we begin the Song at any
other Note, we could not proceed. This will
be plain from Examples, in order to which,
view the Scale cxprehed by Letters, in which
I make a Colon (': ) betwixt Two Letters, the
Sign of a greater Tone 8:9, a Semicolon Sjj
the Sign of a leffer Tone 9:10, and a Point
(.) the Sign of a Semitone 15 : 16. And thefe
Letters I fuppofe reprefent the fevcral Notes of
an Inftrument, tuned according to the Relations
marked by thefe Tones and Semitones--^
C. :D j E . F ' : G ; A : B . c : d ; e .f ; g ; a : b ,cc
•
Here we have the diatonick Series with the.
3d and 6th greater, proceeding from C ; and
therefore, if only this Series is exprefted, fome
Songs compofed with.ajto Melody, i. e. whofb
Key has a leffer 3d, &c, could not be performed
on
284 ^Treatise Chap. X.
on this Inftrument, becaufe none of the OtJaves
of this Series has all the natural Intervals of
the diatonick Series, with a 3d lelfer, as they
have been fiiewn in Chap. 8. For Example, the
Octave proceeding from E has a id I. but in-
flead of a tg. next the Fundamental, it has a
Semitone, Again, the O clave A has a 3^ /.
but it has afalfe 4th from A to ^ being Two
greater Tones and a Semitone in the Ratio of
20 : 27. Let us then fuppofe, that a Note is
put betwixt r and J, making a true qth with
y^ 3 to make the Odfave A a true diatonick Se-
ries. By this Means we can perform upon this
Inftrument moft Songs, that are fo iimple as to
be limited within one Key, I mean that make
Clofes or Cadences only in one Note ; for every
Piece of diatonick Melody being regulated by
the Intervals of that Scale, and every Key or
Mode being either the greater or lejjer (u e. ha-
ving either a 3 d greater or leffer, with the o-
ther Intervals that properly accompany them,
which have been already fhewn ) 'tis plain,
that beginning at A or E on this Inftrument,
we can find the true Notes of any fuch iimple
Song, as was fuppofed ; unlefs the Melody in the
flat Key is fo contrived, as to ufe the 6th and
yth greater, as I have faid it may do in fome
Circumftanccs, for then there will be fliil a De-
fect, even as to fuch Iimple Songs.
But there are many other conliderable Rea-
fons why this Inftnament is yet very imperfect.
And into. Conflder what has been already faid
concerning the Variety of Keys ox 0ofes } which
§ i. of MUSIC K. 28 j
may he in one Piece of Melody j and then we
(hall find that this fixt Series will be very
inefficient for a Song contrived with fuch Va~
riety $ for Example, a Song whofe principal
Key is Cwith its yd g. may modulate or change
into F \ but on this Inftmment F has a falfe
qth at B, and if a true qth is required in the
Song, 'tis not here ; or if it modulate into D,
then we have a falfe ^d at F, and a falfe $th at
ji, which are altogether inconfiftent with right
Melody ; 'tis true that the Errors in this Jaft
Cafe are only the Difference of a greater and
lefler Tone, as you'll find by considering how
many, and what Kind of Degrees the true 3^
and $th contains ; or by confidering their Pro-
portions in Numbers, in the Tables of Chap. 8.
And this Difference is in the common Account
neglected, tho' it has an Influence, of which I
fhall fpeak afterwards ; but where the Error is
the Difference of a Tone and Semitone, it is fo
grofs, that it can in no Cafe be neglected; as
the falle /\th betwixt F and B; or when a Se-
mitone occurs where the Melody requires a
Tone j for Example, if from the Key C there
is a Change into E, to which a tg. is required,
we have in the Inftmment only a Semitone. And,
to fay it all in few Words, imo. The harmoni-
ca! and concinnous Intervals of which all true
Melody confifts, may be fo contrived, or taken
in Stwc ejfion, that there is no Letter or Note of
this Inftrument at which we can begin, and find
all the reft of the Notes in true Proportion
which yet is not the Fault of the Scale, that not
being
i86 A Treatise Chap. XT
being the Office of it. ido. When the fame
Song is to be performed by an Inftrument arA
a Voice, or by Two Inftruments in Uhjjfbj?, it
may be required > for accommodating the "one
to the other, either to alter the Pitch of the
Tuning, fo as the whole Notes may be equally
lower or higher ; or, becatife this is in fome
Cafes inconvenient, and in others impofflble, as
when any Wind-inftrument, as Organ or Flute y
is to accompany a Voice, and the Note at
which the Song is begun on the Inftrtiment is
too high or low for the Voice to carry it thro"
in ; in iuch Cafes the only Remedy is to begin
at another Note, from which, perhaps, you can-
not proceed and find all the true Notes of the
Song, for the Reafons fet forth above ; or let it.
be yet further illuftrated by this Example, A
Song is contrived to proceed thuSji^V/?, upward
a tg. then & tL then a Senu &c. fuch a Pjxh
grefs is melodious ', but is not to be found from
any Note of the preceeding Scale j except c - 3 and
therefore we can begin only there,unlefs the Iii-
flrument has other Notes than in the Order of
the diatonick Scale.
We fee then plainly the Defect of Ihjlru-
mentSy whole Notes are fixt ; and if this is cu-
rable, 'tis as plain that it can only be effected
by inferring other Notes and Degrees betwixt
thefe of the diatonick Series : How far this i?,
or may be obtained, fliaJl be our next Enquiry ;
and the firft Thing I fliall do, is, to demonflrate
that there cannot poffibly be a perfect Scale
fixed upon Inftruments, u e* fuch as from any
Note
§,i. of MUSIC K. *8 7
Note upward or downward, fliall contain any
harmonic at or continuous Interval required in
their exact Proportions.
Since the Inequality ofthc Degrees into which
the natural Scale is divided, is theReafon that
Inftruments having fixt Sounds are impcrfeel ; for
hence it is that all Intervals of an equal Num-
ber of Degrees, or whofe Extremes comprehend
an equal Number of Letters, are not equal; fo
from C to E has TwoDcgrees, and EXo G wM
as many • but theDegrees, which are the com-
I ponent Parts of tliefe Intervals, differ, and fo
imift the whole Intervals: Therefore it is ma-
, nifeft, that if there can be a perfetJ Scale (as
. above defined) fixt upon Inftruments, it muft bo
fuch as fhall proceed from a given Sound by
equal Degrees falling in with all the pivifions
j or Terms of the natural Scale, in order to
I preferve all its harmonious Intervals, which
, would otherwife be loit, and then it could bo
f no mufical Scale,
I f fuch a Series can be found, it will be ab-
| folutely perfect, becaufe its Divilions falling in
| with thefe of the natural Scale, each Degree
, and Interval of this will contain a certain
; Number of that new Degree ; and therefore w$
ji fliould have, from any given Note of this Scale,
j any other Note upward pr downward, which
i fliall be to the given Note in any Ratio of tk§
• diatonick Scale ; and confequently any Piece
I of Melody might begin and proceed from any
Note of this Scale indifferently : But fuch a I)i*
l vifion is impoffiblc, which I fliall demonftmt©
X
*88 ^Treatise Chaf.X
thus. imo. If any Series of Sounds is expreffe
by a Series of Numbers^ which contain betwix
them the true Ratios or Intervals of thefi
Sounds, then if the Sounds exceed each othd
by equal Degrees or Differences of Tune, thM
Series of Numbers is in continued geometrica
Proportion, which is clear from what has beef
explained concerning the Expreffion of the Ir\
tervals of Sound by Numbers. 2do> Since it i]
required that the new Degree fought, fall
with the Divifions of the natural Scale, 'tis evi
dent that this new Degree muft be an exa£
Meafure to every Interval of that Scale -, tha,
is, This Degree muft be fuch, that each o
thefe Intervals may be exactly divided by it, o
contain a certain precife Number of it withou
a Remainder ; and if no fuch Degree or com
mon Meafure to the Intervals of the natura
Scale can be found, then we can have no fuel
perfect Scale as is propofed. But that fuch ;
Degree is impoffible is eafily proven ; confider i
muft meafure or divide every diatonick Interval
and therefore to prove the Impolfibility of i
for any one Interval is fufficient ; take for Ex
ample the Tone 8 : 9, it is required to divid
this Interval by putting in fo many geometrica
Means betwixt 8 and 9 as {hall make the Whol!
a continued Series, with thefe Qualifications, viz
That the common Ratio, (which is to be tin
firft and common Degree of the new Scale) ma]
be a Meafure to all the other diatonick Inter
vals : But chiefly, ido. 'Tis required that i
be a rational Quantity, expreffible in rational o
knoWi
i§ i. of MUSIC K. r% 9
■ known Numbers. Now fuppofe one Mean, it
is the fquare Root of 72 (viz.oi 8 multiplied by
p.) which, not being a fquare Number, has no
j fquare Root in rational Numbers ; and univer-
fally, let n reprefent any Number of Means, the
firft and leaft of them,is by ammwer/al Theorem
(as the Mathematici ans know) thus expreft
§VPJ £r t9 equal to this T^J -? t X 9 §?: But fup-
pofe 11 to be any Number you pleafc, fince 9 is
a figurate Number of no Kind but a Square,
\ therefore this Mean will in every Cafe be f'urd
^or irrational, and confequently the Tone 8 : 9
cannot be divided in the Manner propofed ; and
} ifo neither can the diatonick Scale.
Again, if the Di virion cannot be made in
j rational Numbers, we can never have a mufical
\ { Scale ; for fuppofe that by fome geometrical
I Method we put in a certain Number of Lines,
\mean Proportionals betwixt 8 and 9, yet none
1 of tlicfe could be Concord with any Term or
Note of the diatonick Scale ; becaufe the Coin-
cidence of Vibrations makes Concord, but Chords
that are not as Number to Number, can never
LJ coincide in their Vibrations, fince the Number
\t of Vibrations to every Coincidence are reciprc-
. cally as the Lengths, which not being as Num-
6 ber to Number, they could not make a mufical
t Scale. In the laft Place, Let us fuppofe the In-
u tewal 8 : 9 divided by any Number of . fuch
1 geometrical Means, and fuppofe (tho' abfurd)
that they make Concord with the rational Terms
of the Scale, yet it is certain. We could never
, find a common Meafure to the whole Scale;
2 V; . T for
tpo ^Treatise Chap- X.
for every Term of a geometrical Series multi- ;
plied by the common Ratio? produces the next
Term ; but the Ratio here is a furd Quantity,
viz. 8 n "x9~j £- : 8, and therefore, tho' it were
multiplied in infinitum with any rational Nun>
ber, could never produce any Thing but a Surd ;
and confequently never fall in with the Terms
of the natural Scale : Therefore, fuch a perfect
Series or Scale of fixt Sounds is impoffible.
Tho' the Defects of Internments cannot be
perfectly removed, yet they are in a good Mea-
sure cured, as we fliall prefently fee -, in order to
which let me premife, that the nearer the Scale
in fixt Sounds, comes to an Equality of the De-
grees or Differences of every Note to the next,
providing always that the natural Intervals be
preferved, the nearer it is to abfolute Perfection;
and the Defects that ftill remain after any Di-
Vifion, are lefs fenfible as that Divifion is grea-
ter, and the Degrees thereby made fmaller and
more in Number -> but by making too many we
render the Inftrument impracticable ; the Art is
to make no more than that the Defects may be
infenfible, or very nearly fo, and the Inftrument
at the fame Time fit for Service.
I know that fome Writers fpeak of the Di-
vifion of the Oc~iave into i<f>, 18, 20, 24, 26,31^
and other Numbers ok Degrees, which, with,
the Extremes, make 17, 19, 21, 25, 27, and
32 Notes within the Compafs of an Off ate $
but 'tis eafily imagined how hard and difficult
a Thing it muft be to perform upon fuchan In-
ftrument j fuppofe a Spinet, with 21 or 32
i
§ i. of MUSIC K. 291
Keys within the Compafs of ah Obla<ve; what
■an EmbarafTment and Confufion rnuft this occa-
fton efpecially to a Learner. Indeed if the
Matter could not be tolerably rectified another
Way, we fhould be obliged patiently to wreftle
with fo hard an Exercife ': But 'tis well that
wc are not put to fuch a difficult Choice, either
to give up our Hopes of fo agreeable Entertain-
ment as mufical Inftruments afford, or refolve to
acquire it at a very painful Rate ; no, we haVe
ft eaiier, and a Scale proceeding by 1 2 Degrees,
that is, 1 3 Notes including the Extremes, to an
clave, makes our Inftruments fo perfect that
we have no great Reafon to complain. This
{ therefore is the-prefent Syftem for Inftruments,
'&iz. betwixt the Extremes of every Tone of
""the natural Scale is put a Note, which divides
It into Two unequal Parts called Semitones % and
the whole may be called the femitonick Scale,
containing 12 Semitone s betwixt 13 Notes with-
in the Compafs of an Off ewe : And to preferve
the diatom ck Series diftincl:, thefe inferted Notes
take the Name of the natural Note next be-
low, with this Mark % called a Sharp, as
CI or Cfijarp, to iignify that it is a Semitone
above C ( natural ;) or they take the Name of
! the natural Note next above, with this Mark f/ y
called a Flat, as Z)b or I) flat, to fignifie a Se-
mitone below D (natural;) and tho' it he indiffe-
rent upon the main which Name is ufed in
any Cafe, yet, for good Reafons, fometimes the
one Way is ufed, and fometimes the other, as
I fhajl have Oc canon to explain : But that I
T a may
%$i ^Treatise Chap. X.
may proceed here upon a fixt Rule, I denomi-
. nate them from the Note below, excepting that
betwixt A and i?, which I always mark j, {im-
ply without any other Letter $ underftand the
fame of any other Character of thefe Letters ;
as always when I name any Letters for Exam-
ples, I fay the fame of all the other Characters
of thefe Letters, i, e. of all the Notes through
the whole Scale that bear thefe Names ; and
thus the whole ObJave is to be expreffed, 'viz.
a c%. n. m e. f m G- <?* a. ],. % c—
The Keys of a Spinet reprefent this verydiftincl:-
ly to us ; the f oremoft Range of continued Keys
is in the Order of the diatonic]?. Scale, and the
other Keysfct backward are the artificial Notes.
Why we don't rather ufe 1 2 different
Letters, will appear afterwards. The Two na-
,' tural Semitones of the diatonick Scale being be-
. twixt E F and A B fliew that the new Notes
fall betwixt the other natural ones as they are £et
down. Thefe new Notes are called accidental
or fiblitious, becaufe they retain the Name of
their Principals in the natural Syftem : And
this Name does alfo very well exprefs their De-
fign and Ufe,- which is not to introduce or ferve
any new Species of Melody diftincl: from the
diatonick Kind; but, as I have faid in the Be-
ginning of this Chapter, to ferve the Modula-
tion from one Key to another in the Courfe of
any Piece,' of the Tranfpofition of the Whole
to a different Pitch, for accommodating Inftru-
ments to a Voice, that beginning at a conve-
nient Note, yieijnilrument may accompany the
"Voice
§ z. of MUSICK. 293
Voice in Unifbn. How far the Luxury, if I may
jl fo call it, of the prcfent Mufick is carried, fo as
j to change the Species of Melody ', and bring in
, fomething of a different Character from the
i true Diatonick) and for that Purpofe have Ufe
\ for a Scale of Semitones, I (hall have Occafion
j to fpeak of afterwards : But let us now pro-
j ceed to fhcw how thefe Notes are proportioned
j to the natural Ones, u e. to fhew the Quantity
of the Semitones occafioned by thefe accidental
Notes, and then fee how far the Syfiem is per-
fected by them.
(j 2. Of the true Proportions of the Semitonick
Scale, and how far the Syfiem is perfettedhyit*
, HP HERE is great Variety, or I may rather call
r*~ it Confufion, in the Accounts that Writers
upon Mufick give of this Matter ,• they make
i different Divifions without* explaining the Rea-
fons of them. But fmce I have fo clearly ex-?
plained the Nature and Defign of this Improve-
ment, it will be eafy to examine any Divifion,
and prove its Fitnefs, by comparing it with the
End : And from the Things above laid, we
have this general Rule for judging of them,
viz. That, the Diviiion which makes a Series,
from, whofe every Note we can find any diato-*
nick Interval, upward or downward, with
leaft and feweft Errors, is moil: perfedt.
There are Two Divisions that I propofe to
explain here •> and after thefe I fliall explain the
T 3 ordi-
294 /f Treatise Chap.X.
ordinary and moft approven Way of bringing
Spinets and fuch kind of Inftrunients to Tune,
and (hew the true Proportion that fuch Tuning
makes among the feveral Notes.
The firfi Dwifion is this : Every Tone of
the diatonick Series is divided into Two Parts
or Semitones, whereof the one is the natural
Semitone i $ : 16, and the other is the Re-
mainder of that from the Tone, viz. 128 : 135
in the tg. and 24 : 25 in the f I. and the Semitone'
15 : 16 is put in the Joweft Place in each 5 except
the tg. betwixt/ and g, where 'tis put in the
upper Place ; and the whole OBave ftands as
in the following Scheme, where I have written
the Ratios of each Term to the next in a Fracti-
on fet betwixt them below.
S C A L E of SEMITONES.
c . c% . d , dk e ./ .f%. g . g% . a . I . b . cc°
15 128 15 14 15 128 15 15 24 15 128 15
16 135 16 25 16 J35 16 16 25 16 135 \6
It was very natural to think of dividing each
Tone of the diatonick Scale, fo as the Semi-
tone 15: 16 fliould be one Part of each Divifion^
becaufe this being an unavoidable and neceffary
Part of the natural Scale, would moft readily
occur as a fit Degree in the Divifion of the
Tones thereof ; eipecially after considering that
this Degree 15 : 16 is not very far from the
exacl: Half of a Tone, Again there muft be
fome Reafon for placing thefe Semitones in one
Order rather than another, i, e. placing 15:16
nppermoft in the Tone f ' : g, and undermoft in
al!
§ 2. of MUSIC K. 19 y
all the reft ; which Reafon is this, that here-
by there are fewer Errors or Defeats in the
Scale; particularly, the 15 : 16 is fet in the up-
per Place of the Tone f : g^ becaufe by this
the greatcft Error in the diatonick Scale is
perfectly corrected, viz* the falie qth betwixt
/ and b upward, which exceeds the true harmo*
uical A(th by the Semitone 128 : 135, and this
Semitone being placed betwixt/ and /&, makes
from f% to b a true qth ; and corrects alfo
an equal Defect in the Interval b-f taken up-
ward, which inftead of a true $th. wants 128 :
135, and is now juft, by taking f% for/, that
is^ from 1/ up tof% is a juft 5th. There were
the fame grofs Errors in the natural %ve pro-
ceeding from/, which are now corre&ed by the
.altered b viz, b, which is' a true qth above f y
whereas b (natural) is to the / below as 32 : 45
exceeding a true 4^ by 128 : 135 ; a:fo from b
(natural) up to/ is a falfe 5^Z?,as 45 : ^4,but from
^ to /is a juft $th 2:3; and therefore re-
fpe&ing thefe Corrections of fo very grofs Er-
rors, we fee a plain Reafon why the greater
Semitone 1$ : 16 is placed betwixt f% and g,
and betwixt a and \f ; For the Place of it in the
) other Tones ^ I {hall only fay, in general, that
there are fewer Errors as I have placed them
than if placed otherwife ; and I {hall add this
Particular, that we have now from the Key c
both the diatonick Series with the %d I. and %d
g. and their Accompanyments all in their juft
Proportions, only we have 9 : 16, viz. from c
to i for the leffer nth> which tho' it make not
T 4 fo
1^6 ,4 Treatise Chap. X.
fo many harmonious Relations to the other dia-
tonick Notes as % : 9 would do, yet considering
a *jth is ftill but a Difcord,and for what Reafon
^ was made a greater Semitone 15 : *6 above a.
This jth ought to be accounted the beft here ,-
yet the other 5 : 9 has Place in other Parts of
the Scale j I (hall prefently ftiew you other
Reafons why 9 : 16 is the beft in the Place
where I have put it, viz. betwixt c and ^.
Concerning this Scale of Semitones, Ob"
ferve imo, From any Letter to the fame again
comprehending Thirteen Notes is always a true
$ve, as from c to c, or from c% to c%. 2 Jo.
We have Three different Semitones 15 : 16 the
great eft) 128 : 135 the middle ', and 24 : 25 the
^4/?, which, when I haveOccafion tofpeakofy
I fhall mark thus, fg. fm, ft. The firft is
the Difference of a 3d g.and /\.th; the fecond the
Difference of t g. and/ 'g. and the Third the
Difference of 1 1, and fg. Cor of %dg. and 3d/.
or 6th g. and 6th I.) pio. We have by this
Divifion alfo Three different Tones, viz. 8 : 9
compofed of fg. and fm. as c : d ; then 9:10
compofed of fg. and//, as J • e ; and 22?:
256 compofed of Two fg. as/$ :g$, which
occurs alfo betwixt band f2?,and no where elfe,
all the reft being of the other Two Kinds which
are the true Tones of the natural Scale. And
tho' we might fuppofe other Combinations of
thefe Semitones to make new Tones, yet their
Order in this Scale affording no other, we are
concerned no further with them. Now obferve,
this laft Tone 225:256 being equal to zfg.
muft
§ u of MUSIC K. i 9 r
i muft be alfo the greateft of thefe Three Tones 5
j fo that what is the greateft of the Two natu-
I ral Tones ', is now the Middle of thefe Three,
i and therefore when you meet with t g. under-
j (land always the natural Tone 8 : 9, unlefs it be
i otherwife faid.
j 4ft). L e t us now confider how the Internals
of this Scale (hall be denominated ,• we have al-
ready heard the Reafon of thefe Names 3 d,
qth, 5th, &c. given to the Intervals of the Scale
of Mufick 1 they are taken from the Number of
Notes comprehended betwixt the Extremes
(inchifive) of any Interval, and exprefs in their
principal Defign, the Number of Notes from the
Fundamental of an Sve concinnoujly divided to
any acute Term of the Series, tho' to make
them of more univerfalUfe they are alfo applied
to the accidental Intervals. See Chap. .8. So
that whatever Interval contains the fame Num-
ber of Degrees is called by the fame Name.;
and hence we have fome Concords fome Dif-
cords of the lame Name $ fo in the diatonick
Scale, from c to e is a 3 d g. Concord, and from
f to g a 3^ /. and from dtof is alfo called a
3d, becaufe/ is the 3^ Note inclufive from d,
yet it is Lif'cord. See Chap. 8. If we confider
next, that the Notes added to the Scale are
not defigned to alter the Species of Melody, but
leave it ftill diatonick, only they correct the
Defccls arifing from fomething foreign to the
Nature and Uib of the Scale of Mufick, viz.
the limiting and fixing of the Sounds ; then we
fee the Reafon why the fame Names are ftill
con-
ip8 A Treatise Chap. X.
continued :. And tho' there are now more Notes
{uanOclave^and fo a greaterNumber of different
Intervals,yct the diatonick Names comprehend
the whole, by giving to every Interval of an c-
qual Number of Degrees the fame Name, and
making a Diflindion of each into greater and lef-'
fer. Thus an Interval of i Semitone is called
a letter Second or zdU of 2 Semitones is a id g,
of 3 Semitones a %dl. of 4, a $dg. and fo on
as in this Table.
Denominations, zdl. zclg. 7,il. $Ag, 4th I. 4th g, tyb. 6th I. 6th g. jthl. -jihg. 8ve.
Num. of item. 1-2- 3-4 - S - 6 - 7 ' $ - 9 ~ 10 - 11 - 12..
In which we have no other Names, than thefe
already known in the diatonick Scale, except
the qth greater, which for equal Reafon might
be called a $th letter, became 'tis a Middle be-
twixt /\th and $th, i. e. betwixt 5 and 7 Semitones ;
and therefore w.ie tnay call all Intervals of 6 Semi*
tones Tr it ones (for 6 Semitones make 3 Tones )
and thefe of 5 Semitones call them limply qths- }
and fo all the Names of the diatonick Scale re-
main unaltered, and we have only the Name
of Tritone added, which yet is not new, for I
have before obferved, that it is ufed in the dia-
tonick Scale, and thus all is kept very cliftin® ;
and if eve proceed above an Oufave, we com-
pound the Names with an Odfave and thefe be-
low. Again take Notice, that as in the pure
diatonick Scale, the Names of id, /\th, &c. an-
fwer to the Number of Letters which are be*
twixt the Extremes ( inclufive ) of any Inter*
vol, whereby the Denomination of the Inter-
val is known, by knowing the Letters by which
the
5 z. of MUSIC K. z 99
:he Extremes of it are expreft, fo in* this new
Scale the fame will hold, by taking any Letter
k vith or without the Sharp or Flat for the lame
Letter, and applying to the accidental Notes., ,in
hme Cafes the Letter of the Note below with
i Sharpy and in others that of the Note above
,;vith a Flat : For Example. d%~g is a 3^, and
ncludes 4 Letters ; but if for d'$ we take e^ 9
■lien ek—g 9 which is the fame individual Inter-
lai/, contains but 3 Letters,- alfo if for k we
:ake a% then a%—c% y which is a true 3d h
ncludes 3 Letters, whereas \f— 0% has but Two.
There is only one Exception, for the Interval
pf 9 which is a qthg. contains 5 Letters, and
cannot be otherwife expreft, unlefs you take e%
vhich is equal to/ natural '; or take r^, which
s equal to b natural ; but this is not fo regular,
md indeed makes too great a Confafion; tho' I
lave feen it fo done in the Compofitions of the
)cft Matters, which yet will not make it reafon-
ible, unlefs in the particular Cafe where 'tis
ifcd, it could not have been fo conveniently or*
lercd otherwife : But if we call the fame In-
erval a $th Jeffer, then the Rule is good ; yet
!f we call every Tritone a 5 th, we fliall ftillhave
in Exception, for then f—b contains only 4 Let-
ers j and therefore 'tis beft to call all Intervals
>f 6 Semitones, Tritones, and then they are not
\xb]etl to this Rule. In this therefore we fee a
leafon, why 'tis better that the accidental Note
J lould be named by the Letter of the natural
Vote, than to make Twelve Letters in an 0-
jtewj belides, the Melody being ftill diatonick 9
thefe
300 ^Treatise Chap. X.
thefe accidental Notes are only in place of the
others ,- and by keeping the fame Names, we
preferve the Simplicity of the Syftem better.
5to. Having thus fettled the Denominations
of the Intervals of this femitonick Scale, we
mull next ohferve, that of each Denomination
there are Differences in the Quantity, arifing
from the Differences of the Semitones of which
they are compofed, as is very obvious in the
Scale : And thefe again may be diftinguifhed
into true andfalfe, i. e. fuch as are either hdr-
monical or concinnous Intervals of the natural
Scale, and fuch as are not ; and in each Deno-
mination we find there is one that is true, and
all the reft arefalfe, except the Tn tones which
are all falfe, tho' they are ufed in fome very
particular Cafes.
6to, Let us next enquire into all the Variety
and the precife Quantity of every Interval with-
in this new Scale, that we may thereby know
what Defects ftill remain. We have al-
ready obferved, that there are Three different
Semitones and as many Tones ; hence it is' j
plain, there are neither more nor lefs than
Three different jths of each Species, i, e. leffer
and greater, which are the Complements of
thefe Semitones and Tones to ObJave, as here.
Semit. -jth g. jth /. Tone.
15 - 16 - 30^ Ti28 - 225 - 256
^28 - 135 - 256 >■< 9 - 16 - 18
24 - 25 - 48J I 5 -. 9 - i°
And
§ *. of MUSI CK. | 301
\ And to know where each of thefe jths lies,
and all the Examples of each in the Scale, 'tis
I but taking all the Examples of thefe Semitones
\ and Tones, which are to be found at Sight in
the Scale marked with the Semitones, as you
fee in Page 294. and you have the correspondent
I 7ths betwixt the one Extreme of that Semitone
or Tone, and the O Stave to the other Extreme.
Then for the other Intervals, viz. %ds, 6ths,
qth s, 5 thsy which are harmonical, I have in the
Table-plate, Fig. fet all the Examples offuch
of them as arefalfe, with their reipective Ra-
tios ; and with the Ratios of the 6th and $th
\l have fet an e or d, to fignify an exceffive,or a
..deficient Interval from the true Concord ; and
Iconfequently their correfpondent $ds and qths
/.will be as much on the contrary deficient or
i.exceflive. All the reft of the Intervals of thefe
feveral Denominations, containing 3,4,5,7,
8 or 9 Semitones, are true of their feveral Kinds,
i whofe Ratios we have frequently feen, and fo
Ithey needed not be placed here. Then for the
j Tritones, you have in the laft Part of the Table
^all their Variety and Examples ; by the Nature
1 of this Interval it exceeds, a true qth, and
( wants of a true 5th; you'll eafily find the Diffe-
rence by the Ratio.
Now we have feen all the Variety of Inter*
vals in this new^Scak; and by what's explain'd
*we know where all the Extremes of each ly;
• and it will be eafy to find the true Ratio of a-
ny Interval, the Letters or Names of whofe Ex-
tremes in the Scale are given, viz. by finding in
the
30i ^Treatise Cha?. X.
the Scale how many Semitones it contains, and
thereby the Denomination of it, by which you'll
find its Ratio in the preceeding Table ', unlefs it
be a true Concord, and then it is not in the Table,
which is a Sign of its being true. And as to
this Table, obferve, that I have no Reipect. to
the different Characters of Letters, and you
mult, fuppofe every Example to be taken up-
ward in the Scale, from the firft Letter of the
Example to the fecond, counting in the natu-
ral Order of the Letters.
7 mo. We are now come to confidcr how far
the Scale is perfected ; and firft obfcr-ve, that
there are no greater or leffer, and precisely ho
other Errors in it, than the Differences of the
Three Semitones, which are thefe following ;
of which
■ f/5>" and/"///. - 2025" : 2048] the up
^ </';;/. and/"/. - 80 : 81 !> perm oft is
£3 [fgi and//. : 125 : 128 J the leafr,
and the
lower the greateft Error. In the diatonick Scale
f:me Intervals erred a whole Semitone, and all
the reft only by a Comma 80 : 81 ; here we
have one Error a very little greater, and another
leffer: All the $ths and qths except Three,
are j lift and true ; of the $dL and 6th g. there
arc as many true asfal/e; and of the %dg. and
6th L we have Five fall { e and Seven true. Thefe
Errors are fo fmall, that in a fingle Cafe the
Ear will bear it, efpecially in the imperfect Con-
Cords of %d and 6th ; but when many of thefe
Errors happen in a Song, and efpecially in the
prin-
§ 2. of MUSIC K. 305
principal Intervals that belong to the Key, it
will interrupt the Melody, and the Inftr anient
will appear out of Tune ( as it really is with
refpeft to that Song : ) But then we muft oh-
ferve, that as the Order of thefe Semitones is
different in every Octave, proceeding from each
of the Twelve different Keys or Letters of the
Seals $ io we find that fome Songs will proceed
better, if begun at fome Notes, than at others.
If we compare one Key with another, then we
muft prefer them according to the Perfection of
their principal Intervals, viz. the id, $th and
•6th, which are Effentials in the Harmony of
Wery Key : And let any Two Notes be propo-
sed to be made Keys of the fame Species, viz,
*both with the $dl b &c. or 3^, &c. We can
feafily find in the proceeding Table what Inter-
nals in the ^ cale arc true or falfe to each of
Ithem; and accordingly prefer the one or the
other: But I fhall proceed to
T h e fecond Divijton of the %ve into Semi-
tones which I promifed to explain, and it is
this: Betwixt the Extremes of the t g. and/-/,
the natural Scale is taken an harmonica} A [5 an
vhich divides it into Two Semitones nearly
equal, thus, the t g. 8 : 9 is divided into Two
"Semitones which are \6 1 17 and 17 : 18, as
fiere 16 : 17 : 18, which is an arithmetical Di-
Virion, the Numbers reprefenting the Lengths
yi Chords ; but if they reprefent the Vibrations,
".he Lengths of the Chords are reciprocal, 0/2. as
\ : ,s : I which puts the greater Semitone ] 6 7 next
fee lower Part of the Tone p and. the Jeflcri&next
the
304 ^Treatise Chap. X.
the upper, which is the Property of the harmO'
«0ftf/ Divifion : The fame Way thej /, 9 : io
is divided into thefe Two Semit. 18 : 19, and
ii$ : 2 o, and the whole $ve ftands thus.
c.cm. d. d% . e .f.fn. g.gM.a . I .b .&
16 17 18 19 15 16 17 18 19 16 17 15
17 18 19 20 16 17 18 19 20 17 18 16
In this Scale we have thefe Things to ob-
ferve,i7/zo.That every Tone is divided into Two,
Semit, whereof I have fet the greater in the
loweft Place. 2 do. We have hereby Five diffe-
rent Semitones ; out of which as they ftand in
the Scale we have Seven differentjTo/z^as here.
Sem. Tones. Considering how, by
j^ ' g' a harmonic al Mean y the $th y
w >}< i s ^- ' 5^ D and 3 dg. were divided in-
x 7 x ^ to their harmonic al or <wz-
IZ >i( !§ ~ IZ cinnons Parts, it could not
18 19 "19 but readily occur to divide
18 19 9 the To7Z£j- the fame Way, when
5"'^ P ^ tS a ^i y ^ on was found necef-
fary j but we are to confider
2 jl, 11 t- 2 what Effetf: this Divifion has
20 16 64 for perfecting of Inftruments.
15 16 15 It would be more troublefom
16 > ij *t 17 tnan difficult to calculate a
x T z • T^/e of all the Variety of
— >J< ~ t- ^- Ratios contain'd in this $7^/^
20 17 85 j fl la j] j eave y OU t0 thisExer-
I 7 V I 5 ^ $5 cite for vour Diverfion, and
18 T 16 96 only tell you here, that ha-<
vin§
% ».'■ of MUSI CK. 30 j
Ving calculate all the $ths and qths, I find
I there are only Seven true $ths<> and as many
1 qths, whereas in the former Scale there were*
I Nine i and then for the Errors, there are none*
of them above a Comma 80 : 81 ,- in fhorr,- there
! .is one falfe 5th and §th. whofe Error is a Com-
1 ma, and the reft are all very much lefs ; and,tho*
there are fewer true $ths and fyhs here, yet the'
Errors being far lefs and more various, compen-
fate the other Lofs : As to the ^ds and 6th gg
there are alfo here more of them falfe than \ in
the proceeding Scale, for of each there are but
I Pour true Intervals, but the Errors are gene-
rally much lefs, the greateft being far lefs than
the greateft in the other Scale*
I fliall fay no more upon this, only let yoti
know, That Mr. Salmon in the Philofophical
\Tranf actions tells us, That he made an Expe-
riment of. this Scale upon Chords exactly in thefe
Proportions, which yielded a perfect Confort
with other Inftruments touched by the beft
Hands : But obferve, that he places the leffer
Seinit* loweft, which I place uppermoft j and
when I had examined what Difference* this
Would produce, I found the Advantage would
rather be in the Way I have chofen* And this
brings to mind a Queftion which Mr* Simgfon
.makes in his Compend of ' Mufich, viz* Whether
the greater or lefler Semitone lies from a to fa
tie fays 'tis more rational to his Underftanding>
:hat the leffer Semitone ly next a -, but he does
lot explain his Reafoit ; he fpeaks dnly of the
mthmetical Divifion of a Chord intd equal
U Parts*
306 ^Treatise Chap. X.
Parts, but has not minded the harmonic al Di-
vifion of an Interval^ by which we have feen
the diatonick Scale fo naturally constituted,
whereby the greater Partis always laid next
the graveft Extreme : But in ftiort, when we
fpeak of the Reafon of this, we muft coniider
the Defign of thefe Semitones^ and which one
in fuch a Place anfwers the End beft, and then
I believe there will be no Reafon found why it
fhould be as Mr. Simpfon fays, rather than the
other Way.
§ 3* Of the common Method of Tuning Spinets,
demonjlrating the Proportions that occur in
it \ and of the Pretence of a nicer Method
confidered.
*Tp RE laft Thing I propofed to do upon this
-*• Subject, was to explain the ordinary Way of
tuning Spinets and that Kind of Inftruments -, for
whether it be 5 that the tuning them in accurate
Proportions in the Manner mentioned is not ea-
fily done, or that thefe Proportions do not fuf-
ficiently,c6rre& the Defects of the Inurnment,
there is another Way which is generally follow-
ed by practical Mujicians °> and that is Tuning
by the Ear, which is founded upon this Suppo-
fition, that the Ear is perfectly Judge of an $<ve
and 5th, The general Rule is, to begin at a
certain Note as r ? taken toward the Middle of
the
§ 3- of MU SICK. Z o7
the Inftrtiment, and tuning all the %<ves up and
'down, and alfo the sths 9 reckoning Seven Semi*
Jones to every $th> whereby the whole will be
Alined ; but there are Differences even in the
Way of doing this, which I ftiall explain.
Some and even the Generality who deal
vith this Kind of Inftrument, tune not only
f:heir OBaves, but alfo their $ths as perfectly
Concord as theirEar can judge 3 and confequently
make the qths perfec% which indeed makes
i great many Errors in the other Intervals of
\\d and 6th (for the difcord Intervals^ they are
| lot fo confiderable; ) others that affecl; a greats
1 Nicety pretend to diminifh all. the f ths, and
aiake them deficient about a Quarter of a Com*
W-> in order to make the Errors in the reft
imaller and lefs fenfible : But to be a little
ore particular,' I fhall fhew you the Progrefs
hat's made from Note to Note; and then con-
ider the Effecl: of both thefe Methods* In or-
ler to this, let us view again the Scale wkh its
2 Semitones in an Oblave ; but we have Ufe for
Fwo 05fayes to this Purpofe. Then imo. Be-
ginning at c take it at a certain Pitch, and
;une all its Offiaves above and below • then
'ido. Tune g a $th above c, and next tune all
ie Odfaves of g ; itio. Take d a stb above
>, and then tune all the Offiaves of d. a^o*-
IFake a a 5th above d, then tune all the 0-
Waves of a, 5 to. Take e sl $th above a> and
1 une aft the O&aves of e : Then, 6to. Take
h (natural) a 5th above e b , and tune all the 0-
%0ms ePlh 7mt>. Take ftit a $tfy above by
U z theii
*3o8 ^Treatise Chap/ X.
then tune -all the OSiaves of /&. $<vo* r$ a
.5^/7 above /*, and then all the Staves of fl&
■9.«o. Take g% a 5^ above c$, then all its
Offiaves} and having proceeded fo far 3 we have
all the f\.eys tuned except /> <&?,- and (, ; for
•which, 10700. Begin again at <v an d take /a
$th -downward 5 then tune all the /s. 11 mo*
-Take [^a 5^ downward to/, and tune all the
£s. I*aflly. Take d ^a 5^? below],, and then
tune all the Staves of ^ $ and fo the whole
Inftrtiment is in Tune. And obfer®e, That hav-
ing tuned all the Qftaves of any Key, the next
Step being to take a $th to it, you may take
that from any of the Keys of that -Name.
. Now fuppofing all thefe Offiaves and sths
to be in perfect Tunc, we (hall examine the
Effects it will have upon the reft of the Inter-
vals \ and in order, to it,; I have expreft this
Tuning in Plate 1* Fig. 6* by drawing Lines
betwixt every Note, and another, according to
the Method of Procedure • but I have only
•marked the $ths> fuppofing the Offiaves to be
.tuned all along as you proceed,- then I have
marked the Frogrefs from $th to $th by Num-
bers fet upon them to lignify the i/?, 2^, &c,
Step.; and in the Method there taken you fee
all the Notes tuned from c to fM above its
051 ape V W^ fuppofe all the other Notes above
and beloW: in the Inflrument to have been tim-
ed by OSiaves to thefe, but for the Thing in
Hand we have Ufe for -no more of the Scale*
Obferve next, That I have marked the Semi*
tones betwixt every ; Note by the Letters g> /.
"r ■ viz*
§ 3- of MUSICK. 309
1 viz, greater arid lcffer; for there are only Two
\ Kinds in this Scale, as we fhall prefently fee^
»j and alfo what they are, for the natural Senn
\ 15 : 16 is not to be found here ; and while I
i| fpeak of this Scale and of Semitones greater and
\ letter, I mean always thefe Two, urrkfs it be
j laid otherwife.
i I f we find the Degrees of this Scale in the
j Tones or Se?nitones, we fhall by thefe eafily
) find the Quantity of every other Interval ; and
in the following Calculations I take all the Ex-
amples upward from the firft Letter named, and
therefore I have made no Diftinction in the
Character of the Letters : To begin, from c to
g is a 5th 2 : 3, and from g t® d a $th^. there-
fore from c td d is Two $ths 4.: 9 ; out of this
take an OcJave, the Remainder is 8 : 9 a tg.
and confequently c-d is a tg. 8 : p j by this
Method you'll prove that each of thefe Inter-
) vals marked in the following- Table is a tg.
8 : 9. In the next Place, confider, from a to: e
is a $th therefore from e to a is a qth : But
from f to a there are Two tg
^ c - d as in the preceeding Table,
qq d - e whofe Sum is 64 : 8i, which.
§*$?-/ taken from a qth 3 : 4, leaves
& ' e ■■'■■'- j% this Semitone 243 : 256 for tf :'/
^jf - g (which is lefs than 15 .-: 16 by.a
o f% - g% Comma) then if we fubftracl: this
§ g - a from a To/^ 8 : ?,it leaves 2048:
©o^ - Z> 2.187, a' greater Semitone than
v ^ - £ the former, and if we mark the
one /. and the other %. all the
U 3 &mz-
310 A Treatise Chap. X.
Semitones from d to a, will be as I have marked
them in the Fig. referred to; for fince e :/$is a
tg. and e ,/is a/7, therefore/. /$ is a/'g. and
fo of the reft, every Two Semitones from d to
a being a £g. Again fince / - c is a 5/i?, and
alfo £ f £, taking away what's common to both,
viz. f - b, there remains on each Hand thefe
equal Parts e ./ and b . c, fo that Z> . c is alfo
a//, and fince f, : c is sl tg. and Z> . c a /7. [, ,
b muft be a/£. and alfo a , I sl fl. becaufe
a : b is a £g. J5&i^ from cM to g3£ is a $tb,
alfo from <$£ to l, and taking away d%-g%
out of both, there remains c% : M equal to
g%-J„ which contains Twofl, out d: M is
already found to be a/7, therefore f# , J is//,
and c : d being sl tg. c . c^ muft be a /g.
Thus we have difcovered all the Semitones
within the Oftave-, of which as they ftand in
the Scale, we have only Two different Tones,
viz. the t g. 8:9 and another which is kffer
5*9049 : 65536 compofed of Two of the leffer
Semitones, as you fee betwixt c% : M, and
alfo betwixt g% : {/ j in every other Place of
the Scale it is a tg.
Lit us next confider the other Intervals, and
firft, We have all the Offiaves and stbs perfect
except the 5th «•- M which is 531441 :
786432, wanting of a true 5th more than a Com-
ma, viz. the Difference of the fg. and//, as is
evident in the Scheme, for g - d is a true 5/7.J
but the Interval g% - dis common to g - d,
and g% - d%, and being taken from both,
Jeavei
§ 3 . of MUSIC K. 3 xx
leaves in the firft the /g. g . g%, and in laft the
fl. d .d% ; then all the /\ths are of confe-
quence perfect, except d% - g%^ which ex-
; ceeds as much as its correfpondent %fh is defi-
I cient. But Lafily^ For the ids and 6ths they
i are all falfe, plainly for this Reafon, that in the
whole Series there is no leffer Tone 9 : 1 o,
which with the tg. 8 : 9 makes a true id g.
nor any of the greater Semitone 15 : 16, which
with tg makes a 3 dl. And for the Errors they
are eaiily difcovered, in the id g. (and the Cor-
refpondent 6 1.) the Error is either an Excefs
of a Comma 80 : 81 the Difference of tg. and
t I. of the natural Scale ; which happens in
thefe Places where Two tg. ftand together, as
in the id g. from c to e ,• or it is a Deficiency
equal to the Difference of the lelTer Tone 9 :
io, and the 7 one above mentioned 59049 :
655363 which Tone is lefs than 9:10 by this
Difference 32768, 32805 (as in the id g. r* :
f) which is greater than a Comma ; and for
the 3d I. (and its 6th g.) it has the fame Er-
rors, and is either deficient a Comma, viz. the
Difference of the fg. 15 : 16. and the/*/. 243 :
256, as in the 3d I. c \ <$?, or exceeds by the
Difference of the new fg. 2048 : 2187 and the
fg. 15 : 16 which is lefs than the other by this
Difference 32768 : 32805 which is greater than
a Comma.
. Now the $ths and qths are all perfect but
one, yet the ids and 6ths being all falfe,
there is no Note in all the Scale from which
We have a true diatonick Series $ and the Er-
V 4 rors
$i% j4 Treatise Chap. X,
Jprs being equal to a Comma in fome and
greater in others, makes this Scale lefs perfect
than any yet described ; at leaft than the firft
Divifion explained, in which there were only
3 falfe 5ths, whereof Two err by a Com-
ma, and the other by a leffer Difference \ and
having many true -$ds and 6th s, feems plainly a
more perfect Scale. Thefe Errors may ftill be
made lefs by multiplying the artificial K.ej>s,
and placing them betwixt fuch Notes of the
preceeding Scale as may correct the greateft
Errqrs pf the moft ufual Keys of the diatonick
Series^ aud of fuch Divifjons you have Accounts
in Merfennus and Kircher •> but a greater Number
than 1 9 Keys in an Ofiave is fo great a Difficulty for
Prac~tice,tbat they are very rare,and our beft Com-
pofitions are performed on Inftruments with 13
Notes in the OcJave, and as to the tuning of thefe^
jL e T us now confider the Pretences or
the nicer Kind of Muficians; they tell us.
That in tuning by OtJaves and $ths, they
diminifh all the gtks by a Quarter of a
Comma, or near it ( for the Ratio 80 : 81 can-
not be divided into 4 equal Parts, and expreft in
rational Numbers) in order to make the Er-
rors through the whole Initrument very fmall
and infenfibJe. I fhall not here trouble you.
with Calculations made upon this Suppoiition,
becaufe they can be eafily done by thofe who
understand what has been hitherto explained
upon this Subject ; therefore I fay no more but
this, That it muft be an extraordinary Ear that
can judge exactly of a Quarter Comma, and I
{hall
§ y. of MUSI CK, 313
[hall add, That fome Praclifers upon Harpfi-
chords have told me they always tunc their
$ths perfect, and find their Inftrument anfwer
very well. 'Tis true they cannot deny that the
fame Song will not go equally well from every
Kj\ which argues full the Imperfection of the
Inftrument ; but there is no Song but they can
find fome Key that will anfwer. If a very juft
land accurate Ear can diminifh the Errors, fo as
to make them yet fmaller and more equal thro*
the whole Inftrument, I will not fay but they
may make more of the OSfaves like other, and
confequently make it an indifferent Thing
which of thefe Keys, that are brought to fucn
1 Likenefs, you begin your Song at ; but even
thefe cannot deny that a Song will do better
from one Key than another ; fo that the De-
fects are not quite removedeven as to Senfe.
D r. Wcillls has a Difcourfe in the Philofo-
fhical l^ranf actions concerning the Imperfecti-
on of Organs, and the Remedy applied to it ;
the Imperfection he obferves is the fame I have
already fpoken olyjiz.Th.zt from every Note you
ffcnnbt find any Internal in its juft Proportion.
'Tis true indeed the Doctor only confiders the
(Imperfection of a Scale of 'Semitone 's, and parti-
cularly one conftituted in the Ratio of the id
• Kind of Divifion abovementioned ; he does not
I fay dire&Iy for what Reafcns a Scale of Semi-
1 tones was neceffary ; but, as if he fuppofed that
plain enough,he fays there are (till fome Defects $
and therefore, fays he, Inftead of thefe Propor-
tions ( of the- Semitones) it is Co ordered, ifl
miftake
314 A Treatise Chap. X.
mi flake not the Practice, that the 13 Pipes
within an Octave, as to their Sounds, with
refpect to acute and grave, JJjall be in continual
Proportion, whereby it comes to pafs that each
Pipe doth not exprefs its proper Sound, but
fomething 'varying from it, which is called
Bearing - y and this, fays he, is an Imperfection
in this noble Inflrument. Again, lie fays, That
the Semitones being all made equal, they do
indifferently anfwer all Pofitions of mi (i.e. of the
Two natural Semitones in an OStave ; of the
Ufe of this Word;;//, we fliallhear again) andtho'
not exactly to any, yet nearer to fome than to
others ; whence it is that the fame Song ftands
better in one Key than another. I have fliewn
above,that a Scale of Degrees accurately equal, |
which will coincide with the Terms of the na-
tural Scale is not poffible ; and now let me fay,
That tho' the ObJave may be divided into 1 2
equal Semitones by geometrical Methods, that
is, 1 3 Lines may be conftrutted, which (hall be
in continued geometrical Proportion, and the
greateft to the leaft be as 2 to i, yet none of
thefe Terms can be expreft by rational Num-
bers, and fo 'tis impolfible that fuch a Scale
could exprefs any true Mufick, and hence I
conclude, that this Bearing does not make the
Semitones exa&ly equal, tho' they may be fen-
(iby fo in a fingle Comparifon of one with an-
other; and fuppofmg them equal, the Doctor
fays the fame Song will ftand better at one
Key than another ,• which may be very true,
becaufe none of the Terms of fuch a Scale can
poffiblj;
§ 3 . of MUSIC K. 315
pciTibly fall in with thefe of the natural Scale,
which are all expreft by rational Numbers, and
the other are all Sums ; whereas had we a
Scale of equal Degrees, coinciding with the
Statural Scale^ every Key would neceffarily be
alike for] every Song. Thefe Imperfections,
(fays the Do£tor, might be further remedied by
(multiplying the Notes within an OtJave, yet not
{without fomething of bearing,unlefs to every Key
3 (he means of the Seven natural ones) befitted
\a diftindl Scale or Set of Pipes rifing in thetrue
'Proportions, which would render the Inftrument
j impracticable : But even this I think would
*nct do j for let us fuppofe that from any one
it Key as c, we have a Series of true diatonick
Notes, in both the Species of Jharp and flat Key,
let a Song be begun there as the principal Ke)\
jiand fuppofe it to change into any or all of the
I confonant Keys within that Otlace, then 'tis
5 plain that if a Series is fitted to all thefe natu-
\ ral Notes of the Key c, the Inftrument is fo per-
fected for r, that any Piece of true diatonick
\Mnfick may begin there; but fuppofe, for the
■[Accommodation of one Inftrument to another,
\ we would begin the Piece in g, 'tis plain this
I cannot be done with the fame Accuracy as froni
\ c perfected as we have fuppofed, nnlefs to thefe
•Notes that proceed concinnoufly from g, and
are now confidered as the natural Notes of that
J Key, be alfo fitted other Scales for anfweriftg
- the Modulations of the Song from the principal
i Key (which is now g) to the other confonant
'■ Keys, And if we fliould but perfect Two Keyf
of
$i6 ^Treatise Chap. X.
of the whole Inftrument in this Manner, what
a Multitude of Notes muft there be ? But I have
done with this.
$ <\- A h'ief 'Recapitulation of the preceeding
Sections.
HT HE Amount of all that has been faid upon this
-*- Subject of the Syftem of M^c&,withrefpe£fc
to Inftruments having fixt Sounds, is in ihort this.
17116, Becaufe the Degrees of the true natural
diatonick Scale are unequal -, fo that from every
Note to its Ofiave contains a different Order of
Degrees i therefore from any Note we cannot
find any Intervhl^ in^ a Series of fixt Sounds
conftituted in thefe Ratios ; which yet is ne-
ceffary, that all the Notes of a Piece of Mufick
which is carried thro' feveral Keys, may be
found in their juft Tune ; or that the fame
Song may be begun indifferently at any Note,
as will be neceuary or at leaft very convenient
for accommodating fome Inftruments to others,
cr thefe' to the human Voice, when it is requir-
ed that they accompany each other in Unifon,
ido. 'Tis impoflible that fuch a Scale can be
found i yet Inftruments are brought to a toler-
able Perfection, by dividing every Tone into
Two Semitones^ making of the whole Offiave
12 Semitones, which in a fingle Cafe are fenfibly e-
qual
§4- of MUSIC K. 317
. qual. 3^'o.Thefe Semitones may be made in exacl:
il Proportions,according to the Methods above ex-
I plained^or the Inftrument tun'd by the Ear,as is al-
| fo explained,which reduces all to the particular
Kinds of Degrees and Order alfo (hewn above.
qto. The diatonick Series, beginning at the low-
eft Note, being firft fettled upon any Inftrument,
and diftinguifned by their Names a . b . c . d .
e . f .■ g. the other Notes are called -fifctitioys
Kates, taking the Name or Letter of the Note
below with a $ as f$, fignifying that 'tis a
Semitone higher than the Sound of c in the' na-
tural Series, or this Mark j/ with the Name of
the Note above fignifying a Semitone lower, as
d r ; which are neceflary Notes in a Scale of
fixt Sounds, for the Purpofes mentioned in the
laft Article ; what Reafons make them to be
named fometimes the one, fometimes the other
Way ftial] be {hewn afterwards ,- and obferve y
that fince there is no Note betwixt e and /,
which is the natural Semitone ', therefore /can-
not, be marked (/, for with that Mark it would
be e ; nor can e be marked $ r which would
raiie ; it to f; but e is capable of a (/, as f is of a
$. So b . c being the other natural Semitone^
b is incapable of a % which would make it
coincide with c, but it properly takes a ^, and
when this Mark is fet alone it expreffesjto b ;
again c receives not a ^, for e b is equal to b
natural^ but it takes a $. All the reft of the
Notes d.g. a are made either fr or;& becaufe
they have a Tone on either Hand above and be-
low. Hence it is, that b and e are faid to be
naturally
Ji8 ^Treatise Chap. X,
Haturaly Jharfr as c and /"naturally fiat ; and
yet infome Cafes I havefeen c and /marked m
tmd b and e marked ;#, which makes thefe
Letters fo marked coincide with the natural
]Notes next below and above, pio, Becaufo the
Semitones are very near equal, therefore in Pra*
tike (upon fuch Inflruments at leaft) they are
all accounted equal, fo that no Diftinc~tion is
made of Tones into greater and leffer ; and (at
the other Intervals they are alfo considered here
without any Differences,every Number of Semi-
tones having a diftinct Name, according to the
Bule already laid down -, and therefore when a
true id or 4^, &c. is required from any Note*
we muft take fo many Semitones as make an
Interval oi that Denomination in general^ which
will in fome Cafes be true, and in others a falfe
Interval, and cannot be otherwife in fuch In-
ftruments. 4^0. The Differences among the Se*
mitones, in the beft tuned InftrumentSj is the
Reafon that a Song wilt go better from one
Note or Key of the Inftrument than another j
becaufe the Errors occur more frequently in
fome Combinations and Succeflions of Notes
than in others ■> and happen alfb in the more
principal Parts of one Key than another.
And becaufe the Defign of thefe new Notes
is not to alter the Species of the true diatoniek
Melody, but to correft the Defects arifing not
from the Nature of theSyftem of Mujiek it Mi,
but the Accident of limiting it to hxt Sounds;
therefore beginning at any Note, i£ we take
ftfi %ve ConcUmoujlj divided- by Tbnes emd Semi-
toms
I 4 . of music a: 319
tones in the diatonick Order ( which will be
found more exacl: from fome Notes than others
oecaufe of the fmall Errors that ftill remain)
jthat may be juftly called a natural Series*, and
all thefe Notes natural Notes with refpecl: to
'the Firft or Fundamental from which they pro-
ceed ; and yet in the common Way of fpeak-
jing about thefe Things, no 8ve is called a na-
tural Key that takes in any of thefe Notesmar-
ked % or (/, in order to make it a concinnous
Series. And, as I have obferved in another
Place, there is no Key called natural in the
whole Scale but C and A I have alfo explain-
ed that there are properly but Two Kinds of
Keys or Modes ', the greater with the 3d g, &e.
as in the %®e C, and the leffer with the 3d /, &c.
ias in A ; but whenever in any Syftem of fixt
founds we can find a Series that is a trueKey(or
ifo near that we take it for one)there is no other
[Reafon of calling that an artificial Key^ than
the arbitrary Will of thofe who explain thefe
Things to us, unlefs they make the Word arti-
ficial include the Imperfections of thefe Keys y
which I believe they don't mean, becaufe they
fuppofe the Errors are inconfiderable ; for with
refpecl: to the Tune or Voice, 'tis equally a na-
tural Key, begin at what Pitch you will ; and
we can fuppofe one Inftrument fo tuned as to
play along Unifbn with the Voice, and be ir a
natural Key-> and in another fo tuned as that,
to go unijon with the fame Voice, it muft take
an artificial Key : But I fliall have Occafion to
consider this again in the next Qhapter^ wbre
320 ^Treatise Chap. %
I fhall alfo fhew you what Letters or Notes
muft be taken in to make a true diatonick Scale
of either Species proceeding from any one of the
Twelve different Letters in this new Scale*
The diatonick Series upon all Inftruments^
being kept diftincl by the Seven diftinel: Let-
ters, is always firft learned ; and becaufe in eve-
ry ive of the diatonick Scale, there are Two
Semitones diftaht one from another by 2 Tones
or 3, therefore if the firft %ve of the diatonick
Series upon any Inftrument is learned, by the
Place of the Two Semitones, we fhall eafily
know how we ought to name the firft and low-
eft Note ; for if the id and jth Degrees are
Semitones, then the firft Note is c, if the id and
6th then it is d, and fo of the reft, which are
eaiily found by Infpeclion into a Scale Carried to
Two $ves. And different Inftruments begin at
[ i. e* their loweft Note is named by ] different
Letters ; in fome Cafes becaufe the natural Se-
ries, which is always moft confiderable, is more
eafily found if we begin with one particular
Order of the Degrees; and in other Cafes the
Reafc-n may be the making oneJnftrument con-
cord to another. So Flutes begin in /, Haut-
boys, Violins, and fome Harpjichords begin in
gs tho' the laft may be made to begin in any
Letter. As to the Violin, let me here obferve,
that it is a Kind of mixt Inftrument,. having
itSjSouncts partly fixed and partly unfixed : It
W only Four fixt Sounds, which are the Sounds
of the Four Strings untouched by the Finger*
and are called g-d~a-e> and can with very 'final!
Trouble
§ 4- of MUSIC K. jit
Trouble be altered to a higher or lower Pitchy
which is one Conveniehcy ; all the reft of the
Notes being made by fliortning the String with
one's Finger, are thereby unfixed Sounds, and a
good Ear learns to take them in perfect Tune
with refpeft to the proceeding Mote ; fo that
from any Note up or down may be found an^
Interval propofed $ and therefore we may begin
a Song at any Note, with this Provifion that it
be moft eafy and convenient for the Hand j yet
a Habit of Practice in every Key may make
this Condition unneceffary ; There is only 1 this
one Variation to be obferved,that by making the
Four open Strings true gthsi all continuous^ d-a
is here a true 5^, which in the diaiomck Series
wants a Comnia ; from this follow other Varia-
tions from the Order of the diatonick Scale 5
as here^ fromg (the firftNote of the qtb String)
to a is made a greater Tbne y that it may be a
true %ve below a the firft Note of the 2d String,
Which is occasioned by making d-a a true $th r
whereas in the Scale g-a is a lefler Tone : And
fo from a to b will be made a leffer Tone± tho'
'tis igi in the Scak y that g-b may be made a
true idg. Which are Advantages When We begin
in g. The fame happens in the 3d Strings whofe
firft Note is d, from Which to the next Note e
will be made a tg* that it may be art %ve to
the firft Note of the firft String, yet di e in the
Scale is at I. Again* if having made d-f on
the 3d String a true 3d I. We Would rife to a
true $th above d$ *tis plain / ; g muft be a th
to make g» a true qth to d t and then g : a will
T\ ' X be
$%% ^Treatise , Char. X.
.be &jgy becaufe cl-a is a 5th "in- this Tuning
which is plainly inverting the Order of the Scdle^
for there/. • g is tg, and g ■: a a tL but ftill
this is an Advantage, that we can exprefs any
Order of Degrees from any Note j'fo thatfome-
times we can make that a t g. which at other
times the Melody requires to be a t L Yet let
ine obferve in the lafi Place, that if all thefe in-
termedia e Notes betwixt the open Sounds of
the Four Strings, be conftantly made in the
fame Tune, they become thereby fixt Sounds ;
and this Instrument will then have as great Im-*
perfections as any other ; and indeed confidering
that the flopping of the String to take thefe
Notes in Tune is a Very mechanical Thing, at
leaft the doing of it right in a quick Succeilion
of Notes mult proceed altogether from Habit,
''tis probable we take them always in the fame
Tune ; nor do I believe that any Praclifer on
this Inftrument dare be very positive on the con-
trary i yet I don't fay 'tis impoifible to do other-
wife, for I know a Habit of playing the fame
Piece in leveral Keys might make one fenfible
of the contrary, if obferved with great Atten-
tion ; and Upon the larger Inftruments of this
Kind, that have Frets upon the Neck for di-
recting to the right Note, it would be very fen-
fible j and even upon the ^7o//V/,we find that fome
Songs go better from one Key than another j
which proves that thofe at leaft to whom this
happens, take thefe Notes always in the fame
L Tune,
HaV-
§ i. of MUSIC K. 323
Having done what I propofed for explain-
ing the Theory of Sounds with refpeft to Ttine y
the Order feems to require, that I fliould iiext
eonfider that of Time > but tho' this be very
confiderable in Practice, yet there is much lefs
to be faid about it in Theory $ and therefore I
chufe to explain next the Art of writing Mu-
Jick, where I fliall have Occafion to fay what is
needful with refpe& to the Time*
G H A & XL
The Method and hit of Writing Mufick^
particularly how the Differences 0/Tune
are reprefentedi,
I 1* A general Account tf the Method*
7 HAT this Title imports has been Jk-
plained in Chap. 1. § 2. And to conie
to the Thing it felf, let us eonfider.
I t was not enough to have difcovered fo
much of the Nature of Sound, as to make it
ferviceable to our Pleafure, by the Various C6m-
X 2 binations
3*4 4 Treatise Chap. XL
binations of the Degrees of Tune , and Meafnres
of Time ; it was neceffary alfo, for enlarging
the Application, to find a Method how to re-
prefent thefe fleeting and tranlient Obje&s, by
fenfible and permanent Signs ; whereby they are
as it were arrefted j and what would othenvife
be loft even to the Compofer, he preferves for
his own Ufe, and can communicate it to others
at any Diftance; I mean he can direcl: them
how to raife the like Ideas to themfelves, fup-
pofing they know how to take Sounds in any
Relation olTiine and Time directed ,♦ for the
Bufinefs of this Art properly is, to reprefent the
various Degrees and Meafures of Tune and
Time in fuch a Manner, that the Connection
and Succeflion of the Notes may be eafily and
readily difcovered, and the skilful Pra&ifer may
at Sight find his Notes, or, as they fpeak, read
any Song.
A s the Two principal Parts of Mufick are
the Tune and Time of Sounds, fo the Art of
writing it is very naturally reduced to Two
Parts correfponding to thefe. The firft, or the
Method of~fe~prefenting the Degrees of Tune^ I
fhall explain in this Chapter ,* which will lead me
to fayfomething in general of the other, a more
full and particular Account whereof you fhall
hay in the next. .Chapter.
We have alreajdy feen how the Degrees of Tune
or the Scale oi~„ Mufick may be expreft by 7 Letters
repeated as oft as we pleafe in a different Chara-
cter; but thefe, without fome other Signs,do not
gxprefs the Mgaferes of ilw^unlefswe fuppofe aU
~ • " ■ ~" " ~ ' |h§
§ x. of MUSIC K. 3 2y
the Notes of a Song to be of equal Length.
Now, . fuppofing the Thing to be made not
much more difficult by thefe additional Signs of
Time<> yet the Whole is more happily accom-
plished in the following Manner.
If we draw any Number of parallel Lines, as
in Plate i. Fig. 7. Then, from every Line to
the next Space, and from every Space to the
next Line up and down, reprefents a Degree
of the diatonick Scale ; and confequently from
every Line or Space to every g other at
greater Diftance reprefents fome other Degree
of the Scale, according as the immediate De-
grees from Line to Space, and from Space to
Line are determined. Now to determine thefe
we make Ufe of the Scale expreft by 7 Letters,
as already explained, viz. c : d} e .f :g;a : b ,
c~- where the Tone greater is reprefented by a
Colon (: ) the Tone leffer by a Semicolon (;)
and the Semitone greater by a Point (.). If
the Lines and Spaces are marked and named
by thefe Letters, as you fee in the Figure, then
according to the Relations afflgned to thefe
Letters ( i. e, to the Sounds expreft by them)
the Degrees and Intervals of Sound expreft by
the Diftances of Lines and Spaces are deter-
mined.
A s to the Extent of the Scale of Miifick^ it
is infinite if we confider what is {imply poifible,
but for Pradice, it is limited • and in the pre-
fent Practice 4 Odlaves, or at moft 4 Oblaves
with a 6th) comprehending 34 diatonick Notes,
is the greateft Extent. There is fcarcely any
•X 3 one
$i6 ^Treatise Chap.XI.
one Voice to be found that reaches near fo far,
tho' feveral different Voices may; nor any one
{ingle Piece of Melody, that comprehends fo
great an Interval betwixt its higheft and loweft
Note : Yet we muft confider not only what
Melody requires, but what the Extent of feve^
ral Voices and Inftruments is capable of, and
what the Harmony of feverals of them requires $
and in this refpeft the whole Scale is neceffary,
which you have reprefented in the Figure di-
rected to ; I ihall therefore call it the iiniverfal
Syflem, bec^ufe it comprehends the whole Ex-
tent of modern Practice.
But the Queftion full remains, How any
particular Order and Succeffion of Sounds is re-
prefented ? And this is done by fetting certain
Signs and Characters one after another, up and
down on the Lines and Spaces, according to
the Intervals and Relations of Tune to be ex-
preft ; that is, any one Letter of the Scale, or
the Line or Space to which it belongs, being
chofen to fet the firft Note on, all the reft are
fet up and down according to the Mind of the
Compofer, upon fuch Lines and Spaces as are
at the defigned Diftances, i. e. which exprefs
the defigned Interval according to the Number
and Kind of the intermediate Degrees ; and
mind that the firft Note is taken at any con-
venient Pitch of Time ; for the Scale, or the
Lines and Spaces, ferve only to determine the
Time of the reft with relation to the firft, leav-
ing us to take that as we pleafe: For Example »,
if the firft Note is placed on the Line c, and
the
§ i. of MU SICK. 3i7s
the next defigned a Tone or 2d g. above, it, it
fet on the next Space above, which is d; or i
i^ is defigned a 3 d g. it is fet on the Line above
which is e - y or on the fecond Line above, if it
was defigned 5/^, as you fee reprefented in. the
id Column of the Scale in the proceeding . Fi-
gure, where I have ufed this Character O for a
Note. And here let nie obferve in general, that
thefe Characters ferve not only to direcl: how
to take the Notes in their true Tune^ by the
Diftance of the Lines and Spaces on which
they are fet ; but by a fit Number and Variety
of them, (to be explained in the next Chapter)
they exprefs the Time and Meafure of Durati-
on of the Notes ; whereby 'tis plain that thefe
Two Things are no way confounded $ the re-
lative Meafures of Tune being properly deter-
mined by the Diftances pi Lines and Spaces,
and the Time by the Figure of the Note or
Character.
'T 1 s eafy to obferve what an Advantage
there is in this Method of Lines and Spaces, .
even for fuch Mufick as has all its Notes of
equal Length, and therefore needs no other
Thing but the Letters of the Scale to exprefs
it ; the Memory and Imagination are here
greatly aififted, for the Notes Handing upward
and downward from each other on the Lines
and Spaces, exprefs the rifing and falling of the
Voice more readily than different Characlers of
Letters j and the Intervals are alfo more readily
perceived.
' . • X 4
328 -^Treatise Chap. XI,
Observe in the next Place, That with
refpe£t to Inftruments of Mufick, I fuppofe their
Notes are all named by the Letters of the Scale,
having the fame Diftances as already ftated in
the Relations of Sounds expreft by thefe Let-
ters j fo that knowing how to raile a Series of
Sounds from the loweft Note of any Inftrument
by diatonick Degrees (which is always firft
learned) and naming them by the Letters of
the Scale, 'tis eafily conceived how we are di-
^ re&ed tq play on any Inftrument, by Notes fct
upon Lines and Spaces that are named by the
iame Letters. It is the Bufineis of the Mafters
and Profeflors of feveral Inftruments to teach the
Application more exprefly. And as to the
human Voice^ obferve, the Notes thereof, be-<
ing confined to no Order, are called c or d y &c.
only with refpeft to the Direction it receives
from this Method ; and that Direction is alfo
very plain ; for having taken the firft Note at
any convenient Pitch, we are taught by the
Places of the reft upon the Lines and Spaces
- how to tune them in relation to' the firft, and
tp one another.
% dgMn* as the artificial Notes which divide
the Tones of the natural Series, are expreft by
the fame Letters, with thefe Marks, 8£, [,, al-
ready explained, fo they are alfo plac'd on the
fame Lines and Spaces, on which the natural
Note named by that Letter ftands ; thus c% and
c , belong to the fame Line or Space, as alfo
$< and d. And when the Note on any Line or
Space ought to be the artificial one, it is mar-
ket
§ i. of MUSIC K. 319
kcd $ or l ; and where there is no fuch Mark
it is always the natural Note. Thus, if from
a ( natural ) we would fct a 3d g. upward, it
is c% ; or a ^dh above g, it is b flat or (/, as* .
you lee in the 2d Column of the preceeding
Figure. Thefe artificial Notes are all determi-
ned on Jnftruments to certain Places or Pofitions,
with refpeel to the Parts of the Inftrumcnt and
the Hand; and for the Voice they are taken
according to the Diftance from the laft Note,
reckoned by the Number of Tones and Semi"
tones that every greater Interval contains.
The laft general Obferve I make here is, that
as there are Twelve different Notes in the fe-
mitonich Scale^ the Writing might be fo orde-
red, that from every Line a Space to the next
Space or Line fhould exprefs a Semitone ; but it
is much better contrived, that thefe fliould ex-
prefs the Degrees of the diatonich Scale ( i. e.
fome Tones fome Semitones ) for hereby we
can much eafier chfeover what is the true In-
terval betwixt any Two Notes, becaufe there
are fewer Lines and Spaces interpofed, and the
Number of them fuch as anfwers to the Deno-
mination of the Intervals; fo an Oblave com-
prehends Four Lines and Four Spaces ; a $th
comprehends Three Lines and Two Spaces, or
Three Spaces and Two Lines ; and fo of o-
thers. I have already ftiewn, how it is better
that there fhould be but Seven different Letters,
to name the Twelve Degrees of the Jemitonick
Scale •> but fuppofing there were Twelve Let-
ters, it is plain we fliould need no more Lines
to
330 A Treatise Chap. XI.
to comprehend an ObJave, bccaufe we might
aflign Two Letters to one Line or Space, as
well as to make it, for Ex ample ^ both c% and
c, whereof the one belonging to the diatonick
Series^ fhould mark it for ordinary, and upon
Occafions the other be brought in the fame
Way we now do the Signs $ and )/.
$2, A more particular Account of the Ms*
thodj vohere y of the Nature and Ufe of Clefs.
TH O' the Scale extends to Thirty Four
diatcnickNoteSj which require Seventeen
Lines with their Spaces, yet becaufe no one
lingle Piece nf Melody comprehends near io ma-
ny Notes, whatever feveral Pieces joyned in one
Harmony comprehend among them \ and be-
caufe every Piece or (ingle Song is directed or
written diftin&Iy by it felf ; therefore we never
draw more than Five Lines, which comprehend
the greateft Number of the Notes of any (ingle
Piece ; and for thofe Cafes which require more,
we draw fhort Lines occafionaJly, above or be-
low the 5, to ferve the Notes that go higher or
lower. See an Example in Plate i . Fig. 8 .
Again, tho' every Line and Space may be
marked at the Beginning with its Letter, as has
been done in former Times ; yet, fince the Art
has been improve^ .only one Line is marked,
by which all the reft are eahly known, if we .
reckon up or down in the Order of the Letters ;
the
§z. ofMUSICIC 331
the Letter marked is called the Clef or Key,
becaufe by it we know the Names of all the o-
ther. Line's and Spaces, and confequently the
true Quantity of every Degree and Inters ah
iBut becaufe every Note in the Oflave is called
a Key, tho 1 in another Senfe, this Letter mar-
ked is called in a particular Manner the figned
Clef becaufe being written on any Line, it not
only figns or marks that one, but explains all the
reft. And to prevent Ambiguity in what follows,
by the Word Clef I fliall always mean that Letter,
which, being marked on any Line, explains all
1 the reft ; and by the Word' Key the principal
!Note of any Song, in which the Melody doles,
iin the Senfe explained in the h^ Chapter.
Or thefe figns d Clefs there are Three,
jfe. c, f g; arid that we may know the Im-
provement in having but one figned Clef in one
I particular Piece, alfo how and for what Purpofe
Three different Clefs are ufed in different Pie-
ices, eoniider the following Definition.
A Song is cither fimple or compound. It is a
\ fimple Song, where only one Voice performs ; or,
I tho' there be more, if they are all Unifon or
•Offiave, or any other Concord in every .Note,
' 'tis ffifl but the fame Piece of Melody, perfor-
1 med by different Voices in the fame or different
i Pitches of Tune, for the Intervals of the Notes
i are the fame in them all. A compound Song is
1 where Two or more Voices go together, with
a Variety of Concords and Harmony ; fo that
I the Melody each of them makes, is a
diftinft and different fimple Song y and all toge-
ther
33* ^Treatise Chap. XI.
ther make the compound. The Melody that
each of them produces is therefore called a
Part of the Compofition ; and all fuch Compo-
fitions are very properly called Jymphonetick
Mufick, or Mufick in Parts; taking the Word
Mufick here for the Compofition or Song it felf.
Now, becaufe in this Compofition the Parts
muft be fome of them higher and fome lower,
( which are generally fo ordered that the fame
Part is always higheft or loweft, tho' in mo-
dern Compofitions they do frequently change, )
and all written diftinctly by themfelves, as is
very neceffary for the Performance ; therefore
the Staff of Five -Lines upon which each Par t
is written, is to be confidered as a Part of the
imiverfal Syfiem or Scale, and is therefore called
^particular Syfiem; and becaufe there are but
Five Lines ordinarily, we are to fuppofe as ma-
ny above and below, as may be required for a-
ny {ingle Part; which are actually drawn in
the particular Places where they are neceflary.
The higheft Part is called the Treble,
or Alt whofe Clef is g, fct on the id Line of
the particular Syfiem, counting upward : The
lowed is called the Bass, i. e. Bafts, becaufe
it is the Foundation of the Harmony, and for-
merly in their plain Compofitions the Bafs was
firft made, tho' 'tis otherwife now ; the Bafs-
clefisf on the qth Line upward: All the other
Parts, whofe particular Names you'll learn from
Practice, I fhall call Mean Parts, whofe
Clef is f, fometimes on one, fometimes on an-
other Line $ and fome that are really meam^
Parti
§ i. of MUSIC K. 333
PaVts '-are fet with the g Clef. Sec Plate i.
i*7g. 8. where you'll obferve that the f and/
<?/</j" are marked with Signs no way refembling
thefe Letters ; I think it were as well if we u~
fed the Letters themfelves, but Cuftom has car-
ried it otherwife j yet that it may not feem
altogether a Whim, Kepler in Chap. Book
3d of his Harmony ^ has taken a critical Pains
to prove, that thefe Signs are only Corruptions
of the Letters they reprefent; the curious may
confult him.
W e are next to confider the Relations of
thefe Clefs to one another, that we may know
where each Part lies in the Scale or general Sy-
fiem^ and the natural Relation of the Parts a-
mong themfelves, which is the true Defignand
Office of the Clefs. Now they are taken $ths
to one another, that //, the Clef/ is loweft, c
is a 5th above it, and g sl 5th above c. See
them reprefented in Plate i- Fig. 7. the laft
Column of the Scale ; and obferve^ that tho' in
the particular Syflems^ the Treble or g Clef is
ordinarily fet on the 2 d. Line, the Baft or/
Clef on the qth Line, and the mean or c Clef
on the 3 d Line ( elpecially when there are but
Three Parts ) yet they are to be found on o-
ther Lines y as particularly the mean Clef which
mod frequently changes Place, becaufe there
are many mean Parts^ is fometimes on the ift y
the idy the id or qth Line; but on whatever
Line in the feparate particular Sjfiem any Clef
is (igned, it mufl be underitood to belong to the
|kme Place of the general Syfiem i and to be the
feme
334 A Treatise Chap. XL
fame Individual Note or Sound on the InfTru-
ment which is directed by that Clef as I have
di ftinguiflfd them in the Scale upon the Margin
of the -$d Column ,- fo that to know what Part
of the Scale any particular Syftem is. We muft
take its Clef where it ftands figned in the Scale
( u e. the laft mentioned Fig, ) and take as
many Lines above and belcxvv it, as there are
in the particular Syftem ; or thus, we miift apply
the particular Syftem to the Scale,{o as the Clef
Lines coincide,, and then we fhall fee with what
Lines of the Scale the other Lines of the parti-
cular Syftem coincide : For Example, if we find
the Clef on the Line upward, in a particu-
lar Syftein ; to find the coincident Five Lines
to which it refers in the Scale, we take with
the/ Clef Line, Two Lines above and Two
below* Again, if we have the c Clef on the
qth Line, we are to take in the Scale with the
Clef Line, One Line above and Three below,
and fo of others j fo that according to the diffe-
rent Places of the Clef in a particular Syftem,
the Lines in the Scale correfpondent to that
Syftem may be all different, except the Clef
Line which is invariable : And that you may
with Eafe find in the Scale the Five Lines co-
incident with every particular Syftem, upon
whatever Line of the Five the Clef may be fet,
I have drawn Nine Lines acrofs, which include
each Five Lines of the Schle\ in fuch a Man-
ner, that you have the particular Syftems di-
ftinguifhed for every relative Pofition of any of
the Three figned Clefs*
As
i ■
§ i. of MUSIC K. 33j
As to the Reafon of changing the relative
Place of the Clef, i. e. its Place in the particu-
lar Syftem, 'tis only to make this comprehend
as many Notes of the Song as potfiblc, and by
that Means to have fewer Lines above or below
it ; fo if there are many Notes above the Clef
Note and few below it, this Purpofe is
anfwered by placing the Clef in the firft or fe-
cond Line ; but if the Song goes more below
the Clef then it is beft placed higher in the
Syftem : In Jbort, according to the Relation of
the other Notes to the Clef Note, the particular
Syftem is taken differently in the Scale, the Clef
Line making one in all the Variety, which con-
iifts only in this, <siz. taking any Five Lines
immediately next other, whereof the Clef Line
muft always be one.
B y this conftant and invariable Relation of
the Clefs, we learn eafiiy how to compare the
particular Syftcms of feveral Parts, and know
how they communicate in the Scale, i. e. which
Lines are unifon, and which are different, and
how far, and confequently what Notes of the
feveral Parts are unifon, and ivhat not : For you
are not to fuppofe that each Part has a certain
Bounds within which another muft never come;
no, fome Notes of the Treble, for Example,
may be lower than fome of the mean Parts, or
even of the Bafs ; and that not only when we
compare fuch Notes as are not heard together,
but even fuch as are. And if we would put to-
gether in one Syftem, all the Parts of any Com-
pofition that are written feparately. The Rule
is
l$6 ^Treatis^ Chap. XI.
is plainly this, viz. Place the Notes of each
Part at the fame Diftanccs above and below the
proper Clef as they ftand in the feparate Syfteni.
And becaufe all the Notes that are confonant
(or heard together) ought to ftand, in this De-
fign, perpendicularly over each other, therefore
that the Notes belonging to each Part may be
diftinftly known, they may be made with fuch
Differences as fhall not conrufeor alter their Sig-
nifications with refpeel: to Time, and only fig-
Iiify that they belong to fuch a Part • by this
Means we fhall fee how all the Parts change
3rd. pafs thro' one another, i. e. which of them,
in every Note, is highefl or Joweft or . imifon ■;
for they do fometimes change, tho' more gene-
rally the Treble is higheft and the Bafs loweft,
the Change happening more ordinarily betwixt
the mean Parts among themfelves, or thefe
with the Treble or Bafs: The Treble and Bafs
Clefs are diftant anO&ave and Tone, and their
Parts do feldom interfere, the Treble moving
more above the Clef Note 5 and the Bafs be-
low*
We fee plainly then^ that the Ufe of parti-
cular fign'd Clefs is an Improvement with re-
fpecl: to the Parts of any Compofition ; for un-^
lefs fome one Key in the particular Syftems were
diftinguifhed from the reft, and referred invari^
ably and conftantly to one Place in the Scale j
the Relations of the Parts could not be diftind^
ly marked j and that more than one is neceffa-
ty, is plain from the Diftance there muft be a-
jnong the Parts : Or if one Letter is chofen for
___. all,
■§ i; of MUSIC K. 33?
all, there muft be fome other Sign to fhew
what Part it belongs to, and the Relation of
the Parts. Experience having approven the Num-
ber and Relations of the figned Clefs which are
explained, I fliall add ho more as to that, but
there are other Things to be here obferved.
The choofing thefe Letters/, c .g for fign-
ed Clefs, is a Thing altogether arbitrary ; f6r
any other Letter within theSyftem, will explain
the reft as well ; yet 'tis fit there be a conftant
Rule, that the feveral Parts may be right di-
ftinguifhed $ and concerning this obferve agairi^
that for the Performance of any fingle frece the
Clef ferves only for explaining the Intervals a-
mong the Lines and Spaces, fo that we heed
not mind what Part of any greater Syftem it is*
and we may take the firft Note as high or low
as We pleafe : For as the proper Ufe of the Scale
is not to limit the abfolute Degree of Tone, fo
the proper Ufe of the figned Clef is not to limit
the Pitch, at which the firft Note of any Part
is to be taken, but to determine the Tune bf
the reft with relation to the firft, and^ consi-
dering all the Parts together, to determine the
Relations of their feveral Notes,by the Relations
of their Clefs in the Scale : And fo the Pitch of
Tune being determined in a certain Note of one
Part, the other Notes of that Part are deter-
mined, by the conftant Relations of the Letters
of the Scale ) and alfo the Notes of the Other
Parts, by the Relations of their tiefs. To
fpeak particularly-^f the Way of tuning th.6 tn-
ftrumefits that are Employed in executing the
Y ieyerat
338 ^Treatise Chap. XI.
feveral Parts, is out of my Way $ I {hall only
fay this, that they are to be fo tuned as the
Clef Notes, wherever they ly on the Inftruments
which ferve each Part, be in the foremention-
ed R elations to one another.
As the Uarpfichord or Organ (or any other
of the Kind) is the mod extenfive Inftrument,
-we may be helped by it to form a clearer Idea
of thefe Things : For confider, a Harpflchord
contains in itfelf all die Parts oiMufick, I mean
.the whole Scale or Sjftem of the modern Pra-
ctice i the foremoft Range of Keys contains the
diatonick Series beginning, in the largeft Kind,
in g, and extending to c above the Fourth %<ve\
which therefore we may well fuppofe reprefen-
ted by the preceeding Scale, In Practice, upon
that Inftrument, the Clef Notes are taken in the
Places reprefented in the Scheme ; and other In-
, ftruments are fo tuned, that, confidering the
Parts they perform, all their Notes of the fame
Name are imifon to thofe of the Harpfichordth&t
^elong to the fame Part, I have faid, the
Jfarpjichord contains all the Parts of Mufick ;
. and indeed any Two diftinft Parts may be per-
formed upon it at the fame Time and no more;
yet upon Two or more Harpfichords tuned uni-
Jons, whereby they are in Effect but one, any
Number of Parts may be executed : And
in this Cafe we fhould fee the feveral Parts ta-
ken in their proper Places of the Inftrument, ac-
cording to the Relations of their Clefs explain-
ed : And as to the tuning the Inftrument, Ifhall
only add 3 that there is a certain Pitch to which
it
§ t; of MUSIC K. 339
it is brought, that it may be neither too high
nor too low, for the Accompaniment of other
Inftruments, and efpecially for the human Voice,
whether in Unifon or taking a different Part ;
and this is called the Consort Pitch. To
have done, you muft confider, that for perform-
ing any one fingle Part, we may take the Clef
Note in any $ve, i- e. at any Note of the fame
Name, providing we go not too high or too
low for finding the reft of the Notes of the Song:
But in a Confbrt of feveral Parts, all the Clefs
muft be taken, not only in the Relations, but
alfo in the Places of the Syftem already mentio-
ned, that every Part may be comprehended in
it : Yet ftill yon are to mind, That the Time
of the Whole, or the abfolute Pitch, is in it felf
an arbitrary Thing, quite foreign to the Ufe of
the Scale ; tho* there is a certain Pitch general-
ly agreed upon, that differs not very much in
the Practice of any one Nation or Set of Muft^
cians from another. And therefore,
When I Ipeak of the Place of tfye Clefs in
the Scale or general Syftem, you muft underftand
it with refpe& to a Scale of a certain determi-
ned Extent i for this being undetermined, fo
muft the Places of the Clefs be : And for any
Scale of a certain Extent^ the Rule is, that the
mean Clef c be taken as near the Middle of the
■Scale as poffible, and then the Clef g a $th a-
bove, and/ a $th below, as it is in the prefent
general Syftem of Four .%<oes and a 6th, repre-
sented in the preceeding Scheme, and actually
determined upon Harpfichords,
Y 2 N
34° ^ Treatise • ' Chap. XI.
Ik the I aft Place confider, that fince the
Lines and Spaces of the Scale, with the Degrees
ftated among them by the Letters, fufficiently
determine how far any Note is diftant from a-
. nether, therefore there is no Need of different
Characters of Letters, as would be if the Scale
■ were only expreft by thefe Letters : And when
we fpeak of any Note of the Scale, naming it
; by a- qr b, &c. we may explain what Part of
■the Scale it is in, . either by numbring the %ves
•from the loweft Note, and calling the Note fpo-
ken of (for Example) c in the loweft %ye or in
the id fiktei and fo on : Or, we may determine
its Place by a Reference to the Seat of any of
the Three figned Clefs ; and fo we may fay of
any Note, as/ or g, that it is fuch a Clef Note 3
or the firfl or fecond, &c f or g above fuch a
Clef. Take this Application, fuppofe you ask
me what is the highefl Note of my Voice, if ,
I fay d, you are not the wifer by this Anfwer,
till I determine it by faying.it is d in the fourth
Oclave, or the firfl d above the Treble Clef. But
again, neither this Quedion nor the Anfwer is
fofficiently determined, unlefs it have a Refe-
rence to fome fuppofed Pitch of Tune in a cer-~
tain fixt Inilrument, as the ordinary Confort
Pitch of a Harpfichordy becaufe, as 1 have fre-
quently laid, the Scale of Mufick is concerned
.only with the Relation of Notes and the Order
of Degrees, which are ilill the fame in all Dif-
ferences of Time > in the whole Series.
% i\
§ 3 . ofMUSlCK. 341
§ 3. Of the Reafon, Ufc, and Variety of the
Signatures of Clefs.
I Have already faid, that the natural and arti-
ficial Note cxpreffcd by the fame Letter,
as c and c$, are both fet on the lame Line or
Spare. When there is no M or ]/ marked on any
Line or Space, at the Beginning with the Clef
then all the Notes are natural ; and if in any
particular Place of the Song, the artificial Note
is required, 'tis fignified by the Sign $ or (/, fet
upon the Line a Space before that Note ; but if
a $ or \t is fet at the Beginning in any Line or
Space with the Clef then all the Notes on that
Line or Space are the artificial ones, that is,
are to be taken & Semitone higher or lower than
they would be without fuch a Sign ; the lame
arfecls all their %ves above or below, tho' they
are not marked fo. And in the Courfe of
the Song, if the natural Note is fometimes re-
quired, it is fignified by this Mark ^. And the
marking the Syftem at the Beginning with
Sharps or Flats, I call the Signature of the
Clef
I n what's faid, you have the plain Rule for
Application ,• but that we may better conceive
the Reafon and Ufe of thefe Signatures, it will
be neceflary to recollect, and alfo make a little
clearer, what has been explained of the Nature
of Keys or Modes, and of the Original and Ufe
of the Jharp and flat Notes in the Scale, I have
Y 3 in
34- ^Treatise Chap. XL
in Chap. p. explained what a Key and Mode \
in Mujick is $ I have diftinguifhed betwixt thefe
Two, and (hewn that there are and can be but
Two different Modes, the greater and the leffer,
according to the Two continuous Divi&ons of the
%<ve y viz.by the sdg. or the 3d I. and their proper
Accompanyments$ and whatever Difference you
may make in the abfolute Pitch of the whole
Notes, or of the firft Note which iimites all the
reft, the fame individual Song rnnft ftill be ' in
the fame Mode ; and by the Key I understand
only that Pitch or Degree of Tune at which
the fundamental or clofe Note of the Melody \
and confequently the whole $®e is taken ; and
becaufe the Fundamental is the principal Note
of the %ve which regulates the reft, it is pecu-
liarly called the Key. Now as to the Variety
of Keys, if we take the Thing in fo large a
Senfe as to fignify the abfolute Pitch of Tune
at which any fundamental Note may be taken,
the Number is at leaft indefinite; but in Practice
it is limited, and particularly with refpecl: to
the Denominations of Keys, which are only
Twelve, viz. the Twelve different Names or
Letters of the femitonick Scale ; fo we fay the
Key of a Song is c or d, &c. which Signifies
that the Cadence or Clofe of 'the Melody is upon
the Note of that Name when we fpeak of any
Inftrument ; and with refpect to the human
Voice, that the clofe Note is Unifon to fuch a Note
on an Inftrument > and generally, with refpeft
both to Inftruments and Voice, the Denomina-
tion of the Key is taken from the Place of the
clofe
§ 3 < of MUSIC K. 343
clofe Note upon the written Muftck, i. e. the
Name of the Line or Space where it (lands :
Hence we fee, that the-' the Difference of Keys
refers to the Degree o£Tune y at which the Fun-
damental^ and confecjuentJy the whole %ve is
taken, in Diftinction from the Mode or Conten-
tion of an Otfavey yet thefe Denominations de-
termine the Differences only relative 'y, with
refpect to one certain Series of fixt Sounds, as a
Scale of Notes upon a particular Inflrument, in
which all the Notes of different Names are diffe-
rent Keys, according to the general Definition,
becaufe of their different Degrees of Tune; but
as the tuning of the whole may be in a different
Pitch, and the Notes taken in the fame Part of
the Inftrument, are, without refpect to the tun-
ing of the Whole, ftill called by the fame Names
C or d> &c. becaufe they ferve only to mark
the Relation of Time betwixt the Notes, there-
fore 'tis plain, that in Practice a Song will befaid
to be in the fame Key as to the Denomination,
tho' the abfolute Time be different, and to be in
different Keys when the abfolute Tune is the
fame ; as if the Note a is made the Key in one
Tuning, and in another the Note d unifon
to a of the former. Now, this is a Kind of Li-
mitation of the general Definition, yet it ferves
the Defign beft for Practice, and indeed can-
not be otherwife without infinite Confufion. I
{hall a little below make fbme more particular
Remarks upon the Denominations of Sounds or
Notes raifed from Inftruments or the human
yoke: But from what has been explained, you'll
X 4 eafity
344 ^Treatise Chap, XL
eafily underftand what Difference I put betwixt
a Mode and a Key ; of Modes there are only-
Two, and they refpeCt what I would call the
• Internal Conftitution of the 8w, but Keys are
indefinite in the more general and abftract Senfe,
and with regard to their Denominations in
Practice they are reduced to Twelve, and have
relped to a Circumftance that's, external and
accidental to the Modejaxid therefore a Key may
be changed under the fame Mode, as when the
fame Song, which is always in the fame Mode^ is
taken up at different Notes or Degrees of Tune y
and from the fame Fundamental or. Key a Series
may proceed in a different Mode, as when dif-
ferent Songs begin in the fame Note, But then
becaufe common Ufe applies the Word Ky in
both Senfes, i. e. both to what I call a Kjy and
a Mode.) to prevent Ambiguity the Word /harp
or flat ought to be added when we would ex-
prefs the Mode-, fo that a Jharp Key is the fame
as a greater Mode, and aflat Key a leffer Mode ;
and when we would exprefs both Mode and
Key*, we joyn the Name of the Key Note, thus,
we may fay fuch a Song is for Example in the
fharp or flat Key c, to fignifie that the funda-
mental Note in which the Clofe is made is
the Note called c on the Inflrument, or uni-
fon. to it in the Voice ; or generally, that
it is fet on the Line or Space of that Name
in Writing $ and that the 3d g. or 3d /. is ufed
in the Melody, while the Song keeps within that
Key; for I have alfo obferved, that the fame
Song may be carried thro' different Keys r df
make
§ 3 . of MUSIC K. 34 5
make fuccetfive Cadences in different Notes,
which is commonly ordered by bringing in fome
Note that is none of the natural Notes of the
former Key, of which more immediately : But
when we hear of any Key denominated c or d
without the Word Jharp or flat, then we can
understand nothing but what I have called the
Key in Diftinction from the. Mode, i. e. that the
Cadence is made in fuch a Note.
AG A IN, I have in Chap. io. explained the
Ufe of the Notes wecall/Jjarp and flat, ox arti-
ficial Notes, and the Diftin&ion of Keys in that
refpeel: into natural and artificial ; I have fhewn
that they are neceffary for correcting the De-
fects of Inftrurnents having fixt Sounds, thatbe~
ginning at any Note We may have a true con-
einnous diatonick Series from that Note, which,
in a Scale of fixt Degrees in the 8ce we cannot
have, all the Orders of Degrees proceeding from
each of the Seven naturalNotes being different,
of which only Two are concinnous, $fe, from
c which makes aJJjarp Ky, and from a which
makes aflat Key ; and to apply this more par-
ticularly, you muft underftand the Ufe of thefe
fiarp or flat Notes to be this, that a Song,
which, being fet in a natural Key or with-
out Sharps and Flats, is either too high
or too low, may be tranfpofed or fet in ano-
ther more convenient Key ; which neceflfarily
brings in fome of the artificial Notes, in or*
der to make a diatonick Series from this
new l{,ey, like that from the other ; and when
pie Song changes the Key before it come to the
final
A Treatise Chap. XI.
final Clofe, tho' the principal Key be natural,
yet feme of t[hefe into which it changes may
require artificial Notes., which are the effential
and natural Notes of this new Key ; for tho*
this be called an artificial Key, 'its only fo with
reipecl: to the Names, of the Notes in the fixt
Syftem, which are (till natural with refpect to
their proper Fundamental^ viz. the Kjey *into
which the Piece is tranfpofed, or into which it
changes where the principal Key is natural.
And even with refpect to the human Voice,
which is under no Limitation, I have fliewn the
Neceffity of thefe Names,for the fake of a regu-
lar, diftincl: and eafy Reprefentation of Sounds,
for directing theVoiceinPerformance. Iftiallnext
more particularly explain by fome Examples, the
Bufinefs of keeping in and going out otKeys. Ex-
ample. Suppofe a Song begins in r, or at leaft
makes the firft Clofe in it \ if all the Notes
preceeding that Clofe are in true mufical Rela-
tion to c as a Fundamental in one Species, fup-
pofe as 'XjJoarp Key\ u e. with a 3 d g. the Me-
lody has been 'ftill in that Key (See Example 5«
Plate 3.) But if proceeding, theCompofer brings
in the Note f% he leads the Melody out of the
former Key, becaufe/^ is none of the natural
Notes of the 8ve c, being a falfe qth to c. A-
gain, he may lead it out of the Key without
any falfe Note, by bringing in one that belongs
not to the Species in which the Melody was
begun: Suppofe after beginning in the jlmrp
Key'c, he introduces the Note g%, which is a
6th % tor, and therefore harmonious^ yet it be-
longs
j 3. of MUSICK. 347
angs to it as a flat Key, and confequently is
*ut or" the Key as ajharp one: And becaufethe
lame Song cannot with any good Effect be made
clofc twice in the fame Note in a different
ipecies, therefore after introducing the Noteg$,
i:he next Clofe muft be in fome other Note as a,
ind then the Key in both Senfes will be chan-
ged, becatile a has naturally a %dl j and there-
fore when any Note is faid to be out of a Key,
i'tis underftood to be out of it either as making
a falfe Interval, or as belonging to it in another
Species than a fuppofed one, /, e, if it belong to
it as a (harp Key, 'tis out of it as aflat one jfo in
.Example 3. Plate 3. the firft Clofe is in a as a
'./harp Key, all the proceeding Notes being natu-
ral to it as fuch; then proceeding in the fame
.Key, you fee g (natural) introduced, which
i belongs not to a as a Jharp K^ey, and al-
fo a $, which is quite out of the former
Key : By thefe Notes a Clofe is brought on
in b, and the Melody is faid to be out of the
firft Key, and is fo in both Senfes of the Word
Key, for b here has a sdl - y then the Melody is
carried on to a Clofe in d, which is a Third
Key, and with refpeft to that Piece is indeed
the principal Key, in which alfo the Piece be-
gins ; but I (hall confider this again ; it was e-
nough to my Purpofe here, that all the Notes
from the Beginning to the firft Clofe in a were
natural to the Odfave from a with a 3 d g , and
tho' the $dg. above the Ciofe is not ufed in the
Example, yet the 6th I, below it is ufed, which
is the fame Thing in. determining the Species,
I
348 ^Treatise Chap. XI.
I have explained already, that with the 3d/.
the 6th I. and jth l,ov 6th g. and jthg. are ufed
in different Circumftances $ and therefore you
are to mind that the 6th gm jthg. being intro-
duced upon ^ flat 8j&y, does not make any
Change of it ; fo that tho' the 6th I and jthL
is a certain Sign of a flat Key, yet the 6/-^ g.
and 7^g» belong to either Species ; therefore
the Species is only certainly determined by the
3d in both Cafes j and fo in the preceeding Ex-
ample, where I fuppofe g% is introduced upon
the Jharp Key c, the next Clofe cannot be in
c, becaufe g% being a 6th I, to c, requires a 3d
/. which would altogether deftroy that Unity of
Melody which ought to be kept up in every
Song; therefore when I fay the fame Song can-
not clofe twice in one Note in different Species,
the Determination of that Difference depends on
the 3d, which being the greater, muft always have
the 6th g. and jthg. but the 3d/. takes fometimes
the 6th I. and jthl. fometime the 6th g. and
jthg. See Ex. 6. Mate 3* where the whole keeps
within the flat %ey a, andclofes twice in it; the
flrft Clofe is brought on with the 6th Land jthl.
the next Clofe in the Qblave above is made
with the 6thg. and %thg. but a Ciofe in a, u-
ling the idg. would quite mine the Unity of
the Melody ; yet the fame Song may be carried
into different Keys, of which fome are JJoarp,
fome flat, without any Prejudice ; but of all
thefe there mult be one principal K.ey, in
which the Song fets out, and makes moft fre-
quent Cadences, and at lealt the final Cadence.
The
I 3. ofMUSICK. 340
The laft Thing I fhall obfervc upon thisSub-
jjecl of Keys is, that iometimcs the Key is
changed, without bringing the Melody to a Ca-
dence in the Key to which it is transferred,
[thai is, a Note is introduced, which belong \
properly to another Key than that in which
the Melody cxifted before, yet no Cadenc
jmade in that Key ; as if after "a Cadence in the
\jharp Key c t the Note g% is brought in, which
Ifhould naturally lead to a Clofe in a, yet the
Melody may be turned off without any formal
, and perfect Clofe in r/, and brought to its next
Clofe in another Key.
I return now to explain the Reafon and Ufe
■of the Signatures of Clefs. And firft, Let us
fuppofe any Piece of Melody confined ftricUy to
one Mode or Key, and let that be the natu-
ral Jharp Key c, trom which as the Relation
of the Letters aredetermined in the Scale, there
is a true miifical Series and Gradation of Notes,
and therefore it requires no $ or I/, confequent-
ly the Signature of the Clef muft be plain : But
let the Piece be tranfpofed to the Key d, it
muft neceffarily take/$ inftead of/, and c% for
c y becaufe/^ is the true idg. and c% the true
ytbg. to d. See an Example in Plate 3. Fig*
5, Now if the Clef be not figned with a ^ on
the Seat of/ and c, we muft fupply it wherever
theie Notes occur thro' the Piece, but 'tis plain-
ly better that they be marked once for all at
the Beginning.
Again, fuppofe a Piece of Melody , in which
fkere is a Change of the Key or Mode; if the
fame
3Jo A Treatise Chap. XL
fame Signature anfwer all thefe Keys, there is
no more Queftion about it $ but if that cannot
be 5 then the Signature pught to be adjufted to
the principal Key, rather than to any other,
as in Example 3* Plate 3. in which the priiicfa
pal Key is d with a idg. and becaufe this de-
mands/* and c$ for its 3 d and 7th, therefore
the Signature expreffeth them. The Piece actual-
ly begins in the principal Key, tho' the rirft
Clofe is made in the 5th above, viz. in a, by
bringing in gft ; which is very naturally mana-
ged, becaufe all the Notes from the Beginning
to that Clofe belong to both the Jharp Keys d
and a, except that gff which is the only Note
in which they can differ* then you fee the Me-
lody proceeds for fome time in Notes that are
common to both thefe Keys, tho* indeed the
Impreffion of the laft Cadence will be ftrongeft j
and then by bringing g ( natural ) and aM, it
leaves both the former Keys to clofe in b> and
here again there is as great a Coincidence with
the principal Key as po(fible 5 for the flat Key
b has every one of its effential Notes common
with fome one of thefe k of the Jharp Key d, ex-
cept a% andg% the 6th g. and ythg. which that
flat Key may occafionaJly make ule of > but as
it is managed here, the 6th I. is ufed, fo that it
differs from the principal Key only in one Note
a% j then the Melody is after this Clofe imme-
diately transferred to the principal Key, ma-
king there (the final Cadence. In what Notes every
Key differs from or coincides with any other,
you may learn from the 8c ah oi Semitones °>
but
§ 3 . of MUSI CK, 351
but you fhall fee this more eafily in a following
Table.
To [proceedwith our Signatures^ you have, in
what's (aid, the true Ufe and Reafon of the Signa-
tures of Clefs ; in refpeft of which they are diftin-
guiflied into natural, and artificial ox tranfpofed
Clefs > the firft is when no % ovb is fetatthe Be-
ginning j and when there are, it is faid to be
tranfpofed. We fhall next confider the Variety of
Signatures of Clefs, which in all are but 12. and
the 1110ft reafonable Way of making the artifi-
cial Notes, either in the general Signature, or
where they occur upon the Change of the Key.
I n the femitonick Scale there are 1 2 different
Notes in an Octave ( for the 1 \th is the fame
with the ift ) each of which may be made the
Fundamental or Key of a Song, i. e. from each of
them we can take a Series of Notes, that fhall pro-
ceed concinnoujly by Seven diatonick Degrees of
Tones and Semitones to an Octave, in the Spe-
cies either of ajbarp or flat Key, or of a grea-
ter or lejfer Mode ( the fmall Errors of this
Scale as it is fixt upon Inftruments, being in all
this Matter neglected. ) Now, making each of
thefe 1 2 Letters or Notes a Fundamental or
Key-note, there mult be in the Compafs of an
QUave from each, more or fewer, or different
Sharps and Flats neceffarilv taken in to make
a concinnous Series of the lame Species, i. e.
proceeding by the greater or leffer 3d ( for thefe
fpecify the Mode, and determine the other Dif-
ferences, as has been explained )> andfince from
every one-of the 12 Keys we may proceed con-
cinnouf-
352 ^ Treatise Chap; XL'
Cinnotif]} 1 , either with a greater or letter -$d, and
their Accompanyments, it appears at firft Sights
that there mnft be 24 different Signatures of
Clefs, but you'll eafily underftand that there am
kit 12. For the fame Signature ferves Two
different A^j-, whereof the one is a j5SW/> and
the other a flat Key, as you fee plainly in the
Nature of the diatonick Scale, in whicfi. the 0-
ffiave from c proceeds concinnoufy by a %dg>
£nd that from # ( which is a d/^g. above, Or a
$dl. below c) by a 3d/. with the #&/« and jth I
/. for its Accompanyments, which I fuppofehere
eflential to ah flat Keys $ coiifequently, if we"
begin at any other Letter, and by the Ufe of $
or \t make a continuous diatonick Series of ei-
ther Kind, we , {hall have in the fame Series*
continued from the 6th above or id below* an
061 /We ofthe other Species; therefore there can be
but 1 2 different Signatures of Clefs, whereof 1 is
plain or natural, and 1 1 tranfpofed or artificial*
What the proper Notes of thefe tranfpofed
Clefs are, you may find thus ; let the Scale of
Semitones be continned to Two OcJaves, then
begin at every Letter, and, reckoning Two Se-
mitones to every Tone, take Two Tones and
one Semitone, then Three Tones and one Se-
mitone, which is the Order of a Jharp Key or
of the natural clave from c, the Letters whidh
terminate thefe Tones and Semitones, are the
eflential or natural Notes of the Key or clave,
whofe Fundamental is the Letter or Note you
begin at .' By this you'll find the Notes be ong-
ing to every Jharp Key^ and thefe being conti-
nued.,
$ 3 . if MUSIC K. m
nued, you'll have aJfo the Notes belonging to e-
Very flat Key, by taking the 6th above the
\Jharp Key for the Fundamental of the flat: But
i to fave you the Trouble, I have collected them
an one Table. See plate 2. Fig. 1. The Table
ilhas Two Parts, and the upper Part contains 16
! Columns : From the 3 to the 14 inc!ufive,you have
j expreft in each an Offiave, proceeding from fome
ithe 12 Notes of different Names within the
\femitonick Scale, the Fundamental whereof you
jtake in the lower End of the Column, and read-
ing it upward, you have all the Letters or
INames belonging to that Offiave in a diatbnick
!Scale, in the Species of &Jbarp Key : In the
I tft Column on the left Hand yop have the De-
crees marked in Tones and Semitones ± without
any Diftindion of greater and leffer Tone: In
ithe Fifth Column, you have the DenOminati-
)ons of the Intervals from the Fundament ah
Then for the 12 flat Keys take, as I faid be-
Ifore, the 6th s above the other* and they are
ithe Fundamentals of the flat Keys, whole
I Notes are all found by continuing the Scale
j upward : But as to finding the Note where
jany Interval ends, 'tis as well done by counting
(downward; for fince 'tis always anOffiave from
;any Letter to the fame again, and alfo fince a.
jth upward falls in the fame Letter with a 2d
(downward, a 6th upward in the fame With a 3d
(downward, and a 3 d upward in the fame with
a 6th downward, alio a qth or $th upward
iin the fame with a 5th or qth downward, 1
therefore in the 16th Column, you fee Key flat
Z written
3 1 4 ^Treatise Char XL
written againft the .Line in which the 6th s of
the \%z,flyatf Keys (land ,- and the Denomi-
nation of the Intervals are written againft thefe
Notes where they terminate $ and becaufe the
Scale in that Table is carried but to one
05fave y 4b that we have only a ^d L above the
Fundamental of the flat Key y therefore the reft
of the Intervals are marked at the Letters be-
low, which wi]J be eafier underftood if yoirll
fuppofe the Key to (land below., and thefe In-
tervals to be reckoned upwards. In the id
Part of the Table you have a Syftem of 5 Lines
marked j with the Treble or g Clef in 1 3 Divifions
each anfwering to a Column of the upper Part,-
and thefe exprefs all the various .Signatures of
the Clef that is^ all the accidental or JJjarp
andflat Notes that belong to any of the 1 2 Keys
of the Scale, pi
WiTH : Refped to the Names, and Signatures in
the Table, there remain fome Things to be ex-
plained : I told you in the laft Chapter that
upon the main it was an indifferent Thing whe-
ther the -artificial Notes in the Scale were nam-
ed from the, Note below with a $, or from that
above with a f/ : Here you have each of them
marked^ in fome Signatures $ and in others f/ 5
but in every particular Signature the Marks are
all of one Kind % of ]/ y tho' one Signature is %>
and another (/ ; and thefe are not fo order-
ed at random; the Reafon I fhall explain to
you : In the firft Place there is a greater Har-
mony with refpeCbto the Eye; but this is a
fmall Matter., a better Reafon . follows $ confi-
§ 3- of MUSIC K. 3?j
dcr, every Letter has two Powers, i. e. is ca-
pable of reprefenting Two Notes^ according as
you take it natural or plain, as c$ d^ &c. or
travfpqfcd as c% or ffl ; again } every Line
and Space is the Seat of one particular Letter :
Now if we take Two Powers of one Letter in
the fame 061 ave or Kej^ the Line or Space to
which it belongs muft have Two different Signs i
and then when a Note is fet upon that Line or
Space, how ftiaJl it be known whether it is to
bo taken natural or tranfpofed? This can on-
ly be done by fetting the proper Signs at every
inch Note j which is not only troublefom, but
renders the general Signature ufelefs as to that
Line or Space : This is the Reafon why fome
Signatures are made % rather than (^ and con-
trarily ; for Example, take for the Fundamental
c#, the reft of the Notes to make ajharp Key
are dM .f : fM : g% : a% : c. where you fee /
and c are taken both natural and tranfpofed^
which we avoid by making all the artificialNote
(r, as in the Table, thus $f : e\* ./ : gb : df : I :
I . dK 'Tis true that this might be helped an-
other Way, viz. by taking all the Notes
% i.e. taking for/, and ft$ Cove; but the
Inconveniency of this is vifible, for hereby We
force Two natural Notes out of their Places^
whereby the Difficulty of performing by fuch.
Direction is inereafed : In the other Cafes
where I have marked all \/ rather than %, the
fame ReafonS obtain : And in fome Cafes, fome
Ways of iigning with % would have both thefe
Inconveniencies. The fame Reafons. make it
Z 2 -neceflfary
3j4 A Treatise Chap. XI.
neceffary to. have fome Signature & rather than
f/ ; but the Qffiave beginning in gl is fingular
in this Reipe6t, that it is equal which Way it
is Signed, for in both there will be one natural
Note difplaced unavoidably ; as I have it in
the Table b natural is figned c^ and if you
make all the Signs $, you muft either take in
Two Powers of one Letter, or take e% for/.
Now neither in this, nor any of the other
Cafes will the mixing of the Signs remove the
Inconveniencies ; and fuppofe it could, another
follows upon the Mixture, which leads me to
{hew why the fame Clef is either all $ or all j/ y
the Reafon follows.
The Quantity of an Interval expreft by
Notes fet upon Lines and Spaces marked fome
&, fome l/ y will not be fo eafily difcovered, as
when they are all marked one Way, becaufe
the Number of intermediate- Degrees from Line
to Space, and from Space to Line, anfwers not
to the Denomination of the Interval ; for Ex-
afnple^ if it is a 5th, I (hall more readily dis-
cover it when there are 5 intermediate Degrees
from Line to Space, than if there were but 4 ;
thus 5 fromg$ to M is a 5th, and will appear
as fucli by the Degrees, among the Lines and
Spaces [j out If we mark itg$, e^ it will have
the Appearance of a qth ; alfo from f% to a%
Is a 3<s?, and appears fo, whereas from/* to fr
looks like a qth --$■ and for that Reafon Mr.
JSimpfon in his Compend of Mufick calls it a lef-
fet-qthy which I think he had better called am
apparent qth± andfo by making the Signs of the
§ 3- of MUSIC K. 3 T7
[Clef all of one Kind, this Inconveniency is fav-
ed with refpe£t to all Intervals whofe both
Extremes have a tranfpofed Letter ; and as to
fuch Intervals which have one Extreme a na-
' tural Note, or expreft by a plain Letter, and
I the other tranfpofed, the Inconveniency is pre-
vented by the Choice of the $ in fome Keys,
I and of the \/ in others ; for Example, from d
I to/K is a %dg. equal to that from d to gfcj but
i the firft only appears like a 3^, and fo of other
Intervals from d, which therefore you fee in
the Table are all figned ^. Again from/ to (^
or /to aM is a 4^?, but the firft is the beft Way
of marking it,- there are no more tranfpofed
Notes in that Offiave, nor any other OUave,
whofe Fundamental is a natural Note, that is
marked with ]/.
I t muft be owned, after all, That whate-
ver Way we chufe the Signs of tranfpofed
Notes, the Sounds or Notes ' themfelves on an
Inftrument are individually the fame ,- and
marking them one Way rather than another,
refpe&s only the Conveniences of reprefenting
them to the Eye, which ought not to be ne-
glected ; efpecially for the Direction of the hu-
man Voice, becaufe that having -no fixt Sounds
(as an Inftrument has,whofe Notes may be found
by a local Memory of their Seat on the Inftru-
ment) we have not another Way of finding the
true Note but computing the Interval by the
intermediate diatonick Degrees, and the more
readily this can be done, it is certainly the
better.
Z 3 Now
08
A Treatise Chap, XI.
Now you are to ohferve, that, as the Mg4
nature or the Clef is defigned for, and can ferve
but one Key, which ought rather to be th6
principal Key or Otlave of the Piece than any
other, fhewing what tranfpofed Notes belong to
it, fo the Inconveniency laft mentioned is re-
medied, by having the Signs all of one KM, on-
ly for thefe Intervals one of whofe Extremes is
the Key-note^ or Letter : But a Song may
modulate or change from the principal into
other Keys, which may require oth^r Notes
than the Signature of the Clef afford/, \ fo we
find $ and \/ upon fome particular Notes con-
trary to the Ckfy which mews that* the Melo-
dy is out of the principal Key, fuch Notes be-
ing natural to fome other fubprincipal Key into
which it is carried ; and thefe Signs are, or
ought always to be chofen in the moft conve-
nient Manner for expreffmg the Interval; for
Example, the principal Key being C with a *$d
p. which is a natural Q&ave (i. e. expreffed all
with plain Letters) fuppofe a Change into its
4th f ; and here let a qth upward be required,
we muft take it in j/ or a% ; the firft is the beft
Way, but either of them contradicts the Clef
which is natural; and we no fooner find this
than we judge the Key is changed. But again,
a Change may be where this Sign of it cannot
appear, viz. when we modulate into the 6th
ol&fljarp principal Key, or into the 3 d of a
fiat principal Keys ; becaufe thefe have the fame
Signature;, as has been already fhown, and have
fuch
§ 3. of MUSIC K. 3 yp
ifuch a Connection that, unlefs by a Cadence,
I the Melody can never be fiid to be out of the
t principal l\cy. And with refpect to a flat prin-
cipal Key ', obferve^ That if the 6f^? g. and jth
g. are ufed, as infomcGrcumftances they may,
efpccially towards a Cadence, then there will
be nccefiarily required upon that 6th and jth y
another Sign than that with which its. Seat is
marked in the general Signature of the Clef,
which marks all flat Keys with the leffcr 6th ^
and -jths ; and therefore in fuch Cafe (i. e.
where the principal Key is flat) this Difference
from the Clef is not a Sign that the Melody
leaves the '%jey^ becaufe each of thefe belong to
it in different Circumftances ; yet they , cannot
be both marked in the Clef therefore that
which is of more general Ufe is put there
and the other marked occasionally.
From what has been explained, you learn
another very remarkable Thing, viz. to know
what the principal Key of any Piece is, without
feeing one Note of it; and this is, done by know-
ing the Signature of the Clef : There are *but
Two Kinds of Keys (or Modes of Melody) dif-
tinguiihed into jharp and flat*, as already ex-
plained j each of which may have any of the
12 different Notes or Letters of the Jemitomck
Scale for its Fundamental ,• in the ifi' and 6th
Line of the upper Part of the preceeding Table
you have all thefe Fundamentals or ./u^-notes,
and under them refpe&ively (land tiie Sig-
natures, proper to each, in which, as has been
Z 4 the
36*0 ./f Treatise Chap. XL
often faid, the flat Keys have their 6th and 7th
marked of the leffer Kind; and therefore as by
the Key j or fundamental Note, we know the
Signature, fo reciprocally by the Signature we
can know the Key ; but 'tis under this one Li-
mitation that, becaufe one Signature ferves Two
Keys, ajharp one, and a flat , which is the 6th
above or 3d below the ffiarp one, therefore we
only learn by this, that it is one of them, but
not which ; for Example, if the Clef has no
tranfpofed Note but /3?, then the Key is g with
a %d g. or e with a 3d I. If the Clef lias ^
and ev> the Key is (/ with a 3 J g: or g. with a
3d /. as fo of others, as in the Table : I know
indeed, for I have found it fo in the Writing
of the beft Matters, that they are not ftricl: and
conftant in obferving this Rule concerning the
Signature of the C ef, efpecially when the prin-
cipal Key is aflat one ; in which Cafe you'll
find frequently, that when the 6th /. or qth I.
to the Key, or both, are tranfpofed Notes, they
don't fign them fo in the Clef ,but leave them to
be marked as the Courfe of the Melody requires ,-
which is convenient enough when the Piece is
fo conducted as to ufe the leffer 6th and pi
feldomer than the greater.
J 4- Of
§ 4- of MUSIC K.
§ 4, Of Tranfpofition*
HP HERE are Two Kinds of Tranfpofition,
-*- tjie one is, the changing the Places or
Seats of the Notes or Letters among the Lines
and Spaces, but fo as every Note be fet at the
fame Letter j which is done by a Change with
refpecl to the Clef : The other is the chang-
ing of the Key, or letting all the Notes of the
Song at different Letters, and performing it con-
fequently in different Notes upon an Jnjirument :
Of thefe in Order.
1. Of Tranfpofition with refpeffi to the Clef
This is done either by removing the fame
Clef to another Line 5 or by ufing another Clef;
but ftill with the fame Signature, becaufe the
Piece is ftill in the fame Key : How to fet the
Notes in either Cafe is very eafy : For the ift y
You take the firft Note at the fame Diftance
above or below the Clef-note in its new Por-
tion, as it was in the former Pofition, and then
all the reft of the Notes in the fame Relations
or Diftances one from another ,- fo that the
Notes are all fet on Lines and Spaces of the
fame Name. For the 2 d, or fetting the Mujlck
with
$6t ^4 Treatise Chap. XI.
with a different Clef, you muft mind that the
Places of the Three Clef-notes are invariaole in
the Scale, and are to one another in thefe Re-
lations, viz. the Means. $th above the Bafs ;
and the Treble • a cth above the Mean, and
confequenfcly Two zths above the Bafs : Now
when we would tranfpofe to a new C/c^fuppofe
from the Treble to the Mean, whereveer we
fet that new Clef, we fuppofe it to be the fame
individual Note, in the fame. Place of the Scale,
as if the Piece were that Part in a Compofiti-
011 to which this new Clef is generally appro-
priated, that fo it may direct us to the fame
individual Notes we had. before Transposition :
Now from the fixt Relations of the Three
Clefs in the Scale, it will be eafy to find the
Seat of the hrft tranfpofed Note, and then all
the reft are to be fet at the fame mutual Dis-
tances they were at before ; for Example, fup-
pofe the nrft Note of a Song is d, a 6th above
the Bqfs-clef the Piece being fet with thatCfe^,
if it is tranfpofed and fet with the Mean-clef,
then wherever that Clef is placed, the firft Note
muft be the id g. above it, becaufe a id g. a-
bove the Mean is a 6th g. above the Bafs-clef,
the Relation of thefe Two being a $th ; and
fo that firft Note will full be the fame indivi-
dual d: Again, let a Piece be fet with
the Treble-clef, and the firft Note be e, a
%d I. below the Clef, if we tranfpofe this
tq the Mean-clef, the firft Note muft be a
%d g. above it, which is the fame' individual
Note e in that Scale., for a 3 d L and 3d g.
make
§ 4- "f MUSICK. 363
make a $th the Diflancc of the treble and
//z£tf/z Clefs.
T h e Ufe and Dcfign of this Tranfpofitlon
is, That if a Song being fet with a certain Clef
in a certain Poiition, the Notes (hall go fir a-
bove or below the Syftem of Five Lines, they
may,*by the Change of the Place of the fame
Clef in the particular Syftem, or taking a nevv
C/^, be brought more within the Compafs of
the Five Lines : That this may be effected by
fuch a Change is very plain ,• for Example, Let
any Piece be fet with the Treble Qlef onthefirft
Line, (counting upward) if the Notes li& much
below the Clef Note, they are without the Sy-
ft,em, and 'tis plain they will be reduced more
'within it, by placing the Clef on any other
Line above ; and fo in general the fetting any Clef
lower in a particular Syftem reduces the Notes
that run much above it • and fetting it higher
reduces the Notes that run far below. The fame
is effected by changing the Clef it felf in fome
Cafes, tho' not in all, Thus, 11 the Treble Party
or a Piece fet with the Treble Clef, runs high a-
bove the Syftem, it can only be reduced by
changing the Place of the fame Clef ; but if it
run without the Syftem below, it can be redu-
ced by changing to the Mean or Bafs Clef. If
the mean Part run above its particular Syftem,
it will be reduced by changing to the Treble
Clef;, or if it ran below, by changing to the
Bafs Clef. Laftly. If the Bafs Part run with-
out its Syftem below, it can only be reduced by
changing the Place of the fame 'Clef ^but running
above
364- J Treatise Chap. XI.
above, it may be changed into the mean or
treble Clef. Now as to the Pofition of the new
Clef you mull choofe it fo that the Defign be
beft anfwered , and in every Change of the
Clef the Notes will be on Lines and Spaces of
the fame Name, or denominated by the fame
Letter, they refer alfo to the fame individual
Place of the Scale or general Syftem^ differing
only with refpecl: to their Places in the particu-
lar Syftem which depend on the Difference of the
Clefs and their Pofitions, and therefore will al-
ways be the fame individual Notes upon the
fame Inftrument.
As to both thefe Tranfpofitions I muft ob-
ferve^ that they increafe the Difficulty of Pra-
ctice, becaufe the Relations of the Lines and
Spaces change under all thefe Tranfpofitions ^
and therefore one muft be equally familiar with
all the Three Clefs^ and every Pofition of them,
fo that under any Change we may be able with
the fame Readinefs to find the Notes in their
true Relations and Diftances : And as this is
not acquired without great Application, I think
it is too cruel a Remedy for the Inconveniency
to which it is applied : It is better, I ftiould
think,to keep always the fame Clef for the fame
Party and the fame Pofition of the Clef -, but
if one will be Maft er of feveral Inftruments, and
be able to perform any Part, then he mufl be
equally well acquainted with all their proper
Clefs., but ftill the Pofition of the Clef in the
particular Syfie m may be fixt and invariable.
§ 4 . of MUSIC K. 3 <*5
2. Of Tranfpofition from one Key to another.
The Defign of this Tranfpofition is, That
a Song} which being begun in one Note is too
high or low, or any other way inconvenient,
as may be in fome Cafes for certain Inftruments,
may be begun in another Note, and from that
carried on in all its juft Degrees and Intervals.
The Clef and itsPofition are the fame, and the
Change now is of the Notes themfelves from one
Letter and its Line or Space to another. In
the former Tranfpofition the Notes were expreP
fed by the fame Letters, but both removed to
different Lines and Spaces > here the Letters
are unmoved, and the Notes of the Song are
transferred to or expreffed by other Letters,and
confequently fet alfo upon different Lines and
Spaces, which it is plain will require a diffe-
rent Signature of the Clef Now we are eaiily
directed in this Kind of Tranfpofition, by the
preceeding Table, Plate 2. Fig. 1. For there
we fee the Signature and Progrefs of Notes in
either Jharp or flat Keys beginning at every
Letter : The lower Line of the upper Part of
the Table contains the fundamental Notes of
the Twelve Jharp Keys ; and under them are
their Signatures, ftiewing what artificial Notes
are neceffary to make a concinnous diatonick
Series from thefe feveral Fundamentals : In
the 6th Line above are the fame Twelve Let-
tersjconfidered as Fundamentals of the Twelve
Jkt Ac?S;Which have the fame Signatures with
' . ' the
366 ^Treatise Chap. Xt
the (Jjarp Keys (landing in the under Line, and
in the fame Column : So that 'tis equal to make
any of thefe Twelve Notes the Key Note,
changing the Signature according to the Table:
And objerve, tho' the Fundamentals of the
Twche flat Keysft&nd in the Table as 6th s to
the Twelve Jharp Keys, yet that is not to be
underltood as if the flat Keys muft all be a 6th
above (or in their %<veS a 3 d below ) the Jlmrp
Keys i it happens fo there only in the Order
mid Relation of the Degrees of the Scale : But
as the Fundamentals of the Twelve flat Keys
are the fame Letters with thofe of the Jharp
Keys j they fhtw us that the fame Key may ei-
ther be the JJjarp or flat , with a different Sig-
nature.
But to make this Matter as plain as poflfible,
I fhall confider the Application of it in Two di-
ftinct Queftions. imo> Let the Fundamental
or Key Note to which you would tranfpofe a
Song be given, to find tile proper Signature*
Rmsi In the firft or 6th Line of the upper Part,
according as the Key is Jharp or flat, find the
given Key to which you would tranfpofe, and
under it you have the proper Signature. For
"Example, Suppofe a Song in the Jharp Key c,
which is natural, if you would tranfpofe it to
g, the Clef muft be figned with/*, or to d and
it muft have./* and c%. Again, fuppofe a Song
in aflat X^ey as d whofe Signature has b flat j£ you
tranfpofe it to e the Signature has f% or to g
and it has \/ and eh 2do* Let any Signature
be affigned to find the Key to which we. muft
tranf-
§ 4 . of MUSIC K. ^67
tranfpofe. Rule An the upper Part of the Table fn
the fame Column with the given Signature you'll
find the Key fought, cither in the iff or 6th
Line according as the Key hjharf or flat. But
without confidering the Key, or whether the
Signature be regular or not, we may know
how to tranfpofe by confldering the Signature as
it is and the rirft Note, thus, find the Signa-
ture witli which it is already fet, and in the
fame Column in the upper Part find the Letter
of the firft Note; in that fame Line (betwixt
Right and Left) find the Letter where you de-
fire to begin, and under it is the proper Signa-
ture to be now ufed : Or having chofen a cer-
tain Signature you'll End the Note to begin at,
in the fame Column, and in the fame Line with
the Note it began in formerly. Having thus
your Signature, and the Seat of the firft Note,
the reft are eafily fet up and down at the fmie
mutual Diftances they were in formerly j and
where any 2?, j/ or ^ is occafionally upon any
Note, mark it fo in the correfpondent Note in
the Tranfpofitioiiibut mind that if a Note with
a % or V is tranfpofed to a Letter which in the
new Signature is contrarily ^ or % then mark
that Note tj ; and reciprocally if a Note marked
tj is tranfpofed to a Letter, which is natural in
the new Signature, mark it % or k according
as the tf was the removing of a V or M in the
former Signature. In all other Cafes mark the
tranfpofed Note the fame Way it was before.
For Examples of this Kind of Tran/pofitionfos
Plate 3. Examples 3 and 5.
5 5. 0/
3^8 ^Treatise Chap* XL
§ 5' Of Sol-fa-ing, with fotiie ether particular
Remarks about the Names of Notes*
TN the fecdnd Column of the proceeding
* Table , you have thefe Syllables written a-
gainft the feveral Letters of the Scale, viz* fa,
fol, la, fa, fol, la, mi, fa, &c. Formerly thefe
Six were in ufe> viz. ut, re, mi, fa, fol^ la j
from the Application whereof the Notes of the
Scale were called G fit re ut, A la mi re, &c.
and afterwards a 6th was added, viz. fi ; but
thefe Four fa, fol, la, mi being only in Ufe a -
mong us at prefent, I (hall explain their Ufe
here, and fpeak of the reft, which are ftill in
Ufe with fome Nations, in Chap.14* where you
fhall learn their Original. As to their Ufe 5 it is
this in general ; they relate chiefly to Singing
or the human Voice, that by applying them to
every Note of the Scale it might not only be
pronounced more eafily, but principally that by
them the Tones and Semitones of the natural
Scale may be better marked out and diftingui-
fhed.
This Defign is obtain'd by the Four Syllables
fa, fol, la, mi, in this Manner $ from fa to fol
is a 7 one, alfo from fol to la, and from la to mi,
without diftinguifhing the greater m& feffo
§ y. vfMUSlCK. 369
Tone ; but from la to fa, alfo from mi to fa is
a Semitone : Now if thefe are applied in this
Order, fa, fol, la, fa, fol, la, mi, fa, &c.they
cxprefs the natural Series from c, as in the
Table i and if it is repeated to another 8 ve, we
fee how by them to exprefs all the Seven diffe-
rent Orders of Tones and Semitones within the
diatonick Scale. If the Scale is extended to
Two 8 yes, you'll perceive that by this Rule 'tis
always true, tho' it were further extended in in-
finitum, that above mi ftands fa, fol, la, and
below it the fame reverfed la, fol, fa ; and that
1 *me mi is always diftant from another by an
1 Offiave, (which no other Syllable is) becaufe
after mi afcending comes always fa, fol, la 9 fa>
fol, la, which are taken reverfe descending.
But now you'll ask a more particular Account
of the Application of this ; and that you may
underftand it, confider, the firft Thing in teach«
jng to fing is, to make one raife a Scale of
"Notes by Tones and Semitones to an Octave,
and defcend again by the fame Notes, and then
to rife and fall by greater Intervals at a Leap,
as a 3d, qth and 5th, dec. And to do all this
by beginning at Notes of different Pitch ; then
thefe Notes are reprefented by Lines and Spaces,
as above explained, to which thefe Syllables are
applied ; 'tis ordinary therefore, to learn a
Scholar to name every Line and Space by thefe
Syllables : But mil you'll ask, to what Purpofe ?
The Anfwer is, That while they are learning
to tune the Degrees and Intervals of Sound ex-
preft by Notes fet upon Lines and Spaces, or
A a learn-
370 ^Treatise Chap. XL
learning a Song to which, no Words are applied,
they may do it better by an articulate Sound;
and chiefly that by knowing the Degrees and
Intervals expreft by thcfe Syllables, they may
more readily know the true Diftance of their
Notes. I fhall firft make an End of what is to
be faid about the Application, and then {hew
>yhat an ufelefs Invention this is.
The only Syllable that is but once applied
in Seven Letters is mi, and by applying this to
different Letters, the Seat of the Two natural
Semitones in the 8<z^,expreffed by la-fa and mi-fa,
will be placed betwixt different Letters (which
is all we are to notice where the Difference of
the greater and leffer Tone is neglected, as in all
this) But becaufe the Relation of the Notes ex-
preft by the feven plain Letters, c, d, e, f, g,
a,, b, which we call the natural Scale, are fup-
pofed to be fixt and unalterable, and the De-
grees expreft by thefe Syllables are alfo fixt,
therefore the natural Seat of mi is faid to be b y
becaufe then mi fa and la-fa areapplied to the
natural Semitones b.c and e.f, as you fee in the
Table : But if mi is applied to any other of the
Seven natural Notes, then fome of the artifi-
cial Notes will be neceflary, to make a Series
anfwering to the. Degrees which we iuppofe are
invariably expreft by thefe Syllables ; but mi
may be applied not only to any of the Seven na-
tural Notes, it may alfo be applied to any of the
Five artificial Ones' : And now to know in any
Cafe (V. e, when mi is applied to any of tha
Twelve Letters of thefemitonick Scale)to what
Notes
§ j. cf MUSIC & ift
Notes the other Syllables are applied^ you need
but look into the proceeding Table, where if
yoii fuppofe mi applied to any Letter of that
line where it ftands, the Notes to which fa,
(bl> la are applied are found in the fame Ccf-
lumn with that Letter, and in the fame Line
with thefe Syllables. By this Means I hope you
have an eafy Rule ion Jbl-fa-ing, or naming the
Notes by fol, fa 9 &c. in any Clef and with an£
(Signature.
But now let us confider of what great Im-
portance this is, either to the underftanding or
practifing of Mufich In the firft Place, the
I Difficulty to the Learner is increafed by the
'Addition of thefe Names, which for every dif-
ferent Signature of the Clef are differently ap-
i plied ; fo that the fame Line or Space is in one
Signature called fa, in another fol, and fo oni
And if a Song modulates into a riew Key-> then
for every fuch Change different Applications
of thefe Names may be required to the fame
Note, which will beget much Confufion and
Difficulty : And if you would conceive the
whole Difficulty, confider, as there are 12 difc
fbrent Seats of mi in the Offiave, therefore the
naming of the Lines and Spaces of any partial-*
lar fyftem and Clef has the fame Variety ; and
if one muft learn to name Notes in every Clef
and every Pofition of the Clef then as there is
one ordinary Pofition for the Trehle-chf one
for the Ba/S) and Four for the meqtu if we ap-
ply to each of thefe the 12 differ at Signatures^
and c©nfequent Ways of fbl-j -mg$ we have
A a 1 hi
17% ^Treatise Chap. XL
in all 72 various Ways of applying the Names
of/0/, fa, &c. to the Lines and Spaces of a par-
ticular Syftem ; not that the fame Line can fea^e
72 different Names, but in the Order of the
Whole there is fo great a Variety : And if we
fuppofe yet inore'Pofitions of the Clefs, the Va-
riety will mil be increased, to which you muft
add what Variety happens upon changing the
Key in the Middle of any Song. Let us next
fee what the Learner has by this troublefom
Acquifition : After confidering it well, we find
nothing at all ; for as to naming the Notes,
pray what want we more than the Seven Let-
ters already applied, which are conftant and cer-
tain Names to every Line and Space under all
different Signatures ,the C/^being the fame and
in the fame Pofition ; and how much morefimple
and eafy this is any Body can judge. If it be
complained that the Sounds of thefe Letters are
harfh when ufed in railing a Series of Notes,
then, becaufe this feems to make the Ufe of
thefe Names only for thefofter Pronounciation
of a Note, let Seven Syllables as foft as poflible
be chofen arid joyned invariably to the Letters
or alphabetical Names of the Scale ; fo that as
the fame Line or Space is, in the fame Clef 'and
Pofition, always called by the fame Letter,
whether 'tis a natural or artificial Note, fo let
it be conftantly named by the fame Syllable ,-
and thuis we leave the true Diftance or Internal
to be found by the Degrees among the Lines
find Spaces, as they are determined by the Let-
ters appHedto them ; or ^ fince the In-
. § y. of MUSICK. J73
i. tervals are fufficiently determined by the alpha-
betical Names applied to the Lines and Spaces,
there is no Matter whether the fyllabical Names
be conftant or not, or what Number there be
of them, that is\ we may apply to any Note |t
random any Syllable that will make the Prc-
nounciation foft and eafy, if this be the chief
End of them, as I think it can only be, becaufe
the Degrees and Intervals are better and more
regularly expreft by the Clef and Signature :
Nay, 'tis plain, that there is no Certainty of a-
ny Interval expreft fimply by thefe Syllables,
without confidering the Lines and Spaces with
their Relations determined by the Letters ; for
Example, If you ask what Diftance there is
betwixtyb/ and /# ,the Queftion has different An-
{Wers/or 'tis either a Tone or a 5?/?,or one of thefe
compounded with 8#?,and fo of other Examples,
as are eafily fcen in the preceeding Scheme : But
if you ask what is betwixt /o/ in fuch a Line or
Space, and la in fuch a one above or below,
then indeed the Queftion is determined ; yet 'tis
plain, that we don't find the Anfwer by thefe
Names /tf,/o/, but by the Diftances of the Lines
and Spaces, according to the Relations fettled
among them by the Letters with which they
are marked.
I know this Method has been in Credit, and
I doubt will continue fo with fome People, who,
if they don't care to have Things difficult to
themfelves, may perhaps think it an Honour
both to them and their Art, that it appear my-
ftariouSi and fome fhrewd Gueflfers may poflibly
A a 3 ailed ^e
A Treatise Chap. XI,
alledge fomethiug elfe j but I (hall only lay that^
for the Reafons advanced, I think this an im-
pertinent Burden upon Muftck*
flirt her Reflections upon the Names of Notes*
A s there is a Neceffity, that the Prqgreflion
of the Scale of Mufick^ and all its Internals y
with their feveral Relations, fhould be diftiȣlh
ly marked, as is done by means of Letters re^
prefenting Sounds; fn it is neceffary for Practice,
that the Notes and Intervals of Sound upon
Inftruments fhould be named by the fame Let-
ters, by which we have feen a clear and eafy
Method of expreiling any Piece of Melody ', for
directing us how to produce the fame upon a
mufical lnftrument : But then obferve, that as
the Scale of Mufick puts no Limitations upon
the abfolute Degree of jTune, only regulating
the relative Meafures of one Note to another,
fo the Notes of Inftruments are called r, ^, &c.
not with refpe& to any certain Pitch of Tune y
but to mark diftin£tly the Relations of one Note
to another j and, without refpe£t to the Pitch
of the Whole, the fame Notes, 'k e. the Sounds
taken in the fame Part of the lnftrument, are
always named by the fame Letters, becaufe the
Whole makes a Series, which is conftantly in
the fame Order and Relation of Degrees. For
Exam fie ', Let the Four Strings of a Violin be
tuned as high or low as you pleafe, being al-
ways $ths to one another, the Names of the
Four open Notes are ftill called gya\3ft andfo of
• ~ J the
§ j. 'of MUSIC K. 37 y
I the other Notes ,• and therefore, if upon hearing
(any Note of an Inftrument we ask the Name of
[it 3 as whether it is c or d, &c. the Meaning
| can only be, what Part of the Inftrument is it
taken in, and with what Application of the
Hand ? For with refpecl to the abfo T ute Time
it cannot be called by one Letter rather than a-
nother, for the Note which is called c, accor-
ding to the forefaid general Rule, may in one
! Pitch of Tuning be equal to the Note called
d, m another Pitch.
But for the human Voice, confider there is
no fixt or limited Order of its Degrees, but an
Odfave may be raifed in any Order ,• therefore
the Notes of the Voice cannot be called c or d,
&c. in any other Senfe than as being unifon to
the Note of that Name upon a fixt Inftrument:
Or if a whole Offiave is raifed in any Order of
Tones and Semitones, contained within the dia-
tonick Scale, fuppofe that from c, each of thefe
Notes may be called c, d, &c. in fo far as they
exprefs the Relations of thefe Notes one to a-
nother. And laftly, With refpect to this Me-
thod of writing Mufick, when the Voice takes
Direction from it, the Notes muft at that Time
be called by the Letters and Names that di-
rect it in taking the Degrees and Intervals that
compofe the Melody ; yet the Voice may be-
gin ftill in the fame Pitch of Tune, whatever
Name or Letter in the Writing the nrft Note
is fet at, becaufe thefe Letters ferve only to
mark the Relations of the Notes : But in In-
ftrqments, tho the Time of the Whole may be
A a 4 highe*
37^ ^Treatisi Chap. XI.
higher or lower, the fame Notes in the Writing
direct always to the fame individual Notes with
refpect to the Name and the Place of the Inftru-
ment, which has nothing parallel to it in the
human Voice. Again, tho' the Voice and In-
ftruments are both dire&ed by the fame Me-
thod of Writing Mufick, yet there is one very
remarkable Difference betwixt the Voice and
fuch Inftruments as have fixt Sounds ; for the
Voice being limited to no Order of Degree s y
has none of the Imperfections of an Inftrument,
and can therefore begin in Unif'on with any
N6te of an Inftrument, or at any other conve-
nient Pitch, and take any Interval upward or
downward in juft Tune : And tho' the unequal
Ratios or Degrees of 'the Scale \ when the Sounds
are fixt, make many {mall Errors on Inftru-
ments, yet the Voice is not fubjected to thefe :
But it will be objected, that the Voice is directed
by the fame Scale^ whofe Notes or Letters have
been all along fuppofed under a certain deter-
minate Relation to one another,, which feems
to lay the Voice under the fame Limitations
with Inftruments having fixt Sounds, if it follow
the precife Proportions of thefe Notes as they
ftand in the Scale : The Anfwer to this is, That
the Voice will not, and I dare fay cannot pof-
fibly follow thefe erroneous Proportions g be-
caufe the true harmoniousDiftances are much ea-
fier takep,to which a good Ear will naturally lead:
Confider again, that becaufe the Errors arefmall
In a fingle Cafe, and the Difference of Tones
qx of Semitones fcarcs fenfible^ , therefore^ they
are
§ y. of MUSIC K. 377
are confidcred as all equal upon Inftruments %
and the fame Number or Tones or Semitones is,
every where thro' the Scale 9 reckoned the fame
or an equal Interval,and fo it muft pafs with fome
fmall unavoidable Errors. Now that the Voice
may be directed by the fame Scale or Syftem of
Notes, the Singer will alfo confider them as e-
qual, and in like manner take the fame Num-
ber for the fame Interval; yet, bytheDire&ion
of a well tuned Ear, will take every Interval in
its due Proportion, according to the Exigences
of the Melody ; fo if the Key is d p and the Three
rirft Notes of a Song were fet in d, e, £, the
Voice will take d-e a tg. and e-fafg. in order
to make d-f a true 3 d /. which is defective a
Comma in the So ale 9 becaufe d-e is a tL In a-
nother Cafe the Voice would take thefe very
Notes according to the Scale*, as here, fuppofe
the Key c y and the firft Three Notes f, d 9 f 9 the
Voice will take c-d sl tg, becaufe that is a more
perfect Degree than tL and then will take /not
a true id I to d, but a true 4^ to the Key c ?
which the Melody requires rather than the other,
whereby d-f is made a deficient 3d I ,• and \i
we fuppofe eis the third Note, and /the Fourth,
the Voice will take e a tl above J, in order to
make c-e a true 3d g. I cjon't pretend that thefe
fmall Differences are very fenfible in a Jingle
Cafe 3 yet 'tis more rational to think that a good
Ear leit to itfelf will take the Notes in the beft
Proportions, where there is nothing to deter-
mine it another Way, as the Accompany ment
of an Inftrument i and then it is demonfirated
by
378 ^Treatise Chap. XI.
by this, that in the beft tuned Inftruments ha-
ving fixt Sounds, the fame Song will not go
equally well from every Note ; but let a Voice
directed by a juft Ear begin unifon to any Note
of an Inftrument, there (hall be no Difference :
I own, that by a Habit of finging and ufing the
Voice to one Pitch of Tune^ it may become
difficult to fing out of it, but this is accidental
to the Voice which is naturally capable of fing-
ing alike well in every Pitch within its Extent
of Notes, being equally ufed to them all.
APPENDIX.
Concerning Mr. Salmon' j- Prof of al for reducing
all Mulick to One Clef.
*TTIS certainly the Ufe of Things that makes
•*• them valuable j and the more univerfal the
Application of any Good is, it is the more to
their Honour who communicate it : For this
Beafon, no doubt, it would very well become
the Profeffors of fo generous an Art as Mufick^
and I believe in every refpetfc would be their
Intereft, to ftudy how the Pra&ice of it might
be made as eafy and univerfal as poflible j and
to encourage any Thing that might contribute
towards this; End.
It will be eafily granted that the Difficulty
of Practice is much increafed by the Difference
of Clefs in particular Syftems, whereby the fame
Line or Space, 4', e, the firft or.fecond Line 3 (je*
§ ?. of MUSICK. 379
is fometimes called c, fometimes g : With re£
jpect to Infiruments 'tis plain ; for if every Line
and Space keeps not conftantly the fame Name,
the Note fet upon it muft be fought in a diffe-
rent Place of the Inftrument : And with refpcd
to the Voice, which takes all its Notes accor-
ding to their Intervals betwixt the Lines and
Spaces, if the Names of thefe are not conftant
•neither are the Intervals conftantly the fame in
every Place ,• therefore for every Difference
either in the Clef or Pofition of it, we have a new
Study to know, our Notes, which makes difficult
Practice, efpecially if the Clef fhould be chang-
ed in the very middle of a Piece, as is frequent-
ly done in the modern Way of writing Mufick.
Mr, Salmon reflecting on thefe Inconveniencies,
and alfo how ufeful it would be that all ihould
be reduced to one conftant CUj\ whereby the
fame Writing of any Piece of Mufick would e-
qually ferve to direct the Voice and all Inftru-
ments, a Thing one fhould think to be of very
great Ufe, he propofes in his E$ay to the Ad-
vancement of Mufick^ what he calls an univer-
fal Character, which I fhall explain in a few
Words. In the ift Place, he would have the
loweft Line of every particular Syftem conftant-
ly called g, and the other Lines and Spaces to
be named according to the Order of the 7 Let-
ters ; and becaufe thefe Pofitions of the Let-
ters are fuppofed invariable, therefore he thinks
there's no Need to mark any of them ,* but
then, 2do. That the Relations of fevera] Parts
of a CompofitiqiTi may be diftin&ly known ; he
marks
380 ^Treatise Chap. XI.
marks the Treble with the Letter T at the
Beginning of the Syftem ; the Mean with M.
and the Bafs with B. And *the gs that are on
the lowed Line of: each of thefe Syftems, he
fuppofes to be Offiaves to each other in Order.
And then for referring thefe Syftems to their
correfponding Piaces in the general Syftem, the
Treble g, which determines all the reft, muft
be fuppofed in the fame Place as the Treble
Clef of the common Method ,• but this Dif-
ference is remarkable, That tho' the g of the
Treble and Bafs Syftems are both on Lines in
the general Syftem, yet the Mean g, which is
on a Line of the particular Syftem, is on a Space
in the general one, becaufe in the Progreflion
of: the Sca.{e, the fame Letter, asg, is alternate-
ly upon a Line and a Space,- therefore the
Mean Syftem is not a Continuation of any of
the other Two, fo as you could proceed in Or-
der out of the one into the other by Degrees
from Line to Space, becaufe the g of the Mean
is here on a Line, which is neceffarily upon a
Space in the Scale; and therefore in referring
the mean Syftem to its proper relative Place in
the Scale, all its Lines Correfpond to Spaces,
of the other and contrarily ; but there is no
Matter of that if the Parts be fo written fe-
parately as their Relations be diftin&Iy known,
and the Pra&ice made more eafy ,- and when
we would reduce them all to one general Syftem,
it is enough we know that the Lilies of the
mean Part muft be changed into Spaces, and its
Spaces into Lines. $tio. If the Notes of any
Part
§ ;. of MUSIC K. 381
Part go above or below its Syftem, we may
let them as formerly 011 fhort Lines drawn on
Purpofe : But if there are many Notes together
above or below, Mr. Salmon propofes to reduce
them within the Syftcm by placing them on the
Lines and Spaces of the lame Name, and pre-
fixing the Name of the Offiave to which they
belong. To underftand this better, confider, he
has cholen three diftincl: Odfaves following one
another j and becaufe one O&tave needs but
4 Lines therefore he would have no more in
the particular Syftem j and then each of the
three particular Syftems expreflmg a diftincl:
Offiave of the Scale, which he calls the proper
Odfaves of thefe feveral Parts, if the Song ran
into another OEiave above or below, 'tis plain,
the Notes that are out of the OSfave peculiar
to the Syftem, as it ftands by a general Rule
marked T or M or B, may be fet on the fame
Lines and Spaces ,• and if the Octave they be-
long to be diftin&ly marked, the Notes may be
very eafily found by taking them an Odlave
higher or lower than the Notes of the fame
Name in the proper O Stave of the Syftem* For
J?xample> If the Treble Part runs into the;
middle or Bafs Offiave, we prefix to thefe Notes
the Letter M or B, and fet them on the fame
Lines and Spaces, for all the Three Syftems,
have in this Hypothefis the Notes of the fame
Name in the fame correfpondent Places ; if the
Mean run into the Treble or Bafs 06laves %
prefix the Signs T^or M. And hftly, Becaufe
£ta Pqrtj may comprehend more than 3 Ch
$ayc$
382, A Treatise Chap.
BaveS) therefore the Treble may run higher
than mQWci$fy and the BafS lower; in luch
. Cafes, the higher Offiave for the Treble may
be marked Tt. and the lower for the Bqf's
Bb. But if any Body thinks there be any con-*
fiderable Difficulty in this Method, which yet
I'm of Opinion would be far lefs than the chang-
ing of Clefs in the common Way, the Notes
may be continued upward and downward upon
new Lines and Spaces, occafionally drawn in
the ordinary Manner, and tho' there may be
many Notes far out of the Syfiem above or be*
low, yet what's the Inconveniency of this } Is
the reducing the Notes within 5 Lines, and
faving a little Paper an adequate Reward for
the Trouble and Time fpent in learning to per*
form readily from different Clefs ?
A s to the Treble and Bafs> the Alteration by
this new Method is very fmall j for in the com*
mon Pofition of the BaJ's-clef^ the loweft Line
is already g ; and for the Treble it is but re-
moving the g from the id Line, its ordinary Po-
rtion, to the firft Line; the greateft Innova-
tion is in the Parts that are fet with the c Ckf*
. And now will any Body deny that it is a
great Advantage to have an univerfal Character
in Mufick) whereby the fame Song or Part u of
any Compofition may, with equal Ea£S and
Readinefs be performed by the Voice or any
Inftrument $ and different Parts with alike
Eafe by the lame Inftrument ? 'tis true that each,
Part is marked with its own Q&ave^ but the
Deiign of this is only to mark the Relation of
the
§ 5 . efMVSlCK. , 385
the PiiYtSy that feveral Voices or Inftruments
performing thefe in a Concert may be directed
to take their firft Notes in the true Relations
which the Compoier defigned j but if we fpeak
of any one fingle Part to be fnng or per-
formed alone by any Inftrument, the Performer
in this cafe will not mind the Diftinction of
the part) but take the Notes upon his Inftru-
ment, according to a general Rule, which
teaches him that a Note in fuch a Line or Space
is to be taken in fuch a certain Place of the
Inftrument. You may fee the Propofal and the
Applications the Author makes of it at large
in his Effay, where he has confidered and an-
fwered the Objections he thought might be
raifed ; and to give you a fhort Account of them,
conlider, that beiides the Ignorance and Super-
ftition that haunts little Minds, who make a
Kind of Religion of never departing from recei-
ved Cuftoms, whatever Reafon there may be
for changing ; or perhaps the Pride and Vani-
ty of the greateft Part ofProfeffors of this Art,
joyned to a falfe Notion of their Intereft in
making it appear difficult, for the rational Part
of any Set and Order of Men is always the
leaft ; befides thefe, I fay, the greateft Difficul-
ty feems to be, the rendring what is already
printed Ufelefs in part to them that fhall be
taught this new Method, unlefs they are to
learn both, which is rather enlarging than leflen-
ing their Task: But this new Method is fo eafy,
and differs fo little in the Bafs and Treble Parts^
from what obtains already, that I think it would
add
384 ^Treatise Chap. XI.
add very little to their Task, who by the com-
mon Method, muft learn to fing and pJay from
all Clefs and Variety of Portions ; and then Time
would wear it out, when new Mufick were
printed, and the former reprinted in the Man^
ner propofed. Mr. Salmon has been a Prophet
in guefling what Fate it was like to have ; fork
lias lain Fifty Years negle&ed : Nor do I revive
it with any better Hope. I thought of nothing
but confidering it as a Piece of Theory, to ex-
plain what might be done, and inform you qt
what has been propofed. I cannot however
hinder my felf to complain of the Hardfhips of
learning to read cleverly from all Clefs and Po-
rtions of them : If one would be fo univerfally
capable in Mufick as to fing or play all Parts,
let him undergo the Drudgery of being Mafter of
the Three Clefs ; but why may not the Politions
be fixt and unalterable ?' And why may not the
fame Part be conftantly fet with the tame Clef
without the Perplexity of changing, that thole
who confine themfelves to one Inftrument, or
the Performance of one Part, may have no
more to learn than what is neceflary ? This
would favea great deal of Trouble that's but for-
rily recompensed by bringing the Notes within
or near the Compafs of Five Lines, which is all
can be alledged, and a very {illy Purpofe con-
fidering the Confequence.
CHAP,
§ i. of MUSIC K. 38;
c h a p. xir;
Of the Time or Duration of Sounds irk
Mufick.
§ 1. Of fhe Time in general, and its Suhdivi-
fion into abfolute and relative j and .particu-
larly of the Names, Signs, and Proportions,
or relative Meafures of Notes, as to Time,
WE are now come to the fecond general
Branch of the Theory of Mufick, which
• is to confider the Time 6*r Duration of Sounds
in the fame Degree of Tune*
TUNE "and TIME are the Affe&ions or Pro-
perties of Sound, upon whofe Difference or Pro-
portions Mufick depends. In each of thefe
iingly there are very powerful Charms : Where
the Duration of the Notes is equal, the Diffe-
rences of Tune are capable to entertain us with
an endlefs Variety of Pleafure, either jn an art-
■ B b full
$$6 ^Treatise Chap. XII.
fill and well ordered Succeffion of fimple Sounds,
which is Melody ', or the beautiful Flarmony of
Parts in Confonance: And of the Power of Time
alone, i.,e. of the Pleafure arifing from the va-
rious Meafures of long and JJoort, or fwift and
m floiv in the Succeffion of Sounds differing only in
Duration, we have Experience in a Drum,
which has no Difference of Notes as to Tune.
But how is the Power of Mufick heightned,
when the Differences of Tune and Time are art-
fully joined : 'Tis this Compofition that can
work fo irrefiftibly on the Paffions, to make
one heavy or cheerful ; it can be fuited to Occa-
iions of Mirth or Sadnefs ; by it we can raife,
and at leaft indulge, the folemn compofed
Frame of our Spirits, or fink them into a trifling
Levity : But enough for Introduction.
I n explaining this Part there is much lefs to
do than was in the former ; theCaufes and Mea-
fures of the Degrees of Tune, with the Inter-
vals depending thereon : And all their various
Connections and Relations, were not fo eafily
difcovered and explained,as we can do what re-
lates to this,which is a far more fimple Subject.
The Reafon or Caufe of a long or fhorfe
Sound is obvious in every Cale ; and I may fay,
in general, it is owing to the continued Impulf@
of the efficient Caufe, for a longer or fhorter
Time upon the fonorous Body,- for I fpeak her©
of the artful Duration of Sound. See Page 17.
where I have explained the Diftinction betwixt
natural and artificial Duration, to which I fhall
here
§ i. of MUSIC K. %tf
here add the Confederation of thofe Inftrumehts
that are ftruck with a Kind of inftantaneous Mo-
tion, as Harffichords and Bells, where the
Sounds cannot be made longer Or fhorter by
Art ,- for the Stroke cannot be repeated fo oft
as to make the Sound appear as« one continued
Note ; and therefore this is fupplied by the
Paufe and Diftance of Time betwixt the ftrik-
ing one Note and another, i, e. by the Quick-
nefs or Slownefs of their Succeflion; fo that long
and feort, quick and/Jozv are the fame Things
in Mufick ; therefore under this Title of the
Duration of Sounds., muft be comprehended
that of the Quicknefs or Slownefs of their Suc-
ceffion, as well as the proper Notion of Length
and Shortnefs : And fo the Time of a Note is
not computed only by the Uninterrupted Length
of the Sound, but alfo by the Diftance betwixt
the Beginning of one Sound and that of the next,
And mind that when the Notes are in the ftrift
Senfe long and fhort Sounds, yet fpeaking of
their SuccefTion we fay alfo, that it is quick or
flow, according as the Notes are fliort Or long %
which Notion we have by confidering the Time
from the Beginning of one Note to that of ano-
ther.
Next, as to the Meafiire of the Duration
of a Notej if we chufe any fenfibly equal Mo-
tion, as the Pulfes of a well adjtifted Clock or
Watch, the Duration of any Note may be mea-
fured by this, and we may juftly fay, that it
is equal to 2, 3 or 4^ (jc. Puifes ; andrif any o-
ther Note js compared to the fame Motion^
B b ? we
388 ^Treatise Chap. XII.
we fhall have the exacl: Proportion of the Times
of the Two, expreft by the different Number of
Pulfes. Now, I need give no Reafon to prove,
that the Time of a Note is juftly meafured by
the fucceOfive Parts of an equable Motion ; for
'tis felf-evident, that it cannot be better done ;
and indeed we know no other Way of mea-
furing Time, but by the Succefiion of Ideas in
our own Minds.
We come new to examine the particular
Meafures and Proportions of Time that belong
to Mufick ; for as in the Matter of Time, eve-
ty Proportion is not fit for obtaining the Ends
of Mufick T fo neither is every Proportion of
Time ; and to come clofe to our Purpofe, ob-
fcrve,
Time in Mufick is to be confidered either
with refpeel: to the abfolute Duration of the
Notes, u e. the Duration confidered in every
Note by it feJf, and meafured by fome external
Motion foreign to the Mufick; in refpeft of
which the Succeflion of the whole is faid to be
quick or flow : Or, it is to be confidered with
refpedl to the relative Quantity or Proportion
of the Notes, compared one with another.
Now, to explain theie Things, we muft 'firft
know what are the Signs by which the Time
of Notes is reprefented. The Marks and Cha-
racters in the modern Practice are thefe Six,
whofe Figures and Names you fee in Plate 2.
Fig. §#• And obfer've, when Two or more
Quavers or Semiquavers come together, they
are made with one or Two Strokes acrofs their
Ml
§ i. of MU SICK. 38?
Tails, and then they are called tied Notes.
Thefe Signs exprefs no abfolute Time, and are
in different Cafes of different Lengths, but their
Meafures and how they are determined, we
fhall learn again, after we have confidered.
The relative Quantity or Proportions of Time.
This Proportion I have fignified by Num-
bers written over the Notes or Signs of Time;
whereby you may fee a Semibreve is equal to
Two Minims, a Minim equal to Two Crotch-
ets, a Crotchet equal to Two Quavers, a Qiia-
ver equal to Two Semiquavers, a Semiquaver
equal to Two Demi-femi quavers. The Pro-
portions of Length of each of thefe to each o-
ther are therefore manifeft : I have fet over each
of them Numbers which exprefs all their mutu-
al Proportions j fo a Minim is to a Qiiaver as
\6 to 4, or 4 to i, i. e. a Minim is equal to
Four Quavers, and fo of the red. Now
thefe Proportions are double, (/. e. as 2:1) or \
compounded of feveral Doubles,fo 4 : 1 contains
2 : 1 twice ; but there is alfo the Proportion of
3 : 1 ufed in Mufick : Yet that this Part may be
as fimple and eafy as polfible, thefe Proportions
already ftated among the Notes, are fixt and in-
variable j and to exprefs a Proportion of 3 to 1
we add a Point (.) on the right Side of any
Note, which is equal to a Half of it, where
by a pointed Semibreve is equal to Three Mi-
nims, and fo of the reft, as yon fee in the
Figure. From thefe arife other Proportions, as
<pf 2 to 3, which is betwixt any Note ( as a
B b 3 Crotchet)
3po A Treatise Chap. XII.
Crotchet) plain, and the fame pointed » for the
plain Crotchet is Two Quavers, and the poiri-
ted is Three, Alfo we have the Proportion of
3 to 4, betwixt any Note pointed, and the
Note of the next greater Value plain, as betwixt
a pointed Crotchit and a plain Minim, And of
thefe arife other Proportions, but we need not
trouble our felves with them, fince they are not
diredly ufeful; and that we may know what are
fo, fuller me to repeat a little of what I have
faid elfewhere, viz, that
Things that are defigned to affeQ: our Senfes
muft bear a due Proportion with them ; and fo
where the Parts of any Object are numerous,
and their Relations perp r ext, and not eafiiy per-
ceived, they can raife no agreeable Ideas $ nor
can we eafily judge of the Difference of Parts
where it is great ; therefore, that the Proporti-
on of the Time of Notes may afford usPleafure,
they muft be fuch as are not difficultly percei-
ved : For this Reafon the only Ratios fit for
Mufick) befides that of Equality, are the double
and triple, or the Ratios of 2 to i and 3 to 1 $
of greater Differences we could not judge, with-
out a painful Attention ; and as for any other
Ratios than the multiple Kind ( i. e. which are
as 1 to fome other Number) they are ftill more
perplext. 'Tis true, that in the Proportions of
Tune the Ratios of 2:3, of 3 : 4, &c. produce
Concord i and tho' we conclude thefe to be the
Proportions, from very good Reafons, yet the
Ear judges of them after a more fubtil Manner $
pr rather indeed we are confcious of no fuch
Thing
§ i. of MVSICK. 391
Thing as the Proportions of the different Num-
bers of Vibrations that constitute the Interval!
of Sound, tho' the Agreeablenefs or Difagrec-
ablenefs of our Senfations feem to depend upon
it, by fc-me fecret Conformity of the Organs of
Senfewith the Impulfemade upon them in thefe
Proportions ; but in the Bufinefs of Time, the
good Effe£t depends entirely upon a diftincl;
Perception of the Proportions.
Now, the Length of Notes is a Thing
merely accidental to the Sound, and depends al-
together upon our Will in producing them: And
to make the Proportions diftin£t and perceiv-
able, fo that we may be pleafed with them,
there is no other Way but to divide the Two
Notes compared into equal Parts • and as this
is eafier done in multiple Proportions, becaufe
the fliorter Note needs not be divided, being
the Divifor or Meafure of the imaginary Paits
of the other, fo 'tis i till eafier in the firft and
more fimple Kind as 2 to 1, and 3 to 1 j and
the Neceflity of fuch fimple Proportions in the
Time is the more, that we have alio the In-
tervals of Tune to mind along with it. But ob-
ferve, that when I fay the Ratio of Equality,
and thofe of 2 to 3 and 3 to i, are the only
Ratios of Time fit for Mufick^ 1 do not mean
that there mufl not be, in the fame Song, Two
Notes in any other Proportion j but you muft
take it this Way, viz. that of Two Notes im-
mediately next other, thefe ought to be the Ra~
tios, becaufe only the Notes in immediate Suc-
ceffion are or can be directly minded, in pro-
B b 4 por-
39^ ^Treatise Chap. XII.
portioning the Time, whereof one being taken
at any Length, the other is meafured with rela-
tion to it, and fo on : And the Proportions of
other Notes at Diftances I call accidental Pro-
portions. Again cbferve^ that even betwixt Two
Notes next to other, there may be other Pro-
portions of greater Inequality, but then it is be-
twixt Notes which the Ear does not directly
compare, which are feparate by fome Paufe, as
the cn^ being the End of one Period of the
Song, and the other the Beginning of another g
or even when they are feparate by a lefs Paufe,
as a Bar ( which you'll have explained prefent-
ly.) Sometimes alfo a Note is kept out very
long, by connecting feveral Notes of the fame
Value,and directing them to be taken all as one,
but this is always fo ordered that it can be eafily
fubdivided in the Imagination, and efpeciaily by
the Movement of fome other Part going along,
which is the ordinary Cafe where thefe long
Notes happen, and then the Melody is in the
moving Party the long Note being designed on-
ly for Harmony to it ; fo that this Cafe is no
proper Exception to the Rule, which relates to
the Melody of fucceffive Sounds, but here the
Melody is transferred from the one Part to ano^
ther. And lajily, confider that it is chiefly in
brisk Movements, where neither of the Two
Notes is long, that no other Proportions betwixt
them than the fimple ones mentioned are admit-
ted.
§ i. of MSUICK. 393
5 2. Of the abfolutc Time; and the various
Modes, or Conftitution of Parts of a Piece
@f Melody, en which the different /firs in
Mufick depend, and particidnrlv of the Df-
tinblion of common and triple Time, and the
Dtfcription of the Chronometer j or meafur-
ing it,
ROM the Principles mentioned in the laft
Article, we conclude that there are cer-
tain Limits beyond which we muft not go 5
either in Swiftneft or Slovvnefs of Time, i.e.
Length or Shortnefs of Notes ; and therefore
let us come to Particulars, and explain the vari-
ous Quantities, and the Way of meafuring them.
In order to this we muft here coniidcr ano-
ther Application of the preceeding Principles,
which is, that a Piece of Melody being a Com-
position of many Notes fucceffively ranged, and
heard one after another, is divisible into fe-
veral Parts ; and oudxt to be contrived fo as the
levcral Members may be eafily diftingmfSied 5
that the Mind, perceiving this Connection of
Parts conftituting one Whole, may be delighted
with it ,• for 'tis plain where we perceive there
are Parts, the Mind will endeavour to Hiftin-
ginfh them, and when that cannot be eafily
done, we muft be fo far difappointed of our
Pleafure. Now aDivifion into equal Parts i^ of
all others, the moft fimple and eafily perceived ;
and in the prefent Cafe, where fo many othe/
JThings require our Attention, as the irious
Com-
394 <A Treatise Chap. XII.
Combinations of Tune and Time, no other Di-
vision can be admitted: Therefore,
Every Song is actually divided into a cer-
tain Number of equal Parts, which we call
Bars ( from a Line that feparates them, drawn
flratght acrofs the Staff, as you fee in Plate
i. ) or Meafure s, becaufe the Meafure of the
Time is laid upon them,or at leajl by means of
their Subdivifions we are affiftcd inmeafuring it,-
and therefore you have this Word Meafure ufed
fometime for a Bar, and fometime for the ab-
foliite Quantity of Time •> and to prevent Am-
biguity, I fhall afterwards write it in Italick
when I mean a Bar*
• B y faying the Bars are all equal I mean
that, in the fame Piece of Melody > they contain
each the fame Number of the fame Kind of
Notes, as Minims or Crotchets, &c. or that the
Sum of the Notes in each (for they are vari-
oufly fubdivided) reckoned according to their
Ratios one to another already fixt, is equal $
and every Note of the fame Name, as Crot-
chety &c. muft be made of the fame Time
through the whole Piece, confequently the
Times in which the feveral Bars are performed
are all equal ; fee the Examples of Plate 3.
But what that Time is, we don't yet know >
and indeed I muft fay it is a various and unde-
termined Thing. Different Purpofes, and the
Variety which we require in our Pleafures,
make it neceffary that the Meafures of a Bar±
or the Movement with refpe£t to quick and
flow, be m fome Pieces greater, and in others
lefler 2
§ i. of MU SICK. 39f
leflcr ; and this might be done by having the
Quantity of the Notes of Time fixt to a certain
Meafure, fo that wherever any Note occurred
it fhonld always be of the fame 7 me; and then
when a quick Movement were deiigned, the
Notes of fhorter lime would ferve, and the
longer for a ilow Time ; and for determining
tilde Notes .we might ufe a Pendulum of a cer-
tain Length, whole Vibration being the fixt
Meafure of any one Note, that would determine
the reft ; and it would be beft if a Crotchet
were the determined Note, by the Subdivibon
or Multiplication whereof, we could eaiily mea-
fure the other Notes • and by Practice we might
eaiily become familiar with that Meafure ,' but
as this is not the Method agreed upon, tho*
it fecms to be a very rational and eafy one, I
pall not infift upon it here.
In the prcfent Practice, tho' the fame Notes
of Time are of the fame Meafure in any one
Piece, yet in different Pieces they differ very
much, and the Differences are in general mark-
ed by the Words Jloiv, brisk^ fmft^ &c. writ-
ten at the Beginning ; but ftill thefe are uncer-
tain Meafures, fincc there are different Degrees
onflow and fwift ; and indeed the true Deter-
mination of them mult be learnt by Experience
from the Practice of Muficians - r yet there are
fome Kind of general Rules commonly delivered
to us in this Matter, which I fliill (hew you,
and at the fame Time the Method ufed for
affifting us to give each Note its true Propor-
$ion<> according to the Meafure or determined
Quantit
3$6 A TkEATisfe Chap. XIT.
Quantity of Time, and for keeping this equal
thro* the Whole. But in order to this, there
is another very confiderable Thing to be learnt)
concerning the Mode or Conflitution of the
Me af ure , and firft dbferve, That I call this Dif-
ference in the abfolute Time the different
Movements of a Bteee, a Thing very difnnd
from the different Meafure or Conflitution of
the Bar, for feveral Pieces may have the fame
Meafure, and a different Movement. Now by
this Conflitution is meant the Difference with
refpecl: to the Quantity of the Meafure, and the
particular Subdivifion and Combination of its
Parts ; and by the total Quantity, I underftand
that the Sum of all the Notes in the Meafure
reckoned according to their fixt Relation, is e-
qual to fome one or more determined Notes,
as to one Semibreve or to Three Minims or
Crotchets ; &c. which yet without fome other
Determination is but relative : And in the Sub-
divifion of the Meafure the Thing chiefly con-
fidered is, That it is diviiible into a certain
Number of equal Parts, fo that, counting from
the Beginning of the Meafure, each Part fhall
end with a Note, and not in the Middle of one
(tho* this is alfo admitted for Variety ;) for -Eyy-
ample, if the Meafure contain 3 Minims, and
ought to be divided into Three equal Parts,
then the Subdivifion and Combination of its
lefler Parts ought to be fuch, that each Part,
counting from the Beginning, fhall be compofed
of a precife Number of whole Notes, without
breaking in upon any Note ; fo if the firft Note
were
§ 2. of MUSIC K. 397
were a Crotchety and the fecond a Mining we
could not take the firft 3 J Part another Way
than by dividing that Minim,
We confidered already how neceffary it
is that the Ratios of the Time of fuccetfivc
Notes be fimpJe, which for ordinary are only
as 2 to 1, or 3 to i, and in any other Cafes
are only the Compounds of thefe Ratios^ as 4
to 1 ; 10 in the Conftitntion of the Meafure>
we are limited to the lame Ratios^ i. e. the
Meafures are only fobdivided into 2 or 3 equal
Parts ; and if there are more,they muft be Mul-
tiples of thefb Numbers as 4 to 6, is compol-
fed of 2 and 3 ; again obferve, the Meq/ures of
feveral Songs may agree in the total Quantity,
yet differ in the Subdivifion and Combination
of the leffer Notes that fill up the Meafure \ alfo
thofe that agree in a fimilar or like Combination
or Subdivifion of the Mea/ure y may yet differ in
the total Quantity. But to come to Particulars*
Of common and triple Time.
These Modes are divided into Two general
Kinds, which I {hall call the common and trifle
Mode, called ordinarily common and triple Time,
1 . COMMON TIM Eis of Two Species ,• the
ift where every Meafure is equal to a Semibreve^
or its Value in any Combination of Notes of a
leffer relative Quantity ; the 2 d> where every
Meafure is equal to a Mining or its Value in
leffer Notes. The Movements of this Kind of
Meafure are very various ,- but there are Three
comnioa Diftin&ions, the firft is flow 7 iignified
at
398 ^Treatise Chap.XIT*
at the Beginning by this Mark C, the id is
frisk, fignified by this (£, the 3d is very quick
figniried by this .J),- but what that flow, brisk,
and quick is, is very uncertain, and, as I have
faid already, muft be learned by Practice .* The
neareft Meafure I know, is to make a Pjiavefr
the Length of the Pulfe of a good Watch^ and
fo the Crotchet will be equal to 2 Pulfes, a Mi-
nim equal to 4, and the whole Meafure or
Semibrcve equal to. 8 Pulfes; and this is very
near/ the Meafure of the brisk common Time,
the flow Time being near as long again, as the
quick is about half as long. Some propofe to
meafure it thus, viz. to imagine the Bar as actu-
ally divided int04 Crotchets in the firft Species,
and to make the whole as long as one may dif-
• tindly pronounce thefe Four Words, One, two,
three, jMr, .all of equal Lengthy fo that the
firft Crotchet may be applied to One, the id to
Two, &c. and for other Notes proportionally ;
and this they make the brisk Movement of
common Timei and where the Bar has but
Two Crotchets, then 'tis meafured by one, two :
But this is ftiil far from being a certain Mea-
fure. I fhall propofe feme other Method pre-
sently, mean while
Let us fuppofe the Meafure or Quantity
fixt, that we may explain the ordinary Method
praCtifcd as a HeJp for perferving it equal thro'
the whole Piece.
The total Meafure of common Time is e-
qual to a Semibre-ve or Minim, as already faid •
but thefe are varioufly fubdjvided into Notes of
lefler
§ i. of MUSIC K. i99
lciTer Value. Now to keep the Time equal, we
make ufe of a Motion of the Hand, or Foot (if
the other is employed,) thus ; knowing the true
Time of a Crotchet, we (hall fuppofe the Mea-
fare actually fubdivided into 4 Crotchets for the
firft Species, and the half Meafure will be 2
Crotchets,thei'cibre the Hand or Foot being up,
if we put it down with the very Beginning of
- the firft Note or Crotchety and then raife, it
with the Third, and then down with the Be-
ginning of the next Meafure, this is called
Beating the Time ; and by Practice we acquire
a Habit of making this Motion very equal, and
confequently of dividing the Meafure in Two
equal Parts : Now whatever other Subdivifiofi
the Meafure confifts of, we muft calculate, by
the Relation of the Note?, where the firft Half
ends, and then applying this equable Motion of
the Hand or Foot, we make the tirftas long as the
Motion down (or as the Time betwixt its being
down and raifed again,for the Motion is frequent-
ly made in an Ihftant ; and the Hand continues
down for fome Time,) and the other Half as long
as the Motion up (or as the Hand remains up J
and having the half Meafure thus determined.
Practice very focn learns us to take ail the
Notes that compofe it in their true Proportion
one to another, and fo as to begin and end
them precifely with the beating. In the Meafure
. of Two Crotchets, we beat down the firft and
the fecond up.
OBSERVE, That fome call each HaJf
©f the Meq/ure 9 in common Time, ATimr;
and
4oo A Treatise Chap. XII.
and fo they call this the Mode or Meafure of
Two Times, or the Dupla-meafure. Again
you'll find fome mark the Me aj tire of Two
Crotchets with a 2 or J, fignifying that 'tis e-
qual to Two Notes, whereof 4 make a $?m/-
iretf? ; and fome alfo marked | which is the'
very fame Thing, i, e. 4 Quavers.
2. TR IPLE TIME confifts of many dif-
ferent Species, whereof there are in general 4,
each of which have their Varieties under it ,-
and the common Name of Triple is taken
from this, that the Whole or Half Meafure
is diviiible into 3 equal Parts, and fo beat.
The ift Species is called the fimple Triple?
whofe Meafure is equal either to 3 Semibreves?
to 3 Minims ?ot to 3 Crotchets?or to 3 Quavers?
or laftly to 3 Semiquavers > which are mark-
ed thus, viz. \ or \ or \ | ^ but the laft is not
much ufed, nor the firft, except in Church-mu-
fick. The Meafure in all thcfe, is divided into
3 equal Parts or Times*, called from that pro-
perly Triple-time, or the Meafure of 3 Times?
whereof 2 are beat down, and the 3^ up.
The %d Species is the mixt Triple: its
Meajure is equal to 6 Crotchets or 6 Quavers
or 6 Semiquavers, and accordingly marked f or
I or ,|, but the laft is feldom ufed. Some Au-
thors add other Two, viz. 6 Semibreves and
6 Minims, marked f or - but thefe are not in
ufe. The Meafure here is ordinarily divided
into Two equal Parts or Times? whereof one is
beat down, and one up ; but it may alfo be di-
vided into 6 Times? whereof the iirft Two are
beat
i, of MUSIC K, 40 1
beat down> and the yd up, then the next Two
c^own and the I-iit up, that is, beat each Half
of the Meafure like the fimple Triple ( upon
"which Account it may alfo be called a compound
Triple,) and becaufe it may be thus divided
either into Two or 6 Times (V. e. Two Triples)
tis called mixt, and by fome called the Mea-
fure of 6 Times.
The %d Species is the compound Triple*, con*
lifting of 9 Crotchets, or Quakers or Semiqua-
vers marked thus -J, -f, 4 * the ^ r ^ afi d the
3aft are little ufed, and fome add $ 4 which are
never ufed. This Meqfure is divided either in-
to 3 equal Parts or Times, whereof Two are
beat down and one up $ or each Third Part
of it ,may be divided into 3 Times, and bea,t
like the fimple Triple, and for this *tis called the
Meafure of 5? Times*
The 4th Species is a Compound of the 2d
Species, containing 1 2 Crotchets or Quavers or
Semiquavers marked $ ^ %, to which fome
add ~ and ~ that are not ufed; nor are the
ift and 3^ much ija Ufe> elpecially the 3^
The Meafure here may be divided into Two
Times, and beat one down and one up ; Or eacfo
Half may be divided and beat at the id Species*
either by Two or Three* in which Cafe it will
make in all 1 2 Times, hence called the Meafure
of 1 2 Times. See Examples of the moll or-
dinary Species in Plate 3d*
Now as to the Movement of thefe feveral
Kinds of Meafure s both duple and triple*, *tis
various and as I have faid, it muft be learned
C c by
40 2 A T r e a t;A s e Cha p. XIL
by Practice"; yet ere I leave this Part, I fhall
make, thefe general Obf er vat ions, Firfi, That the
■Movement in every Piece is ordinarily marked by
fuch Words' as JIozv,Jb:ift, &c. But becaufe the
Italian Compofitions are the Standard and Model -
of the better Kind of modern Mufick, I fhall ex-
plain the Words by which they mark their Move-
ments, and which are generally ufed by all others
in Imitation of them : They have 6 common Dif-
tinfitions of Tl'w^cxprefTed by thefe Wovds,grave,
adagio, largo,, vivace, allegro, prejio, and fome-
times preftiffimo. The firft exprelfes the floweft
Movement, and the reft gradually quicker ; but
indeed they leave it altogether toPraclice to de-
termine the precife Quantity, ido. The Kind
of Meafure influences the Time expreft by thefe
Words, in refpect of which we find this gene-
rally, true,, that the Movements of the fame
Name, as adagio or allegro, &c. are fwifter in
trifle than in common Time, ^tio. We find
common Time of all thefe different Movements;
but in the triple, there are fome Species that
- are more ordinarily of one Kind of Movement
than another : Thus the triple f is ordinarily a-
dagio, fometimes vivace ; the f is of any Kind
from adagio to allegro ; the § is allegro, or vi-
• vace , the \A i are more frequently allegro ;
the '•£ is fometimes adagio but oftner allegro. Yet
after all, the allegro of one Spescies of triple is
. a quicker Movement than that of another, fo
very uncertain thefe Things are.
There is another very eonfiderable Thing !
to be minded here, viz, that the Air or Hu-
mour
§ a if MUSIC K 40$
mour of a Song depends very much upon thefo
different Modes of Time, or Con flit ut ions of the,
Meafure, which joined with the Variety of
Movements that each Mode is capable of,makes
this Part of Mufick wonderfully entertaining ;
but we muft be acquainted with practical Mu-
fick to underftand this perfectly ; yet the follow-.
ing general Things concerning the Species of
Triple, may be of fome Ufe to remark.
1 mo. A s to the Differences in eachSpccies,fucri
a$|j !'* \ in the jfimple triple, there is more Ca-
price than Reafon ; for the fame Piece of Me-
lody may be fet in any of thefe Ways without
loling any Thing of its true Air, fince the Rela-
tion of the Notes are invariable, and there is
no certain Quantity of the abfoluteTime, which
is left to the arbitrary Direction of thefe Words^
ndagio, allegro^, &c.
2 do. Of the feveral Species of triple, there
are fome that are of the fame relative Mea-,
Jure, as |. |i F | ; and f. §• thefe are fo far of
the fame Mode as the Meafure of each contains
the fame total Quantity ; for Three Minims and
Six Crotchets and Twelve Quavers are equals
and fo are Three Crotchets equal to Six Quaver s\
but the different Conftitutions of the Meafure,
with refpeet to the Subdivifions and Connections
of the Notes, make a fnbft remarkable Diffe-
rence in the Air : For Example, The Time of
(eonfifts generally o£ Minims, and thefe fome-
times mixt with Semibreves or with Crotchets^
and fome Ears will be all Crotchets ; but
contrived. fo that the Air requires the
• ;i C c z Meaj uve
404 ^Treatise Chap. XII.
Me a fur e to be divided and beat by Three Times ^
and will not do another Way without mani-
festly changing and fpoiling the Humour of the
Song: Suppofe we would beat it by Two Times,
the firft Half will always ( except when the
Meafure is actually divided into Six Crotchets,
which is very feldom) end in the Middle, or
within the Time of fomeNote; and tho' this is
admitted fometimes for Variety (whereof after-
wards) yet it is rare compared with the general
Rule, which is, to contrive the Divifion of the
Meafure fo that every Down and Up of the
Beating fliall end with a particular Note ; for
upon this depends very much the Diftinctnefs
and, as it were, the Senfe of the Melody ; and
therefore the Beginning of every Time, or Beat-
ing in the Meafure, is reckoned the accented
Part thereof. For the Time ~ it confifts of
Crotchets fometimes mixt with Quavers, and
even with. Minims, but fo ordered that 'tis ei-
ther dupla or tripla, as above explained, which
makes a great Difference in the Air. The
Time $ is alfo mixt of dupla and tripla, and
confifts generally of Quavers, and fometimes of
Crotchets, but thefe are tied always by Three ;
and we have the Bar frequently compofed of
Twelve Quavers tied Three and Three 5
which, if we fhould ty Two and Two, would
quite alter the Air : The Reafon is, That m
this Mode there are in each Bar Four remark-
ably accented Parts, which are diftant from each
other by Three Quavers; and the true Reafon
©f tying the Quavers in that manner, feems to
me
§ 2. of MUSIC K. 40 y
me to be, the marking out thefe diftincl: Parts of
the Meafure -, but when the Quavers are tied in
even Numbers by Two or Four, or by Six, it
fuppofes the Accent upon the \ft, 3^, and $th
Quaver > which gives another Air to the Melo-
dy, and always a wrong one, when the skilful
Compofer defigned it otherwife. The fame
Reafons take place in the Difference of thefe
Times %•%; the fiirfi confifts more ordinarily of
Crotchets, and Quavers tied in even Numbers,
becaufe 'tis divided into Three Parts or Times ;
but the other is rnixt of duplet and tripla, and
therefore 'tis tied in Threes, unlefs it be fub-
divided into Semiquavers, and then thefe are ti-
ed in even Numbers, becaufe Two Semiqua-
vers make a Quaver.
Again, there is another Queftion to be con-
fidered here, viz. What is the real Difference
betwixt ■* and |, and betwixt |, { and '| ? The
Lengths of the feveral Strains, or more general
Periods of the Song, depend upon thefe, which
make a' considerable Difference ; but their
principal Difference lies in the proper Move-
ments of each, and a certain Choice of the fuc-
ceffive Notes that agree only with that Move-
ment ; fo - is always allegro, and would
have no agreeable Air if it were performed a-
dagio or largo : Another Thing is, that the Be-
ginning of each Bar is a more diftinct and accen-
ted Part than the Beginning of any Time in the
Middle of a Bar, and therefore if we (hould
take a Piece fet ~, and fubdivide its Bars to
make it ~ 3 there would be Hazard of feparat-
C 2 3 ing
406 -^Treatise Chap. XII.
ing Things that ought to ftand in a clofer Con-
nexion ; and if we put Two Bars in one of a
Piece fet & to make it ~, then we fhould joyn
Things that ought to be diftinct : But I doubt I
have already faid more than can be well under-
{lood withe ut fome Acquaintance with the
Practice ; yet there is one Thing I cannot omit
here, ®iz. that in commonTime we have in fome
Cafes Quavers tied by Threes, and the Num-
ber 3 written over them, to fignify that thefe
Three are only the Time of other Two Qua-
kers of that Meafure.
Observe, in explaining what a Bar or
Meafure is, I have faid that all the Meafures
of the fame Piece of Melody or Song, are of
equal relative Value ; and the Differences in this
refpeft are brought under theDiftin&ionof diffe-
rent Modes and Species ; but that is taking the
Unity of the Piece in the ftricteft Senfe. We have
alfo a Variety of fuch Pieces united in one prin-
cipal Key, and fuch an Agreement of Air as is
confident with the different Modes of Time-, and
fuch a Compofition of different Airs is called, in
a large Senfe, one Piece o£ Melody, under the
general Name of Sonata if 'tis defigned only for
Inftruments, or Cantata if for the Voice j and
thefe feveral leffer Pieces have alfo different
Names,fuch as Allemanda, Gavottajkc. (which
are always common Time) Minuet, Sarabanda,
@iga> Corrante, Siciliana, &c. which are triple
Time.
Of
§ 2. of MUSICK. 407
Of the CHRONOMETER.
I have fpoken a little already of the meafur-
ing the ahfoluie Time^ or determining the Move-
ment of a Piece by means of a Pendulum^ a Vi-
bration ofwhich being applied to any one Note,
as a Crotchety the reft might be eaiily determi-
ned by that. Monfieur Loulie in his Elemens^
ou PrincipeS' de Mufique^ propofes for this Pur-
pofe a very limple and eafy Machine of a Pen-
dulum^ which he calls a Chronometer; it
confifts of one large Ruler or Piece of Board,
Six Foot or Seventy Two Inches long, to be fet
on End ; it is divided into its Inches, and the
Numbers fet ib as to count upward ; and at ever
ry Divifion there is a fmall round Hole, thro'
whofe Center the Line of Divifion runs. At
Top of this Ruler, about an Inch above the
Divifion 72, and perpendicular to the Ruler is
infertedafmall Piece of Wood, in the upper Side
of which there is a Groove,hollowed along from
the End that (lands out to that which is fixt in
the Ruler, and near each End of it a Hole is
made : Thro' thefe Holes a Pendulum Chord is
drawn, which runs in the Groove ; at that End
of the Chord that comes thro' the Hole furtheft
from the Ruler the Ball is hung, and at the o-
ther End there is a fmajl wooden Pin which can
be put in any of the Holes of the Ruler ; when
the Pin is in the upmoft Hole at 72, then
the Pendulum from the Top to the Center of
C c 4 , the
4o8 ^Treatise Chap. XII
the Ball, niuft be exaclly Seventy Two In-
ches j and therefore whatever Hole of the
Ruler it is put in, the Pendulum will be juft fo
many Inches as that Figure at the Hole de-
notes. The Ufe of this Machine is j the
Vtompofer lengthens or fhortens his Pendu-
lum till one Vibration be equal to the de-^
figned Length of his Bar, and then the Pin
{rands at a certain Divifion, which marks the
Length of the Pendulum ,• and this Number
being fet with the Clef, at the Beginning of the
Song, is a Direction to others how to ufe the
Chronometer in meafuring the Time according
to the Compofer's J2eiign ; for, with the Num-
ber is fet the Note (Crotchet or Minim) whofe
Value he would have the Vibration to be $
which in brisk common Time is beft a Minim
or half Bar , or even a whole Bar when that is
but a Minim, and in flow Time a Crotchet :■'
In triple Time it will do well to be the id Part a
or Half or qth Part of a Bar ; and in the
Jimple Triples that are allegro, let it be a whole
Bar. And if in every Time that is allegro, the
Vibration is applied to a whole or half Bar,
Practice will teach us to fubdivide it juftly and
equally. And mind, to make this Machine of
univerfal Ufe, fome canonical Meafure of the
Divisions mult be agreed upon, that the Figure
may give a certain Dire&ion for the Length of
the Penduttim A
§?. Cm-
§ §. of MUSIC fC 409
i§ 9. Concerning Refts or Paufes o/Time ; rW
fome other neceffary Marks in writing Mu-
fick.
AS Silence has very powerful Effects in Ora-
tory, when it is rightly managed, and
brought in agreeable to Circumftances, fo in
Muftvk) which is but another Way of exprelfmg
and exciting Paflions, Silence is fometimes ufed
1 to good Purpofe : And tho it may be neceffary
iiri a fingle Piece of Melody for exprelfmg fome
I Paifibn, and even for the Pleafure depending on
Variety, where no Paffion is directly minded,
1 yet it is ufed more generally in Jymphonetick
i Compoiitions ; for the fake of that Beauty and
I pieafnre we find in hearing one Part move on
while another refts, and this interchangeably;
which being artfully contrived, has very good
Effects. But my Bufinefs in this Place is only to
let you know the Signs or Marks by which this
$ilenee is exprefted.
T a e s e Refts are either for a whole Bar,
1 or more than one Bar, or but the Part of
; a Bur ■: When it is for a Part of a Bar,
1 then it is expreffed by certain Signs corre-
fponding to the Quantity of certain Notes of
Time, as Minim, Crotchet, &c. and are ac-
cordingly called Minim-refts, Crotchet-reft s, &c.
See their Figure in Plate 2. Fig* 3. where the
Note and cbrrefponding Kelt are put together j
and
410 -^Treatise Chap. XII.
and when any of thefe occur either on Line or
Space, for 'tis no Matter where they are fet,
that Part is always filent for the Time of a Mi-
nim or Crotchet, &c. according to the Nature of
the Reft. A Reft will be fometimes for t a
Crotchet and Quaver, or for other Quantities of
Time.) for which there is no particular Note;
in this Cafe the Signs of Silence are not multi-
plied or made more difficult than thofe of Sound,
but fnch a Silence is marked by placing toge-
ther as many Refts of different Time as make
tip the whole defigned Reft; which makes the
Practice more eafy, for by this we can more rea-
dily divide the Meafure, and give the juft Al-
lowance of Time to the Refts : But let Practice
fatisfie you of thefe Things.
When the Reft is for a whole Bar, then the
Semibreve Reft is always ufcd, both in common
and triple Time. If the Rei\ is for Two Mea-
tures^ then it is marked by a Line drawn crofs
a whole Space, and crofs a Space and an Half
for Three Meajures, and crofs Two Spaces for
Four Meafures; andfo on as you fee marked in
the Place above directed. But to prevent all
Ambiguity, and that we may at Sight know the
Length or the Reft,- the Number of Bars is
ordinarily written over the Place where thefe
Signs ftand.
IJknow fome Writers fpeak differently about
thefe Refts, and make fome of them of different
Values in different Species of triple Time : For
Examp le, they fay, that the Figure of what is
the Minim-reft in common Time y expreflfes the.
Reft
§ 3- of MUSIC K. 4"
Reft of Three Crotchets ; and that in the
Trifles | ,| l \ || it marks always an half MeqfurL
however different thefe are among thcmfclvcs :
Again, that the Reft of a Crotchet in common
JTime is a ite/? of Three Quavers in the Triple
|, and that the Quaver-reft of common Time is
equal to Three Semiquavers in the trifle J. But
this Variety in the Ufe of the fame Signs is
pow generally laid afide, if ever it was much in
Fafhion •> at leaft there is a good Reafon why
it ought to be out, for we can obtain our End
ealier by one conflant Value of thefe Marks
of Silence, as they are above explained.
There are fome other Marks ufed in writ-
ing of Mufic% which I fhall explain, all of
which you'll rind in Plate 2. A Jingle Bar is
a Line acrofs the Staff, that feparates one Mea-
sure from another. A double Bar is Two pa-
rallel Lines acrofs the Staff, which feparates the
greater Periods or Strains of any particular or
Jimple Piece. A Repeat is a Mark which fig-
nines the Repetition of a Part of the fiece,-
which is either of a whole Strain, and then the
Rouble Bar, at the End of that Strain, which is
repeated, is marked with Points on each Side
pf it ; and fome make this the Rule, that if
there are Points on both Sides, they direct to a
Repetition both of the proceeding and following
Strain, i. e, that each of them are to be "j^ay'd
or fung twice on End j but if only one of thefe
' Strains ought to be repeated, then there yrroft
be Joints only on that Side, i. e* on tlje left, if
it
412 A Treatise Chap. XII.
it is the preceeding, or the Right if the fol-
lowing Strain: When only a Part of a Strain is to
be repeated, there is a Mark fet over the Place
where that Repetition begins, which continues
to the Jlnd or the Strain.
A Direct is a Mark fet at the End of a Staff,
cfpecially at the Foot of a Page, upon that Line
or Space where the firft Note of the next Staff
is fet.
You'll find a Mark, like the Arch of a
Circle drawn from one Note to another, com-
prehending Two or more Notes in the fame
or different Degrees ; if the Notes are in dif-
ferent Degrees, it (ignifies that they are all to be
fung to one Syllable, for Wind-inftruments that
they are to be made in one continued Breath,
and for ftringed Inftruments that are ftruck with
a Bow, as Violin, that they are made with one
Stroke. If the Notes are in the fame Degree,
it (ignifies that 'tis all one Note, to be made
as long as the whole Notes fo conneded ,• and
. this happens moft frequently betwixt the laft
Note of one Bar and the firft of the next,
which is particularly called Syncopation, a Word
alfo applied in other Cafes : Generally, when
>&ny Time of a Mea/hreends in the Middle of a
Note, that is, in common Time, if the Half or
any of the qth Parts of the Bar, counting from
Beginning, ends in the Middle of a Note, in the
jtmpk Treble if any 3^ Part of the Meafure
..ends within a Note, in the compound Treble if
any 9th Part, and in the Two mixt Triple s^
if any 6th or 1 ith Part ends in the Middle of
any
§ 3. of MUSIC K. 413
any Note, 'tis called Syncopation^ which pro-
j periy fignifies a ftriking or breaking of the Time y
\ becaufe the Diftin£tnefs of the feveral Times or
! Parts of the Me of lire is as it were hurt or in-
terrupted hereby, which yet is of good Ufe in
Mufick as Experience will teach.
You'll find over fome fingle Notes a Mark
like an Arch, with a Point in the Middle of it
which has been ufed to fignifie that that Note
is to be made longer than ordinary, and hence
called a Hold; but more commonly now it iig-
nifies that the Song ends there, which is only
ufed when the Song ends with a Repetition of
the firft Strain or a Part of it ; and this Repe«
tition is alfo directed by the Words, Da capo,
u e. from the BeginniHg.
Over the Notes of the Bafs-part you'll
find Numbers written, as 3 . 5, (jc» thefe dire&
to the Concords or Difcords^ that the Com-
pofer would have taken with the Note over
which they are fet, which are as it were the
Subftance of the Bqfs> thefe others being as
Ornaments, for the greater Variety and Plea-
fure of the Harmony*
CHAP,
*
414 ^Treatise Chap. XIII.
CHAP. XIIL
Containing the general Trinclples and Rule.
^/Harmonick Composition.
i i. D EFINITIQ N&
1* Of Melody and Harmony and their Ingre-
dients*
TH O' thefe, and alfo the next definition
concerning the Key^ have been already
largely explained \ yet 'tis neceffary
they be here repeated with a particular View
to the Subjea of this Chapter*
MELOBT is the agreeable Effeft of dif-
ferent mufical Sounds, fucceifively ranged , and
dilpofedj fo that Melody is the Effect only of
one tingle Part ; and tho' it is a Term chiefly
applicable to the Treble^ as the Treble is moftly
to be diftinguifhed by its Mir? yet in fo far as
the Bafs may be made airy, and to ting well, it
may be alfo properly faid to be melodious,
BAR-
§ i. ofMUSLCK. 41*
HJRMONT is the agreeable Refult of the
Union of Two or more mufical Sounds heard at
one and the fame Time ; fo that Harmony is the
Effect of TwoParts at leaft: As therefore 7 a con-
tinued Succeffion of mufical Sounds produces
Melody * fo does a continued Combination of
thefe produce Harmony.
Of the Twelve Intervals o£ mufical Sounds,
known by the Names of Second leffer* Second
greater* 'third leffer* Third greater* Fourth^
falfe Fifth* (which is called Tritone or Semi"
diapente in Chap. 8. § 4.) Fifth* Sixth leffer ,
Sixth greater* Seventh leffer, Seventh greater
and Octave* all Melody and Harmony is com-
pofed , for the Octaves of each of thefe are but
Replications of the fame Souads,- and whatever
therefore is or (hall be faid of any or of all of
thefe Sounds, is to be undcrftood and meant as
faid aJfo of their O Staves.
These Intervals* as they are expreffed by
Notes, ftand, as in Example 1. C being t\\o fun-
damental Note from which the reft receive their
"Denominations : Or they may ftand as in the
Second Example* where g is the fundamental
Note j for whatever be the Fundamental* the
Diftances of Sound are to it, and reciprocally to-
each other the fame.
O f thefe Intervals Two, viz. the Octave and
' Fifth* arc called perfect Concords ; Four, viz,
the Two ids and Two 6ths* are called imper-
fect Concords 5 Five viz. the faife Fifth* the
: Two Seconds and Two Sevenths* are Difcords,
yhe Fourth is in its own Nature a perfect Con-
cord
416 -^Treatise Chap. XIII
cord ; but becaufe of its Situation, lying betwixt
the 3d and the $th, it can never be made ufe
of as a Concord, but when joined with the 6th
with which it (lands reciprocally in the Rela-
tion of a 3d; it is therefore commonly clafled
among the Difcords, not on account of the Na-
ture of the Interval, but becaufe of its little
Ufe in the Harmony of Concords*
2. Of the principal Tone or Key.
The Key in every Piece and in every Part
of each Piece of mufical Composition is that
Tone or Sound which is predominant and
to which all the reft do refer (See above
Chap. 9)
Every Piece of Mujick, as a Concerto, So*
nata or Cantata is framed with due regard to
one particular Sound called the Key, and in
which the Piece is made to begin and end ; but
In the Courfe of the Harmony of any fuch
Piece, the Variety which in Mujick is fo necef-
fary to pleafc and entertain, requires the intro-
ducing of feveral other K^eys.
It is enough here to confider, that every
the leaft Portion of any Piece of Mujick has its
Key -, which rightly to comprehend we are to
take Notice, that a well tuned Voice, tho* un-
accuftomed to Mujick, afcending by Degrees
from any Sound affigned, will naturally proceed
from fuch Sound to the 2d g. from thence to
the
§ i. of MUSIC K. 417
(the '3^//. or to the %d g. indifferently from ei-
1 ther of thefe to the 4^, from thence to the pkj
(from thence to the 6th l- or 6th g. accordingly
I as it has before cither touched at the id I. or
f id ?. from cither of thefe to the jth g. and
(from thence into the Ofiave : From which it is
1 inferred, that of the 12 Interval* within the
fCompafsof the Oftave of any Sound afligned,
Ifeyen are only natural and melodious to that
Sound, mz* the id g* id g. tyh, 5th) 6th g,
jth g. and 8#<?, if the proceeding be by the
id g. but if it is by the 3 J /. the Seven natural
'Sounds are the id g* id I* qth, $th^ 6th 7. qth g.
and 8w, as they are exprefs'd in the Examples^
id and qth.
A s therefore the id and 6th may be either
greater or leffer, from thence it is that the Key
is denominated Jharp or Jlat; the JJ:arp Key
'being diftinguifhed by the id g. and the Flat by
the 3d L
I n fuch a Progreflion of Sounds, the funda-
mental one to which the others do refer, is the
'principal Tone or Key-, and as here C is the
Kej'j 10 may any other Note be the Key^ by be-
ing made the fundamental Note to fuch ]ikc
Progreffion of Notes, as is already exempli-,
fied* •
Whatever be the Key^ none but the
Seven natural Notes can enter into the Com-
petition of its Harmony .The Five other Notes
that are within the Compafs of . the Otlave of.
the Key, viz. the id I. id I falfe 5^, .6th h
" D d nth I
4i8 ^Treatise Chap. XIII.
jth I. in a/harp Key; and the 2d L zdg,falfe $th,
6th g. and jth I. in &flai 0112, are always extra-
neous to the Key.
When thefs Seven Notes {hall happen to be
mentioned in the Bafs as Notes, I (hall for Di-
ftinclion's fake exprefsthem by the Names of id
Fundamental or id f. 3 J /. 4th f. 5th f. 6th f.
jthf the Off ace being a Replication of the
Key, will need no other Name than the K^ey f
But when any of the Offiaves of thefe Seven
Notes {hall happen to be mentioned as Ingredi-
ents of the Treble, I fliall defcribe them by the
{ImpJe Names of id, 3d, qth,. 5th, &c. Thus,
when the %df or its Octave, which is the fame
Thing, fhall happen to be confidered as a Treble
Note, it is to be marked {imply thus (%d) as be-
ing a Third to the K^ey Fund. Thus the $th
f or its05lave, when confidered as a Note -in
the Treble, is to be {imply marked thus ($th)
as being a 5th to the Key f: Or thus (3d) as be-
ing a 3d to the idf: Or thus (6th) as being a
6th to the "jthf, and fo of the reft.
Each of the Seven natural Notes therefore
in each Key, coniidered as fundamental, or as
Notes of the Bafs, have their refpe&ive 3d*
$ths, 6th s, &c. which refpeclive %ds, %ths,
6ths, &c. muft be fome one, or Offiaves to fome
one or other of the 7 fundamental Notes that are
natural to the Key; becaufe, as was faid before,
nothing can enter into the Harmony of any
Key, but its Seven natural Notes and their 0-
ftaves. .
*• on
§ i. of MUSIC K. 4 i 9
3. Of Compofition*
Under this Title of Compofition are juftly
comprehended the practical Rules, imo. 0(
Melody > or the Art of making a fingJe Partj. e*
contriving and difpofing the lingle Sounds, fo
that their Socceifion and Progrefs may be agree-
able ; and ido. Of Harmony ^ or the Art of
difpofing and conferting feveral fingle Parts fo
together, that they may make one agreeable
Whole. And here ob/erve, the Word Harmony
is taken fomewhat larger than above in Chap. 7.
for Di/'cords are ufed with Concords in the Com*
pofition of PartS) which is here expreft in gene-
ral by the Word Harmony ; which therefore
is diftinguimed into the Harmony of Concords in
which no Dif cords are ufed, and that of DiJ cords
i which are always mixt with Concords. Obferve
alio that this Art of Harmony has been long
known by the Name of Counterpoint ,• which
arofe from this, That in the Times when
* Parts were firft introduced, their Mufick being fo
iimple that they ufed no Notes of different
Time, that Difference depending upon the
> Quantity of Syllables of the Words of a Song>
they marked their Concordsby Points fet againft
one another. And as there were no different
r Notes of Time, fo the Parts were in every
t Note made Concord : And this afterwards was
called fimple or plain Counterpoint ', to diftin-
guifh it from another Kind, wherein Notes of
, Eifferent Value were ufed, and 2)if cords brought
D d i in
42.0 A Treatise Ghap. XIII.
in betwixt the Parts, which was called figu-
rate Counterpoint. '
OBSERVE again, Melody is chiefly the
Bufihefs of the Imagination ,- fo that the Rules
.of Melody ferve only to prefcribe certain Limits
to it, beyond which the Imagination, in fearch-
ing out the Variety and Beauty of Air, ought
not to carry us : But Harmony is the Work of
Judgment ; fo that its Rules are more certain,
extenfive, and in Practice more difficult. In
the Variety and Elegancy of the Melody, the
Invention labours a great deal more than the
Judgment ; but in Harmony the Invention has
nothing to do, for by an exact Obfervation of
the Rules of Harmony it may be produced
without that Afliftance from the Imaginati-
on.
I t may not be impertinent here to obferve,
that it is the great Bufinefs of a Compofer not
to be fo much attach'd to the Beauty of Air,
as to neglect the folid Charms of Harmony ;
nor fo fervilly fubjected to the more minute
Niceties of Harmony, as to detract from the
Melody: but, by a juft Medium, to make his
Piece confpicuous, by preferring the united
Beauty both of Air and Harmony.
§ 2*' Rules of Melody.
... ■ • ■ .. . ,
I. ANY Note being chofen for the Key, and
~* its Quality oijharf or flat determined,
no Notes muft be ufed in any Part but' the fta*
'■" t u U tufa l
§ i. of MU SICK. 421
tural and effential Notes of the Key, as" thefe
i are already {hewn : And for changing or modu-
lating from one Key to another, which may
j ? alfo be done, you'll find Rules below in
II. Concerning the Succeflion of Intervals in
ithe feveral Parts, you have thefe general
(Stales.
1. The Treble ought to proceed by as little
^Intervals, as is poffibly confiftent with that Va^
jriety of Air, which is its diftinguifliing Cha-
j rafter.
2. The Bafs may proceed either gradually
i or by larger Intervals, at the Will of the Com-
] pofer.
3. The afcending by the Diftance of a falfe
igth is forbid, as being harfh and difagreeable ;
J but defcending by fuch a Diftance is often
1 pra&ifed efpecially in the Bafs. '
4. T o proceed by the Diftance of a fpurious
% d, that is,£mm any Note that is 2?, to the Note -
immediately above or below it that is f/; or from
any Note {/ to the Note immediately above or
below it $,is very offenfive. As we are in great-
eft Danger of tranfgrefling this Rule in a flat
Key, becaufe of the 6th I. and jth g. which
are Two of the natural Notes of the Harmo-
ny, we are therefore to take Care, that defcend-
ing from the Key we may proceed by the ytfa
h to the 6th I. and afcending to it we may pro-
ceed by the 6th g. to the 7th g. For altho the
6th g* and 7th L are not of the Seven Notes of
D d 3 a
4%t A Treatise Chap. XIIJ.
a fiat Key, yet they may be thus made Ufe of
as Tran(itioh% without any Offence.
5. The proceeding by the Diftance of a 7th.
I. in any of the Parts, is very harfh.
Thus far may Rules be given to correct th$
Irregularities of Invention in point of /&r\ but
to acquire or improve it, nothing lefs is nec?f-
fary than to be acquainted with the Melody of
the more celebrated Compofers, fo as to have
the more ordinary, and,, as it were, common
Places of their ^/o^ '/ami liar to the Ear ; and
what is further neceflary will, in due Time, na-
turally follow a Genius turned that Way.
§ 3. Of the Harmony of Concords, or fimpk
Counterpoint.
THE Harmony of Concords is compofed
of the imperfe6fy as well as of the
ferfetl Concords -, and therefore may be faid to
e perfebl and imperfeSf> according as the Con-
cords are of which it is compofed ; thus the
Harmony that arifes from a Conjunction of any
Note with its $th and Odfave is perfect, but
with its %d and 6th is imperfect,
I t has been already fliewn what may enter
into the Harmonj of any Key, and what may
not* I proceed to fliew how the Seven natural
Notes, and their Otlcwes in any Key, may
ftand together in a Harmony of Concord * and
bow
:§ 3 . of MUSIC K. 423
how the feveral Concords may iucceed other ;
and then make fome particular Application,
which will finifh what is deli g d on this
I Branch.
I. How the Concords may ft and together.
1. T o apply, firftj the precceding Diftindi-
On of perfeB and imperfect Harmony, take
this general Rule,ciz. to the Key f, to the 4th f.
and to the %thf, a perfect Harmony muft be
Joyned. To the idf. to the id f. and to the
jtbf. an imperfect Harmony is in all Cafes in-
difpenfably required. To the 6th f. a perfect or
imperfect Harmony is arbitrary.
OBSERVE, In the Competition of Two
Tarts, tho' a 3d appears only in the Treble
upon the Key f. the 4th f. and the $th f. yet
the perfect Harmony of the $th is always fup-
pofed, and muft be Supplied in the Accompany-
ments of the thorough Rafs to thefe fundamen-
tal Notes.
2. But more particularly in the Compofttion
of Two Parts.
The Rules are,
1. The Key f. may have either its 05fave>
its id or its $th.
2. The qth f, and 5th f. may have either
their refpeftive %ds or $th$^ and the firft may
have its 6th ; as, to favour a contrary Motion,
the laft may have its Ociave.
D d 4 3. The
424 A Treatise Chap. XIII;
3. The 6th f* may have either its 3 J, its
5th or its 6th.
4. The 2 df. id f and jthf. may have
either their respective 3 ds or 6ths ; and the kit
may, on many Occafions, have itsfalfe Kth.
These Kuhs are mil the fame whether the
the Key Isjharp or flat, as they are exemplified
in Example 5, 6, 7, 8, <?, 10, 11,
. After having confidercd what are the feve-
ral Concords jthat may be harmonioufy applied
to the fev 'en fundamental Notes ; it is next to be
learned, how thefe feveral Concords mayfucceed
each other, for therein lies the greater!: Difficul-
ty of mufical Compojition.
II. The general Rules of Harmony, refpedfing
the Succejfion of Concords.
1. That as much as can be in Parts may
proceed by a contrary Movement//^ is, when
the Bafs afcends, the Treble may at the fame
Time defcend, & vice verfa ; but as it is impof-
fible this can always be done, the Rule only
prefcribes the doing fo as frequently as can be,
Exam. 12.
2. The Parts moving the fame Way either
upwards or downwards. Two OtJaves or Two
Sths muft never follow one another immediate-
ly. Exam. 13.
3. Two 6th s I. muft never fucceed ^ach o-
ther immediately \ the Danger of tranfgrefiing
which lies chiefly in a Jharp Key, where the 6th
to the 6th f. and to the jthf are both leffer,
JZxam, 14.
4. Whenever
§ 3 - of MUSIC K. 4zy
4. Whenever the Stave ov^th is to be made ufc
i d'i the P# m muft proceed by a contrary Move-
iment to each other,' except the Treble move in-
i to fuch Octave or $th gradually jwhich Rule muft
I be carefully obferved, becaufe the Occafions of
i tranfgreffing it do moft frequently occur 5J &. 15.
5. If in sifiarp Key, the Bafs defcends gradu-
;; ally from the $th f. to the 4th f ; the laft muft
i never in that Cafe have its proper Harmony
applied to it, but the Notes that were Harmony
to the preceeding $th f. muft be continued upon
the qthf. Exam. 16.
6. THIRDS and 6th s may follow one
another immediately^ often as one has a Mind,
Exam. 17.
Here then are the Rules of Harmony plain-
ly exhibited, which tho' few in Number, yet
the Beginner will find the Obfervance of them
j a little difficult, becaufe Occafions of tranfgref-
1 fmg do moft frequently offer themfelves,
I n the former Article it is (hewn what Con-
cords may be applied to each Fundamental or
: Bafs-note ; and here is taught how the Parts
; may proceed joyntly, the SecJion 2d (hewing
how they may proceed fingly, and what in
either Cafe is to be avoided. It remains there-
fore now to make the Application*
III. A particular Application of the preceeding
Rules, to tzvo Parts,
WftEREAsitis natural to Beginners, firft to
imagine the Treble, and then to make a JZafs
to
4i6 -^Treatise Chap. XIII.
to it, the Treble being the Alining Part,in which
the Beauty of Melody is chiefly to appear ; in
Compliance therewith, I fliall, by inverting as
It were the Rules in the foregoing Se&ion, fet
forth ,in the following Rules, which of the Seven
fundament W Notes, in the fjarp and flat Keys,
can properly be made ufe of to each of theSeven
natural Notes that may enter into the Treble ;
of which an exa£t Remembrance will very
much facilitate the attaining a Readinefs in the
Practice oifingle Counterpoint.
RULES for making a Bafs to a Treble, in
the fharp as well as flat Key.
i. The Key may have for its Bafs, either
the Key f. the qth f to which it is a
$th, the 3 df to which it is a 6th, or the 6th
ft to which it is a 3d.
2. Th e 2 d may have for its Bafs, either the
ythf. to which it is a 3^, or the 5th f. to which
is is a 5th, and fometimcs the qthf. to which
it is a 6th.
3. T h e 3 d can rarely have any other Bafs
but. the Keyf. tho' fometimes it may have the
6th f. to which it is a $th.
4. The qth may have for its Bafs either
the idf. to which it is a 3 d, or the 6th f. to
which it is & 6th, and fometimes, to favour a
contrary Movement of the Parts, it may have
the jth f. to which it is a falfe 5th, which
ought to refolve in the 3 d> the Bafs afcending
to
§ j. of MU SICK. 4 i 7
to the Key, and the Treble dcfcending to the
5. The 5th may have for its Bqfs, either the
3^ /. to which it is a 3 J, the Key to which it
is a $tky the 7/-/; /. to which it is a 6th ; or,
fbmetimes, to favour a contrary Movement of
the Parts, it may have the 5?hfi to which it
is an Offiave.
6. T h e eV& may only have for its i?tf/j the
qthf. to which it is a 3^.
7. The 7/-^ may have for its Bafs, either
the $thf* to which it is a 34 or the id f. to
which it is a 67&.
I have carefully avoided the mentioning
the ~$ds and 6th's, particularly as they are
greater or kjjer, which would inevitably
puzzle a Beginner : According to the Plan I
have followed, there is no need to be fo parti-
cular, becaufe when a 3d and 6th are mentioned
here in general, one is always to underiland fuch
a 3^ and fuch a 6th as makes one of the Seven
natural Notes of the Key; thus when I fay that
in ajharp K.ey the $th is a 3d, to the 3d f. I
muft neceifarily mean that it is a 3d I. to it, be-
caufe the 3d g* to the 3d f. is one of the Five
extraneous Notes ; juft fo when I fay that in a
flat Key the 5th is a 3d to the 3 d f. I muft
needs mean that it is a 3dg. to it, becaufe the 3d I.
to it is one of the Five extraneous Notes: Thus
when 1 fay that the 3d f. in either Key may
have a 3 d or a 6th for its Treble Note, it muft
be underftood as if I faid that fuch 3d and 6th in
428 ^Treatise Chap. XIII.
ajbarp Key muft be both leffer, and in 2. flat
Key, they muft be both greater, becaufe in the
firft or Jharp Key the ^dg. and 6th g. of the 3d
f are extraneous, and fo are the 3d /. and the
6th L of the 3d/. in a flat Key : But consider-
ing how much it would embarafs and multiply
the Rules,to have characterized the 3d/ and 6ths
fo particularly, I have therefore contrived the
Plan I proceed upon, fo as to avoid both thefe
Inconveniencies, and by being general make the
fame Rules rightly underftood, ferve both for a
Jharp and a flat Key.
But now that the Contents of the foregoing
Rules may be the more eafily committed to the
Memory, I fhall therefore convert them into this
Scheme, where the After if m is intended to de-
note what is but ufed fometimes.
Scheme drawn from the preceeding Rules.
^ ■
c
Q
CO
r
id
id
qth
Sth
6th
7th
3d,5th,6th,ox%<ve.
3d, 5th, 6th*
3d, 5th I* 6th.
3d, 5th, 6th, %ve.
34
6th.
See this exemplified, Example 1 8.
6f.4f.3f.Kf
ifsf^f
Kf.6f
2f.7f.6f
ifKfqfsf
qf
Sf if
Thefe Rules being well underftood, and ex-
actly committed to the Memory, the Treble in
Ex. 19. is fuppofedto be aflign'd, and the Bafs
compofed to it according to. thefe and the for-
mer Rules.
The
§ 3 . of MU SICK. 419
The firft Thing I am to obferve in the
Treble is, that its Key is c natural, i. e. with
the $d g. becaufe it begins and ends in c with-
out touching any Note but the Seven that be-
long to the Harmony of that Key.
The fccond Note in the Treble is the fecond
in the Harmony of the K^ey ; which, according
to the Rules, might have flood as a %d to the
Bafs 9 as well as a 5 th; to which therefore the
Bafs might have been b, as well as g. but I ra-
ther chufed the latter, becaufe having begun
pretty high with the Bafs, I forefaw I fhould
want to get down- to c below, for a Bafs to
the id Note in the Treble^ and therefore I chu-
fed g here rather than b, being a more natural
and melodious Tranfition to c below.
The third Note in the Treble, and id in
the Harmony of the Key, has c the Keyf. for
its Bafs, becaufe it is almoft the only Bafs it
can have : And I chufed to take the Key be-
low for the Reafon I juft now mentioned.
The fourth Note in the Treble and tfh in the
Harmony of the Key, has the id/, for its BafL
which here is d ; it is capable of having for
its Bafs the 6th f. but eonfidering what beho-
ved to follow, it would not have been fo na-
tural.
The fifth Note in the Treble and $tk in the
Harmony of the Key, has for its Bafs the id
f. which is here e. it might have had c the Key
for its Bafs, and the going to f afterwards
^ould have fung as well ; but I chufed to afcend
gradually
430 -^Treatise Chap. XIII.
gradually with the Bafs^ to preferve an Imita-
tion that happens to be between the Parts, by
the Bafs afcending gradually to the 5 thf. from
the Beginning of the fecona Bar, as the Treble
does from the Beginning of the firft Bar,
The fixth Note in the Treble^ and Key in
the Harmovy^ (lands as a $th ; and has for its
Bafs the /\thf. rather than any other it might
have had, for the Reafon juft now mentio-
ned.
The feventh Note in the Treble^ and -jth in
the Harmony of the Key, has the 5th f. rather
than the 2d f. for its Bafs y not only on ac-
count of the Imitation I took Notice of, but to
favour the contrary Movement of the Parts, ;
and befides- considering what behoved . to fol-
low in the Bof's, the idf. would not have done
fo well here ; and the Tranfition from it to the
Bafs Note * that muft neceifarily follow, would
not have been fo natural. As to the following
Notes of the Bafs I need fay nothing ,• for the
Choice of them will appear to be from one of
thefe Two Considerations, either that they are
the only proper Bafs Notes that the Treble
could admit of, or that one is chofen rather
than another to favour the contrary Movement
of the Parts,
I chufed rather to be particular in fetting
forth one Example than to perplex the Begin-
ner with a Multitude of them; I have therefore
only added a fecond, which I refer to the Stu-
dents o\vn Examination ; both which are {o
Contrived, as to be capable of being tranfpofed
into
•
§ 3 . ofMUSICK. 431
into a flat Kjy y with the Alteration of the 3^
and 6th.
When thefe Examples are thoroughly exa-
mined, the next Step I would advife the Be-
ginner to make, would be to tranipofe thefe
Trebles into other Keys -, and then endeavour to
make a Bafs to them in thefe other Keys : For
to him, the fame Treble in different Keys will
be in fome Meafure like fo many ^different
Trebles, and will be equally conducive to his
Improvement* And when he has fmifhed the
Bafs in thefe other Keys, let him call: his Eyes
on the Example, and tranlpofe the Bafs here
into the fame Keys*, that he may obferve where-
in they differ, and in what they agree j by
which Comparifon he will be able to difcovej?
his Faults, and become a Mailer to himfelf.
And by the Time that . he can with Facility
write a Bafs to thefe Two Treble s, in all the
ufual Keys, which upon Examination he flialj
find to coincide with the Examples, I may ven-
ture to allure him that he has conquered the
greateft Difficulty,
Notwithstanding the infinite Variety
•of Air there may be in Mufick, I take it for
granted, that there are a great many common
Places in point of Air, equally familiar to aii
Compofers, which necenarily produce corrc*
fpondent common Places in Harmony ; thus it
molt frequently happens that the Treble do
fcends from the 3d to the Key, as at the Ex-
ample 20. as often will the Treble defcend from
tke
43* ^Treatise Chap.XIIL
the jth to tlie $th. Examples 21$ 22, and in
this Cafe the Bafs is always the 5/. as in that
the Bafs is always the Key f. Thus frequently
in the Treble ', after a Series of Notes the Air
will terminate and come to a Kind of Reft or
Chfe upon the id or 7?^ ; in both which the
Bafs muft always be the $thf. as in Examples
23, 24. Some other common Places will ap-
pear funiciently in the Examples^ and others,for
the Beginner's Inftruttion, he will b?ft gather
liimfelf from the Works of Authors, particularly
ofCorelli.
Asa thorough Acquaintance with fuch com-
mon Places, will be a great Affiftance to the Be-
ginner, I would firft recommend to him the
Pradice of thofe here fet forth, in all the ifual
Keys floarp as well as fat, till they are become
very familiar to him : But in tranfpofing them
to flat Keys, the Variation of the 3d and 6th is
to be carefully adverted to.
After fimple Counterpoint , wherein nothing
but Concords have Place, the next Step is to
that Counterpoint wherein there is a Mixture of
Difcord j of which there are Two Kinds, that
wherein the Difcords are introduced occahonal-
ly to ferve only as Tranfitions from Concord f to
Concord, or that wherein the Difcord bears a
chief Part in the Harmony,
U-Qf
§ 4 . */ MUSIC K, 43 3
(j 4. Of the life of Difcords^ or Figurate Coun-
terpoint.
ii Of the trarifient Difcords that are fub ferm-
ent to the Air, hut make no Part of the Har-
mony.
TC VERY Bar or Meafure has its accented
*wf 1 and unaccented Parts : The Beginning and
Middle, or the Beginning of the firft Half of the
Bar, and Beginning of the latter Half thereof
in common Time&vA the Beginning,or the firft of
the Three Notes in triple Time, are always the
accented Parts of the Meafure. So that in com-
mon Time the firft and third Crotchet of the
Bar, or if the Time be very (low, the ifl, 3d,
$th and jth Quavers are on the accented Par^ts
of the Meafure, the reft are upon the unaccen-
ted Parts of it. In the various Kinds of Triple
whether f 4<4 tif \% the Notes go always Three
and Three, and that which is in the Middle of
every Three is always unaccented, the firft and
laft accented ,• but the Accent on the firft is fo
much ftrongerj that, in feveral Cafes, the laft
is accounted as if it had no Accent ; fo that a
Difiord duly prepared never ought to come up-
on it.
The Harmony muft always be full upon the
accented Parts oF the Meafure, but upon the
unaccented Parts that is notforequifite: Where-
fore Difcords may tranfiently pafs there with-
E e out
434 ^Treatise Chap. XIII.
out any Offence to the Ear : This the French
call Suppofition, becaufe the tranfient Difcord
fuppofes a Concord immediately to follow it,
which is of infinite Service in Mufick^ as con-
tributing mightily to that infinite Variety of
Air of which Mufick is capable.
Of SUPPOSITION there are feveral
Kinds. The firft Kind is when the Parts pro-
ceed gradually from Concord to Difcord^ and
fromWifcord to Concord as in the Examples
25 and 26. where the intervening Li/cord
ferves only as a Tranfition to the following Con-
cord.
E y imagining all the Crotchets in the Treble
to be Minims^nd all thcSemihreves in the Bafs
of the Example 25. to be pointed, it will ferve
as an Example of this Kind of Suppofition in
triple lime.
There is another Kind, when the Parts do
not proceed gradually from the Difcord to the
Concord^ but defcend to it by the Diftance of a
3 J. as in the Examples 27 and 28. where the
Difcord is efteem'd as a Part of the preceeding.
Concord,
There is a third Kind refembling the-fe-'
cond, whea the riimg to the Difcord is gradu-
al, but the defcen ding from it to the. following.
Concord is by the Diftance of a 4^, as in Ex~\
ample 29. m which the Difcord ,fs alfb conftV
dered as a Part or Breaking of the preceeding;
Concord. '; .:.
There
§ 4 > »f MUSICK, 43 f
Th ere is a fourth Kind very different from
the Three former, when the Difcord falls upon
the accented Parts of the Meafure^ and when
the riling to it is by the Diftance of a /\th ; but
then it is abfolutely necefTary to follow it im-
mediately by a gradual Defcent into a Concord
that has ]uft been heard before the Harmo-
ny fay which the Difcord that preceeds gives no
Offence to the Ear,ferving only as a Tranfition
into the Concord^ as in Example 30.
Thus far was necefTary to be taught by
Way of Institution upon the Subject of Suppo-
sition; what further Liberties may be taken
that Way in making Divifiohs upon holding
Notes, as in Example 31. may be eafily gather-
ed from what has been faid ; obferving this as
a Principle never to be departed from, that the
lefs one deviates from the Rules, for the fake of
y^/r, the better*
1. Of the Harmony (/Discords.
The Harmony of Difcords is, that wherein
the Difcords are made ufe of as a folid and
fubftantial Part of the Harmony ; for by a pro-
per Interpontion of a Difcord the fucceeding
Concords receive an additional Luftre; Thus
the Dif cords are in Mufick what the ftrong
Shades are in Painting; for as the Lights there,
fo the Concords here, appear infinitely more
beautiful by the Oppofition.
The Discords are imoi the $th when joyn'd
ivith the 6th) to which it flands in relation as
£ e 2 a
43<S >4 Treatise Chap. XIII.
a Difcord, and is therefore treated as a Dif-
cord in that Place ; not as it is a 5^^ to the Bat's
in which View it is a perfect Concord, but as
being Joyn'd with the Note immediately above
it, there arifes from thence a Senfation of Dif-
cord.
ido. The 4^, tho' in its own Nature it is a
Concord to the Bafs, yet being joyn'd with the
$th, which is immediately above it, is alfo ufed
as a Difcord in that Cafe.
%tio. The Ninth which is in eflfecl: the 2 d y
and is only called the Ninth to diltinguifh it
from the 2^, which under that Denominati-
on is ufed in a different Manner, is in its own
Nature a Difcord.
qto. The yth is in its own Nature a Dif-
cord,
$to. The 2 d and 4^ is made ufe of when
the Bqfs fyncopates, in a very different Man-
ner from that of ufing thofe above mentioned,
as will appear in the Examples.
A s I treat only of Compofition in Two Parts,
there is no Occafion to name the Concords withf
which, in Compofition of Three or more Paris A
the Difcords are accompanied ,* theie, I take!
for granted, are known to the Performer of the!
thorough Bafs \ and tho' in Compofition ofl
Two Tarts they cannot appear, yet they are|
always fitppofed and fupplied by the Accom<
panyments of the Bafs,
§ 4 . of MUSIC K. 437
Of Preparation and Refoktion of Difcords.
The Difcords here treated of are introdu-
ced into the Harmony with due Preparation ;
and they muft be fucceeded by Concords, com-
monly called the Refolntion or the Difcord,
The Difcord is prepared, by fubiifting firft
in the Harmony in the Quality of a Concord,
that is, the fame Note which becomes the Dif-
cord is firft a Concord to the Bafs Note imme-
diately preceeding that to which it is a Difcord;
the Difcord is refolded, by being immediately
fucceeded by a Concord defcending from it by
fhs Diftance only of id g. or id I.
A s the Difcord makes a fulpftantial Part of
the Harmony, fo it muft always polTefs an ac-
cented Part of the Meafure : So that in com-
mon Time it muft fall upon the \ft and ^d Crot-
chet j or, if the Time be extremely flow, up-
on the ift, id, 5th or -jth Quaver of the Bar -,
and in triple Time it muft fall on the firft of e-
very Three Crotchets, or of every Three Mi-
nims, or of every Three Qiiavers, according as
the triple Time is, there being various Kinds
of it.
I n order then to know how the Difcords
may be properly introduced into the Harmony,
I (hall examine what Concords may ferve for
their Preparation and Refohtion ; that is,
Whether the Concords going before and follow-
ing fuch and fuch a Difcord may be a 5th, 6th,
3d or Qffiave.
E e 3 The
43$ -^Treatise Chap. XIII.
The $th may be prepared, by being either
an 8#£, 6th or 3d; it may be refolved either in-
to the 6£& or 3^ but mod commonly into the
3d, Example 32.
The /{th may be prepared in all the Co/z-
ror<^j ; and may be refolved into the 6^, 3^ or
$ve, but moft commonly into the 3d. Exam-
fk 33-
The stf/:? may be prepared in all the Co/z-
rori/ except the %ve^ and may be refolded into
the 6z^, 3d or 8w, but moft commonly into the
%ve. Example 34.
The 7th may be prepared in all the Co/z-
corix ; and may be refolved into, the 3^, 6/^ or
5?^, but moft commonly into the 6th or 3d:
Example 35.
The 2^ and 4^ are made ufe of after a
quite different Manner from the other Difcords,
being prepared and refolved in the Bafs. Thus,
when the Bafs defcends by the Diftance of a
id y and the firft Half of the Note falls upon an
unaccented Part of the Meafure, then either the
^th or the id may be applied to the laft or ac-
cented Half of the Note ; if the 2 J, it is conti-
nued upon the following Note in the Bafs, and
becomes the 3d to it ; if the fifth is applied, the
Treble rifes a Note, and becomes a 6th to the
Bafs. Example 36.
From all which I muft obferve, that the
$th and jth are Dif cords of great Ufe, becaufe,
even in Two Parts, they may be made ufe of
fucceffively for a pretty long Series of Notes
without Interruption^ efpeciaily the 7th, as pro-
ducing
§ 4 . of MUSIC K. 439
ducing a moft beautiful Harmony. The qth is
not ufeful in Two Parts in this fucceflive Way,
but is otherwife very ufeful. The $th in the
fame Manner is only ufeful as the /\th is.
Having once diftin£tly underftood how the
Difbords are introduced and made a Part of the
Harmoii)^ by the Examples that I have exhi-
bited in plain Notes, it may not be amifs to
take a View, in the Examples here fet forth,
how thefe plain Notes may be broke into Notes
of lefs Value ; and being fo divided, how they
may be difpofed to produce a Variety of Air :
Which Examples may fuffice to give the Begin-
ner an Idea how the Difcords may be divided
into Notes of fmall Value, for the fake of Airi
Of the Manner of doing it there is an infinite
Variety, and therefore to have fiiewn all the
poflible Ways how it may be done, would have
required an infinite Number of Examples : I
fhall therefore only give one Caution, that in
all fuch Breakings the firft Part of the difccJrd-
ing Note muft diftin&Iy appear, and after the re-
maining Part of it has been broke into a Divi-
fion of Notes of lefs Value, according to the
Fancy of the Compofer, fuch Divifion ought
to lead naturally into the refolving Concord that
it may be alfo diflinclly heard. See Example
37-
Having now confidered the Matter ofcHar-
moiiy as particularly as is neceffary to do by
way of Infiitutioiiy to qualify the Student for
reading and receiving Inttruttion . from the
E e 4 Works
44° ^ Treatise Chap. XIII.
Works of the more celebrated Compofers,
which is the utmoft that any Treatjfe in my
Opinion ought to aim at, I proceed to defcribe
the Nature of Modulation, and to give the
Rules for guiding the Beginner in the Practice
of it.
§ 5. 0/ MODULATION; and
1 mo. What it is,
A LTHO' every Piece of Mufick has one
■* * particular Key wherein it not only begins
and ends, but which prevails more through the
whole Piece ; yet the Variety that is fo neceflary
to the Beauty of Mufick requires the frequent
changing of the Harmony into feveral other
Keys-, on Condition always that it return again
into the Key appropriated to the Piece, and ter-
minate often there by middle as well as final
Cadences, efpecially if the Piece be of any
Length, elfe the middle Cadences in the Y^ey
are not fo neceffary.
These other Keys, whether Jharp or flat
into which the Harmony may be changed, muft
be fuch whofe Harmonies are not remote to the
Harmony of the principnl Key of the Piece ;
frecaufe otherwife the Tranfitions from the prin-
cipal K^ey to thofe other intermediate ones,
Would be unnatural and inconfiftent with that
Anah~
§ ,, of MUSIC K. 441
jfnah?y which ought to be prcfcrved between
all the Iviembers ef the fame Piece. Under the
Term of Modulation may be comprehended
the regular Progreffion of the feveral Parts thro'
the Sounds that are in the Harmony of any
particular Key as well as the proceeding natu-
rally and regularly with the Harmony from one
Key to another : The Rules of Modulation
therefore in that Senfe are the Rules of Melody
and Harmony ', of which I have already treated^
fo that the Rules ok Modulation only in this laft
Senfe is my prefent Bufinefs.
Since every Piece muft have one principal
Key ', and fince the Variety that is fo neceifiry
in Mufick to pleafe and entertain, forbids the
being conhn'd to one Key, and that therefore it
is not only allowable but requifite to modulate
into and makeCadences upon feveral other Key s 9
having a Relation and Connection with the
principal K.ey, I am firft to confider what it is
that conftitutes a Connection between the Har-
mony of one Key and that of another, that from
thence it may appear into what Keys the Har-
mony may be led with Propriety : • And in order
to comprehend the better wherein this Con-
nection between the Harmony of different Keys
may coniift, I ftiall firft jQiew what it is that oc-
calions an Inconfiftency between the Harmony
of one Key and that of another.
2. Of the Relation and Connection of Keys.
I t has been already fet forth, that each Kty
has Seven Notes belonging to it and no more.
In
44* ^Treatise Chap. XIII.
In a Jharp Key thefe are fix'd and unalterable ;
but in a flat Key there is one that varies, viz.
the jth. Hitherto I have accounted the jth g.
one of the Seven natural Notes in a flat Key,
and I behoved to do fo in the Matter of Har-
mony^ becaufe the jth g- is the id g. to the $th,
without the Help of which there would be no
Cadence on the Key ; and befides, it is alone by
the Help of it that one can afcend into the Key.
But here when Iconfider not the particular Exi-
gencies of the Harmony in aflat Key, but the
general Analogy there is between the Harmony
of one Key and that of another, I muft reckon
that the jth which is effential in a flat Key is
the 'jth I. becaufe both the id and 6th in a
flat Key are leffer, therefore as to our prefent
Enquiry the 'jth g. in aflat Key muft be hence*
forth accounted extraneous.
The diftinguifhing Note in each Key, next
to the Key-note it feli, is the 3d; any Key
therefore that has for its id any one of the Five
extraneous Notes of another Key, under what
Denomination foever of $ or j/ is difcrepant with
that other Key to which fuch 3 d is extraneous.
Thus the extraneous Notes of the Jharp Key c
' being c%, d% f%, g% y a%, or as the fame Notes
may happen to be differently denominated <$ ?
e K gK ^K 1/ : Thejharp Key a therefore ha^
ving c% for its id, the Jharp Key b having d%
for its id 3 the harp Key e having g% for its id,
the Jloarp Key fM having a% for its 3 d, or the
flat Key (/ having $f for its id, the flat Key c
having ek for its id, the flat Key e^ having gb
for
§ f, of MUSIC K. 445
for its id, the flat Key f having ab for its id,
and the flat Key g having 1/ for its 3d, are all,
I fay, difcrepant with tliefljarp K.ey c, bceaufe
the ids which are the diftinguiiliing Notes of
thefe other Keys are all extraneous Notes to r,
with a jdg. and fince any 7Q; 7 which has for
its id any one of the Five extraneuus Notes of
another Key* is difcrepant with that other Key,
a fortiori therefore any one of the Five extra-
neous Notes of a Key being a Key it lelf^ is ut-
rerly difcrepant with a Key* to which fuch Kcy~
note it felf is extraneous ; thus therefore c$ 9 d%,
f%, 0, a%, or, dfr, eb, gb, a^, (/ being confidered
as Keys, whether with idg. or id I. are utterly
difcrepant to c with a idg. becaufe they are all
extraneous to it,
A Key then being affign'd as a principal Key,
as noae of its five extraneous Notes can either
be Keys themfelves, or ids to Keys that can
have any Connexion with it, fo it will from
thence follow, that the Seven natural Notes of
the Key affigned, being conftituted Kys with
fuch ids as are one or other of the Seven natu-
ral Notes of the laid Key affign'd, may be ac-
counted confonant to it ; provided they do not
effentially introduce the principal Key or its id
under a new Denomination, that is, the Key
aifign'd being for Example the floarp Key c, no
Key can be confonant to it, that introduces ne-
ceffarily and eilentially cM, which is the Key
under a new Denomination, or eft, which is its
id under a new Denomination, and different
from what they were la tne Key affign'd ; there-
fore
444 A Treatise Chap. XIIL
fore to the Jharp Key r, which I ffiall take for
the principal Key amgn 6, the flat Keys d y e and
a y alfo the Jharp l\,eys f and g are confonant ;
font the flat Key &, altho' both it felf and its
3 d are Two of the Seven natural Notes of the
Key affigned, is not confonant to it y becaufe it
would effentially introduce c% for its. 2 J, which
being the Key affign'd under a new Denomina-
tion, would produce a very great Inconfiftency
with it. And here, left from thence the Begin-
ner may form this Objection again!! theflatKey
4 being reckoned confonant to the Jharp Key f,
as I have done, becaufe that Key d does intro-
duce c% for its ythg. I muft inform him, as I
have before obferved, that the ythg. to a flat
Key is only occasionally made Ufe of; and that
the yth I. is ] the jth that is effential in a flat
Key.
Tkk flat Key c being the principal flat Key
ndgned, the flat Keys f and g, alfo the Jharp
Keys eby aS/ and \/ are confonant to it, but the
flat Key d> tho' both it felf and its 3 J are ofthe
natural Notes of the Key affigned, yet as this
flat Key d being conltituted a Key, behoved to'
have e for its Second, which is the 3d of the
Key affigned, under a different Denomination,
therefore it cannot be admitted as a confonant
Key to it.
To the Harmony therefore of & flat princi-
pal Key, as well as of a Jloarp one, there are
Five Keys that are confonant, that, with all
the Elegancy and Property imaginable, may be
introduced in the Coarfe of the Modulation of
any
§ j. of MUSICK. 445
any one Piece of Mufich To all fljarp princi-
pal Keys the Five confonant Kjeys are the 2cl>
3 d, /\th, $th and 6th to the principal Key, with
their respective ids, viz. with the id, the 3^/.
3^, 3d/. 4th, {dg. 5th, ?>dg. 6th, idh To ail
fiat principal Keys the Five confonant K,eys arc
the ^d, qth, sth, 6th and jth to the principal
Key, with their refpective 3^jy viz. with the ji,
the idg. $th, id I 5th, id I 6th, idg. Jth, idg.
each of which confonant K.eys, tho' reckoned
dependent upon their principal Key with regard
to the Structure of the whole Piece, yet with
refpecl to the particular Places where they pre*
vail, they are each of them principal fo long as
the Modulation continues in them, and the
Rules of Melody and Harmony are the fame
way to be obferved in them as in the principal
Key; for aWK^eys of the fame Kind are the fame,
and this Subordination here difcourfedof is only
accidental^ for no K^ey in its own Nature is
more to be accounted principal than another*
The feveral Keys then that may enter into
the Composition of the fame Piece being known,
it is material next to learn in what Order they
may be introdue'd; and herein one muft have
Recourfe to the current Practice of the Mafters
of Compofition -, from which, tho' indeed no
certain Rules can be gathered, becaufe the Or-
der of introducing the confonant Keys is very
much at the Discretion of the Compofer, and
in the Work of the fame Author is often vari-
ous, yet generally the Order is thus.
44^ /^Treatise Chap, XIII.
In ajfjarp principal Key, the firft Cadence is
upon the principal Key it felf often ; then fal-
low in Order Cadences on the $th 3 3d, 6th, id,
rfih, concluding at laft with a Cadence on the
principal Key. In a flat principal Key the in-
termediate Cadences are on the id, 5^, 7//^
4?£ and 6th. Now, whatever Liberty may be
taken in varying from this Order, yet the be-
ginning and ending with the principal Key h a
Principle never to be departed from,- arid as far as
I have obferved, it ought to be a Rule alfb 5
that in ajharp principal Key, the $th, and in a
flat one the 3d, ought to have the next Place to
the principal Key.
Itio. How the Modulation is to be perfor-
med.
It now remains to {hew, how to modulate
from one KeyXo another, fo that the Tranfiti-
ons may be eafy and natural; but how to teach
this Kind of Modulation by Rules is the Diffi-
culty • for altho' it is chiefly performed by the
Help of the ythg. of the Key into which we are
refolved to change the Harmory, whether it be
Jharp or flat; yet the Manner of doing it is fo
various and extenfive, as no Rules can circum-
fcribe : Wherefore in this Matter, as well as in
other Branches of my Subject, I muft think it
enough to explain the Nature of the Thing fo,
and to give the Beginner fuch general Notions
of it, as he may be able to gather by his own
Obfervation, in theCourfe of his Studies of this
Kindj what n.o Rules can teach.
HB
§ j. of MUSI CK. 447
.The jthg. in either floarp ox flat K.ey is the
■$dg. to the 5th f. of the Key, by which the Ca-
dence in the Key is chiefly perform'd ; and by
being only a Semitone under the Key, is there-
fore the nioft proper Note to lead into it, which
it does in' the moft natural Manner that can be
imagin'd j infomuch that the jth g. is never
heard in any of the Parts, but the Ear expects
trie Key fliould fucceed it > for whether it be
ufed as a %d or as a 6th, it doth always affect
us with fuch an imperfect Senfation, that we
naturally expect fomething more perfect to fol-
low, which cannot be more eafily and fmoothly
accomplifhed, than by the (mail Interval of a
Semitone,, to pafs into the perfect Harmony of
the Key ; from hence it is that the Traniition
into any Key is beft effected, by introducing its
Jthg. which fo naturally leads to it ; and how
this jthg. may be introduced, will beft appear
in the Examples.
In Ex. 38. the Key is firft the Jharp Key
c, but/$, which is the jth g. to g, introduces
and leads the Harmony into the firft confonant
Key of c with a $dg. In this Example f%
(lands in the Treble a 6th; but it may alfoftand
a 3dg. as in Ex. 39. or it may be introduced
into the Bafs with its proper Harmony of a
id or 6th, as in Examples 40 and 42, or
it may, as a 6th g. or -$dg. in the Treble, be the
refolving Concord of a preceeding Difcord, as in
Examples 41 and 44. or it may (land in
the Treble as a qthg. accompanied alfo in that
Cafe with a zd y or fuppofed to be fo as in Ex.
46.
^Treatise Chap. XIII.
46. or otherwife ufed as in Examples 45
and 47. The Modulation changes from the
fiarp Key c into the flat Key a, one of itsconfo-
nant Keys, whole jth g. is introduced in the
Quality of a 6th g. and ^dg. ferving as the Re-
folutions of preceeding Bifcords. In Examples
48 and 51. the 6th is applied to the Key y
which is always a good Preparation to lead
the Harmony out of it ; for a Key can be no
longer a Key when a 6th is applied. The re-
maining Examples (hew how the Harmony may
pafs through feveral Keys in the Compafs of a
few Notes.
From thefe Examples I fhall deduce fome
few Obfervations, that may ferve as fo many
Rules to guide the Beginner in this firft At-
tempt.
lfl. The jthg. of the Key into which WO
intend to lead the Harmony \ is introduced into
the Treble either as a ^dg» or 6th g. or as a
qthg. with its luppofed Accompany men ts ot
qth and 6t% and as %dg. or 6th g. it]is common-
ly the Refolution of a preceeding Di/cord.
id. When this ythg. comes into the Treble
in what Quality foeveiy as $dg. 6th g. &c. it is
either fucceeded immediately by that Note
which is the K.ey whereto it immediately leads,
or immediately preceeded by it, and moft com-
monly the laft i in which Cafe the Treble muft
of confequence defcend to it by the Diftance of
a Semitone. Thus, when we are to change the
Harmony from the Jharp Key c to the flat Key
a, that is, from afiarp principal Key into its
6thi
§ j. of MUSIC K. '449
6th ) we life it in the Treble as the 6th to the
principal Key c^ or as the $th to d, or as the
3d tofy and being once upon the Note which
we defign to be the Key> the falling half a Note
to its jthg. for fixing the Harmony fairly in the
Key*, is mod eafily performed ,- thus were we to
go from a principal Key into the 3^, we fhotild
life a 6th on the 5/. ; or were we to go into the
2^ 3 wefliouldufea6^onthe 4/. and the rather,
becaufe in the Key whereto we defign to go, a
6th is the proper Harmony , for that stbf. of the
principal Key becomes the 3 df. of the 3 J, when it
i isconftilute a /C^.5 and fodoes the dthf. of the
1 principal Key become the 3 df. of the 2^ when
i conftitute a Key*
Itio. When the 7th g. of the Key, into
1 which we defign to change the Harmony ', isin-
Itroducedin the Baft, it is always immediately,
ifucceeded by the Key; and then the Tranfitioii
: to the 'jthg. is moft part gradual, by the Inter-
: ml of a Tone or Semitone, or by the Interval
:of a 3<i/. But moft commonly it is introduced
i into the Bafs % by proceeding to it from the na*
tnral Note of the fame Name, that is, from a
t Note that is natural in the Key, as from/ to /$ in
the j& # rp Key c^or from ^ to b in the flat Key d*
Ato. When the qth g. of the Key to which
we defign to lead the Harmony ', is one of the
Seven natural Notes of the Key wherein the
Harmony already is, the introducing it into the
Bafs is meft natural, as being of courfe $ this
happens when we would modulate from a Jharp
Key into its Ath> or from a flat Key into it«
Ff 3d.
4jo -^Treatise Chap. XIII.
3d. In which Cafes the 7th g. is introduced in-
to the Bafs\ and in the Treble the falfe $th is
applied to it, which refolves into the %dg.
5 to. When this 7th g. comes into the Bafs y
it niuft of neceffity h ave either a 3d I. 6th I. ov falfe
$th in the Treble \ if a %dl. it refolves into the
8^ if a 6th I. it commonly paffes into thefalfe
^th) and from thence refolves into the %d of the
Key.
6to, B y applying the 6th "to any Note of the
K.ej'> to which the $th is a more natural Har-
mony^ as for Example, to the Key it felf^ to
the qth f. or 'sthf. a Preparation is thereby
made for going into another Key, viz. into that
Note which is fo made Ufc of, as a 6th to any
of thefe fundamental Notes, as in the Examples.
Having thus explained the Nature of Mo-
dulation from one Key to another, it may feem
natural to treat now of Cadences-, but of thefe I
cannot fuppofe a Performer of the Thorough-bafs
ignorant, they being fo frequent in Muficki all
I fliall therefore fay of them is, that they muft
always be finiflied with an accented Part of the
Meafnre. As to what concerns Fugues and ./-
Hiitations I am to fay nothing* becaufe thefe are
to be learnt more by a Courfe of Obfervation
than by Rule. What I propofed was, to fet
forth the Principles of Compofition in Two
Parts, by way of Inflitution only, not daring to
proceed any further than the fmall KnowledgQ
I have oiMitfick would lead me with Safety.
Ci H A Po
§ i. of MUSIC K. 4jt
0OQQQQGQQ0O0 8OQ0QQGQ0 0Q00QQ
CHAP, xm
Of the Ancient Musigk*
§ i. Of the Name., with the 'various Definitions
and Divifions of the Science*
"'HE Word M u s i c k comes to us from
the Latin Word Mufica, if not immedi-
ately from a Greek Word of the fame
Sound, from whence the Romans, probably took
theirs ; for they got much of their Learning from
the Greeks* Our Criticks teach us, that it
comes from the Word Mufa± and this from a
Greek Word which fignifies to fearch or find
out, becaufe the Mufes were feigned to be In-
ventrefTes of the Sciences^ and particularly of
Poetry and thefe Modulations of Sound that
conftitute Mufich But others go higher, and
tell us, the Word Mufa comes from a Hebrew
Word, which fignifies Art or Difcipline;
hence Mufa and Mufica anciently fignified
F f a Jjearn*
4Jt ^Treatise Chap. XIV.
Learning in general, or any Kind of Science ;
in which Senfe you'll find it frequently in the
Works of the ancient Philofophers. But Kir-
ch er will have it from an Egyptian Word ; be-
cause the Reiteration of it alter the Flood was
probably there, by reafon of the many Reeds
to be found in their Fens, and upon the Banks
of the Nile. Hcfy chins tells us, that the Athe-
nians gave the Name of Mufick to every Art.
From this it was that the Poets and Mytholo-
gies feigned the nine Mufes Daughters offapi-
r^r, who invented the Sciences, andprefide over
them, to affift and infpire thefe who apply to
ftudy them, each having her particular Province.
In this geneal Senfe we have it dehVd to be, the
orderly Arangement and right Difpofition of
Things; in fhort, the Agreement and Harmony 'of
the Whole with its Parts, and of the Parts among
themfelves. Hermes Trifmegifius fays, That
Mufick is nothing but the Knowledge of 'the Order
cf all Things > which was alfo the Docltrine of the
Pythagorean School, and of the Platonicks>
who teach that every Thing in the Univerfe is
Mufick. Agreeable to this wide Senfe, fome
have diftinguifhed Muhck into Divine and Mun-
dane ,• the nrft refpe&s the Order and Harmony
that obtains among the Celeftial Minds ; the b-
ther refpe&s the Relations and Order of every
other Thing elfe in the Univerfe* But Plato by
the divine Mufick understands, that which exifts
in the divine Mind, viz. thefe archetypal Ideas
of Order and Symmetry, according to which
God formed all Things } and as this Order
: - / £ - - exifts
§ i. of MU SICK. 4jj
exifts in the Creatures, it is called Mundane
Mufick : Which is again fubdivided, the re-
markable Denominations of which are, Firfl\
Elementary or the Harmony of the firft Ele-
ments of Things ; and thefe according to the
Philofophers, are Fire, Air, Water, and Earth,
which tho' feemingly contrary to one another,
are, by the Wifdom of the Creator, united and
compounded in all the beautiful and regular
Forms of Things that fall under our Senfes,
id, Celeftial^ comprehending the Order and
Proportions in the Magnitudes, Diftances, and
Motions of the heavenly Bodies, and the Har-
mony of the Sounds proceeding from thefe Mo-
tions : For the Pythagoreans affirmed that they
produce the moft perfect Confort ; the Argu-
ment, as Macrobius in his Commentary on Cice-
ro's Somnium Scipionis has it, is to this Purpofe,
viz. Sound is the Effect ofMotion,and (nice the
heavenly Bodies mud be under certain regu-
lar and (rated Laws of Motion, they muft pro-
duce fomething mufical and concordant,- for
from random and fortuitous Motions, governed
by no certain Mea'fure, can only proceed a gra-
ting and unpleafant Noife : And the Reafon,
fays he, why we are not feniible of that Sound,
is the Vaftnefs of it, which exceeds our Senfb of -
Hearing $ in the fame Manner as the Inhabi-
tants near the Cataracts of the Nile^ are infen-
fible of their prodigious Noife. But fome of
the Hiftorians, if I remember right, tell us that
by the Exceffivenefs of the Sounds, thefe Peo-
ple are rendred quite deaf, which makes that
F f 3 Demon-
454 ^Treatise Chap. XIV.
Demonftration fomewhat doubtful, fince we hear
every other Sound that ^reaches to us. Others
alledge that the Sounds of the Spheres, being
the firft we hear when we come into the World,
and being habituated to them for a long Time,
when we could fcarcely think or make Re-
flection on any Thing, we become incapable
of perceiving them afterwards. But Pythago-
ras faid he perceived and underftood the Ce-
JefKal Harmony by a peculiar Favour of that
Spirit to whom he owed his Lifers jtamhlichus
reports of him,whofays,Thattho' he never fang
or played on any Inftrument himfe!f,yet by an in-
conceivable Sort of Divinity, he taught others
to imitate the Celeftial -Muflck of the Spheres,
by Inftruments and Voice : For according to
him, all the Harmony of Sounds here below, is
but an Imitation, and that imperfect too, of the
other. This Species is by fome called particu-
larly the Mundane Mnfich 3d. Human,
which confifts chiefly in' the Harmony of the
Faculties of the human Soul, and its various
Paifions ; and is alfo confidered in the Proportion
and Temperament, mutual Dependence and
Connection, of all the Parts of this wonderful
Machine of our Bodies. $th. Is what in a more
limited and peculiar Senfe of the Word was
called Mufich, which has for its Object Motion,
confidered as under certain regular Meafures
and Proportions, by which it affects the Senfes
in an agreeable Manner. All Motion belongs
to Bodies, and Sound is the Effedt of Motion,
and cannot be without itj but all Motion does
not
§ i. of MUSIC 'K. 4 jy
not produce Sound/ therefore this was again
fubdivided. Where the Motion is without
Sound, or as it is only the Object of Seeing, it
was called Mufica Orcheftria or Sanatoria,
which contains the Rules for the regular Mo-
tions of Dancing -, alfo Hjpocritica, which
refpedfcs the Motions and Geftures of the Pan-
tomimes. When Motion is perceived only by
the Ear, i. e. when Sound is the Objccl; oiMu-
fick, there are Three Species ; Harmonica,
which confiders the Differences and Proportion
of Sounds, with refpecl to acute and grave;
Rythmica, which refpe&s the Proportion of
Sounds as to Time, or the Swiftnefs and SJow-
nefs of their Succeifions ; and Metric a,
which belongs properly to the Poets, and ref-
pecls the verifying Art : But in common Accep-
tation 'tis now more limited, and we call no-
thing Mufick but what is heard ,- and even then
we make a Variety of Tones ncceflary to the
Being of Mufick.
Aristides Q[u intilianus, who writes
a profeft Treatife upon Mufick) calls it the
Knowledge of ringing, and of the Things that
are joyned with linging (s7ri<T'/j{jiy] fiiXxq Kcd tuv
nspl y^koq av^ouvorru^ which Meibomius tranf-
ates, Scientia cantus, eorumq; qua circa cant um
contingunt ) and thefe he calls the Motions of
the Voice and Body, as if the Cantus it felf
confifted only in the different Tones of the
Voice. Bacchius who writes a fliort Introducti-
on to Mulick in Queftion and Anfwer, gives
the fame Definition. Afterwards, Ariftides con-
F f 4 fidcrs
^(J ^Treatise Chap. XIV.
fiders Mufick in the larger!: Senfe of the Word,
and divides it into Contemplative and A&ive.
The firft, he fays, is either natural or artificial ;
the naturalis arithmetical, becaufe it confiders
the Proportion of Numbers, or phyfical which
ttifputes of every Thing in Nature ; the Ar-
tificial is divided into Harmonica, Rythmica
(comprehending the dumb Motions) and Metri-
ca : The a&ive >vA\\ch. is the Application of the
artificial, is either emmciatwe (as in Oratory,)
Organic l al '(or Inftrumental Performance,) Odical
(for Voice and fmging of Poems, ) Hypocritical
(in the Motions of the Pantomimes^) To what
Purpofe fome add Hydraulical I do not under-
ftand, for this is but a Species of the Organical,
in which Water is fome way ufed for producing
or modifying the Sound. The mufical Facul-
ties, as they call them, are, Melopocia which
gives Rules for the Tones of the Voice or In-^
ftrument, Rjthmopocia for Motions, and Poefis
for making of Verfe. Again, explaining the
Difference of Rythmus and Metrum, he tells
ns, That Rythmus is applied Thee Waysj
either to immoveable Bodies, which are called
Ewythmoi, when their Parts are right propor-
tioned to one another, as a well made Statue $
or to every Thing that moves, fo we fay a
Man walks handfomly ( compofite,) and under
this Dancing will come, and the Bulinefs of the
Pantomimes ; or particularly to the Motion of
Sound or the Voice, in which the Rythmuf
confifts of long and fhort Syllables or Notes,
(which lie calls Times) pyned together (in
Sue-*
§ r. of MUSIC K. 4? 7
Succeflion) in fome kind of Order, fo that their
Cadence upon the Ear may be agreeable ;
which conftitutes in Oratory what is called a
numerous Stile, and when the Tones of the
Voice are well chofen 'tis an harmonious Stile.
Rythmus is perceived either by the Eye or
the Ear,and is fomething general,which may be
without Metrum ; but this is perceived only by
the Ear ? and is but a Species of the other, and
cannot exift without it : The firft is perceived
without Sound in Dancing ; and when it exifts
with Sounds it may either be without any Dif-
ference of acute and grave, as in a Drum, or
with a Varitey of thefe, as in a Song, and then
the Harmonica and Rythmic a are joyned ; and
' if any Poem is fet to Mufick, and fung with a
Variety of Tones, we have all the Three Parts
of Mufick at once. Porphyria in his Com-
mentaries on Ptolemejs Harmonicks, inftitutes
the Diviiion of Mufick another Way; he takes
it in the limited Senfe, as having Motion both
dumb and fonorous for its Object j and, without
diftinguifliing the fpeculat we and practical, he
makes its Parts thefe Six, viz. Harmonica,
Rythmica, Metrica, Organica, Poetica, Hypo-
critica ; he applies the Rythmica to Dancing,
Metrica to the Enunciative, and Poetica to
Verfes.
All the other ancient Authors agree in the
fame threefold Divifion of Mufick into Harmo-
nica, Rythmica and Metrica: Some add the
Organica, others omit it, as indeed it is but-an
gqeidenta] Thing to Mufick, in what Species of
Sounds
458 ^Treatise Chap. XIV.
Sounds it is expreft. Upon this Divifion of
Mufic'h the more ancient Writers are very
careful in thelnfcription or Titles of their Books,
and call them only Harmonica, when they con-
fine themfelvcs to that Part, as Ariftoxenus,
Euclid, Nicomachv.s, Gaudentius, Ptolomej,
Bryennius ; but ' Ariftides and Bacchms call theirs
Mu fie a, becatife they protefs to treat of all the
Parts. The Latinesxie not always fo accurate,
for they inferibe all theirs Mujica, as Boethius,
tho' he only explains the Harmonica ; and
St. Auguftin, tho' his Six Books de Mufica
Ipeak only of the Rythmus and Metriwi; Mar-
ti anus Capella has a better Right to the Title,
for he makes a Kind of Compend and Tranfla-
tion of Ariftides Qiiintil. tho' a very obfeure one
of as obfeure an Original. Aurelius Cajfiodorus
needs fcarcely be named, for tho' he writes a
Book de Mufica, 'tis but barely fome general
Definitions and Divifions of the Science.
T h e Harmonica is the Part the Ancients have
left us any tolerable Account of, which are at
leaft but very general and Theoriccd \ fuch as it
is I purpofe to explain it to you as diftinctly
as I can j but having thus far fettled the Defi-
nition and Divifion of Mufick as delivered by
the Ancients, I chufe next to conhder hifto
rically.
§ 2. The
§ i. of MUSICK. 459
§ 2. The Invention and Antiquity of Mufick,
with the Excellency of the Art in the vari-
ous Ends and Uf'es of it,
f~^p a ll human Arts Mufick has jufteft Pretences
%-s to the Honour of Antiquity: Wc fcarceneed
any Authority for this Aflertion ,• the Reafon of the
Thing demonftrates it, for the Conditions and
Circumftances of human Life required fome
powerful Charm, to bear up the Mind under
the Anxiety and Cares that Mankind foon af-
ter his Creation became fubjecl: to; and the
Goocfnefs of our bleffed Creator foon difcovcred
it felf in the wonderful Relief that Mufick
affords againft the unavoidable Hardships which
are annexed to our State of being in this Life ;
fo that Mufick muft have been as early in the
World as the mod neceflary and indilpenfable
Aits. For
I f we confider how natural to the Mind of
Man this kind of Pleafure is^ as conilant and
univerfal Experience funiciently proves, we can-
not think he was long a Stranger to it. Other
Arts were revealed as bare Neceifrty gave Occa-
lion, and fome were afterwards owing to Luxury;
but neither Neceifity nor Luxury are the Pa-
rents of this heavenly Art; to be pleafed with
it feems to be a Part of our Constitution ; but
'tis made fo, not as abfolutely neceilary to our
Being, 'tis a Gift of G o d to us for our more
happy and comfortable Being,- and 'therefore we
can make no doubt that this Art was among
the very firft that were known to Men. It is
reafon-
4^0 ^Treatise Chap. XIV.
reafonable to believe, that as all other Arts, fb
this was rude and fimple in its Beginning, and
by the Induftry of Man, prompted by his natu-
ral Love of Pleafure, improven by Degrees. If
we confider, again, how obvious a Thing Sound
is, and how manifold Occafions it gives for In-
vention, we are not only further confirmed in
the Antiquity of this Art, but we can make very
flirewd Guefles about the firft Difcoveries of it.
J/bcal Mufick was certainly the firft Kind - 3 Man
had not only the various Tb/z^j of his own Voico
to make his Obfervations upon, before any other
Arts or Inftruments were found, but being daily
entertained by the various natural Strains of the
winged Choirs, how could he not obfervethem,
and from hence take Occaiion to improve his
own Voice, and the Modulations of Sound, of
which it is capable ? *Tis certain that what-
ever thefe Singers were capable of, they pofTeft
it actually from the Beginniug of the World ;
we are furprifed indeed with their fagacious I-
mitations of human Ait in Singing, but we
know no Improvements the Species is capable
of; and if we fuppofe that in thefe Parts where
Mankind firft appeared, and eipecially in thefe
firft Days, when Things were probably in their
greateft Beauty and Perfection, the Singing of
Birds was a more remarkable Thing, we fhall
have lefs Reafon to doubt that they led the
Way to Mankind in this charming Art : But
this is no new Opinion ,- of many ancient Au-*
thors, who agree in this very juft Conjecture, I
(hall only let you hear Lucretius Lib, 5,
§ i. of MUSIC K. 4<Ji
jit liquidas avium voces imitarier ore
Ante fuit multo^ quant lama carminacanta
Concelebrare homines pojfent^aureif que juvare.
The firft Invention of Wind-inftruments he
afcribes to the Obfervation of the Whiffling of
the Winds among the hollow Reeds.
Et Zephyr i cava per calamorumfibilaprimum
Agrefleis docuere cavas inflare cicutas,
hide minutatim dukeii didicere querelas*,
Tibia quas fiindit digitis pulfata canentum,
or they might alfo take that Hint from fome
Thing that might happen accidentally to them
in their handling of Corn-ftalks^ or the hollow
Stems of other Plants. And other Kinds of Inftru-
ments were probably formed by fuch like Acci-
dents : There were fo many Ufes for Chords
or Strings, that Men could not but very foon
obferve their various Sounds, which might give
Rife to ftringed Inftruments : And for the pul-
fatile Inftruments, as Drums and Cymbals, they
might arife from the Obfervation of the hollow
Noife of concave Bodies. To make this Ac-
count of the Invention of Inftruments more pro-
bable., Kircher bids us confider, That the firft
Mortals living a paftoral Life, and being con-
ftantly in the Fields, near Rivers and among
Woods, could not be perpetually idle ; 'tis pro-
bable therefore, fays he 5 That the Invention of
Pipes arid ^Vhiftleswas owing to their Diverfions
and
461 ^Treatise Chap. XIV.
and Exercifes on thefe Occasions ; and becaufe
Men could not be long without having Ufe for
Chords of various Kinds., and variously bent,
. thefe, either by being expofed to the Wind, or
neceflarily touched by the Hand, might give
the firft Hint of ftringed Inftruments ; and be-
caufe, even in the firft fimple Way of Livings
they could not be long without fome fabrile
Arts, this would giveOccaficn to obferve various
Sounds of hard and hollow Bodies, which might #
raife the firft Thought of the pulfatile - Inftru-
ments j hence he concludes that Mufitik was a-
mong the firft Arts.
I f we confider next, the Opinion of thofe
that are Ancients to us, who yet were^too far
from the Beginning of Things to know them
any other way than by Tradition and probable
Conjecture ; we find an univerfal Agreement in
this Truth, That Mufick is as ancient as the
World it felf, for this very Reafon, that it is
natural to Mankind. It will be needlefs to
bring many Authorities, one or Twofhall ferve:
Plutarch in his Treatife of Mufick y which is
nothing but a Converfation among Friends, a-
bout the Invention, Antiquity and Power of
Mv.ficli^ makes one afcribe the Invention to
Amphion the Son of Jupiter and Antiopa^ who
was taught by his Father ; but in the Name of
another he makes Apollo the Author, and to
prove it, alledges all the ancient Statues of this
God, in whofe Hand a mufical Instrument was
always put. He adduces many Examples to
prove the naturaMnfluence Mufick has upon
the
§ i. of MUS1CK. 463
the Mind of Man,and fince he makes no lefsthan
a God the Inventor of it, and the Gods exifted
before Men, 'tis certain he means to prove,
both by Tradition and the Nature of the Thing,
that it is the molt ancient as well as the moil
noble Science. Quintilian (Lib. 1. Cap. 11.)
alledges the Authority of Timagenes to prove
that Mufick is of all the mod ancient Science ;
and he thinks the Tradition of its Antiquity is
■' Efficiently proven by the ancient Poets, who
reprefent Muficians at the Table of Kings ^
finging the Praifes of the Gods and Heroes.
Homer flicws us how far Mufick was advan-
ced in his Days, and the Tradition of its yet
greater Antiquity, while he fays it was a Part
of his Hero's Education. The Opinion of. the
divine Original and Antiquity of Mufick, is al-
io proven by the Fable of the Mufes, fo univer-
fal among the Poets ,- and by the Difputes among
the Greek Writers concerning the firft Authors,
fome for Orpheus ', iome for Amphion, fome for
Apollo, &c. As the beft of the Philofophers
own'd the Providence of the Gods, and their
particular Love and Benevolence to Mankind,
fo they alfo believed that Mufick was from the
Beginning a peculiar Gift and Favour of Heaven ;
and no Wonder, when they looked upon it as
neceffary to afliit the Mind to a raifed and ex-
alted Way of praifing the Gods and good
Men.
I /hall add but one Teftimony more, which
is that of the /acred Writings; where Jfy-
hal the Sixth from Adam^ is called the Fa-
ther
46*4 d Treatise Chap. XIV*
ther of fuch as handle the Harp and Organ i
whether this {ignifies that he was the Inventor,
ot one who brought thefe Inftruments to a good
Perfe£tion,or only one who was eminently skil^
led in the Performance, we have fufficient Rea^
fon to believe that Mufick was an Art long be-
fore his Time ; fince It is rational to think that
vocal Mufick was known long before Inftrumsn^
talj and that there was a gradual Improvement
in the Art of modulating the Voice $ unlefs j£-
dam and his Sons were infpired with this Know-
ledge, which Suppofition would pro re the Point
at once. And if we could believe that this Art
was loft by the Flood, yet the fame Nature re-
maining in Man, it would foon have been re-
vered jj and we find a notable Inftance of it in
the Song of Praife which the Ifraelites railed
with their Voices and Timbrels to GOD,
for their Deliverance at the Red Sea ; from
which we may reafonably conjecture it was an
Art well known, and of eftablifhed Honour long
before that Time.
It may be expected I ffiould, in this Place,
give a more particular Hiftory of the Inventors
of Mufick and mufical Inftruments, and other
famous Muficians fince the Flood. As to the
Invention, I think there is enough faid already
to (how that Mufick is natural to Mankind ;
and therefore inflead of Inventors, the Enquiry
ought properly to be about the Improvers of it ,*
and I own it would come in very naturally here?
But the Truth is 3 we have fcarce any Thing
left
§ z. of MUSIC K. 46 i
left us we can depend upon in this Matter ; or
at lead: we have but very general Hints, and ma-
ny of them contrary to each other, from Au-
thors that fpeak of thefe Things in a tranfient
Manner : And as we have no Writings of the
Age in which Mufick was firft reftored after the
Flood, fo the Accounts we have are fuch un-
certain Traditions, that no Two Authors
agree in every Thing. Greece was the Country
in Europe where Learning firft ilourifhed ; and
tho' we believe they drew from other Fountains,
as Egypt and the moreEaflern Parts, yet they
are the Fountains to us, and to all the Weftern
World : Other Antiquities we neither know fo
well, nor fo much of, at leaft of fuch as have
any Pretence to a greater Antiquity ; except the
yezmjh ; and tho' we are fure they had Mufick y
yet we have no Account of the Inventors a-
mong them, for 'tis probable they learned it
in Egypt ; and therefore this Enquiry about the
Inventors of Mufick fince the Flood, muft be li-
mited to Greece. Plutarch, Julius Pol-
lux, Atheneus, and a few more, are the
Authorities we have principally to truft to, who
take what they fay from other more ancient
Authors of their Tradition. I hope to be for-
given if I am very fliort in the Account of Things
of fuch Uncertainty.
A m p h 1 o n, the Thebmij is by fome reckoned
the moil ancient Mufician in Greece, and the
Inventor of it, as alfo of the Lyra. Some lay ■
Mercury taught him, and gave him a Lyre of
Seven Strings* He is laid to be the firft who
G g tau^lrt
^66 A Treatise Chap. XIV.
taught to play and fing together. The Time
he lived in is not agreed upon.
Chiron the Pelithronian, reckoned a
Lemigod,thc Son of Saturn and Phyllira, is the
next great Matter ; the Inventor of Medicine ;
a famous Philofopher and Mufician, who had
for his Scholars JEfcul aphis, Jafon, Hercules,
Thefeus, jichilles, and other Heroes.
Demodocus is another celebrated M ufi-
ciaiij of whom already.
Hermes, oi-MercuryTrismegistus,
another Demigod, is alfo reckoned amongft the
Inventors or Improvers of Mufick and of the
'Lyra*
Linus was a' famous Poet and Mufician.
Some fay he taught Hercules, Thamyris and
Orpheus, and even Amphion. To him fome as-
cribe the Invention of the Lyra.
Olympus the Myjian is another Benefaclor
to Mufick j he was the Difciple of Marjyas the
Son of Hyagnis the Phrygian ; this Hyagnis
is reckoned the Inventor of the Tibia, which
others afcribe to the Mufe Euterpe, as Horace
infinuates, — Si lie que tibia s Euterpe cohibet.
Orpheus the Thracian is alfo reckoned
the Author, or at leaft the Introducer of various
Arts into Greece, among which is Mufick ; he
paclifed the Lyra he got from Mercury. Some
fay he was Matter to Thamjris and Linus.
Phemius of Ithaca. Ovid ufes his Name
for any excellent Mufician ; Homer alfo names
him honourably.
Ter-
§ 2. of MU SICK. 467
Terpakder the JOejbianfiv'd in the Time
of Lycurgus, and fet his Laws to Mufick. He
was the firft who among the Spartans applied
Melody to Poems, or taught them to be fung in
regular Meafures. This is the famous Mufician
who quelled a Sedition at Sparta by his Mufick*
He and his Followers are laid to have firft in-
stituted the mufical Modes>ufed in finging Hymns
to the Gods -, and fome attribute the Invention
of the lyre to him.
Thales the Cretan was another great Ma- :
fter, honourably entertain'd by the Lacedemoni-
ans, for inftrucling their Youth. Of the Won-
ders he- wrought by his Mufick^ we ftiall hear
again.
Thamyris the Thracian was fo famous, 1
that he is feigned to have contended with the
Mufes, upon Condition he fhould poflefs all their
Power if he overcame, but if they were Victors
lie confented to lofe what they pleafed; and be-
ing defeat, they put out his Eyes, fpoiled his
Voice 3 and ftruck him with Madnefs. He was
the firft who ufed injlrumental Mufick without
Singing.
These are the remarkable Names of Mu-
ficians before Homer's Time, who himfelf was
a Miifician ; as was the famous Poet Pindar*
You may find the Characters of thefe mentioned
at more large, in the firft Book of Fabritius\
Bibliotheca Gr<eca.
W e find others of a later Date, who were
famous in Mufick, as Lafus Jffermionenfis^Me-
lanippideS} Philoxenus, Timotbeus^ Phrjnnis,
4<£8 ^Treatise Chap. XIV.
JZpigonius, Lyfander, Simmicus^ Diodorus the
Thehan j who were Authors of a great Variety
and luxurious Improvements in Mufich Lcif'us,
who lived in the Time of Darius Hyficif'pes,
is reckoned the flrft who ever wrote a Treatife
upon Mujick. Lpigonius Was the Author of an
Inftrument called JTpigonium, of 40 Strings;
he introduced Playing on the Lyre with the
Hand without a Plectrum \ and was the flrft
who joyned the Clihara and Tibia in one Con-
cert, altering the Simplicity of the more anci-
ent Muficki as Lyfander did by adding a great
many Strings to the Cithara, Simmicus alfo
invented an Inftrument called Simmicium of
35 Strings. Diodorus improved the Tibia,
which at firft had but Four Holes, by contriving
more Holes and Notes.
T j motheus, for adding a String to his
'Lyre was fined by the Lacedemonians, and the
String ordered to be taken away. Of him and
Phrynnis, the Comic Poet Pherecrates makes
bitter Complaints in the Name of Mufic'k, for
corrupting and abufing her, as Plutarch reports :
For, among others, they chiefly had completed
the Ruin of the ancient fimple Mufick, which,
lays Plutarch, was nobly ufeful in the Educa-
tion and forming of Youth, and the Service of
the Temples, and ufed principally to thefe Pur-
pofes, in the ancient Times of greateft Wifdom
and Virtue ; but was ruined after theatrical
Shews came to be fo much in Fafhion, fo that
fcarcely the Memory of thefe ancient Modes
remained in his Time. You (hall have fome
Account
§ 2. of MU SICK. 469
Account afterwards of the ancient Writers of
Mufich
As we have but uncertain Accounts of the
Inventors of mufical Inftruments among the An-
cients, fo we have as imperfect an Account of
what thefe Inftruments were, fcarce knowing
them any more than by Name. The general
Divifion of Inftruments is into Jlringed Inftru-
ments, Wind Inftruments and the pulfa tile Kind;
of this laft we hear of the Tympanum or Cym-
balum^ of the Nature of our Drum • the Greeks
gave it the laft Name from its Figure, refem-
bling a Boat.
There were alfo the Crepitaculum y Tinti-
nabulum, Crotalutn, Siftrum; but, by any Ac-
counts we have, they look rather like Chiidrens
Rattles and Play Things than mufical Inftru-
ments.
O f Win ^-inftruments we hear of the Tibia^
fo called from the Shank-bone of fome
Animals, as Cranes, of which they were ftrft
made. And Fiftula ma$e alfo of Reeds. But
thefe were afterwards made of Wood and alfo
of Mettal. How they were blown, whether as
Flutes or Hautboys or otherwife, and which
the one Way, and which the other, is not fufii-
ciently manifeft. 'Tis plain, fome had Holes,
which at firft were but few, and afterwards in-
creafed to a greater Number ; fome had none.
Some were lingle Pipes, and fome a Combina-
tion of feverais, particularly Pan's Syr'wga y
which confuted of Seven Reeds joyned together-
G g 3 fide-
470 ^Treatise Chap. XIV.
Tideways ; they had no Holes, each giving but
one Note, in all Seven diftinct Notes ; but at
what mutual Diftances is not very certain, tho'
perhaps they were the Notes of the natural or
diatonick Scale ; but by this Means they would
want an %<oe 9 and therefore probably otherwife
conftituted. Sometimes they played on a fingle
Pipe, fometimes on Two together, one in each
Hand. And left we fhould think there could
little Mufick be expreft by one Hand, If. Vojfius
alledges, they had a Contrivance by which they
made one Hole exprefs feveral Notes, and cites
a Paffage of Arcadius the Grammarian to
prove it : That Author fays, indeed, that there
were Contrivances to flint and open the Holes,
when they had a Mind, by Pieces of Horn he
calls Bombyces and Opholmioi ( which Julius
Pollux alfo mentions as Parts of fome Kind of
itibia) turning them upwards or downwards,
inwards or outwards : But the Ufe of this is not
clearly taught us, and whether it was that the
fame Pipe might have more Notes than Holes,
which might be managed by one Hand: Per-
haps it was no more than a like Contrivance in
our common Bagpipes, for tuning the Drones
to the Key of the Song. We are alfo told that
Hyagnis contrived the joyning of Two Pipes,
fo that one Canal conveyed Wind to both,
which therefore were always founded together.
W e hear alfo of Organs, blown at firft by a
Kind of Air-pump, where alfo Water was fome
way ufed, and hence called Organum Hydrauli-
cum-, but afterwards they ufed Bellows, Vitru-
vius
§ i. of MUSIC K. 471
mas has an obfcnre Defcription of it, which If
Vojfms and Kircher both endeavour to clear.
There were 7Vba y and Comua^ and Li-
tui, of the Trumpet Kind, of which there
were different Species invented by different Peo-
ple. They talk of fome Kind of Tnhj^ that
without any Art in the Modulation^ had fuch
a prodigious Sound, that was enough to terrify
one.
O f ftringed Inflruments the firft is the Lyra c r
Cithara (which fome diftinguilh :) Mercury is faid
to be Inventor of it, in this Manner ; after an Inun-
dation of the Nik he found a dead Shell-fifb,
which the Greeks call Chelone, and the Latins
Tefiudo ; of this Shell he made his Lyre^ mounting
it with Seven Strings, as Lucian fays ; and added
a Kind o£jugum to it, to lengthen the Strings,
but not fuch as our Violins have, whereby one
String contains feveral Notes ,* by the common
Form this jugum feems no more than Two di-
ftincl: Pieces of Wood, fet parallel, and at fome
Diftance, but joyn'd at the farther End, where
there is a Head to receive Pins for ftretching
the Strings. Boethius reports the Opinion of
fome that lay, the Lyra Mercurii had but Four
Strings, in Imitation of the mundane Mujick of
the Four Elements : But Liodorus Siculus
fays, it had only Three Strings, in Imitation of
the Three Seafons of the Year, which were all
the ancient Greeks counted, <wz. Spring, Sum-
mer and Winter. Nicomachus, Horace^ Luci-
an and others fay, it had Seven Strings, in Imi-
tation of the Seven Planets. Some reconcile Dio-
G g 4 dcru-i
4?x A Treatise Chap. XIV.
cdorus, with the laft, thus, they lay the more
ancient Lyre had but Three or Four Strings,
and Mercury added other Three, which made
up Seven. Mercury gave this Seven-ftringed
lyre to Orpheus , who being torn to Pieces by
the Bacchanals* the Lyre was hung up in A-
f olios Temple by the Lesbians: But others
fay, Pythagoras round it in forne Temple of E-
gy.pt} and added an eighth String. Nicomachus
fays, Orpheus being killed by the Thracian
Women, for contemning their Religion in the
Bacchanalian Rites, his Lyre was caft into the
Sea,- and thrown up at Antiffa a City of Lef-
bos-y the Fifhers finding it gave it to ferp under ,
who carrying it to Egypt* gave it to the Priefts,
and call'd himfelf the Inventor. Thofe who
call it Four-ftring'd, make the Proportions thus,
betwixt the \ft and id, the Interval of a qthy.
3 : 4, betwixt the 2 d and 3^, a Tone 8 : 9, and
betwixt the 3 d and qth String another qth ': The
Seven Strings were diatonically difpofed by
Tones and Semitones* and Pythagoras"^ eighth
String made up the Oc~iave.
The Occafion of afcribing the Invention of
this Inftrument to fo many Authors, is probably,
that they have each in different Places invented
Inftruments much refembling other. However
fimple it was at firfr, it grew to a great Number
of Strings ; but 'tis to no Purpofe to repete the
Names of thefe who are fuppofed to have ad-
ded new Strings to it.
From this Inftrument, which all agree to be
firft of the if ringed Kind in Greece* arofe a Mul-
titude
§ i. . of MUSIC K. 475
titude of others, differing in their Shape and
Number of Strings, of which wc have but indi-
ftinel: Accounts. We hear of the Pfalteriuni y
TrigoUy Sambuca\ PeSfis^ Magadis y Barbiton y
Teftudo ( the Two Jaft'ufcd by Horace promif-
ciiouily with the Lyra and Cithara ) Lpigoni-
tim y Simmicium y Pandura y which were all
(truck with the Hand or a PWclrum ; but it
does not appear that they ufed any Thing like
the Bows of Hair we have now for Violins,
which is a raoft noble Contrivance for making
long and iliort Sounds, and giving them a thou-
fand Modifications 'its impoffible to produce by
a Pleffirum.
Kircher alfo obferves, that in all the ancient
Monuments, where Inftruments are put in the
Hands of Apollo and the Mufes, as there are
many of them at Rome fays he, there is none
to be found with fuch a jugum as our Violins
have, whereby each String has feveral Notes, but
every String has only one Note : And this he
makes an Argument of t'he Simplicity and Im-
perfection of their Inftruments. Befides feveral
Forms of the Lyra Kind, and fome Fiftnla y he
is poiitive they had no Inftruments worth na-
ming. He confiders how careful they were to
tranfmit, by Writing and other Monuments,
their moft trifling Inventions , that they might
not lofe the Glory of them; and concludes, if
they had any Thing more perfect, we fhould
certainly have heard of it, and had it preierv-
ed, when they were at Pains to give us the Fi-
gure
474 ^Treatise Chap. XIV.
gure of their trifling Reed-pipes, which the
Shepherds commonly ufed. But indeed I find
fome PalTages, that cannot be well understood,
withont fuppofing they had Inftruments in which
one String had more than one Note : Where
Pherecrates ( already mention'd ) makes Mufick
complain of her Abufes from Timotheus^s Inno-
vations j fhe fays, he had deftroyed her who
had Twelve Harmonies in Five Strings ; whe-
ther thefe Harmonies fignify (ingle Notes or
Confonances, 'tis plain each String mull have
afforded more than one Note. And Plutarch
afcribes to Terpander a Lyre of Three Chords,
yet he fays it had Seven Sounds, i. e. Notes.
I have now done as much as my Purpofe re-
quired. If you are curious to hear more ofthis,
and fee the Figures of Inftruments both ancient
and modern, go to Merfennus and Kircher.
§ 3. Of the Excellency and various Ufes of
Mufick.
HP Ho' the Reafons alledged for the Antiqui-
* ty of Mufak, fnew us the Dignity of it,
yet I believe it will be agreeable, to enter into
a more particular Hiftory of the Honour Mu-
fick was in among the Ancients, and of its va-
rious Ends and Ufes, and the pretended Virtues
and Powers of it.
The
§3- of MUSIC K. 47 j
The Reputation this Art was in with the
jfczmfh Nation, is I fuppofe wdl known by the
/'acred Hiftory. Can any Thing (hew the Excel-
lency of an Art more, than that it was reckoned
ufefuj and neceffary in the Worihip of God;
and as fuch, diligently praclifed and cultivated
by a People, feparated from the reft of Man-
kind, to be Witnelfes for the Almighty, and
preferve the true Knowledge of God upon the
Earth ? I have already mentioned the Initance
of the Ifraelites Song, upon their Delivery at
the Red Sea, which* feems to prove that Mufick
both vocal and inftr anient al, was an approven
and ftated Manner of worfhipping God: And
we cannot doubt that it was according to his
Will, for Mofes the Man of G o d, and Miriam
the Prophetefs, were the Chiefs of this facred
Choir ; And that from this Time to that of the
Royal Prophet David, the Art was honoured
and encouraged by them both publickly and
privately, we can make no Doubt; for
when Saul was troubled with an evil Spirit from
the Lord, he is advifed to call for a cunning
Player on the Harp, which fuppofes it was a
well known Art in that Time; and behold, Da-
ma\ yet an obfeure and private Perfon, being
famous for his Skill in Mufick, was called ; and
upon his playing, Saul was refrejhed and was
welly and the evil Sprit departed from him.
Nor when David was advanced to the King-
dom thought he this Exercife below him, Spe-
cially the religious Ufe of it. When the Ark
was brought from Kirjath-jearim y David and
4.76 A Treatise Chap. XIV.
all Ifrael played before GOD with all their
Might, and with Singing, and with Harps, and
with Pfalteries, and with Timbrels, and with
Cymbals, and with Trumpets, 1 Chron. 13. 8.
And the Ark being fet op in the City of David,
what a folemn Service was infticuted for the
publick Worfhip and Praife of G o d ; Singers and
Players on all Manner of Inftruments, to\minifter
before the Ark of the DO RD continually, to
record, and to thank) and praife the Lord GOD
o/Israel. Thefe feem.to have beeen divided
into Three Choirs, and over them appointed
Three Choragi or Mafters, Afaph, Heman
and Jeduthun, both to inftru6t theni, and to
prefide in the Service : But David himfelf
was the chief Muftcian and Poet of 'Ifrael. And
when Solomon had flniilied the Temple, behold,
at the Dedication of it, the Levites which were
the Singers, all of them of Afapn, of Heman,
of Jeduthun, having Cymbals, and Pfalteries,
and Harps, flood at the Haft-end of the Altar ^
praifing and thanking the LO RD. And this
Service^ as David had appointed before the
Ark, continued in the Temple ; for we are told,
that the King and all the People having dedi-
cated the Houfe to G o D,-—Ttie Priefis waited
on their Offices ; the Levites alfo with Inftru-
ments of Mufick of the LORD, which Da-
vid the King had made to praife the LORD.
The Prophet Elifha knew the Virtue of
Mufick, when he called for a Minftrel to coni-
pofe his Mind ( as is reafonably fuppofed ) ' be-
fore the Hand of the LORD came upon him*
To
§ 3- of MUSICK. 477
\ T o this I fiiall add the Opinion and Tefti-
(;mony of St. Chryfqftom, in his Commentary on
the /\oth Pfalm. He lays to this Purpofe,
c That God knowing Men to be fiotlifiil and
i C . backward in fpiritual Things, and impatient
6 of the Labour and Pains which they require,
i c willing to make the Task more agreeable,
c and prevent our Wearinefs, he joyn'd Melody
€ or Mufick with his Woiihip ; that as we are
c all naturally delighted with harmonious Num-
c bers, we might with Readinefs and Cheerful-
c nefs of Mind exprefs his Praife in facred
c Hymns. For, fays lie, nothing can raife the
c Mind, and, as it were, give Wings to it, free
c it from Earthlinefs, and the Confinement 'tis
c under by Union with the Body, infpire it with
c the Love of Wifdom, and make eveiy thing
c pertaining to this Life agreeable, as well mo-.
6 dulated Verfe and divine Songs harmonioufy
c compofed. Onr Natures are fo delighted with
I Mufick, and we have fo great and neceflaiy
c Inclination and Tendency to this Kind of Plea-
c fure, that even Infants upon the Bread: are
c foothed and lulled to Reft by this means. A-
• gain he fays, c Becaufc this Pleafure fs fo fami-
c liar and connate with our Minds, that we
c might have both Profit and Pleafure, God
I appointed Pfalms, that the Devil might not
I ruine us with prophane and wicked Songs.
And tho' there be now fome Difference of Opi-
nion about its Ufe in facred Things, yet all
Chriftians keep up the Practice of iinging Hymns
and PfalmSj which is enough to confirm the ge=
ncraf
478 ^Treatise Chap. XIV.
neral Principle of Muficlzs Suitablenefs to the
Worfhip of God.
I n St. yohns Villon, the Elders are repre-
fented with Harps in their Hands,- and tho'
this be only reprefenting Things in Heaven, in a
Way eafleft for our Conception, yet we muft
fuppofc it to be a Comparifon to the befi Man-
ner of worfhipping God among Men, with re-
Ipecl: at leaft to the Means of compofing and
railing our Minds, or keeping out other Ideas,
and thereby fitting Us for entertaining religious
Thoughts.
L e x us next confider the Efteem and Ufe of
it among the ancient Greeks and Romans. The
Glory of this Art. among them, eipecially the
Greeks , appears firit, according to the Ob-
fervation of Quint Hi Mi) by the Names given to
the Poets and Muficians^ which at the Begin-
in g were generally the fame Perfon, and their
Characters thought to be fo connected, that the
Names were reciprocal j they were called Sages
or Wifemen^ and the infpired. Salmuth on Pan-
cirollus cites j^rifiophanes to prove^ that by ci-
thar<s calknSj or one that was skilled in playing
on the ditharay the Ancients meant a Wife-
man, who was adorned with all the Graces; as
they reckoned one who had no Ear or Genius
to Miificky ftupid, or whofe Frame was difor-
dered, and the Elements of his Compofition at
War among themfelves. And fo high an Opi-
nion they had of it, that they thought no In-
duftry of Man could attain to fuch an excel-
lent Art 5 and hence they believed tjiisj?aculty
to I
§ 3 . of MUSIC K. 479
to be an Infpiration from the Gods; which alfo
appears particularly by their making Apollo the
Author of it, and then making their moft anci-
eat Muficians, as Orpheus, Linus, and Amphi-
on, of divine Offspring. Homer, who was him-
felf both Poet and Mufician, could have fhppo-
{ed nothing more to the Honour of his Profef-
fion, than making the Gods themfelves deligh-
ted with it ; after the fierce Conteft that hap-
pened among them about the Grecian and Tro-
jan Affairs, he feigns them recreating them-
felves with Apollo's Mufick ; and after this,
'tis no Wonder he thought it not below his
Hero to have been inftrucled in, and a diligent
Practifer of this Godlike Art. And do not the
Poets univcrfally teftify this Opinion of the Ex-
cellency of Mufick, when they make it a Part
of the Entertainment at the Tables of Kings ;
where to the Sound of the Lyre they fung the
Praifes of the Gods and Heroes, and other ufe-
ful Things : As Homer in the Odyffea introduces
Demodocus at the Table of Alcinous, King of
Ph<eacea, ringing the Trojan War and the Prai-
fes of the Heroes : And Virgil brings in ffipas
at the Table of Dido, finging to the' Sound of
his golden Harp, what he had learned in na-
tural Philofophy, and particularly in Aftronomy
from Atlas ; upon which Quintffian makes this
Reflection, that hereby the Poet intends to
ftiew the Connection there is betwixt Mufick
and heavenly Things; and Horace teaches us
the fame Do^rine, when addrefling his Lyre, he
cne5
4§o ^Treatise Chap. XIV.
cries out, decus Phcebt, & dapibus fupr'emu
grata tejiudo^ 'jfovis.
At the Beginning, Mufick was perhaps fought
only for the fake of innocent Pleafure and Re-
creation i in which View Arifiotle calls it the
Medicine of t&at Heavinefs that proceeds from
Labour; and Horace calls his Lyre laborum did-
ce lenimen : And as this is the firft and moft
fimple, fo it is certainly no defpicableUfe of it;
our Circumftances require fuck a Help to make
us undergo the nccenary Toils of Life more
cheerfully. Wine and Mufick cheer the Hearty
faid the wife Man ; and that the fame Power
ftiil remains 3 does plainly appear by univerfal
Experience. Men naturally feek Pleafure, and
the wifer Sort ftudying how to turn this De-
fire into the greateft Advantage, and mix the
utile didci^ happily contrived, by bribing the
Ear, to make Way into the Heart. The fe-
verefi of the Philosophers approved of Mufick,
becaufe they found it a neceflary Means of Ac-
cefs to the Minds of Men, and of engaging
their Paffions on the Side of Virtue and the
Laws ; ancl fo Mufick was made an Handmaid
to Virtue -and Religion.
Jamblichus in the Life of 'Pythagoras tells
us, That Mufick was a Part of the Difcipline by
which he formed the Minds of his Scholars.
To this Purpofe he made, an'd taught them to
make and fmg, Verfes calculated againft the
Paffions and Difeafes of their Minds • which J
were alfo fung by a Chorus, ftanding round one
that plaid upon the Lyre, the Modulations
whereof
§ $; of MUSlC'K. "48 1
whereof were perfectly adapted to[the Defign and
Subject of the Verfes. He ufed alfo to make
them ling fome choice Verfes out of Homer
and HefioL Mufick was the firft Exercife of
his Scholars in the Morning ; as neceffary tg
fit them for the Duties of the Day, by bring-
ing their Minds to a right Temper j particu-
larly he defigned it as a Kind of Medicine a-
gainft the Pains of the Head, which might
be contracted in Sleep : And at Night, before
they Went to reft, he taught them to compofe
their Minds after the Perturbations of the Day^
by the fame Exercife.
Whatever Virtue the Pythagoreans as-
cribed to Mufick, they believed the Reafon of ,
it to be, That the Soul it felf confifted of Har-
| mony ; and therefore they pretended by it to
revive the primitive Harmony of the Faculties
of the Soul. By this primitive Harmony they
meant that which, according to their Doc^rine^
1 was in the Soul in its pre-exiftent State in Hea-
ven. Macrobius^ who is plainly Pythagorean
in this Point, affirms. That every Soul is delight-
ed with mufical Sounds ; not the polite only
but the moft barbarous Nations pra£tife Mu^
ficky whereby they are excited to the Love of
Vertue, or diffolved in Softnefs and Pleafure : The
Reafon is, fays he, That the Soul brings into the
Body with it the Memory of the Mufick which
it was entertained with in Heaven : And there
are certain Nations^ fays he, that attend the
Dead to their Burial with Singing ; becaufe they
believe the Sotil returns to Heaven the Fountain
H h er
482 /* Treatise Chap. XIV.
or Original' of Mufeck, Lib. 2. in Somniinn
Scipionis. And becaule this Seel; believed the
. Gods thcmfelves to have celeftial Bodies of a
moft perfect harmonious Composition, therefore
they thought the Gods were delighted with it;
|and that by our Ufe of it in facred Things,
. we not only compofe our Minds, and fit them
better for the Contemplation of the Gods, but
imitate their Happinefs, and thereby are ac-
ceptable to them, and open for our felves a Re-
turn into Heaven.
Athenaeus reports of one Clinias a Py-
thagorean^ who, being a veiy cholerick and
wrathful Man, as foon as he found his PafTion
begin to rife, took up his Lyre and fung, and by
this means allayed it. But this Difcipline was
older than Pythagoras ; for Homer tells us,
That Achilles was educated in the fame man-
ner by Chiron, and feigns him, after the hot
Difpute he had with Agamemnon, calming his
Mind with his Song and Lyre : And tho' Hc-
.mer fhould be the Author of this Story, it
fhews however that fuch an Ufe was made of
Mufick in his Days ; for 'tis reafonable to think
he had learned this from Experience.
The virtuous and wife Socrates was no lefsa
Friend to this admirable Art; for even in the De-
cline of his Age he applied himfelf to the Lyre,and
carefully recommended it to others. Nor did the
divine Plato differ from his -great Mafter in this
Point j he allows it in his Common-wealth ; and in
many Places of his Works fpeaks with the greateft
I^efpeft of it 3 as a moft ufefuJ Thing in Society ,-
he
§ 3- of MUSIC K. 483
he fays it has as great Influence over the Mind,
as the Air has over the Body; and therefore he
thought it was worthy of the Law to take Care
of it: He underftood the Principles of the Art
fo well that, as Quintilian Juftly obferves, there
are many Paffages in his Writings not to be un-
derftood without a good Knowledge of it.
jiriftotle in his Politicks agrees with Plato in
his Sentiments of Mufich
Aristides the Philofopher and Mufician,
in the Introduction to his Treatife on this Sub-
ject, fays, 'tis not fo confined either as to the
Subject Matter or Time as other Arts and
Sciences, but adds Ornament to all the Parts
and Actions of human Life*: Painting, fays he,
attains that Good which regards the Eye, Me-
dicine and Gymnaftick are good for the Body,
Diale&ick and that Kind helps to acquire Pru-
dence, if the Mind be firft purged and prepared
by Mujick : Again, it beautifies the Mind
with the Ornaments of Harmony, and forms
the Body with decent Motions : 'Tis fit for
young ones, becaufe of the Advantages got by
Singing ; for Perfons of more Age, by teaching
them the Ornaments of modulate Diction, and
of all Kinds of Eloquence; to others more ad-
vanced it teaches the Nature of Number, with
the Variety of Proportions, and the Harmony
tliat thereby exifts in all Bodies, but chiefly the
Reafons and Nature of the Soul. He fays, as
wife Husband-men firft caft out Weeds and
noxious Plants, then fow the good Seed, fo Mu-
jQck is ufed to compofe the Mind, and fit it for
H h 2 receiving
484 ^Treatise Chap. XIV.
receiving Inftru£tion : For Pleafure, fays he, is
not the proper End of Mufick, which affords
Recreation to the Mind only by accident, the
propofed End being the inftilling of Virtue.
Again, he fays, if every City, and almoft every
Nation loves Decency and Humanity, Mufick
cannot poffibly be ufelefs.
I t was ufed at the Feafts of Princes and He-
roes, fays Athenaus-i not out of Levity and
vain Mirth j but rather as a Kind of Medicine,
that by making their Minds cheerful, it might
help their Digeftion : There, fays he, they fung
the Praifes of the Gods and Heroes and other
ufeful and inftructive, Compofures, that their
Minds might not be neglected while they took
Care of their Bodies; and that from a Reve-
rence of the Gods, and by the Example of good
Men, they might be kept within the Bounds of
Sobriety and Moderation.
But we are not confined to the Authority
and Opinion of Philofophers or any particular
Perfons ; we have the Teftimony of whole Na-
tions where it had publick Encouragement,
and was made neceffary by the Law; as in the
moft Part of the Grecian Common-wealths.
Athenaeus affures us, That anciently all
their Laws divine and civil, Exhortations to
Vertuej the Knowledge of divine and human
Things, the Lives and Actions of illuftrious
Men, and even Hiftories and mentions Uerodo-
t j/.r,were written in Verfe and publickly fung by
a Chorus*, to the Sound of Inftruments ; they
found this by Experience an effe&ual means to
im~
§ 5. of MUSIC K. 4 8 j
imprefs Morality, and a right Senfe of Duty :
Men were attentive to Things that were pro-
pofed to them in fuch a fweet and agreeable
Manner, and attracted by the Charms of har-
monious Numbers, and well modulated Sounds,
they took Pleafure in repeating thefe Examples
and* Inftru£tions, and found them eafier retain-
ed in their Memories. Arifiotle alfo in his
Problems tells us, That before the Ufe of Let-
ters, their Laws were lung mufically, for the
better retaining them in Memory. In the
Story of Orpheus and Amphion, both of
them Poets and Muficians, who made a won-
derful Impreifion upon a rude and uncultivated
Age, by their virtuous and wife InftruCtions,
inforced by the Charms of Poetry and Mufick :
The fucceeding Poets, who turned all Things
into Myftery and Fable, feign the one to have
drawn after him, and tamed the moft favage
Beafts, and the other to have animated the
very Trees and Stones, by the Power of Mufick.
Horace had received the fame Traditions of
all the Things I have now narrated, and with
thefe mentions other Ufes of Mufick : The
Paffage is in his Book de arte Poetica, and is
worth repeating.
Siheftres homines, facer interprefq; deorum>
Ctedibus & viffiuftfdo, deterruit Orpheus :
DiSius ob hoc lenire tigres, rabidq/q; leones :
Diclus & Amphion, Thebana conditor arcis,
Saxa mooerefono teftudinis, & prece blanda
Ducere quo vellet, Fuit hac fapientia quondam,
H h 3 FuU
A Treatise Chap. XIV.
Publica pri'vatis fecernere,facra pr-ofanis :
Concubitu prohibere vago : dare/acra metritis :
Oppida moliri : leges ineidere ligno :
Sic honor j & nomen dwinis vatibus, at que
Carminibus venit. Poft hos infignis Homerus,
Tyrta^ufq; mares animos in mania bella
\ Werfibus exacuit. fiitlte per carmina fortes :
\ Et vita monftrata via eft ; & gratia regum
Pleriis tent at a modis : ludifq^ repertus,
Et longorum operum finis : ne forte pudori,
Sit tibi mi fa lyra fokrf, & cantor Apollo'.
From thefe Experiences I fay, the Art was
publickly honour'd by the Governments of Greece.
It was by the Law made a necefTary Part of the
Education of Youth. Plato afTures us it was
thus at Athens -, in his firft Alcibiades, he men-
tions to that great Man, in Socrates's Name,
how lie was taught to read and write, to play
on the Harp, and wreftle. And in his Crito, he
fays, did not the Laws mod reafonably appoint
that your Father fhould educate you in Mufick
and Gymnaftick ? And we find thefe Three
Grammar, Mufick and Gjmnaftick generally
named together, as the known and necefTary
Parts of the Education of Youth, efpecially of
the better Sort : Plutarch and Athena us give
abundant Teftimony to this ; and Terence hav-
ing laid the Scene of his Pkys in Greece, or
rather only tranflated, and at moft but imitated
Menander, gives us another Proof, in the Affi
3. Scene 2. of his Eunuch. Fac periculum in
Uteris, fac in palaftra, in muficis. fhi<e liberum
fcire aquum eft adolefeentem folertem dabo.
The
§ 3 . of MUSIC K. 487
The Ufe of Mufick in the Temples and
folemn Service of their Gods is paft all quefti-
on. Plato in his Dialogues concerning the
Laws, gives this Account of the facred Mufick.
into. That every Song confift of pious Words.
ido. That we pray to God to whom we facri-
fice. pio. That the Poets, who know that
Prayers are Petitions orRequciis to the Gods,
take good Heed they don't ask 111 inftead of
Good, and do nothing but what's juft, hone ft,
good and agreeable to the Laws of the Society , w
and that they (hew not their Compositions to
any private Perfon, before thofe have feeri
and approven them who are appointed Judges
of thefe Things, and Keepers of the Laws :
Then, Hymns to the Praifes of the Gods are
to be lung, which are very well connected
with Prayer ; and after the Gods, Prayers and
Praifes are to be offered to the Damons and
Heroes.
A s they had poetical Compositions upon va-
rious Subjects for their publick Solemnities, fo
they had certain determinate Modes both in the
Harmonia and Rythmus, which it was unlaw-
ful to alter ; and which were hence called Nb m
mi . or Daws, and Mufica Canonic a* They were
Jealous of any Innovations in this Matter, fear-
ing that a Liberty being allowed, it might be
abufed to Luxury ; for tliey believed there was
a natural Connection betwixt the publick Man-
ners and Mufick : Plato denied that the rrivft-
cal Modes or Laws could be changed without a
Change of the publick Laws ,• he meant, the
H h 4 In-
488 ^Treatise Chap. XIV.
Influence of Mufick wasfo great, that the Chan-
ges in it would neceffarily produce a proportion
nal Change of Manners and the publick Confti-
tution.
The Ufe of it in War will eaftly he allow-
ed to have been by publick Authority ; and the
Thing we ought to remark is, that it was not
ufed as a mere Signal, but for infpiring Cou-
rage, railing their Minds to the Ambition of
great Anions, and freeing them from bafe and
cowardly Fear; and this was not done without
great Art, as Virgil {hews when he fpeaks of
MifenuS)
rrr giio noil pr<eftantior alter,
JEre ciere wos, martemque accendere cantiu
From Athens let us come to Lacedemon y
and here we find it in equal Honour, Their
Opinion of its natural Influence was the fame
with that of their Neighbours : And to fhew
what Care was taken by the Law, to prevent
the Abufe of it to Luxury, the Hiftorians tell us
that Timothens was fined for having more than
Seven Strings on his Lyre, and what were ad-
ded ordered to be taken away. The Spartans
were a warlike People, yet very fenfible of the
Advantage of fighting with a cool and delibe-
rate Courage; therefore as Gellius out of Thii-
cydides reports, they ufed not in their Armies,
Inftruments of a more vehement Sound, that
might infkme their Temper and make them
more furious, as the Tuba 3 Qornn and Lituus^
but
§ 3 . of music k: 489
but the more gwitlc ancl moderate Sounds and
Modulations of ihe Tll/ia, that their Minds be-
ing more compiled, j/iey might engage with a
rational Courage, s And Gelllus tells us, the
Cretans ufed the Cithara to the lame Purpofc in
their Armies. We have already heard how this
People entertain'd at great Expence the famous
Thales to inftruft: their Youth in Mufick ; and
after their Mufick had been thrice corrupted,
thrice they reftored it.
If we go to Thebes, Epamlnondas will be a
Witnefs of the Efteem it was in, as Com.
Mepos informs us.
Athenius reports, upon the Authority of
Theopompus, that the Get an AmbafTadors, be-
ing fent upon an Embaffy of Peace, made their
Entry with Lyres in their Hands, ringing and
playing to compofe their Minds, and make them-
felves Mafters of their Temper. We need not
then doubt of its publick Encouragement among
this People.
But the mod famous Inftanc e in all Greece,
is that of the Arcadians, a People, fays Poly-
bius,in Reputation for Virtue among the Greeks;
efpecially for their Devotion to the Gods. Mu-
fick, fays he, is efteem'd every where, but to
the Arcadians it is necerTary, and allowed a
Part in the Eftablifhment of their State, and an
indifpenfable Part of the Education of their
Children, And tho' they might be ignorant of
other Arts and Sciences without Reproach, yet
none might prefume to want Knowledge in Ma-
fick,
490 j4 Treatise Chap. XIV.
(ick, the Law of the Land making it neceffary;
and Insufficiency in it was reckoned infamous
among that People. It was not thus eftabliflied,
fays he, fo much for Luxury and Delight, as
from a wife Confideration of their toiifom and
induftrious Life, owing to the cold and melan-
choly Air of their Climate ; which made them
attempt every Thing for foftning and fweetning
thbfe Aufterities they were condemned to. And
the Neglect of this Difcipline he gives as the
Reafon of the Barbarity of the Cynathians a
People of Arcadia.
\V e (hall next confider the State of Mufick
among the ancient Romans. Till Luxury and
Pride ruin'd the Manners of this brave Nation,
they were famous for a fevere and exact Virtue.
And tho' they were convinced of the native
Charms and Force ofMuftck, yet we don't find
they cherifhed it to the fame Degree as the
Greeks ; from which one would be tempted to
think they were only afraid of its Power, and
the ill Ufe it was capable of ; a Caution that
very well became thofe who valued themfelves
fo much, and juftly, upon their Piety and good
Manners.
Corn. NEPos,in his Preface, takes Notice of the
Differences betwixt thcGreek andRomanCuftoms^
particularly with refpecl; to Mufick > and in the
Life of JEpaminondas^ he has thefe Words, Sci-
mus enim muficum noftris moribus abeffe a prin-
cipis perfona ; fait are etiam in vitiis poni^ qua
omnia apud Gracos (j gratia & laude digna du-
wun'tur.
Cice-
§ 3 . of MUSIC K. 491
Cicero in the Beginning of the hrft Book
of his Tufculan Queftions, tells us, that the old
Romans did not ftudy the more foft and polite
Arts fo much as the Greeks;bc'mg more addict-
ed to the Study of Morality and Government :
Hence Mufick had a Fate fomewhat different
at Rome.
But the fame Cicero {hews us plainly his
own Opinion of it. Lib. 2. de Legibus -,
Affentior enim Platoni, nihil tarn facile in a-
nimos teneros atque molles infiuere quam <va-
rios canendi fonos. Quorum dici mxpoteft quanta
fit vis in utramque partem, namque & incitat
languentes, & languefacit incitatos, & turn re-
mittit animoSj turn contrahit. Certainly he had
been a Witnefs to this Power of Sound, before
he could fpeakfo; and I (hall not believe he had
met with the Experiment only at Athens. A Man
fo famous for his Eloquence, muft have known
the Force of harmonious Numbers, and well
proportioned Tones of the Voice.
Quintilian fpeaks honourably of Mufick.
He fays. Lib. 1. Chap. 11. Nature feems to
have given us this Gift for mitigating the Pains
of Life, as the common Practice of all labouring
Men teftifies. He makes it neceflary to his O-
rator, becaufe, fays he, Jjb. 8. Chap. 4. it is
impoifible that a Thing ftiould reach the Heart
which begins with choking the Ear ; and be-
caufe we are naturally pleafed with Harmo-
ny, otherwife Internments of Mufick that cannot
exprefs Words would not make fuch furprifing
and
49* ^Treatise Chap. XIV.
and various Effects upon us. And in another
Place, where he is proving Art to be only Na-
ture perfe&ed, he fays, Mufick would not o-
therwife be an Art^ for there is no Nation which
has not its Songs and Dances.
Some of the firft Rank at Rome pra£fcifed it.
"Athenaus fays of one Mqfurius a Lawyer,
whom he calls one of the beft and wifeft of Men,
and inferior to none in the Law, that he appli-
ed himfelf to Mufick diligently. And Plutarch
flacesMuJickiVizSmging and playing on thelyr e y
among the Qualifications of Metella the Daugh-
ter of Scipio Metellus.
Macrobius in the i o Chap* Lib. 2. of
his Saturnalia fliews us,that neither Singing nor
Dancing ivere reckoned dishonourable Exercifes
even for the Quality among the ancient Mo*
mans 1 particularly in the Times betwixt the
Two Punick Wars, when their Virtue and
Manners were at the beft, providing they were
not ftudied with too much Curiofity, and too
much Time fpent about them ,• and obferves
that it is this, and not (imply the Ufe of thefe
that Salufi complains of in Sempronia^ when he
lays fhe knew pfalkre & fait are elegantius
quam neceffe erat probx. What an Opinion
Macrobius himfelf had of Mufick. we have in
part fiiewn already ; to which let us add here
this remarkable Paffage in the Place formerly
cited. Ita denique omnis habitus anima catiti-
bus gubernatnr, at & ad helium progreffui
etiam receftui canatur, cantu (j excitante &
ritrfus fedante viriutem > datfomnos adimitque^
nee-
§ 3 . of MUS1CK. 493
necnon curas & immittit & retrahit, tram
fuggerit, dementi am fuadet, corporum quoqae
morbis medetur. Hinc eft quod <£gris remedia
pr<sftantes pr^cinere dicuntur. The Abufe of
it, which 'tis probable Jay chiefly in their idle,
ridiculous and lafcivious Dancing, or perhaps
their fpending too much Time even in the moft
innocent Part of it, and not applying it to the
true Ends, made the wifer Sort cry out, and
brought the Character of a Mufician into fomc
Difcredit. But we find that the true and pro-
per Mufick was ftill in Honour and Practice a-
mong them: Had Rome ever fuch Poets, or
were they ever fo honoured as in Auguftuss
Reign ? Uorace^xho he complains of the Abufe
of the Theatre and the Mufick of it, yet in ma-
ny Places he (hews us, that it was . then the
Practice to fing Verfes or Odes to the Sound of
the Lyre, ox. o£ Pipes, or of both together; Lib.
4. Ode 9. Verba loquor focianda chordis. Lib. 2.
£p. 2. Hie ego <verba lyr* motura fonum con-
nedle're digner ? In the firft Ode, Lib. 1. he gives
us his own Character as a Poet and Mufician,
Si neque tibias Euterpe cohibet, &c. He fliews
us that it was in his Time ufed both publickly
in the Praife of the Gods and Men, and private-
ly for Recreation, and at the Tables of the
Great, as we find clearly in thefe Paffages. Lib.
4. Ode 11. Condifce modos amanda voce quos
reddas, minuentur atr<e carmine curd. Lib. 3,
Ode 28. Nos cantabimus invicem Neptunum,
tu curva recines lyra Latonam, dye. Lib. 4. Ode
,*5- Nofque (j profejiis lucibus (j facris - Rite
JDeo:
494 ^Treatise Chap. XIV-
Deos prius adprecati, mrtute f initios more pa-
trum duces^ Lydis remiflo carmine tibiis Tro-
jamque, (jc. canemus, Epode 9. Qiiando re-
poftum deciibiim adfefias dapef tecum. — Beate
Mecamas bibam ? Sonante mifiis tibiis carmen
lyra. Lib. 3. Ode 11. Tuque tefiudo — Nunc
(j divitum menfis & arnica templis.
For all the Abufes of it, there were fliJI
fome, even of the beft Characters, that knew
how to make an innocent Ufe of it : Suetoti in
Titus's Life,whom he calls Amor ac delicia ge-
neris humani, among his, other Accomplifhments
adds, Sed ne Mufica? quidem rudis y ut qui can-
tar et (j pfalleret jucunde fcienterque.
There is enough faid to fliew the real Va-
lue and Ufe of Mufick among the Ancients. I
believe it will be needlefs to infift much upon
our own Experience ; I fiiall only fay, thefe
Powers of Mufick remain to this Day, and are
as univerfal as ever. We uie it ftill in War and
m f acred Things^ with Advantages that they
only know who have the Experience. But in
common Life almoft every Body is a Witnefs of
its fweet Influences.
W h a t a powerful Impreflion mufical Sounds
make even upon the Brute Animals, eipecially
the feathered Kind, we are not without fome
Inftances. But how furprifing are the Accounts
we meet with among the old Writers ? I have
referved no Place for them here. You may fee
a Variety of Stories in JElians Hiftory of Ani-
malsj
§ 3 . of MUSIC K. 49 j
mals, Straboy Pliny, Marcianus Capella, and
others. ■
Before I leave this, I muft take Notice of
fome of the extraordinary Erfe&s afcribed to
Mufich Pythagoras is laid to have had an
abfolute Command of the human Paffions, to
turn them as he pleafed by Mufick : They tell
us, that meeting a young Man who in great Fu-
ry was running to burn his Rival's Houfe, Py-
thagoras allayed his Temper, and diverted the
Defign, by the fole Power of Mufick. The Story
is famous how Timotheus, by a certain Strain
or Modulation, fired Alexander 's Temper to
that Degree, that forgetting himfelf, in a war-
like Rage he killed one of the Company; and
' by a Change of the Mufick was formed again,
even to a bitter Repentance of what he had
done. But Plutarch fpeaks of one Antigenides
a Tibicen or Piper, who by fome warlike Strain
had tranfported that Hero, fo far that he fell
upon fome of the Company i Icrpander quelled
a Sedition at Sparta by means of Mufich T ha-
les being called from Crete, by Advice of the
Oracle, to Sparta, cured a raging Peftilence by
the fame Means. The Cure of Difeafes by Mu-
fick is talked of with enough of Confidence.
Aulus Gellius Lib. 4. Chap* 13. tells us it was
a common Tradition, that thofe who were
troubled with the Sciatica ( he calls them If-
■chiaci) when their Pain was moft exquifite,
were eafed by certain gentle Modulations of
Mufick performed upon the Tibia ; and fays, he
ta4 rea$ in Theophrafius that, by certain artful
Modu-
49c? ^Treatise Chap. XIV.
Modulations of the fame Kind of Inftrument,
the Bites of Serpents or Vipers had been cured.
Clj'temneftra had her vicious Inclinations to
Unchaftity corrected by the Applications of
Muficians. And a virtuous Woman is faid to
have diverted the wicked Defign of two Rakes
that affaulted her, by ordering a Piece of Mu-
fick to be performed in the Spondean Mode.
The Truth and Reality ofthefe Effects fhallbe
confidered afterwards.
(j 4. Explaining the .HarmoKick Principles
of the Ancients ; and their Scale of Muhck.
Indroduction. Of the ancient Writers on Mufick*
npHESE Principles are certainly to be found
■*- no where, but among thofe who have
written profehedly upon the Subject ,- I ftiall
therefore introduce what I'm to deliver, with a
fhort Account of the ancient Writers upon Mu-
fick.
I have already obferved, that the firft Writer
upon Mufick was Lafus Hermionenfis ; but his
Work is loft, as are the Works of very many
more, both Greek and Latin^ of which you'll
find a large Catalogue in the %d Book of Fa-
britiius Bibliotheca graca 1 where you'll alfo
find an Account of fome others 3 that are pre-
tended to be ftill in IVyinufcript in fome Libra-*
rie?*
§ 4 . of MUSIC K. 497
rics, Here I (hall only fay a few Words con-
cerning thole Authors that are ftill extant and
already made publick.
aristoxenus the Difciple of ' Ariftottejs the
ekleft Writer extant on this Subject $ he calls his
Book Elements of Uarmonicks • and tho' in his
JDiviJion lie fpeaks of the reft of the Parts, yet he
explains there only the Harmonica* . He wrote
a Treatife upon the other Parts, which is loft..
Euclid, the Author of the Elements of
'Geometry, is next to Ariftoxenus, he writes an
' Introduction to Uarmonicks. '
Aristides Qjjintili anus wrote after
'Cicero's Time ; he calls his Book, Of Muficki
becaufe he treats of both the Harmonica and
'Rythmical
Alypius ftands next, who writes only art
Account of the Greek Semeiotica, or of the Signs
by which the various Degrees of Tune were no-
ted in any Song.
Gaudentius the Philofopher makes a
Kind of fliort Compend of Ariftoxenus, which
he calls an Introduction to Uarmonicks.
Nicomachus the Pythagorean writes a
Compend of Uarmonicks, which he fays was
done at the Requeft of fome great Woman, and
promifes a more complete Treatife of Mufich.
'tis fuppofed that Boethius had feen and made
Ufe of it, from feveral Paffages he cites, which
are not in this Compend ; but 'tis loft fince.
Bacchius a Follower of Ahjioxenus^vntes
a very fliort Introduction to the Art of Miifick
in Dialogue.
li Of
A Treatise Chap. XIVJ
f thefc Seven Greek Authors, we have a!
fair Copy, with Tranflation and Notes, by Mei-\
bomius.
Claudius' Ptolomaeus the famous Ma-I
thematician, about the Time of the Emperor
Antoninus Pius, writes in Greek Three Books
of Uarmonicks. He ftrikes a Medium be-
twixt the Pythagoreans and Ariftoxenians>\
in explaining the harmonich Principles.
Of this Author, with his prolix Commenta-
tor Porphyrins, we have a fair Copy with
Tranflations and Notes, by the learned Doctor
Wallis. Vol. III. of his mathematical Works.
And from the fame Hand we have alfo, with
Tranflation and Notes.
Manuel Bryennius, long after any of
the former, who writes of Harmonichs. In his
firft Book he follows Euclid, and in his 2 d and
id Ptolomy.
1 have fpoken of PlutarcFs Book de Mufica,
in the § 1.
O f the Latins we have
B o e t h 1 u s, in the Time of Theodorich the
Goth, he writes de mufica, but explains on-
ly the harmonich Principles - 3 'tis with his other
^Vorks.
Martianus Capella in the 9th Book
of his Treatife de nuptiis Philologi<e & Veneris,
writes de mufica, in which he is but a forry
Copier from Ariftides. We have this Work
with Meibomius\ Collection of the Greek Wri-
ters*
§ 4- of MU SICK. 499
St. Augustin writes de mufica^ but he
treats only of the Rythmi and pedes metrici i
'tis among his Works*
aureliusCassiodorus, m the Time of
Theodoricky among his other Works, and particu-
larly de artibus ac difciplinis liberaliiim literal
rum, treats de mufica \ 'tis a very fhort Sketch,
amounting to no more than fome general Defi-
nitions and Divisions.
There are one or Two more Authors^
which t have not feen : But thefe mentioned
contain the whole Doctrine that's left us by the
Ancients ; and perhaps we might fpare feverals
of thefe without great Lofs, Two or Three of
them containing the Whole ; fo true it is what
Gerhard Vojfms remarks of them, nempe alii
alios illaudato more exfcripferunt.
These then are the Authorities and Origi-
nals, from which I have taken the following
Account of the ancient Sjftem of Mufick* It
will be needlefs therefore, after I have told yoil
this, to make a trotiblefom and tedious Citati-
on for every Thing I mention.
Of the ancient Harmonica.)
H o w the ancient Writers defined and divided
M u s i c k has been explained in § i. of this Chi
and needs not be repeted* My Bufinefs here is
With the Part they called Harmonica^ which
treats of Sounds and their Differences^ with re-
fpecl: to acute and graven Ptolomy calls it d
Power or Famlty perceptive of the Difference
I i i 0f
yoo ^Treatise Chap. XIV.
of Sounds*) with refpeffi to Acutenefs and Gra-
vity; and Bryennius calls it a fpeculative and
practical Science, of the Nature of the harmo-
nick Agreement in Sounds..
• They reduce the Doctrine of Harmonicks
into Seven Parts, viz. ift. Of Sounds. 2 d. Of In-
tervals. 3d. G£ Syfkems. qth. Ot t\\e Genera ov
different Kinds, with refpect to the Conftitution
and Divifionof the Scale. %th. Of the Tones or
Modes. 6th. Of Mutations or Changes. 7th. Of
the Melopccia or Art of making Melody or Songs.
Of thefe in Order.
I, O f S o u n d. This Ptolomy confiders in
a large Senfe, comprehending the whole Object
of Hearing, and calls it by a general Name
tyoCpog, i. e. Strepitus^ or any Kind of Sound.
As it is capable of a Bifferencein -Acutenefs and
Gravity 1 Arifioxenus calls it Ocv??, i. e. Fox,
or Voice. As to the Nature and Caufe of
So^nd, they agree that it is the EfTett of the
Percuflion of the Air, whofe Motion is propa-
gated totlie Ear, and there raifes a Perception.
The principal Difference they confider in Sounds
is of Acutenefs and Gravity y which is produced
by a quicker or flower Motion in the Vibrati-
ons of the Air. A Sound confidered in a cer-
tain determinate Degree of Acutenefs or G ra-
vity\ they call (p^ofyog, i. e. Sonus; and they
define it thus, Ariftox. Qwyjg 7f)£atg S7rl fjrfav. '
tartf, .Q&olyog, i. e. Sonus eft vocis cafus in
imam tenfionem. Ariftides confiders it with re-
gard' to its Ufe, and calls it rdffiv jjishwfojwjv,
tenfionem mekdicam, Ni 'comae h us . defines it,
§ 4. of MU SICK. yoi
<j5oovyjc efAftsXzs ocixXocTYi rdmv, vocis ad cantum
apt£ tenfione?n, latitudinis expert cm. Thus they
diftinguiilied Sounds, according as their Degree
of Acutenefs or Gravity was fit or not for-
Song,- fuch as were fit were alfo called contin-
uous Sounds, and others inconcinnov.s. Thefe
Words zv anting Latitude, were added to con-
tradict a Notion of Lafus and the Epigonians,
that a Voice could not poilibly remain for any
determinate Time in one Degree, but made
continually feme little Variations up and down,
tho' not very fenfible.
Then they confider a Voice as changing
from acute to grave, or from this to that ; and
hereby form the Notion of a Motion of the
Voice, which they fay is Twofold ; the one con-
tinuous, by which we change the Voice in com-
mon Speaking, the other dtferete, as in Singing.
See above Ch. 2. And fome added a Third and
middle Kind, whereby, fay they, we read a Poem.
In Sounds ( ($$otyoi ) they confider Three
Things, Tenfion, which is the P.eft or Standing
of the Voice in any Degree, Intenfion and Re-
mijjlon are the Motions of the Voice upward
and downward, whereby it acquires Acutenefs
or Gravity : And when it moves, all the Di-
ftance or Difference betwixt the firft and laft
Degree or Tenfion^ they called the Place thro'
which it moved. Then there is Diftenfion or
Difference of acute and grave, in which the
Quantity that is the mathematical Objecl: con-
fiits j this they faid is naturally infinite, but with
refpe£t either to our Senfes, or what Sounds .we
I i 3 can
5oz ./f Treatise Chap. XIV.
can poffibly raife by any Means, it is limited ;
and this brings us to the Second Head.
II. Of Intervals. An Interval is the
Difference of Two Sounds, in refpeft of acute
and grave; or, that imaginary Space which is
terminated by Two Sounds "differing in Acute*
nefs or Gravity. Intervals were considered as
differing, 17/20, in Magnitude. 2 do. As the Ex-
tremes were Concord or Di/cord. 3tio. As com-
pofite or incompofite, that is^ fimple or com-
pound. Ato. As belonging to the different ge-
nera ( of which again. ) sto. As rational or
irrational, i. e. fuch as we can difcern and mea-
fure, and which neither exceed our Capacities
in Greatnefs or Littlenefs.
As to the meafuring of Intervals^ and, as
'ptolomy calls it, the Cr iter ions in Harmonkh\
there was a notable Difference among the Phi-
lo/bphers, which divided them into Two Se£ts,
the Pythagoreans and Ariftoxenians ; betwixt
whom Ptolomy (hiking a Midft, made a Third
Pythagoras and his Followers meafured
all the Differences of Aciitenefs and Gravity^
by the Ratios of Numbers. They fuppofed
tnefe Differences to depend upon the different
Velocities of the Motions that caufe Sound; and
thought therefore, that they could only be ac-
curately meafured by the Ratios of thefe Velo-
cities. Which Ratios were firft inveiligate by Py-
thagoras^ as Nicomachus and others inform us,
in this Manner, viz. Palling by a Smith's Shop,
Jae perceived a Concord or Agreement betwixt
the
§ 4 . of MUSIC K. jo 3
the Sounds of Hammers ftriking the Anvil: He
went in, and made feveral Experiments, to
find upon what the Difference really depended ;
and at laft making Experiments upon Strings,
which he ftretchcd by various Weights, he
found, fay they, that if Four Chords., in every
Thing elfe equal and alike, are ftretchcd by
Four Weights, as 6 . 8 . 9 . 12. they yield
the Concord of Offiave betwixt the firft and laft,
a qth betwixt the firft and Second, as alfo be-
twixt the Third and laft, a $th betwixt the firft
and Third, and alfo betwixt the Second and
laft; and that betwixt the Second and Third
was exactly the Difference of qth and $th ; be-
ing all proven by the Judgment of a well tuned
Ear: Hence he determined thefe to be the true
Ratios that accurately exprefs thefe Intervals.
But we have found an Error in this Account,
which Vincenzo Galileo^ in his Dialogues of the
ancient and modem Mufick, is, for what .1
know, the firft who obferves • and from him
Meibomius repetes it in his Notes upon Nico-
machns. We know, that if Four Strings are
in Length, as thefe Numbers 6.8.9. 12. (ce-
teris paribus ) their Sounds make the Intervals
mentioned. But whatever Ratio of Length
makes any Interval^ to make the fame by Two
Chords, in every other Thing equal, butitretcht
by different Weights, thefe Weights mufl be as
the Squares of the unequal Lengths, i. e. for an 0-
dfave 1 : 4, for a $th 4 : 9, and for a qth 9 ' i&
( See above Ch. 2. ) Hence by the Ratios ot
the Lengths of Chords, which are reciprocally as
I i 4 the
yo4 A Treatise Chap. XIV.
the Numbers of Vibrations, all the Differences
of acute and grave are meafured. The Pytha-
goreans juftly reckoned that the minute Diffe-
rences could by no means be trufted to the Ear,
and therefore judged and meafured all by Ra-
tios.
Aristoxenus on the contrary, thought
Reafon had nothing to do in the Cafe ; that
Senfe was the only Judge; and that the other
was too fobtil, to be of any good Ufe: He
therefore took the %ve, $th and qth, which are
the firil and moft fimple Concords by the Ear.
By the Difference of the /\th and '$'th lie found
the Tonus: And this being once fettled as an
Interval the Ear could judge of, he pretended
to meafure every Interval by various Additions
and Subdutfcions made of thefe mentioned, one
with another. Particularly, he calls Diatejfa-
ron equal to Two Tones and a Half ; and ta-
king Two Tones, or Ditonum, out of Diatejfa-
ron, the Remainder is the Hemitonium -, then
the Sum of Tonus and Hemitonium is the Tri-
emitonium. To get an Idea of the Method of
bringing out thefe Intervals, fuppofe Six Sounds
a . ' b : c : d : e : f. If a is the low eft, we can
by the Ear take d a qth and e a $th upward ;
then from e downward we can take b a qth, fo
that a : b and d : e are each the Tonus or Dif-
ference of qth and 5 th ; alfo from b we can
take upward fa 5th, and downward from fa
Aph at c j hence we have other Two Tones b • c
and e : f> alfo a Hemitonium c . d> a Ditonum
a :"- c or d -f 9 a Triemitonium b -d or. c - e.
But
§ 4 . of MUSIC K. joy
But the Inaccuracy of this Method of determi-
ning Intervals is very great.
Ptolomey argues ftronsly againft the laft
Sect, that while they own thefe different Ideas
of acute and grave, which arife from the Relati-
ons of the Sounds among themfelvesj and that
the Differences in the Lengths of Chords which
yield thefe Sounds, are the fame ; yet they nei-
ther know nor enquire into the Relation : But
as if the Interval were the real Thing, and the'
Sound the imaginary, they only compare the
Differences of the Intervals, making by this
Means a Shew of doing fomcthing in Mnjick
by Number and Proportion ; which yet, fays he,
they act contrary to • for they don't deter-
mine what every Species is in it felf ; as we de-
fine a Tone to be the Difference of Two Sounds
which are to one another as 8 : 9 ; but they
fend us to another Thing as indetenriiriate,when
they call it the Difference of a qth and $th.
Whereas if we would raife a Tone exactly, we
need neither qth nor $th. And if we ask how
great that Difference is, they cannot tell us ; if
perhaps they don't fay, 'tis equal to Two fuch
Intervals, whereof Diateffaron contains 5-, or
Diapq/bn 12, and fo of the reft$ but what that
is they determine not. Again, by confidering
the mere Interval, they do nothing at all • for
the mere Diftance is neither Concord nor con-
cinnous, nor any Thing real ; whereas by com-
paring Two Sounds together we determine the
Ratio or Relation, and the Quality of their
Difference, i % e % whether it conftitutes Concord
or
yo6 -^Treatise Chap. XIV.
or Difcord, by the Form of that Ratio.
Next, he fnews the Fallacy of Ariftoxeniis\
Demonstration, whereby he pretended to
prove that a /\th was equal to Two Tones
' and a Half. I need not trouble yon with
it here ; for we have learnt already that a
Tone 8 : 9 is not diviiible into Two equal
Parts. But then he alfo finds fault with the
Pythagoreans for fome falfe Speculations about
the Proportions ; and having too little Re-
gard to the Judgment of the Ear, while they
refufe fome Concords that the Ear approves, on-
ly becaufe the Ratio does not agree with their
arbitrary Rule ; as we (hall hear immediate-
Iy.
Therefore he would have Senfe and Rea-
fon always taken together in all our Judgments,
about Sounds, that they may mutually help
and confirm one another. And of all the Me-
thods to prove and find the Ratios cf Sounds,
he recommends as the mod accurate, this, viz.
to ftretch over a plain Table an evenly well
made String, fixt and raifed equally at both
Ends, over Two immoveable Bridges of Wood,
fet perpendicularly to the Table, and parallel to
each other; betwixt them a Line is to be drawn
on the Table, and divided into as many equal
Parts as you need, for trying all Manner of
Ratios ; then a moveable Bridge runs betwixt
the other Two, which juft touches the String,
and being fet at the feveral Divifions o('the Line,
it divides the Chord into any Ratio of Parts;
whofe Sounds are to be compared together, or
with
§ 4- of MUSIC K. jo/
with the Sound of the Whole. This he calls
Canon Harmonious. And thofe who deter-
mined the Intervals this Way, were particular-
ly called Canonic'^ and the others by the gene-
ral Name of Mujici.
Of Concords. They defined this, An A-
greement of Two Sounds that makes them, ci-
ther fuccclfivcly or jointly heard, plcafant to the
Ear. They owned only thefo Three iimple ones,
viz. the Fourth 3 i 4, and Fifth 2: 3 called
Dld-tejjaron and Dia-pente^ and the Offiavs
1 : 2, which they called Dia-Pa/bn ; the Rea-
fon of thefe Names we fliall hear again. Of
compound Concords], the Pythagoreans owned
only the Sum of the $th and Sve 1 : 3, and the
double 8<ve. 1 : 4 or Dif-dia-pafon> but others
owned alfo the Sum of Ath and 8&% 3 : 3.
The lleafon why the Pythagoreans rejected
the compound Ath^ 3 : 8 was, That they ad-
mitted nothing for Concord but the Intervals
whofe Ratios were multiple or faper particular^
i. e. where the greater Term contained the
other a precife Number of Times, as 3 : 1, or
where the greater exceeded the lefler only by
1, as 3 : 2 or 4 : 3. becaufe thefe are the mofl
ftmple and perfect Forms of Proportion : But
Ptolowy argues againft them from the Perfecti-
on of the Dia-paforiy whereby 'tis impoHible
that any Sound fhould be Concord to its one
Extreme, and Difcord to the other. The Ex-
tremes Bia-pqfon and Difdia-pajbn^ Ptolomy
calls Omophoni or Unifoks, becaufe they a-
gree as one Sound, The Ath and 5th and their
Com-
yo8 A Treatise Chap. XIV.
Compounds he"calls Synphoni or confonant; the <r
thcr Intervals belonging to Mufick he calls Emme-
li or continuous. Others call thofe of equal Degree
Omophoni, the 8o&r Antiphoni, the 4/-&J and 5?/r;x
Par aphonic others call the 5 //?/ only Paraph oni,
and the 4//^/ Synphoni, but all agree to call the
Difcords Diaphoni.
The abftract Reafonings of the Pythago-
reans about the Ratios of the Concords, you
have in Ptolomy ; but more particularly in
Euclid's Seblio Canonis. The fundamental Prin-
ciple is, That every Concord arifes either from
a Multiple or fuperp articular Ratio, The
other neceflary Premifies are. 1 7/zo. That a mul-
tiple Ratio twice compounded, (i. e. multiplied
by 2,) makes the Total a multiple Ratio. Eu-
clid proves it his own Way ; but to our Purpofe
it is fhorter done thus a : ra, and fa : rra, are
both Multiples, and in the fame Ratio ; then
a : rra is the Compound of thefe Two, and is
alfo multiple, ido. The Converfe is true, that
if any Ratio twice compounded makes the to-
tal Multiple, that Ratio is it felf multiple, pio.
A fu per particular Ratio, admits neither of one
■ or more geometrical mean Proportionals : Which
I thus demonftrate, viz. the Difference of the
Terms being i,'tis plain there can be no middle
Term in whole Numbers ; but the firit of any
Number ( n ) of geometrical Means betwixt a
and <tfi, (which reprefents any fuperp articular
Ratio) is the n+i Root of this Quantity a n Xaix
which being a whole Number, if it have
no Root in whole Numbers^ cannot have one in
a
§ 4- of MUSIC K. ■ 509
a mixt Number,, that is*, can have no Root at
all ; and confequently there can be no Mean
betwixt a and cfti. Nor can the Matter be men-
ded by multiplying the Terms of the Ra-
tfa, as if for a : a+t we take ra : ra + r 1 becaufe
if we -have not here a Mean in whole Num-
bers, we cannot havejt at all; and if we have if
in whole Numbers, then all the Series as well
as the Extremes, will reduce to radical Terms
contrary to the laft Demonftr. qtb, From the id
and id follows, that a Ratio not multiple being
twice compounded, the Total is a Ratio, nei-
ther multiple nor fuper particular. Again,
from the id follows, that if any Ratio twice
compofed make not a multiple Ratio, it felf is
not multiple. $to, The multiple Ratio 2 : 1
(which is the leaft and moft fimple of the Kind)
is compofed of the Two greateft fuperparti-
cular Ratios 3 : 2 and 4 : 3, and cannot be com-
pofed of any other Two that are fuperparti-
cular. From thefe Premiffes the Concords are
deduced thus : Diateffaron and Diapente are
Concords ; and they niuft be fuper particular Ra-
tios, for neither of them twice compofed makes
a Concord -yWie Sum therefore not being multiple,
the fimple Ratio is not multiple ; yet this Haifa
being Concord, muft ho J up erp articular. Dia-
pafon and Difdiapafon are both Concords, and
they are alfo multiple : The Difdiapafon can-
not b^fuperp articular, becaufe it has a Mean
(which is the Diapafon,) therefore 'tis multiple;
and diapafon is multiple,becau£c being twice com-
pofed>-it makes a Multiple, viz, the Difdia-
pafon
jio ^Treatise Chap; XIV.
pa/on ; then he proves that 'Diapqfon is duple
ill. Thus, it cannot be any greater Multiple
as i : 3 ; for it is compofed of Two fupzrparti^
cularsjviz. Diatejfaron and fliapente : But 2 : 1
is compofed of the Two greateft Jliperparti^
culars 3 : 2 and 4:3. Now if the Two great-
eft fupef particulars make the leaft Multiple
2:1, no other Two are equal to it, and far
lefs to a greater ,• and the %ve being multiple,
and compofed of Two J uper particulars^ muff
therefore be 2 : 1. From this 'tis alfo conclu-
ded that Diatejfaron is 4 : 3, Diapsnte 3 : 2,
and Difdiapafon 1 : 4 ; and the reft are dedu-
ced from thefe.
Discords are either (JZmmelf) continuous,
i. e. fit for Muftck, which is by fome alfo ap-
plied to Concords , or ( Ecmeli) inconcinnous. Of
the Continuous they numb red thefe, viz. Diefis,
Hemitonium, Tonus, Triemitonium, Ditonum.
There are different Species of each - 3 and of their
Quantities we (hall hear again.
The fimple Internals are called Diaftems,
which are different according to the Genera,
of which below - 3 the Compound are called
Syftems, of which next.
III. Of Systems. A Syftem is an Interval
compofed, or conceived as compofed, of feveral
leffer. As there is no leaft Interval in the Na-
ture of the Thing, fo we can conceive any
given Interval as compofed of, or equal to the
Sum of others ; but here a Syftem is an Inter*
val which is actually divided ki Practice ,• and
where
§ 4. of MUSIC K. yn
where along with the Extremes we conceive
always fome intermediate Terms. As Syfiems
are only a Species of Intervals, fo they have
all the fame Diftin&ions, except that of Com-
pofite and Incompofite, They were alfo di-
itinguifhed fevcral other Ways not worth Pains to
repeat. But there are Two we cannot pafs over,
which arc thefe, viz, into continuous and incon-
cinnous ; the firft compofed of fuch Parts., and in
iuch Order as is fit for Melody ; the other is of
an oppofite Nature. Then into perfect and im-
perfeB : Any Syftem lefs than Difdiapafm was
reckoned imperfecJ ; and that only called Per-
fect^ becaufe within its Extremes are contained
Examples of the limple and original Concords,
and in all the Variety of Order, in which their
continuous Part ought to be taken; which Dif-
ferences conftitute what they call'd the Species
or Figure c on/on ant i arum ; which were alfo
different according to the Genera : It was alfo
called the Syftema maximum, or immutatum, be-
caufe they thought it was the greater!: Extent,
or Difference of Tune, that we can go in mak-
ing good Melody;- tho' fome added a 5th to the
Uifdiapqfon for the greateft Syftem ; and fome
fuppofe Three 8ves; but they all owned the
Biapafon to be the moft perfect, with refpedt to
the Agreement of its Extremes; and that how-
ever many %<ves we put in the Syftema maxi-
mum, they muft all be conftituted or fubdivided
the fame Way as the firft : And therefore when
we know how %ve was divided, we know the
Nature of their Diagramma, which we now
call
ji2 ^Treatise Chap. XIV.
call the Scale of Mufick ; the Variety of which
conftitutes what they called the Genera melodic,
which were alio fubdivided into Species - 3 and
thefe mult next be explained.
IV. Of the Genera. By this Title is meant
the various Ways of fubdividing the confonant
Intervals (which are the chief Principles of Me-
lody) into their concinnous Parts. As the Odface
is the moil perfect Interval, and all other Con-
cords depend upon it $ fo according to the mo-
dern Theory we confider the Divilion of this
Interval^ as containing the true Divifion of the
whole Scale : (See above Chap, 8.) But the An-
cients went to work with this fomewhat dif-
ferently : The Diateffaron or qth was the leaft
Interval they admitted as Concord, and there-
fore they fought fir'ft how that might be moft '
concinnouily divided ; from which they conftitu-
ted the Biapentc or $th, and Diapafon or %ye :
Thus, the Sum of qth and 5/7? is an Octave,
and their Difference is a Tonus ; if therefore to
the fame Fundamental, fuppofc #, we take a
qth b> $th c and %<ve d, then alfo b-d is a $th^
ahd-^ -da qth, and b : c is the Tonus ,• which
they called particularly the Tonus' diazeuHicus^
^becaufe it ■ Separates or Hands in the Middle
hefwixt Two dths, one on either Hand, a - h 3
an#Y - d. This Tonus they reckoned indif-
penfable in rifing to a 5^ : And therefore, the
Divifion of the 4th being made, the Addition
of thi^Tvne made the $th ; and adding another
4?/?, the fame Way divided as the firft, com-
pleted the $yei Now the Diateffaron being
as
§ 4 . of MUSIC & J%$
as it were the Root or Foundation of their
Scale, what they call'd the Genera arofe froni
its various Divifions : Hence they defined the
Genus (inodulandi) the manner, of dividing
the Tetrachord, and difpofing its four
'Sounds (as to their Succeflion : ) And this De-
finition {hews us in general, That the qth was
divided into -J Intervals by two middle Terms^
ib as to contain 4 Sounds betwixt the Extremes:
Hence we have the Reafon of the Name JDiaP
teffaroiij ( i. e. per quatiior ; ) and becaufe from
the qth to the 5th was always the jTone$ th£
$th contained 5 Notes, and hence called J)ia-
' pente ( i. e, per quinque : ) And with refpecl td
the Lyra and its Strings, thefe Intervals were
called Tetrachordum and Pentechordwn. Rut
the %ve was called Diapafon, (as it were per
omnes) becaufe it contains in a manner all the
different Notes of Mufick ; for after one OSiave
all the reft of the Notes of the Scale were
reckoned but as it were Repetitions of it : Yet
with refpeel: to the Lyre, it was alfo called
OcJochordum. The Difdiapafon and all other
Names of this Kind being now plain enough^'
need not be infilled on : And we (hall pro-
ceed;
By univerfal Confent the Genera wereThree^'
viz, the Enharmonichj Chromatick and Dia-
tonick* The Reafons of thefe Names we ftiall
have prefently ; but the two laft were varioufly
fubdivided into different Species ; and even the
firft, tho' 'tis commonly reckoned to be without
any SpeGies^ yet different Authors pronofed difc
K fc. ferent
jr 3t 4 ^Treatise Chap. XIV.
ferent Divifions, under that Name, tho' without
diftinguifhing Names of Species, as were added
to the other Two.
Aristoxenus who meafured all by the
Ear, expreffed his Conftitutions of the Genera
in this Manner : He fuppofes the Tonus (dia-
zeiMicus) or Difference of the qth and 5th,
to be divided into 1 2 equal Parts ; which, to
prevent Fractions, Ptolomy^ when he explains
them, doubles, and makes 24 ; fo that the
whole qth mnft contain 60 of them. A certain
Number of thefe imaginary Intervals he affign-
ed to each of the Three Parts into which the
,4th is to be divided ; and all together -made up
thefe Six following Divifions, which I take with
the common Latin Names.
■Mf
a — b — c — d±
'\EnharmonmfH I I •* * * * 1 . 6 + 6+48 c=s5o
■ {Motte . 3 + 8 -+44 so?
Chroma. <Hemiolion .... 9 + 9 +42 ^o
\T011icum ...... 12 + 1 2 + 36 tno'o
-r.. , !Molk . 12+18 +30^0
^^m^tenmm ..... 11+24+24^0
In the Enharmoninm^ fuppofe 4, (mark^
ed at the Top of the Table) the firft and low-
eft Note of the Tetrachord*, from that to the
id hj is 6 of the Parts mentioned j to the 3d c 9
is other 6, and from the 3d to the acuteftNote
d, is an Interval equal to 48 of thefe Parts : In
this Manner you can explain all the reft. Six
of them he called a Diejis Enharmonic a 5 8 a
JDieftf
.'■
§ 4- of MI/SICK. /r'5
foiefis tricntalisy 9 & Diefis quadrantatis^ 12 £
Hemitonium^ 24 a Tonus ^ %6 a Triemitonium^
and 48 a Ditonum ; but to itieafure all thefe
accurately by the Ear was an extravagant Pre-
tence. Let us confider the Divifions that were
made by Ratios,
Besides fome particular Ratios of A'rchyi
iasy Eratofthenes and Didjmus^ (who were all
Muficians) which I pafs by, Ptolomy gives us
ah Account of the following 8 Divifions of the
Tetrachord ; where the Fractions exprefs the
Ratio betwixt each Sound (marked by the
Letters ftanding above) and the next, in order
from a the loweft, i, e, fuppofe any of the low-
er Notes a, b or c to be 1. the Fraction betwixt
that and the next expreffes the Proportion of
that next to it;
Diateffaronl
*/f-» .
a — b — q — B.
Enbamoniim • : Z ll I "~ X ~ X ~ ta ~
p Molle }
or
40 H 5 4
i • 2 1 v J i v £ - i
t Inteiifum • . — X — X — ss ~
I r 2i> ** 10 ** 8 4
<* ?.*H X t X -: «4
\Pjthagor. ,
S I 2 5T^ TahU continued
A Tr eatise Chap, XIV.
( Intenfum
-\
or > .— a — X " -•*
7)Mfts '"^ &&*&■] g f 6 lo 9 i
V $4*«Wfc S X ,1 X £ ~ 4
These different Species were alfo called the
Color es (Chroai) generum : Molle expreffes a
Progreifion by fmall Intervals, as Intenfum by
greater ,- the other Names are plain enough.
The Two firft Intervals of the Enharmonium^
are called each a Diefis ; the Third is a Dito-
Huptij and particularly the 3d g. already explain-
ed. The Two firft of the Chromatic!*, are cal-
led UemitoneS) and the Third is Triemitoni-
um ; and in the jlntiquum it is the 3 d I. above
explained. The firft in the diatonick is called
Uemitoniam^ and the other Two are Tones;
particularly the "g is called Limma (Pythago-
ricunii ) % is the grcateft of the Tones, and
~ the leaftj but the - and ^ are the Tonus
major and minor above explained.
As to the Names of the Genera themfelves^he
JUnharm. was fo called as by a general Name ;
Or feme fay for its Excellence (tho' where that
lies we don't well know.) The Diatonum, be-
caufe the- Tones- prevail in it. The Chroma-
tick was fo fmled, lay fome, from J(pcot color 3
becaufe as Colpur IsTomething betwixt Black
and White, fo the Chrom. is a medium be-
twixt the other Two.
Bvl
§ 4 . of MUSICK. y i>
But now to what Purpofe all thefc Divifi-
ons were contrived, we cannot well learn by
any Thing that they have told us. The En-
harm, was by all acknowledged to be fo difficult,
that few could pra&ife it, if indeed any ever
could do it accurately; and they own much the
fame of the Chromatkk. Such Inequalities in
the Degrees of the Scale, might be ufed for
attacking the Fancy, and humouring fome dis-
orderly Motions : But what true Melody could
be made of them, we cannot conceive. All
acknowledged, that the Diatonick was the true
Melody which Nature had formed all Mens Ears
to receive and be fatisfied with ; and therefore
it was the general Practice ; tho' in their Specu-
lations of the Proportions they had the Diffe-
rences you fee in the Table. And tho' Diatonick
was the prevailing Kind, yet ftill a Queftion re-
mained among them. Whether it fhould be
Arifioxenuss Tjiatonum intenfum^ox the Pytha-
%brick) whiqh j&ratofthenes contended for: (But
liere obferve, the Pythagoreans departed from
their Principles, by admitting the JLimmaywhich
is neither multiple nor fiiperparticular ; ) or what
Plolonty calls the Syntonum or inten/iim, which
Didymus maintain'd. The Arifiox* could give no
Proof of theirs, becaufe it was impoiTible for the
Ear to determine theDifference accurately:The o-
ther Two might be tried and proven by the Canon
harmonious; but if they tuned by the Ear, they
might difpute on without any Certainty of the
Kind they followed. As to the Species we now
K k 3 make
jit ^Treatise Ghap. XIV.
imake pfe of, the fame may be faid ; but I fhall
jconfider it afterwards.
Now, thefe Parts of the Diateffaron are what
they called the JDiafiems of the feveral Genera*
upon which their Differences depend : Which
are called in the Enharm. the Diefis and Dito-*
num j in the Chromatick, the Hemitonium and
Triemitonium •> in the Diatom the Hemitonium
(or Limmd) and the Tonus ; but under thefe
general Names, which diftinguilh the Genera,
there are feveral different Intervals or Ratios,
which constitute the colores generum, or Species
pi Enharm, Chrom. and Biatonick, as we have
feen : And we are alfo to obfervey that what is
a Jbiafiem in one genus is a Syftem in another :
But the Tonus ^iazeu^licus 8 : $ is effential in
all the Kinds, not as a neceffary Part of every
Tetrachord, but neceffary in every Syftenl of
%<ve, to feparate the qth and 5th, or disjoin the
feveral Tetrachords one from another.
Of the DIAGRAMMA or Scale.
WE have already feen the eflential Prin-
ciples, of which the ancient Scale or
Diagramma, which they called their Syfiema
perfeclum,was compofed,in all its different Kinds.
"Let us now confider the Conftruftion of it ; in
order to which I fhall take the Tetrachords dia-
tonic ally. I have already faid, tha$ the Extent
of it is a Di/didfq/on 5 or Two %<ves ia the Ma-
rio
§* tf MUSIC K. -ji£
tio 1:4: But in that Space they make Eighteen
Chords, tho' they are not all different Sounds.
And, to explain it, they reprefent to us Eighr*
teen Chords or Strings of an Inftrument, as the
Lyre, fuppofed to be tuned according to the
Proportions explained in any one Genus* To
each of thefe Chords (or Sounds) they gave a
particular Name, taken from its Situation in tho
Liagramma,ov alfo in the Lyre-, which Names
are commonly ufed by the Latins without
any Change. They are thefe, Pro/lamb anome~
nos, Hypate-hypaton, parhypate-hypaton, Li-
chanos-hypaton, JLypate-mefon, parhj/pate-me-
fon, Lichanos-mefon, Mefe, Trite-jynemmenon y
Paranete-Jynemmenon, Nete-fynemmenon, Para-
mefe, Trite-die zeugmenon, Paranete-diezeugme-
non, Nete- die zeugmenon, Trite-hyperh,ot<eQn y
Paranete-hyperbol<eon, Nete-hyperboUon.
That you may underftand the Order and
Conftitution of their Scale and theSenfe of thefe
Names, take this fhort Hiftory of it. While the
lyre was Tetra.(ov had but Four Strings) thefe
were called in order from the graveft Sound
Hypate, Parhypate, Paranete, Kete ; which
Names are taken from their Place in the Dia-
gram, in which anciently they fet the graiefi
uppermoft, or their Situation in the Lyre, hence
called Hypate, i. e.fuprema, (Chorda, fcil.)
the next is parhypate, u e.fubfiiprema or juxta
upremam-, then Par anete,i. e.penultima or "juxta
ultimam, and then Kete, ;•*, iiltima, as here.
K k 4 This
.
m
^Treatise Chap.
This refpe&s the ancient
Uypcite \ Lyra, whofe Chords were de-
T; dicate to, or made fyniboiical
Parly pate of the Four Elements: Which
f: > according to fome contained
Paranete an %<ve, but fome fay only a
£; Biateffaron 3 : 4, and the De-
JXete j grees I have marked by ;/ for
Semitone, and t for a Tone,
Without Diftin&ion.
Hypate \ N e x t to this fiicceeded the
f- Septichord lyre of Mercury,
Parhypate which ftands thus. Mefe is me-
fj dia. Lichanos, fb called from
Lichanos the digitus index with which
t: the Chord was ftruck^s fome
Mefe n, fay, or from its being the In-
f: dex of the Genus, according to
Trite itsDiftance from Hy patent was
f: alfo called Hyper mefe, i. e.fu-
Paranete pra medium. Trite fo called
f: as the Third from Nete ; and
N&$e J it is alfo called Paramefe, i. e.
juxta mediant. This contains
Two Tetrachords conjunct in Mefe, which is
common to both, and are particularly cal-
led the Tetrachords Hypaton, and Neton ,-fo that
thefe which were formerly Names of {ingle
Chords, are now Names of whole Tetrachords^
but as yet there was no great Neceflity for the
|)iftinaio% as we ftall fee afterwards,
m
§ 4-
of MUS1CK.
jn
/
Hypate
Parhypate
Lichanos
Mefe
Paramefe
But Pythagoras finding the
Imperfection of this Syflcm, ad-
ded an 8th Chord to complete
an %ve :And this he did by le-
parating the Two Tctrachords
by the Tonus diazeudficus j fo
the # Whole ftood thus. Where
we have Two Tctrachords,
■t:
one from Hypate to Mefe> and
f: * | the other from Paramefe to
Trite \ Nete-, the Tonus diazeufrficus
t : ' coming betwixt them, i. e* be-
Paranete twixt Mefe and Paramefe. So
t : ' here Paramefe and Trite are
j\&/-e ■* different Chords, which were
the fame before.
But there was another ocJi chord Lyre at-
tributed to Terpander; where inftead of disjoin-
ing the Two Tetrachords of the feptichord
Iyre> he added another Chord a Tone lower
than Hypate^ called Hyper-hypate^ i. e* fupef
fupremam^ becaufe it ftood above in the Dia-
gram ; or Proflamhanomenos^ i. e. affumptus 9
becaufe it belonged to none of the Two Te-
trachords : The reft of the Names were un-
changed.
Observe, the feptichord Lyre Was made
fymbolical of the Seven Planets. Hypate repre-
fented Saturn^ with refpecl: to his periodical
Revolution, which is flower than that of any
of the reft, as the graveft Sounds are always
produced by flowcft Vibrations, and fo of the
reft
j22 ^Treatise Chap. XIV.
reft gradually; But others make Nete reprefent
Saturn with refpecl: to his diurnal Motion round
the Earth fin the old Aftronomy) which is the
fwifteft, as the acuteft Sounds are alfo produced
by quickeft Vibrations, and fo of the reft. When
the %th Chord was added, it reprefented the
Coclum ftelliferum.
Afterwards a third* Tetrachord was ad-
ded to the feptichord Lyre ; which was either
conjunct with it, making Ten Chords, or dis-
junct, making Eleven. The Conjunct was
particularly diftinguifhed by the Name Sy-
nemmenon, i. e. Tetrachordum conjundfarum ;
and the other by the Name of DiezeugmenonJ.
e. disjunffiarum. And now the middle Tetra-
chord was called Me/on (mediarum^) and to the
Words Hypate, Parhypate, Lichanos, Trite,
Paranete, Nete, are now added the Name of
the Tetrachord, which is necelfary for Diftin^
ction j and the Whole flands thus,
Tetra. ( Jfypate, hypaton.
\ Parhypate, hyp.
ffyp* i Lichanos, hyp.
\ Hypate, mefbn y
\ Parh, Me/:
Mtf. 1 Lich. Mef.
I Mefe - - r. Mefe. )Tonus
f Trite Synem, $pf Paramefejdiezeudf.
Syn, 1 Paranete,Syn. %% Trite JOiezeug. '
\ Nete, Syn* -^ 1 Paranete Biezeug^
S \ Nete Die zeug.
:*X
■At
§ 4- of MUSIC K. si$
A t length another Tetrachord was added,
palled Hyperbol<eon ( i. e. excelkntium or $%&»
dentium) the acuteft of all ; which being con-
junct with the Diezeugmenon^hc Nete Diezeug-
tnenon was its grayeft Chord, the other Three
being called Trite, Paranete, and Nete Hyper-
bolaon ; and now the Four Tetrachords Hyp a-
foil, Mefon, Diezeugmenon, PtyperboUon, made
in all Fourteen Chords, to which, to complete
the Difdiapafon, a Projhmbanomenos was ad-
ded i all which with the Trite Paranete, and
Nete Synemmenon make up the Eighteen Chords
mentioned ; which yet are but Sixteen different
Sounds, for the Paranete Syn. coincides in the
Trite Diez.as the Nete Syn. with the Paranete
Diez. So that thefe Two differ only in the Trite
Syn. and Paramefe betwixt which there is a
Semitone. And now fee the whole Diagram tc*
gether in the following Page j where to favour
the Imagination more, inftead of marking the
Tone and Semitone by f and t. the Chords that
have a Tone betwixt them are fet further afun-
der than thofe that have a Semitone. At the
fame Time I have annexed the Letters by which
the modern Scale is above explained, that you
may fee to what Part of that this ancient Scale
correfponds. And becaufe we place the graved
Notes' in the lower Part of our Diagram (as the
ancient Latins came at laft to do, tho*
they (till applied Hypate to the gravtft, and
Nete to the acute]}, to prevent Confufion) IfliaH
fb it fp here,
~ * ■ . DlAi
pDiezeugmenon.
J24 A Treatise Chap. XIV.
BIAGRAMMA VETERUM
aa Nete, Hyperbol. 1 Tetfachof.
v Par anete, Hyperbol. \ __ , 7
* >Hyperhol<£on.
f Trite, Hyberbol.
e Jstete, T>iex,eug. }
&f"Sfete 3 Synem. d Paranete Diezeng. J
I >
| J Paranete ,Syn. c Trite, 'Die z,eug.
|*S & Para?nefe.
% \Trite,Synem. $ J
G Lichanos, Mefon, L ^fe/bff*
F Parhypate, Mefon, \
E Hypate, Mefon. )
D Ltehatios* hypaton. I __
yHypaton*
B Hypate, hypaton. J
A Projlaaibanamenos,
You lee, that" by twice applying Efypate,
Parhypate and Uchanos ,- alfo 7Wte 3 Parane-
te and 2\fcte Three Times; the Difficulty of too
many Names is avoided : And by the Diftin-
&ion otTetrachords with thefe particular Names
for the refpe&ive Chords, 'tis eafily imagined in
what Place of the "Diagram any Chord ftands.
But if we confider every Tetrachord hy it felf,
then we may apply thefe common Names to its
Chords, viz. Hypate^ Parhypate ( or Trite )
■ '-T Zdicha^
§ 4 . of MUSIC K. sif
Lichanos ( or Paranete ) and NetC : And then
when Two Tetrachords are conjunct the Hy-
pate of the one is the Nete of the other, as Hy-
pate mefon is equivalent to Nete hypaton ; and
in the Diagram, Mefe is the Nete mefon and
the Hypate jynem. and Paramefe is the Fly pate
diezeiig. And laftly, Nete diezeug. is equal to
Hypate hyperbola on. We fhall know the Ufe
'of the Tetr achord fynemmenon, when we come
.to explain the Bufinefs of their Mutations. The
Reft of the Diagram from Proflamban. is a
concinnous Scries, anfwering to the flat Series
of the diatonick Genus, explained in the Ch. $»
and the Order from Parhypate hypaton con-
tains the floarp Series above explained. Obferve,
tho' there are certain Syftems, particularly di-
ftinguifhed as Tetrachords, yet we have Tetra-
chords ( i. e. Intervals of Four Sounds ) in o-
ther Parts of the Scale, that are true qths 3 ; 4*
Again, if to any true $th a Tonus diazeug, is
added, we have the Diapente, as from Pro-
Jlamb. to Hypate mefon.
I have explained the Diagram in the dia-
tonick genus ; but the fame Names are applied
to all the Three Genera; and according to the
Differences of thefe, fo are the Relations of the
feveral Chords to one another. But fince the
Confutation of the Scale by Tetrachords is the
fame in all, and that the Genera differ only in
the Ratios which the Two middle Chords of the
Tetrachord bear to the Extremes ; therefore
thefe Extremes were called ftanding or immove-
able Sounds (srwTEcfomftantes) and all the middle
ones
jz6 A Treatise Chap. XIY.
ones were called moveable (yjvyjfoljbm mobiles) for
to raife a Series from a given Fundamental or Pro-
Jlambanomenos, the firft and laft Chord of each
Tetrachord is invariably the fame, or common
to every Genus; but the middle Chords vary
according to the Genus. So the Parhypate or
Trite, Lichanos or Paraneie of each Tetra-
chord is variable, and all the reft of the Chords
Of the Diagram are invariable.
The next Thing to be confidered is, what
they called the Figures or Species of the confo*
nant Syftems, viz. of the qth-, 5th and tve ;( for
they extended this Speculation no further than
the fimple Concords. ) The colore s generuih
differed according to the Difference of the cori-
ftituent Parts of the DiateJJaron ; but the figu-
ra or /pedes confonantiarum differ only accor-
ding to the Order andPoiition of the continuous
Parts of the Syfiem : So that in the fame Dia-
gram ( or Series) and under every Difference of
Genus and Color, there are Differences of the
Figure. Now, tho' of a certain Number of
different conftituent Parts, there will be a cer-
tain Number of different Pbfitions or Combi-
nations of the Whole i yet in every Genus there
Is a certain Diaftem agreed upon to be the Cha-
fafcleriftiCk ; and according to the Pofition of
this in the Syflem, fo are the different Figure
reckoned; the Combinations proceeding fr6m
the Differences of the other ^Diaftems being ne-
glected in this Matter. Ptohmy makes the
Char abler iftick of the DiatcJfaron 3 the Ratio b£
the Two acufeji Chords in every Genut; and
oi
§ 4 . cf MUSIC K. S i 7
of the Diapqfoii) the Tonus diezeudficus : But
Euclid reckons them otherwife, and applies
the fame Mark to 4^, and $th and %ve\ thus
in the Enharmonick the Ditonum is the Cha-
raderiftick; in the Chromatick it is the Trie-
mitonium ; and in the Diatonick the Semitone.
If we take Two conjunct Tetrachords, as from
Hypate-hypaton to ,M/<? 3 we {hall find in that
all the Figures of the Diatejfaron, which are
only Three > for there are but Three Places of
the Diateffaron in which the Chara&eriftick can
exift; there are Four Figures of the Diapente
which are to be found in Two disjunct Tetra*
chords^ betwixt Hypate-mefon and Nete-die~
zeugmenon. The $<ve is compofed of the qtb
and 5th, and the Three Species of qth joined to
each of the Four Species of $?h, make in all iz
Species of Svesi but we confider here only thole
Connections of qth and 5/-^, that are actually in
the Syfienty which are only Seven, to be found
from Proflambanomenos to Nete-hyperbolaon,
i. e. in the Compafs of a c Difdiapafon, Pro-
JIambanomenos being the lowed Chord of the
firff $ve, and Lichanos-mefon of the laft 8ve 5
for Mefe begins another Revolution of the J)ia-
pqfon, proceeding the fame Way as from Pro-
JIambanomenos : And becaufethis^/few of Dif~
diapafon contains all the Species of the Concords
it was called perfect. And obferve, that in eve*
ry 8ve Euclid's Char abler iftick occurs twice, and
they are always afunder by Two and Three 2)1*
efes, or Hemitones^ or Tones ( according to the
(Semis) alternatively. Whatwaj the Order they
* .. thought
Ji8 ^Treatise Chap. XlVY
thought" moft continuous and harmonious^ we
fhall fee prefenrly.
•■ V. Of Tones or Modes. They took
the Word Tone in four different Senfes* i . For
"a tingle Sound, as when they faid the Ijra has
Seven Tone's^ i. e. Notes. 2» For a certain In-
terval^ as the Difference of the qth and jffa
3. For the Tenfion of the Voice 5 as when we
fay, One lings with an acute or a grave Voice*
4. For a certain Syftem^ as when they faid, The
borick or Ly&ian Mode^ or Tom ; which is
the Senfe to be particularly confidered in this
Place.
This is the Part of the ancient Harmonica
which we wifh they had explained more clearly to
us; for it muft be owned there is an unaccountable
Difference among the Writers, in their Defini-
tions, Divisions and Names of the Modes. As
to the Definition, I find an Agreement in this,
that a Mode, or Tone in this Senfe, is a certain
Sjfiew or Constitution of Sounds ; and they r.~
gree too, that an O Stave with all its interme-
diate Sounds is iuch a Conftitution : But the fpe-
cifick Differences of them fome place in «the
Manner of Divifion or Order of its continuous
Parts ; and others place merely in the Tenfion of
the Whole, & e. as the whole Notes are acuter
or graver , or ftand higher and lower in the Sea k
of Mufick* as Bryennius fays very exprefly* Bc^
ethius has a very ambiguous Definition^ he firft
tells us, that the Modes depend on the Seven
different Species of the Diapafon> which are al-
io called Trop > and thefe 3 feys he, are Con-*
; • ; - fiitu~
§ 4- of MUSIC K. 529
ftitutiones in totis vocum ordinihis^ vel gram-
tate vel acumine differentes. Again he lays,
Conftitutio eft plenum veluti modulationis cor-*
fiis^ ex confonantiarum conjunt~iione confiftens,
quale eft Diapafon, &c. Has igitur conftitutio-'
nes> ft quis tot as fad at acut tores, vel in gra-
cilis tot as remittat fecundum fupraditlas Dia-
pafon confoiianti<c fpecieS) efficiet modos feptem.
This is indeed a very ambiguous Determination,
for if they depend on the Species of Sees, to
what Purpofe is the laft CJaufe ,- and if they
differ only by the Tenor or Place of the whole
8-0£, i. e. as 'tis taken at a higher or lower
Pitch, what Need the Species of Sves he at all
brought in : His Meaning perhaps is only to fig-
nify, that the different Orders or Species of %ves
ly in different Places, i. e. higher and lower in
the Scale, Ptolomy makes them the fame with
the Species of Diapafon ; but at the fame Time
he fpeaks of their being at certain Diftances
from one another. Some contended for Thir-
teen, fome for Fifteen Modes y which they pla-
ced at a Semitone'' s Diftance from each other;
but 'tis plain, thefe underftood the Differences
to be only in their Place or Diftances one from
; another; and that there is one certain harmoni-
ous Species of Ocfave applied to all, viz, that
Order which proceeds from Pro/Jamb, of the Sy-
ftema immutatum^ or the A of the modern Sy-
ftenu Ptolomy argues, that if- this be all, they
may be infinite, tho' they muft be limited for
Ufe and Practice; but indeed the Generality de~,
fine them by the Species diapafon^ and there-
L 1 fe§
J30 A Treatise Chap. 3£IV,
fore make only Seven Modes ,• but tQ what
they tend, and the true Ufe, is fcarcely well
explained, and we are left to guefs and reafon
about it j J (hall confider them upon both the
Suppoiitions^ and firft as they are the Species of
QBaves,) and here I fliaJl follow Ptolomy.
The Tones have no different Denominations
from the_ Genera ; and what's faid of them in
one Genus is applicable to all ; and I ftiall here take
the diatonich The Syftem of Difdiapafon alrea-
dy explained in the Diagram ( coinciding with
the Series frdm A of the modern Scale ) is the
Syftema itmmitatu'm; which I {hall, in what fol-
lows here, call the Syftem without Diftinftion.
The Seven Species of Qdfaves^as they proceed in
Order from A . B ' , ■€ ' . D . E . F ' . G, arethe
Seven Tones ^ which differ in their Modulations,
i, e. in the pittances of the fucceffive Sounds.,
according to the Ext Ratios in the Syftem. Thefe
SevenPtokmy calls,The i//,Z)ar/V^,tliefame with
the Syjleniy or beginning in ^or Pro/lamb. 2 J,
Hypo-lydian^ "beginning in and following the
Order from R or Hyp-hyp. 3<3 7 , Hypophrygi-
an, beginning at C or Parh-hy. qth, Efypodori*
$n at Z), $th^ Mixolydian in E. 6th, Lydian
in F. 7th y Phrygian in G. The kft Three he
takes in the QEIaves above a for a Keafon will
i>refently appear, Now, every Mode being con-
sidered by it felf as a diftinft Syftem\ may have
the Name 5 Pro/lamb, hyp-hyp* &c. applied to
it 1 for thefe iignify only in genera) the Politions
of the Chords in any particular Syftem ; if they
Hre fo applied^ he calls them the Pojitions -, for.
§ 4. .- of MUSIC K. j 3 i
Example, the firft Chord, or graved Note of
any Mode is called its Pro/lamb, pofitione, and
fo of the reft in Order. But again thefe are
confidered as coinciding, or being unifon, with
certain Chords of t\\Q Syftem; and thefe Chords
are called the potefiates, with refped to that
Mode; for Example, the Ifypodorian begins in
D, or Lichanos hypaton of the Syftem, which
therefore is the pot eft as of its Proflamb. as Hyp-
mefon is the pott ft as of its hyp-hyp. and fo of
others, that is, thefe Two Chords coincide and
differ only in Name ,- and we alfo fay, that fuch
a numerical Chord as Pro/!, pofttione of any
Mode is fuch a Chord, as hyp-hyp. poteftate,
which is equivalent to faying, that hyp-hyp. of
the Syftem is thoPoteftas of the Pro/Iamb, pofi-
tione of that Mode*
You'll eafily find what Chord of the Syftem
or Dorick Mode is the 2d, id, &c. Chord of
any other Mode, by counting up from the Chord
of the Syftem in which that Mode begins. Or
contrarily, to know what numerical Chord of
any Mode correfponds to any Chord of the <S)s
ftem, count from this Chord to that in which
the Mode begins, and you have the Number of
the Chord j to which you may apply the Names
Pro/Iamb. &c. or a, b, &c. And the Chords
of any Mode being thus named to you, yon\l
folve the proceeding Problems cahcii, by finding
what numerical Chord of the- Mode, that is the
Name of j for Example, to find' what Chord
of the Mode Hypwfarvan coincides with the,
J?arhyp4tte-me(bii of the Syftem (or Do+ick Mr.
L 1 z Tb/j
f0 A Treatise Chap. XIV %
The Hypo-dor. Mode begins, or has its Pro*
flamb* pofitione, in D or Lichanos-hyp. of the
Syftem Joetwixt which and Parhy-mef. areThree
Chords ( inclufive ) therefore the Thing fought
is the Third Chord, or Parhyp-hyp. pofitione
of the hyperdorian Mode, Again, to find what
Chord of the Syftem is the potefias of the lych-
hjp or qth Chord of the Hypo-phr, Mode* This
begins in Cor Parhyp-hyp. ofthe Syftem, and
the j\th above is Parhy-mefon or F the Thing
fought. But more univerfally, to find what
Chord of any Mode correfponds to any Chord
of any other Mode,- you may eafily folve this
by the Table Plate 2. Fig, 1. explained above
jn Chap. 11. § 3. Thus, find in the Column of
plain Letters, the Letters at which the Modes
propofed begin, againft which in the fame Lines
you muft find the Letter a, which is the Pro-
Jlamb. pofitione^ or firft, Chord of thefe Modes;
and then thefe refpe£tive Columns compared,
fliew what Chord ofthe one correfponds to any
ofthe other. Qbferve alfo, that were it propo-
fed to begin in any Chord of any Mode ( i. e. at
any Chord ofthe Syftem, or Letter of the plain
Scale ) and make a Series proceeding from that,
in the Order of any other Mode 3 we eafily know
by this Table what Chords of the Syftem muft
be altered to effect this 5 for Example, to begin
In <?,(which is Hyp-mefon ofthe Syftem or dorick
Mode, Proflamb. ofthe Phrygian Mode, &c.)
if we would proceed from this in the Order of
tliepfypo-lydiafiiwluch. begins at b of the Syftem^
we muft find e in the Column of plain Letters^
§4- cfMUSlCK. y 35
and in the fame Line find b • the Signature of
the Letters of that Column where b ftands,
fhews what Chords are to be changed : And by
this Table you folve all thefe Problems, with a
great deal more Eafe, than by the long and per-
plext Schemes which fome of the Ancients give
us : But let us return.
Ptolomy in Chap, 10. Lib. 2. propofes to
have his Modes at thefe Diftances, viz* tone y
tone, limma, tone, tone, limma. The Hypo-
dorian being fet loweft, then Hypo-phr. Hypo*
lyd. JDorick, Phrygian and Mixolydian,jQt ■ac-
cording to the Syftem they won't ftand at thefe
Diftances, nor in that Order, But in the next
Chap, it appears that he means only to take
them fo as their Mefe -pot eft ate (or thefe Chords
of each which is the firft of a Series fimilar to
the Syftema immutatum,) (hall ftand in that Or-
der ; and to this Purpofe he makes the JDorick
the Syftema immut. and the Profl. of the reft
in order as already mentioned ; only he takes
Mixolyd. Lyd. and Phryg. in the 2ve above,
i. e. at Nete diez. Trite hyperbol. Paran-
hjyperbol. whereby their Mefes pot eft ate ftand
in the Order mentioned ; otherwise they had
flood in an Order jufl reverfe of their pro/Jamb,
fojitidne. And now, if we would know at what
Diftances the Mefes pot eft ate of thefe Modes are
let us find what numerical Chord of each Mode
is its Mefe poteftate, and let it be expreft by the
Letters applied pofitione, as already explained :
Then we muft fuppofe that from a of the Syftem
(or Dorich Mode) a Series proceeds in each of
L 1 l the
134 <d Treatise Chap. XIV,
the Seven different Orders ; and by the Table
laft mentioned, we fhall know, in the Manner
alfo explained, what Chords are to be altered
for each ; therefore taking thefe Chords that
are the Mefes potefiate of each Mode, we
(hall fee their mutual Diftances. As Ptolomy
has placed the Prqflambanomenos y ov a^ pofit to-
ne of each Mode, their Mefes potefiate are in
the Chords e : fi%. g : a : b : c%. d. in order from
Hypo-dor. as above mentioned, that is^ when
all the Orders are transferred to the Pro/Iamb.
of the Dorick Mpae 9 the neceffary Variety of
Signatures caufes the / and c to be marked M
for the Pfypo-phr. and Lydian Modes, and thefe
f% and c% are the Mefes potefiate of thefe
Mode's ; all the reft are plain ; therefore the
mutual Diftances of thefe Mefes potefiate are
expreffed in the Scheme by (:) which fignifies a
jTone, (.) a Semitone or limina\ which are diffe-
rent from what he had formerly propofed.
Doctor- JVallis in explaining thefe by the
modern Syftem^ chufes the Signature for the
Lydian Mode, fo that a (its Pro/Jamb.) has a
flat Sign, and the Mefe-poteftate of it is c plain :
But fmce this explained is the only Senfe accor^
ding to which the Diftances of thefe Mefes-po-
t eft ate can be found, and (ince 'tis more ratio-
nal, that when any Mode is to be transferred to
the Prqjl-pofitione of another, that Pr o/L fliould
not be altered ,• for otherwife it is transferred to
another Note ; therefore I was obliged to differ
from the Doctor in that Particular : But neither
does, his Method fet thg Mefes potefiate at the
Diftance*
§4. ofMUSlCK. jjy
Diftances which Ptolomy mentions, and which
by Examination I find Cannot poffibly be done
without changing the Projl, of the Syfiema
immutatutn.
Anciently there were but Three Modes y
the Dorick*, Lydian and Phrygian^ fo called
from the Countries that ufed them, and parti-
cularly called Tones becaufe they were at a
Tones Diftance from each other; and afterwards
the reft were added and named from their Re-
lations to the former, particularly the Hypo-
dorian, as being below the Dorian*, and fo of
the reft ,- for which Reafon *tis by fome placed
firft, and they make its Proflambanomenos the
loweft Sound that can be diftinctly heard. But
we fhould be eafy about their Names or Order,
if we ilnderftood the true Nature and Ufe of
them.
If the Modes are indeed nothing elfe but
the Seven Species olOffiaves^ the Ufe of them
we can only conceive to be this, viz. That the
Profl. of any Mode being made the principal
Note of any Song, there may be different Spe-
cies of Melody anfwering tothefe different Con-
stitutions ; but then we are not to conceive that
the Profl, or Fundamental of any Mode is fixt
to one particular Chord of the Syftem^ for Ex,
the Phrygian to g ; fo that we muft always be-
gin there, when we would have a Piece of Me-
lody of that Species : When we fay in general
that filch a Mode begins in g, 'tis no more than
to fignifie the Species of %ve 7 according as they
L 1 4 &pP ear
5$ 6
'XTreatisj Chap. XIV.
appear in a certain fixt Syftem - } but we may be*
gin in any Chord of the Syflem^ and make it
the Prqfl. of any Mode, by adding new Chords,
or altering the Tuning of the old ( in the Man-
ner already mentioned:) If the Defign is no more,
but that a Song may be begun higher or lower,
that may be done by beginning at the fame
Chord, which is the PrqfL of any Mode in the
Syftem^ and altering the Tune of the Whole,
keeping ftill the fixt Order (which as I have al-
ready faid, is that in our modern natural Scaler
from a) but it Will be eafier to begin in a Chord
which is already higher or lower, and transfer
the Mode in which the Song is, to that Chord.
If every Song kept in one Mode y there was
Need for no more than one diatonick Series, and
by occafional changing the Tune of certain
Chords, thefe Tranfpofitions of every Mode to
every Chord may be eafily performed ; and I
have fpoken already of the Way to find what
Chords are to be altered in their tuning to effect
this,by the various Signatures of $ and j/ : But if
we fuppofe that in the Courfe of any Song a new
Species is brought in, this can Only be effected
by having more Chords than in the fixt Syfiem^
fo as from any Chord of that,any Order or Spe-
cies of %ve may be found.
I f this be the true Nature and Ufe of the
'Tones 1 1 fhall only obferve here, that according
to the Notions we have at prefent of the Prinr
ciples and Rules of Melody ', as they have been
explained in fome of the preceeding Ch after s^
liioft of thefe Modes are imperfect, and inca-
pable
§ 4. of MUSICK. j}7
pable of good Melody ; becaufe they want fome
of thofe we reckon the effential and natural
Notes of a true Mode (or Key) of which we
reckon only Two Species, viz. that from c and
a, or the Parhypate-hypaton and Proflambano*
menos of the ancient fixt Syftem.
Again, if the effential Difference of the Modes
confifts only in the Gravity or Acutenefs of the
whole 8ve; then we muff fuppofe there is one Spe- #
cies or concinnous Divifion of the 8^, which being
applied to all the Chords of the Syftem, makes
them true Fundamentals for a certain Series of
facceffive Notes. Thefe Applications may be
made in the Manner already mentioned j by
changing the Tune of certain Chords in fome
Cafes ; but more univerfally, by adding new
Chords to the Syftem, as the artificial or flmrp
and flat Notes of the modern Scale above ex-
plained. But in this Cafe, again, where we
fuppofe they admitted only one concinnous Spe-
cies, we muft fuppofe it to be correfponding
to the Sve a, of what we call the natural Scale ;
becaufe they all ftate the Order of the Syftema
immutatnm in the Diagram, fo as it aniwers to
that %ve.
But what a fimple Melody muft have been
produced by admitting only one concinnous Se-
ries, and that too wanting fome ufeful and ne-
ceffary Chords ? We have above explained, that
the flat Series, fuch as that beginning in a, has
Two of its Chords that are variable, viz. the
6th and jth, whereof fometimes the greater,
fometimes the kifer is ufed ? and therefore a
Syftem
53* ^Treatise Chap. XfVV
Syftem that wants this Variety muft be fo far
imperfeS : And what has been explained in
Chap. 1 3. fhews how impoffible it is to make
any good Modulation or Change from one Key
to another, unlefs both the Species of _$W/> and
fiat Key be admitted in the Syftem ; which Ex-
perience and all the Reafonings in the preceed-
ing Chapters demonftrate to be neceffary.
Ptolomy has a Paffage relating to the
Modes, with which I fliall end this Head, Lib.
2. Chap. 7. of the Mutations with refpeffi to
what they call Tones, He fays, thefe Mutati-
ons with refpecl: to Tones was not introduced
for the fake of acuter or graver Sounds, which
might be produced by raifing or lowering the
whole Inftrument or Voice, without any Change
in the Song ; but upon this Account, that the
fame Voice beginning the fame Song now in a
higher Note then in a lower, may make a Kind
of Change of the Mode. This^to make any Senfe,
muft fignify that the fame Song might be con-.
• trived fo, as feveral Notes higher or lower
might be ufed as Fundamentals to a certain
Number of fucceffive Notes ; and all together
make one Song ; like what I explained of cur
modern Songs making Cadences in different
Notes, fo aft the Song may be faid to begin
there again. If this is not the Senfe, then
what he fays is plainly a Contradiction*
But this may be the true Ufe of the Tones, m
either of the Hypothefes concerning their eflen-
tial Differences. He fays in the Beginning of
that Chap* % The Mutations which are made
£ by.
§ 4 ; W MUSIC K. j 39
** by whole Syftems, which we properly call
* Tones.) becaufe thefe Differences coniift in
* £ Tenfion^vc infinite with refpecl; to Poifibility,
* 4 as Sounds are, but actually and with refpeci
cc to Senfe they are finite." All thisfeems plain-
ly to put the Difference of the Tones only in the
Acutenefs or Gravity of the Whole, elfe how
do their Differences confift in Tenfion> which
iignifies a certain Tenor or Degree of Tune $
and how can they be called infinite^ if they
depend on the different Constitutions of the" 8 #<?.
Yet ellewhere he argues, that they are no o-
ther than the Species of Sves>> and as fuch makes
their Number Seven ,- and accordingly, in all
his Schemes, fets down their different Modula-
tions : But in Chap. 6. he feems more plainly
to take in both thefe Differences, for he fays,
there are Two principal Differences with relpetl
to the Change of the Tone, one whereby the
whole Song is fung- higher or lower, the other
. wherein there is a Change of the Melody to a-
nother Species than it was begun in ; but this
he thinks is rather a Change of the Song or
Melos than of the Tone^ as if again he would
have us think this depended only on the Acute-
nefs and Gravity of the Whole ; fo obfcurely
has the beft of all the ancient Writers delivered
himfelf on this Article that deferved to have
been moft clearly handled. But that I may
have done with it, I fliall only fay, it muft bo
taken in one of the Senfes mentioned, if not in
both, for another I think cannot be found. Let
me
j4° ^Treatise Ghap. XlV*
me alfo add, that the Moderns who have en-
deavoured to explain the ancient Mufick take
thefe Modes for the Species of 8<ves. If you'J]
except MeibomiuS) who, in his Notes upon A-
riftideS) affirms that the Differences of the
Modes upon which all the different Effects de-
pended, were only in the Tenfion or Acutenefs
and Gravity of the whole Syftem. But there
are Modes I call the Anti quo-modern Modes*,
which fhall be conlidered afterwards.
Observe. The* Tetrachord Synemmenon^
which makes what they called the Syfiema con-
junffium, was added for joyning the upper and
lower Diapafon of the Syfiema immutatum ;
that when the Song having modulated thro'"
Two conjunct Tetrachords, and being come to
Mefe^ might for Variety pafs either into the
disjunct Tetrachord Diezeugmenon or the con-
junct Synemmenon. 'Tis made in our Syftem by
bflat) i.e. putting only a Semitone betwixt a
and b $ fo that from b to d (in 8^,) makes
Three conjunct Tetrachords $ and the Ufe of
that new Chord \t with us is properly for per-
fecting fome $<ve from whofe Fundamental in
the fixt Scale there is not a right eoncinnous
Series.
VI. Of Mutations. This fignifies the
Changes or Alterations that happen in the Or-
der of the Sounds that compofe the Melody.
Ariftox. fays, 'tis as it were a certain Pajfion in
the Order of the Melody. It properly belongs
to the Melopceia to explain this, but is always -
put by it felf as a diftin& Part of the Hdrnu*.
t
§ 4^ »f MUSIC K. y 4 i
nica. Thefe Changes are Four, i. IntheG^-
nus; when the Song begins in one as the Chro-
matic'k, and paries into another as the Diato-
nich i. In the Syfiem^ as when the Song partes
out of one Tetrachord, as Mefon^ into another,
as Biezengmenon ,• or more generally, when it
paries from a high Place of the Scale to a low,
or contrarily, that is^ the Whole is fung fome-
times high, fometimes low j or rather, a Part
of it is high, and a Part of it low. 3. In the
Mode or Tone^ as when the Song begins in one,
as the Doricky and paries into another, as the
Lydian : What this Change of the Mode figni-
fies according to the modern Theory has been
explained already. 4. In the Melopoeia, that
is, when the Song changes the very Air^ fo as
from gay and fprightly to become foft and lan-
guifhing 5 or from a Manner that exprerfes one
Paflion or Subject to the Expreflion of fome
other ; and therefore fome of them call this a
Change in the Manner (fecundum moreni) •
But to exprefs Paifion, or to have what they
called Pathetick Mufick> the various Rjthmus
is abfolutely neceftary to be join'd ; and there-
fore among the Mutations fome place this of
the Rythmus, as from ffimbick to Choraick ;
but this belongs properly to the Rythmica.
Now thefe are at beft but mere Definitions, the
Rules when and how to ufe thefe Changes,
ought to be found in the Melopceia.
VII. O f the Melopoeia, or Ari of ma-
king Melody or Songs. After the End and Prin-
ciples of any Art are fuppofed to be oiftin&ly
V *° u Sh
54* ^Treatise Ghap. XlW
enough fhewn, the Thing to be expe&ed is, that
the Rules of Application be clearly fet forth. But
in this, I muft fay it, the Ancients have left us
little elfe than a Parcel of Words and Names j
fuch a Thing they call fuch a Name ; but the
Ufe of that Thing they leave you to find. The
Subftance of their Doctrine according to Euclid
is this. After he has faid that the Melopxia is
the Ufe of the Parts (or Principles) already ex-
plained. He tells us, it confifls of Four PartSj
firft ccyoyv)^ which the Latins called ductus,
that is, when the Sounds or Notes proceed by
continuous Degrees of the Scale, as a, b„ c.
id. 7TACX/J, nexus, which is, when the Sounds
either afcending or defcending are taken alter-*
nately, or not immediately next in the Scale, as
a, c, Z>, d. or a> dy b, e, c, /, or thefe reverfely
d, by r, a. %d, ) nsr\Mo^ b Petteia, (for the La-
tins made this Greek Name their own) when
the fame Note was frequently repeated toge-
ther, as a, a, a, 4th, rovrj, Lxtenfio, when
any one Note was held out or founded remark- ,
ably longer than the reft. This is all Euclid
teaches os about, it. But Ariftides Quint ilia-
nuSyWho writes more fully than any of them,
explains the Melopxia otherwife. He calls it
the Faculty on: Art of making Songs, which has
Three Parts, wz. tfjtyig, yj&g, )(j$W$, which
the Latins eaH fimtio, miftib, iifiis.
Not to trouble our felves with long Greek
PafTages, I (hall give you the Definitions of thefe
In Meibomius's Words, 1. Sumtio eft per
ejfiim itiufica datur. a qiiali <vo.cis loco Sjyftmafit
§ 4. of MUSIC K. HJ
faciendum, iitrum ah Hypatoide an reliquorum
aliquo. 2.Misrio,per qiiam autfonos inter
fe aut vocis locos coagmentamus, ant modula-
tionis genera, aut modorum Sjftema. 3. U s u s,
certa qiudam modulations confeBio, cujus /pe-
des treSj viz. Duel us 3 Petteia^ Nexus. As to
the Definitions of the Three principal Parts, the
Author of the Ditlionaire de Mufique puts this
Senfeuponthem^-S. to/#/6> teaches the'Compo-
fer in what Syftem he ought to place his Song,
whether high or low, and confequently in what
Mode or Tone, and at what Note to begin and
end. Mi'ftiOj fays he, is properly what we call
the Art of Modulating well, i. e. after hav-
ing begun in a convenient Place, to profecute
or conduct the Song, fo as the Voice be always
in a convenient Tenfion\ and that the effential
Chords of the Mode be right placed and ufed,
and that the Song be carried out of it, and re-
turn again agreeably. Ufus teaches the Com-
pofer how the Sounds ought to follow one ano-
ther, and in what Situations each may and
ought to be in, to make an agreeable Melody ',
or a good Modulation. For the Species of the
Tlfus : Arifiides defines the dutJus and nexus
the fame Way as Euclid does ; and adds, that
the duel us may be performed Three Ways, or
is threefold, viz. duel us reel us, when the Notes
afcend, as <x, Z>, c ; revertens^ when they de-
fcend r, Z>, a ; or circumcurrens, when having
afcended by the Jyftema disjinMum^ they im-
mediately defcend by the fyftema eonjunclum^
91* move downwards betwixt the fameExtremes,
j44 ^ T * E A T r s * Chap. XIV,
In a different Order of the intermediate Degrees,
as having afcended thus, a : b : c ; d> the De-
fcent is d'i c : ]/ : tf, or c : d : e : /, and f ; eb,
d : c. But the Petteia he defines. Of a eognqf-
cimus quinam fonorum omittendi, & qui funt
adfumendi, turn quoties illorum Cinguli • porro
a quonam incipiendum, & in quern defniendum:
atque h<sc quoque morem exhibet. In ftiort, ac~
cording to this Definition the Petteia is the
whole Art.
There were alfo what they called, The
modi melopoeia, of which Ariftides names thefe,
Dithyrambick, Nomick, and Tragick ,- called
Modes for their exprefling the feveral Motions
and Affections of the Mind. The beft Notion
we can form of this is, to fuppofe them fome^
thing like what we call the different Stiles in
Muftcki as the Ecclefiaftick, the Choraick, the
Recitative, &c. But I think the Rythmus
muft have a confiderable or the greateft Share
in thefe Differences.
But now if you'll ask where are the particu-
lar practical Rules, that teach when and how
all thefe Things are to be done and ufed, I muft
own, I have found nothing of this Kind particu-
lar enough to give me a diftincl: Idea of their
Practice in Melody. It is true, that Arijioxe-
nus employs his whole 3d Book very near, in
fomething thatfeems defigned for Rules, in the
right Conduct of Sounds for making Melody*
But Truth is, all the tedious and perplext Work
he makes of it, amounts to no more than fliew-t
§4- ofMUSICK. 545
ing, what general Limitations we arc undcr^
with refpect to the placing of Intervals in Sue-
ceifion, according to the feveral Genera,, and
the Conftitution of the Syfiema immutatum, or
what we call the naturally continuous Series;
You'll underftand it by One or Two Examples •
Firft, in the Diatonick Kind, he fays. That Two
Semitones never follow other immediately, and
that a Hemitone is not to be placed imme-
diately above and below one Tone, but may be
placed above and below Two or Three Tones ;
and that Two or Three Tones may be placed
together but no more. Then as to the Two
other Genera, to underftand what he fays$ ob-
,fer<be\ that the lower Part of the Tetrachord con-
taining Two Diefes in the One, and Two Hk*
mitones in the other Genus ( whofe Sums are
always lefs than the remaining c Ditone or Trie-
mitone that makes up the Diatejfaron) is called
71VKVQV fpiffum, becaufe the Intervals being fmal^
the Sounds are as it were fet thick and near
other,- oppofite to which is oct.vkvov non fpiffum
or rarum : Notice too, that the Chords that
belonged to the fpiffum were called tpjkvqi, and
particularly the loweft or graveft of the Three
in every Tetrachord were called (3<%pv7njx,vpt 3 (£roiin.
fidpvg gravis,) the middle fievoTrJMoi (from (*s-
cog medius) the acuteft o^tcukvol ( trom c^vg
acutus). Thofe that belonged not to the tw-
mqv were called aTtUxvot, extra fpiffum Now
then, with refpeel: to the JZnharmonick and
Chromatich we are told, that Two Spiffed or
M m Twd
546* ^Treatise Chap. XIV,
Two DitoneSj Triemitones^ or Tones cannot
be put together ; but that a Ditone may ftand
betwixt Two fplffa ; that a Tone ( it muft be
the diazeufticus betwixt Two Tetrachords)
may be placed immediately above the Ditone
or Triem, but not below, and below the Spif-
fum but not above. There is a World more of
this kind, that one fees at Sight almoft in the
Diagram^ without long tedious Explications ;
and at beft they are but very general Rules.
There is a Heap of other Words and Names
mentioned by feveral Authors, but not worth
mentioning.
But at laft I muft obferve and own, That
any Rules that can poifibly be given about this
Practice, are far too general, either to teach
one to compofe different Species of Melody ', or
to give a diftinct. Idea of the Practice of others ;
and that 'tis abfolutely neceffary for thefe Pur-
pofes that we have a Plenty of Examples in
a dual Compoiitions, which we have not of the
Ancients. There is a natural Genius, without
which no Rules are fiifdcient : And indeed
what Rules can be given, when a very few ge-
neral Principles are capable of fuch an infinite
Application £ -therefore Practice and Experience
muft be the" liuie ; and for this Reafon we find
both an^;ng the Ancients and Moderns, fo very
few, f.nd thefe very general Rules ipv the Com-
pofition of Melody. Beftdes the Knowledge of
the Syftem^ and what we call Modulation or
keeping in and changing the Mode or Key ;
*licre are other general Principles that Nature
teacheth
§ 4 . of MUSIC K. j 47
tcacliethus, and which muft be attended to,if we,
would produce good Effects, either for the En-
tertainment of the Fancy with the Variety we
find fo indifpcnfable in our Pleafures, or for imi-
tating Nature:, and . moving the Affections;
Thcfc avc^firjh the different Species of Sounds!
aburacl from the Acutenefs, as Drums, Trum-
pets, Vioiins, Flutes, Voice, (jc. which as they
give different Senfations, fo they are fit for ex-
prelfmg different Things, and raifing or humour-
ing different Paffions- to which we may add the
Differences of ftrong and weak, or loud and low
Sounds, ido. Tho 1 a Piece of Melody is ftrici:-
]y the fame, whether it is performed by an a-
cute or grave Voice \ yet *tis certain, That a-
cute Sounds and grave, have different Effects ,*
fo that the one is mere applicable tofome Subjects
than the other; and we know that,in general,
acute Sounds ( which are owing to quicker Vi •
brations) have fomething more brisk and
fprightly than the graver, which are better ap-
plied to the more calm Affections, or to fad
and melancholy Subjects j but there is a great
Variety betwixt the Extremes; and different
Cuftoms and Manners may alfo make a Dif-
ference : We find by Experience a lively Mo-
tion in our Blood and Nerves, under fome
Affections of Mind, as joy and Gladnefs \ and
in the more boifterous Palfionsj as Anger, that
Motion is full greater > but others . are accom-
panied with more calm and flow Motions j and
fince Bodies communicate their Motion, and the
Effect is proportional to the Caufe, we fee a,
Jtyl rfl z natural
548 yf Treatise Chap. XIV.
natural Reafon of thefe different Effefts of acute
and grave Sounds. %ti;o* The Effects of Melo-
dy have a great Dependence on the alternate
Paifage or Movement of the Sounds up and
down, i. e. from acute to grave, and contrarily^
or its continuing for lefs or more Time in one
Place j but the Variety here is infinite ,• yet Ex-
perience teaches fome general Leffons,; for Ex-
ample ^ if a Man in the Middle of a Difcourfe
turns angry, 'tis natural to raife his Voice j this
therefore ought to be expreft by railing the Me-
lody from grave to acute ; and contrarily a
finking of the Mind to Melancholy muff be
imitated by the falling of the Sounds • a more
evenly State by a like Conduct of the Melody.
Again, the taking of the Sounds by immediate
Degrees, or alternatively, or repeating the fame
Note, and the moving by greater or leffer In-
tervals, have all their proper and different Effects :
Thefe, and their various Combinations, muff all
be under the Compofer's Confideration ; but
who can polfibly give Rules for the infinite Va^
riety in the State and Temper of human
Minds, and the proper Application of Sounds
for exprefifmg or exciting thefe ? And when
Compositions are defigned only for Pleafnre in
general, what an infinite Number of Ways may
this be produced $
Again it muft be minded* That the Ryth-
mics is a very principal Thing in MuficL efpe-
cially of the pathetick Kind ; for 'tis this Va-
riety of Movements in the quick or flow Suc-
ce<Tions,o.r Length and Shortnefs of Notes, that's
the
§ 4- of MUSIC K. j 49
the confpicuous Part of the Air^ without which
the other can produce but very weak Effects ,-
and therefore moft of the Ancients ufed to call
the Rythmns the Male^ and the Harmonica
the Female, And as to this I muft take Notice
here. That the Ancients feem to have ufed
none but the long and fiiort Syllables of the
Words and Verfes which were fung,and always
made a Part of their Mufick ; therefore the
Rythmic a was nothing with them but the Ex-
plication of the metrical Feer^ and the various
Kinds of Verfes which were made of them : And
for the RythmopociajoY the Art of applying thefe 9
I am confident no Body will affirm they have
left us any more than very general Hints, that
can fcarce be called Rules : The reading of
Arifiides and St. Augnftin will, I believe, con-
vince you of this ; and all the reft put together
have not faid as much about it. I fuppofe the
ancient Writers, who in their Divifions of Mu-
fickjneke the Rythmica one Part, and in their
Explications of this fpeak of no other than that
which belongs to the Words and Verfes of their
, Songs, I fay thefe will be a fufficient Proof that
they had no other. But you'll fee. it further
confirmed immediately, "when we confider the
ancient Notes or Writing of Mufick. As to the
modem Rythmus, I need fay little about it;
that it is a Thing very different from the an-
cient, is manifeft to any Body who c winders
what I have faid of theirs, and has but the
frnalleft Acquaintance with our Muiick. That
the Meafures an$ Modes of Time explained
M m 3 ill
jyo ^Treatise Chap. XIV.
in Ch. 1 2. and all the poffible Subdivisions and Con-
ftitutions of them,are capable to afford an endlefs
Variety of Rythmus, and cxprefs any Thing that
the Motion of Sound is capable of, is equally cer-
tain to the experienced ; and therefore I fhaJl lay
no more of it here ; Only obferve, That as I
{aid about the Harmonica, fo of this 'tis cer-
tainly true, That the Rules are very general :
We know that quick and flow Movements fuit
different Objects,; when we are gay and cheer-
ful we love airy Motions ; and to different Sub-
jects and Paffions different Movements muft be
applied, for which Nature is our beft Guide :
Therefore the practical Writers leave us to our
own Obferyatipns and Experience, to learn how
to apply thefe Meafures of Time, which they
pan only defcribe in general, as I have done,
and refer us to Examples for perfecting our Idea
pf them, and what they are capable of.
pf the ancient Notes, and Writing of Mufick.
W e learn from Alipius ( md, Meibom. Edi-
tion?) how the Greeks marked their Sounds.
They made ufe of the Letters of their Alpha-
bet : And becaufe they needed more Signs than
there Were Letters, they fupplied that out of
the fame Alphabet ; by making the fame Let-
ter exprefs different Notes, as it was placed up-
right or reverfed, or otherwife put out of the
common Pofition • and alfo making them im-
perfect, by cutting off fomething, or by doubling
feme Strokes, For Example, the Letter Pi
expreffes
§ 4 . of MUSICK. jyr
exprerTes different Notes in all thefc P6fitions and
Forms,s;/.2. n . u • E . !z| P . U^&c. But that
we may know the whole Task a Scholar had to
learn, conflder, that for every Mode there were
1 8 Signs (becaufe they confidered the Tetra-
chordum fynemmenon^ as if all its Chords hrd
been really different from the Diezeugmemn) and
for every one of the Three Genera they were alfo
different ; again the Signs that exprefled the fame
Note were different for the Voice and for the In-
ftruments. Alipius gives us the Signs for 15 diffe-
rent Modes, which with the Differences of the
3 Genera^and the Distinction betwixt Voice and
Inftrument, makes in all 1620 ^ not that thefe
are all different Characters, for the fame Cha-
racter is ufed feveral Times, but then it has
differerent Significations; for Example^ in the
diatonic)?. Genus O is Lichanos hypaton of
the Indian Mode, and Hypate me/on of the
Phrygian*, both for the Voice ; fo that they are
in effect as different Characters to a Learner.
What a happy Contrivance this was for making
the Practice of Mufick cafy, every Body will
judge who confiders, that 15 Letters with fomc
fmall Variation for the Chords mobiles^ in or-
der to diftinguiili the Genera^ was fuflicient for
all. In Boethius\ Time the Romans were
wife enough to eafe thomfelves of this unnece£
fary Difficulty ,- and therefore they mrde. ufe
only of the flirt 1 5 Letters of their Alphabet :
But afterwards Pope Gregory the Great, ccn-
iidering that the %ve was the fame in effect with
the firft, and that the Order of Degrees was the
M m 4 fame
jp *A Treatise Chap. XIV,
fame in the upper and lower %ve of the Dia-
graniyhe introduced theUfeofy Letters, which
were repeated in a different Character. But
hitherto there was no ^uch Thing as any Mark
of Time i thefb Characters expreirmg only the
Degrees of Tune, which therefore were always
placed in a Line, and the Words of the Song
under them, fo that over every Syllable flood a
Note to mark the Accent of the Voice: And
for the Time j that was according to the long
and fhort Syllable of the Verfe ; tho' in feme
very extraordinary Cafes we hear of fome par-
ticular Marks for altering the natural or ordi-
nary Quantity.
I fliall end this Part with obferving that a-
jnong all the ancient Writers on Mujick, there
is not one Word to be found relating to Comr
fofition in Parts, or joining feveral different
Melodies in one Harmony, as what we call
Treble, Tenor, Bafs, &c. But this fhall be more
particularly examined in the next Section.
§ 5. Ajhort HI STO RT of the Improve-
ments in MUSIC K*
OR what Reafons the Greek Muficiansmade
fuch a difficult Matter of their Notes
and Signs we cannot gue£, unlefs they did it
flefignedly to, make their Art myfterious, which
is an odious Suppoution; but one can fcarcely
jhink it was otherwife., who coniiders how ob-
vious
§ j. of MUSIC K. 5 j 3
yious it was to find a more c'afy Method,
This was therefore the firft Thing the Latins
corrected in the Greek Mufick^ as we have al-
ready heard was done* by Boethius^ and further.
improved by Gregory the Great.
The next Step in this Improvement is com-
monly afcribed to Guido Aretinus a BenediUin
Monk, of Arctium in Tufcany\ who, about the
Year 1024, (tho' there are feme Differences a-
bout the Year) contrived the Ufe of a Staff of
5 Lines, upon which, with its Spaces he mark-
ed his Notes, by fetting Points ( .) up and down
upon them, to denote the Rife and Fall of the
Voice, (but as yet there were no different
Marks of Time j ) he marked each Line and
Space at the Beginning of the Staff, with Gre-
gory's 7 Letters, and when he fpakc of the
Notes, he named them by thefe inftead of the
long Greek Names of ' ProJlambanomenos-<kc. The
Correfpondence of thefe Letters to the Names
of the Chords in the Greek Syftem being fettled,
fuch as I have already represented in their Dia-
gram, the Degrees and Intervals betwixt any
Line or Space, and any other were hereby im-
derftood. But this Artifice of Points and Lines
was ufed before his Time, by whom invented
is not known ; and this we learn from Kircher,
who fays he found in the jfefuites Library at
Medina a Greek manufcript Book of Hymns,
more than 7 00 Years old j in which fome Hymns
were written on a Staff of 8 Lines, marked at
the Beginning with 8 Greek Letters j the Notes
pr Points were fet upon the Lines, but no. Ufa
made
554 ^ Treatise Chap. XIV.
made of the Spaces : Vincenzo Galileo confirms
us alfo in this. But whether Guido knew this,
is a QuefHon ; and tho' he did, yet it was well
contrived to ufe the Spaces and Lines both;, by
which the Notes ly nearer other, fewer Lines
are needful for any Interval, and the Diftances
of Notes are eafier reckoned.
But there is yet more of Guido s Contriv-
ance, which deferves to be confidered ; Firfi.
He contrived the 6 mufical Syllables, tit, re, mi,
fa,fol, la, which he took out of this Latin
Hymn.
TJT que ant laxis TLEfonare fibris
MIra geftorum FAmuli tuorum,
SQLve polluti LAbii re at urn,
pater dime.
In repeating this it came into his Mind, by a
Kind of divine InftincTt fays Kircher, to apply
thefe Syllables to his Notes of Mufick : A won-
derful Contrivance certainly for a divine Inftinc~i !
But let us fee where the Excellency of it lies \ :
Kircher fays, by them alone he unfolded all
the Nature of Mufick, diftinguiflied the Tones
(or Modes) and the Seats of the Semitones :
Elfewhere he fays. That by the Application of
thefe Syllables he cultivated Miijlck, and made
it fitter for Singing. In order to know how he
applied them, there is another Piece of the
Hiftory we muft take along, hrz. That finding
the Greek Diagram of too fmall Extent, he ad-
ded 5 more Chords or Notes in this Manner \
having
% f t of MU SICK. jv
having applied the Letter A to the Pro^amba-*
nomenoS) and the reft in Order to Note Hyp:r-
holaon 7 he added a Chord, a Tonus b.low
Pfoflam.&nd called it Hypo-pro fi lambanomcnos,
and after the Latins g. but commonly marked
with the Greek Gamma T ; to (hew by -this,
fay fome, that the Greeks were the Inventors cf
Mufick', but others fay he meant to record him-
fclf (that Letter being the hrft in his Name) as
the Improver o£ Mufick ; hence the Scale came to
be called the Gamm. Above Nete Hyperbolaon
he added other 4 Chords, which made a new
idisjunft Tetrachord, he called Hyper-hyper-*
boltfoiii fo that his whole «Stftz/e contained 20
diatonick Notes •, (for this was the only Genus
now ufed) befides the b flat, which correiponded
to the Trite Synemmenon of the Ancients, and
made what was afterwards called the Series of
b molkj as we (hall hear.
Now the Application of tliefe Syllables to
the Scale was made thus : Betwixt mi and fa is
a Semitone $ tit : re, re : mi\ fa :Jo\ and/o/ ; la
are Tones (without dif ingoifliing greater and
leffer ; ) then becaufe there are but 6 Syllables,
and 7 different Notes or Letters in the %ve ;
therefore, to make mi and fa fall upon the true
Places of the natural Semitones, tit was applied
to different Letters, and the reft of the 6 in
order to the others above,- the Letters to which
tit was applied are g . c .f. according to which
he diftinguiflied three Series, vi%» that which
begun with ut in g y and he called it t lie Series
pf b durum, becaufe b was a whole Tone above
ft
jj<J '^Treatise Chap. XIV.
a; that which begun with ut in c was the Se-
ries of b natural, the fame as the former- and
when ut was in f 3 it was called b molle, where-
in b was only a Semitone above a- See the
whole Scale in the following Scheme, where ob-
ferve, the Series of
G U I D O's Scale. & natural ftands be-
twixt the other two,
and communicates
with both; fo that
to name the Chords
of theScale by thefe
Syllables,if we would
have the Semitones
in their natural Pla-
e e
dd
c c
by
tf
aa
g
f
e
d
c
b
1/
a
G
F
E
B
C
B
A
Tamm
Bdur.
la
fol
fa
mi
re
ut
la
fol
fa
mi
re
ut
la
fol
fa
mi
• re
ut
nat.
mi
re
ut
la
fol
fa
mi
re
ut
la
fol
fa
mi
re
ut
molle
la
fol
fa
mi
re
ut
la
fol
fa
mi
re
ut
other a Tone la : mu
ces, viz. b . r, and
e . f then we ap^
ply ut to g y and af-
ter fe ? we go into
the Series of b natu-
ral at fa, and after
la of thisj we return
to the former at mi 9
and io on ; or we
may begin at ut in
r, and pafs into the
foil: Series at mi 3 and
then back to the o-
ther at fa : By which
I Means the oneTran-
fition is a Semitone,
<viz. la .fa, and the
To follow the Order of
b molle
§ j. of MUSIC K. y J7
b mollis we may begin with ut m c or/, and
make- Tranfitions the fame Way as formerly :
Hence came the barbarous Names of Gammut^
Are^ Bmi 9 &c. with which the Memories of
Learners ufed to be opprefled. Rut now what
a perplext Work is here, with fo many different
Syllables applied to every Chord, and all for
no other Purpofe but marking the Places of the
Semitones, which the fimple Letters, a:b . c, &c.
"do as well and with infinite more Eafe. After-
wards fome contrived better, by making Seven
Syllables,adding Si in the Blanks you fee in the Se-
ries betwixt la and ut b fo that mi-fa and fi-nt are
the two natural Semitones : Thefe 7 completing
the 8c£, they took away the middle Series as of
no Ufe,andfo lit being in g or/, made the Series
of B durum (or natural, which is ail one) and
B molle. But the Engtifh throw out both ut
and fi, and make the other 5 ferve for all in
the Manner explained in Chap* 11. where I
have alfo (hewn, the Unneceflarinefs of the Dif-
ficulty that the beft of thefe Methods occafions^
and therefore fhall not repete it here. This
wonderful Contrivance 'o£Guidos 6 Syllables, is
what a very ingenious Man thought fit to call
Crux tenellorum ingeniorum j but he might
have faid it of any of the Methods ; for which
Reafon, I believe, they are laid aiide with very
many, and, I am fure, ought to be fo with e~
very 13ody.
BuTto go one with Guido ; the Letters he
applied to his Lines and Spaces, were called
Kejs.) and at firit he marked every Line
and
j$ /^Treatise Chap. XIV.
and Space at the Beginning of a Staff with its
Letter $ afterwards marked only the Lines, as
feme old Examples (hew ; and at laft marked
Only one, which was therefore called the figned
Clef j of wliich he diltinguiflied Three different
ones, g , c , / i (the three Letters he had pla-
ced his 2//- in ) and theReafon of this leads us td
another Article of the Hiftory, viz. That Gitido
was the Inventor of Sjmphonetick Compcfltion^
(for if the Ancients had it, it Was loft ; but this
fliall be confidered again) the firft who joyned
in one Harmony feveral diftindt Melodies, and
brought it even the length of 4 Parts, viz:
Bafs, Tenor, Counter, Treble ; and therefore
to determine the Places of the feveral Parts
in the general Syftem, and their Relations to
one another, it was neceffary to have 3 different
figned Clefs \®id. Chap. 11.)
H e is alfo faid to be the Contriver of thofe
Inftruments they call Poly plectra, as Spinets
and Harp} ichor ds ; However they may now dif-
fer in Shape, he contrived what is called the
Abacus and the Palmitic, that is, the Machi-
nery by which the String is ftruck with a Plect-
rum made of Quills. Thus far go the Improve-
ments of Guido Aretinus, and what is called the
Giridoman Syftem ; to explain which he wrote
a Book he calls his Micrologum.
The next considerable Improvement was
about 300 Years after Guido, relating to the
Rythnius, and the Marks by which the Durati-
on of every Note was known ; for hitherto they
had but imitated the Simplicity of the Ancients,^
. *nd
§ f, of MUSIC K. yj9
and barely followed the Quantity of the Syl-
lables, or perhaps not fo accurate in that, made
all their Notes of equal Duration, as fome of the
old Ecclefiaftick Mufick is an Inftance of. To
produce all the Effecls Mufick is capable of,
the Ncccifity of Notes of different Quantity was
very obvious -, for the Rythmus is the Soul of
Muftck; and becaufe the natural Quantity of the
Syllables was not thought fufficient for all the
Variety of Movements, which we know to be
fo agreeable in Mufick, therefore about the
Year 1330 or 1333, fays Kir c her, the famous
Joannes de Muris, Doctor at Paris, invented
the different Figures of Notes, which exprels
the Time, or Length of every Note, at leaft their
true relative Proportions to one another ; yon
fee their Names and Figures in Plate, 2 Fig. 3.
as we commonly call them. But anciently they
were called, Maxima, Longa, Brcvis, Semi^
brews, Minima, Semiminima, Chroma, (or Fu-
fa) Semichroma. What we call the Demifemiqua~
ver is of modern Addition. But whether all thefe
were invented at once is not certain, nor is it pro-
bable they were; at rlrft 'tis like they ufed only
the Longa and Brevis, and the reft were added
by Degrees. Now alio was invented the Divi-
fion of every Song in feparate and diftincl Bars
or Meafures* Then for the Proportion of thefe
Motes one to another it was not always the
fame ,• fo a Long was in fome Cafes equal to
Two Breves, fometimes to Three, and fo of
others^ and this Difference was marked general-
ly at the Beginning ; and fometimes by the
Pofition
y6o ^Treatise Chap. XYSf a
Pofition or Way of joyning them together in
the Middle of the Song; but this Variety hap-
pened only to the firft Four. Jlgain, refped:-
ing the mutual Proportions of the Notes, they
had what they called Modes, Prolations and
Times ; The Two laft were diftingiiifhed'' into
jPsrfe&t and Imperfect \ and the firft into grea-
ter and Iejjer> and each of thefe into perfect and
imperfeB : But afterwards they reduced all into
4 Modes including the Prolations and Times*
I could not think it worth Pains to make a te-
dious Defcription of all thefe, with their Marks
or Signs, which you may fee in the already
mentioned Diffiionaire de Mufique : I fhall on-
ly obferve here, That as we now make little
Ufe of any Note above the Semibreve, becaufe
indeed the remaining 6 are fufficient for all Pur-
pofes, fo we have cart off that Difficulty of vari-
ous and changeable Proportions betwixt the
lame Notes : The Proportions of 3 to 1 and 2
to 1 was all they wanted, and how much
more eafy and limple is it to have one Propor-
tion fixt, sti'%. 2 : 1 ( i. e. a Large equal to
TwoZo/i^J, and fo on in Order ) and if the
Proportion of 3 : 1 betwixt Two fucceifive
Notes is required, this is, without any Manner
of Confufion or Difficulty, expreffed. by annex-
ing a Point (.) on the Right Hand of the great-
eft of the Two Notes, as has been above ex-
plained ; fo that 'tis almoft a Wonder how the
Elements of Mufick were fo long involved in
thefe Perplexities, when a far eafier Way of
coming to the fame End was not very hard to find*
We
§ f of MUSIC & s g I
We ffiall obferve here too, That tiJJ thefe
jSfotes of various Time were invented, in (bu-
rn ental Performances without Song muft have
been very imperfect if they had any >• and what
a wonderful Variety of Entertainments we have
by this Kind of Compofition, I need not tell
you.
There remain Two other very consider-
able Steps, before we come to the prefent State
of the Scale of Mufick. Guido firft contrived
the joyning different Parts in one Concert \ as
has been laid, yet he carried his Syftem no fur-
ther than 20 diatonick Notes: Now . for the
more fimple and plain Compofitions of the Ec-
clefiaftick Stile, which is probable Was themoft
confiderable Application ne made of Mufick^
this Extent would afford no little Variety : But
Experience has fince found it neceffary id en-
large the Syftem even to 34 diatonick Notes,
which are reprefented in the foremoil Range of
Keys on the Breaft of a Harffichord ; for fo
many are required to produce all that admirable
Variety of Harmony, which the Parts in modern
Compofitions confift of, according tb the ma-
ny different Stiles pra&ifed : But a more con-
iiderable Defect of his Syftem is, That except
the Tone betwixt a and b, which is divided in-
to Two Semitones by f/ (flat) there was not a-
nother Tone in all the Scale divided ; and with-
out this the Syftem is very imperfect, with rcf-
pe£t to fixt Sounds* becaufe without«thefe there
Can be no right Modulation or Change from
N rt Key
<j6t A Treatise Chap. XIV.
Key to Key 9 taking Mode or Key in the Senfe
which I have explained in Chap. 9. Therefore
the modern Sjftem has in every %ve 5 artificial
Chords or Notes which we mark by the Let-
ters of the natural Chords, with the Diftin&ion
of % or |/, the Neceflity and true Ufe of which
has been largely explained in Chap. 8. and there-
fore not to be infifted on here $ I fliall only ob«
ferve^ That by thefe additional Chords, we
have the diatonick and chromatick Genera of
the Ancients mixed ; fo that Compofitions may
be made in either Kind, tho' we reckon the
diatonick the true natural Species - 3 and if- at
any Time, Two Semitones are placed immedi-
ately in Succelfion • for Example^ if we fing
c . c%. d) which is done for Variety, tho' fel-
dom, fo far this is a Mixture of the Chroma-*
tick ; but then to make it pure Chromatick^ no
fmaller Interval can be fung after Two Semi m
tones afcending than a Triemitone^ nor defen-
ding lefs than a Tbne; becaufe in the pure .chro-
matick Scale the Spiffiim has always above it a
Triemitone> and below it either a Triemitoneox
a Tone.
T h e laft Thing I lhall confider here is, how
the Modes were defined in thefe Days of Im-
provement ; and I find they were generally cha-
racterized by the Species of 8#? after Ptolomfs
Manner, and therefore reckoned in all 7. But
afterwards they confidered the harmonic al and
arithmetical Divifions of the 8&v, whereby it
refolyes into a qth above a 5/^ or a $th above
a qth
§ s- of MUSIC K. $4$
a qth. And from this they conftituted 12
Modes, making of each %<ve two different Modes
according to this different Divifion ; but becaufe
there are Two of them that cannot be divided
both Ways, therefore there are but fia Modes.
To be more particular, confider, in the natural
Syflem there are 7 different Obi awes proceeding
from thefe 7 Letters, a$ b, c, d, e,/, g; each of
which lias Two middle Chords, which divide
it harmonically and arithmetic 'ally ', except f y
Which has not a true /\.th y ( becaufe b is Three
Tones abo^c it, and a qih is but Two Tones and
a Semitone) and Z>, which confequently Wants
the true $ifo ( becaufe/ is only Two Tones and
Two Semitones above it, and a true $th con-
tains 3 Tones and a Semitone) therefore we
have only 5 Octaves that are divided both
,Ways, mZi a, c, 4> e* g> which make 1 o Modes
according to thefe different Divifions > and the
other Two/ and b make up the 12. Thefe
that are divided harmonically, u e. with the zths
loweft were called authenticity and the other
plagal Modes. See the following Scheme.
To thefe Modes they gave the Names of
the ancient Greek Tones, as Dorian, Phrygian :
But feveral Authors differ in the Application of
thefe Names, as they do about the Order, as,
which they fliall call the firft and fecond, (fyc.
which being arbitrary Things, as far as I can
underftandj it were as idle to pretend to recon-
N n i cile
A Treatise Chap. XIV.
Modes. cile them, as it was
JPlagaL Authentich in them to differ a-
80?. S^. 1 bout it. The mate-
^ Acrizr^ ,v ~ » r | a ] p i n t i Sj if we
4^. j&h. qrb. can find it t know
g — r _._ ^ — c what they meant by
# — ^ — ^35 — J thefe Diftin6tions 3 and
/; _^_ e — y — e what was the real
c — f — c — / Ufe of them in Mu-
$ — _. g — c \ — g fak . but even here
e — a — e „.■„ a where they ought to .
' "*~ : have agreed, we find
they differed. The beft Account I am able to
give you of it is this : They con(idered that an
8<ve which wants a qth or ^th is imperfe<%thefe
being the Concords next to 8^, the Song ought
to touch thefe Chords moft frequently and re-
markably ; and becaufe their Concord is diffe-
rent, which makes the Melody different, they
eftablifhed by this Two Modes in every natural
Oblave, that had a true aph and $th: Then if the
Song was carried as far as the Octave above, it
was called aperfetl Mode ; it lefs, as to the qth
or $th y it was imperfeb~f -, if it moved both a-
bove and below, it was called a mixt Mode :
Thus fome Authors ipeak about thefe Modes,
Others confidering how indifpenfable a Chord
the 5th is in every- Mode^ they took for the fi-
nal or Key-note in the arithmetically divided
ObJaves, not the loweft Chord of that Ociave %
but that very 4th > for Example , the Oblave g
is arithmetically divided thus, g - c - g, c is a
qth above the lower g, and a 5th below the up-
per
§ 5". of MUSIC K, ]6f
per g, this c therefore they made the [final Chord
of the Mode, which therefore properly fpeaking
is c and not g ; the only Difference then in this
Method, betwixt the authentick and plagal
Modes is, that the Authentick goes above its
Final to the O clave, the other afcends a $th 9
and defcends a /\th, which will indeed be attend-
ed with different Effects, but the Mode is effen-
tially the fame, having the fame Final to which
all the Notes refer. We muft next confider
wherein the Modes of one Species, as Authen-
tick or Plagal, differ among themfclves : This
is either by their ftrnding higher or lower in
the Scale, i. e* the different Tenfion of the
whole Oclave; or rather the different Subdivision
of the Offiave into its concinnous Degrees ,-
there is not another. Let us confider then
whether thele Differences are fufficient to pro-
duce fo very different Effects, as have been as-
cribed to them, for Example, one is fa id to be
proper for Mirth, another for Sadnefs, a Third
proper to Religion, another for tender and a-
morous Subjects, and fo on : Whether we are
to afcribe fuch Effects merely to the Conftitu-
tion of the Octave, without Regard to other
Differences and Ingredients in the Compofition
of Melody, I doubt any Body n©w a Days will
be abfurd enough to affirm ; thefe have their
proper Differences,- 'tis true, but which have
fo little Influence, that by the various Combi-
nations of other Giufes, one of thefe ModeS
may be ufed to different Purpofes. The greatr
eft and mod influencing Difference is that of
N n 3 thefe
'56$ ^Treatise Chap.-XIV.
thefe Odfaves, which have the 3 d h or 3d g.
making what is above called ih&Jkafp and fiat
Key :■ But we are to notice, that of all the Sties,
except c and <?, none of them have all their
eifential Chords in juft Proportion, unlefs
we neglect the Difference of Tone greater
and lefler, and alfo allow the Semitone to (land
next the Fundamental in fome flat Keys (which
may beufeful,and is fometimes ufed;)and when
that is done, the Slaves that have a flat 3d
will want the 6th g. and jth g. which are very
necelfary on fome Occafions •> and therefore the
artificial Notes $ and j/ are of abfolute Ufe to
perfect the Syfiem, Again^ if the Modes depend
upon the Species of 8ce\f, how can they be
more than 7? And as to this Diftinclion of au-
thentick and plagal^ I have fliewn thatitisima-
ginary ? with refpecl; to any effential Difference
conftituted hereby in the Kind of the Melody;
for tho' the carrying the Song above or below
the Finals may have a different Effect, yet this
is to be numbred among the other Caufes, and
not afcribed to the Conmtution of the Odfaves.
But 'tis particularly to be remarked, that thefe
Authors who give us .Examples in actual Com-
pontion of their 1 2 Modes, frequently take in
the artificial Notes ^ and 1/ to perfect the Me-
lody of their Key^ and by this Means depart
from the Conftitution of the .8^, as it ftands in
the fixt natural Syftem, So we can find little
certain and confident in their Way of fpeaking
about thefe Things ; and their Modes are all
^R c lkte te'£wo> m%. the Jharp smAjiat ; q~
ther
§ y. of MUSIC K. 5 6 7
ther Differences refpe£ting only the Place of the
Scale where the Fundamental is taken : I con-
clude therefore that the true Theory of Modes
is that explained in Chap. 9. where they are
diftinguifhed into Two Species, JJoarp and flat,
whofe Effecls I own are different ; but other
Caufes (vid. Pag. 547, &c.) muft concur to any
remarkable Effect ; and therefore 'tis unreafon-
able to talk as if all were owing to any one
Thing. Before I have done there is another
Thing you are to be informed of, viz. That
what they called the Series of b molle, was no
more than this. That becaufe the %<oe f had
a 4th above at b, exeeffwe by a Semitone? and
confequently the %ve b had a 5^ above as
much deficient, therefore this artificial Note b
flat or |a, ferved them to tranfpofe their Modes
to the Diftance of a 4th or 5th, above or be-
low; for taking (/ a Semitone above a, the
reft keeping their Ratios already fixt, the Se-
ries proceeding from c with b natural ( i. e. a
Tone above a) is in the fame Order of De-
grees, as that from/ with b flat ( i. e, {/ a Se-
mitone above a \ ) but f is a 4th above c, or a
$th below ; therefore to tranfpofe from the
Series of b natural to b molle we afcend a
4th or defcend a 5^; and . contrarily from
b molle to the other : This is the whole My-
ftery ; but they never fpeak of the other Tranf-
pofitions that may be made by other artificial
Notes.
You may alfo obferve, that what they called
the JjtCckftajlic'k Tones, are no other than cer-
N n 4 tain
j<58 ^Treatise Chap. XIV.
tain Notes in the Organ which are made the
Final or Fundamental of the Hymns ; and as
Modes they differ, fome by their Place in the
Scale, others by the Jharp and flat 3^ but even
here every Author fpeaks not the fame Way :
'Tis enough we know they can differ no other
Way, or at leaft all their Differences can be re-
ducecLto thefe. At firft they were Four in Num-
ber, whofe Finals were d, e, /, g conftituted au-
thentically : This Choice, we are told, was firft
made by St. Ambrofe Biiliop of Mil an \ and for
being thus chofen and approven,they pretend the
Name Authentick was added : Afterwards Gre-
gory the Great added Four Plagals a y &, c, d 9
whofe Finals are the very fame with the firft
Four, and in effect are only a Continuation of
thefe to the 4th below ; and for this Connection
with them were called plagal^ tho' the Deri-
vation of the Word is not fo plain.
But 'tis Time to have done ; for I think I
h ave fliewn you the principal Steps of the Im-
provement of the Syftem of Mujick> to the pre-
sent State of it, as that is more largely explain-
ed in the preceeding Chapters. I have only one
Word to add, that in Guidons Time and long
after, they fuppofed the Divifion of the Tetra-
chord to be Ptoloinys Diatomim diatonicum^ i.
e. Two Tones 8 : 9, and a limma ^-; till
jZarlinus explained and demonftrated, that it
ourht to be the intenjum^ containing the Tone
srw+r f o ' ,. ^ n d Semitone 15 : 16;
t f • ; a c u ? alfo
§ 6. of MUSIC K. 5 6 9
fliews how incontinently they fpake about the
Modes^ where he reduces ail to the Two Sped*
es otjharp and flat, 'Tis true, Galileo approves
the other, as common Practice (hewed that the
Difference was infeniible ,•' yet it muft be ttiemt
only with refpect to common Practice. I have
already expIain<d,how this Difference in hxt In*
ftrumentsis the very Reafon of their Imperfedioa
after the greateft Pains to correct them ,• and
hew the natural Voice will, without any Di-
rection, and even without perceiving it, chcofe
fometimes a greater, fometimes a lefler 2h;ie :
Therefore I think Nature guides us to the Choice
of this Species : If the commenfurate Ratios of
Vibrations are the Caufe of Concord then cer-
tainly 4 : 5 is better than 64 : 81. The firft ar; fbs
from the Application of a fimple general Rule
upon which the more perfect Concords depend;
the other comes in as it were arbitrarily. How
the Proportions happen upon Inftrumcnts de-
pends upon the Method of tuning them ; of
which enough has been already faid.
§ 6. The ancient and modern Mufick compared*
HP HE laft Age was famous for theWai iat
■*" was raifed, and eagerly maintain'd bj r o
different Parties, concerning the anctei ?-hd
modern Genius and Learning, Anv no
difputed Points Muftck was one, I kr<u no-
. iiaa
570 ^Treatise Chap. XIV.
thing new to be advanced on either Side ,- fo
that I might refer you to thofe who have exa-
mined the Queftion already : But that nothing
In my Power may be wanting to make this
Work more acceptable, I fhall put the Subftance
of that Controverfy into the beft Form I can,
and fhall endeavour to be at the fame Time
fhort and diftinct
The Queftion in general is, Whether the
jincients or the Modems beft underftood and pra-
ftifed Mufick ? Some affirm that the ancient Art
of Mufick is quite loft, among other valuable
Things of Antiquity, vid. Pancirollus^de Mufica.
Others pretend, That the true Science of Har-
mony is arrived to much greater Perfection than
what was known or pra&ifed among the Anci-
ents, The Fault with many of the Contenders
on this Point is, that they fight at long Wea-
pons $ I mean they keep the Argument in ge«
nerals^ by which they make little more of it
than fome innocent Harangues and Flourifhes of
Khctorick, or at moft make bold Affertions up-
on the Authority of fome mifapplied Exprefli-
onsand incredible Stories of ancient Writers, for
Fm now Tpeaking chiefly of the Patrons of the
ancient Mufick.
I f Sir William Temple was indeed ferious,
and had any Thing elfe in his View, but to
(hew how he could declaim, he is a notable In-
ftance of this. Says he, " What are become
" of the Charms of Mufick, by which
8 Men and Beafts were fo frequently inchanted,
? and
'§ 6. of MUSIC K. 5 7i
fe and their very Natures changed, by which
jr tjiC Paillons or Men were raifcd to the greateft
ifc Height and Violence, and then as fuddenly
f c appeafed, fo as they might be juftly faid, to
S c be turned into Lions or Lambs, into Wolves
;* c or into Harts, by the Power and Charms of
c this admirable Art ?," And he might have ad-
\ded too,by which the Trees and Stones were &n>
imatedj in Spite of the Senfe which Horace puts
i upon the Stories of Orpheus and Amphion. But
[this Queftion fliall be conftdered prefently. Again
I he fays, " 'Tis agreed by the Learned, that
uc the Science of Mufick, fo admired of the
' C{ Ancients, is wholly loft in the World, and
f cc and that what we have now, is made up out
" of certain Notes that fell into the Fancy or
| Obfervation of a poor Friar, in chanting his
f c Mattins. So that thofe Two divine Excel*-
P lencies of Mufick and Poetry ', are grown in a
cc Manner, but the one Fiddling and the other
cc Rhyming) and are indeed very worthy the Jg-
cc norance of the Friar^ and the Barbaroufnefs
•" of the Goths that introduced them among us;*
JSome learned Men indeed have faid fo ; but as
llearned have faid otherwife ; And for the De^"
tfcription Sir William gives of the modern Mu»
ifick, it is the pooreft Thing eVer was faid, an4
idemonftrates the Authors utter Ignorance of
Mufick : Did he know what Ufe Guido made
of thefe Notes ? He means the Syllables, ut %
me , mi 9 &c. for thefe are the Notes he invented.
If the modern Mufick falls fhort of the ancient,
it
fft A Treatise Chap. XlVi
it muft be in the Ufe and Application - y for the
Materials and Principles of Harmony are the
fame Thing, or rather they are improven • for
Guides Scale to which he applied thefe Syl-
lables, is the ancient Greek Scale only carried to
a greater Extent ; and which is much improven
fince.
A s I have ftated the Queftion, we are hrft to
compare the Principles and then the Praclice.
As to the Principles I have already explained
them pretty largely, at leaft as far as they have
come to our Knowledge, by the Writings on this
Subject that have efcaped the Wrack of Time.
Nor is there any great Reafon to fufpeft that
the beft are loft, or that what we have are but
Sketches of their Writings : For we have not a
few Authors of them, and thefe written at dif-
ferent Times ; and fome of them at good Length;
and by their Introductions they propofe to handle
the Subject in all its Parts and Extent, and
have actually treated of them all.
Meibomius, no Enemy to the ancient
Caufe, fpeaking of Ariftides^ calls him, Incom-
farabilis ant i qua mufica Auffior, & vere exem-
plar unicum^ who, he fays, has taught and
explained all that was ever known or taught
before him, in all the Parts' We have Arifto-
xenus ; and for what was written before him,
he affirms to have been very deficient : Nor do
the later Writers ever complain of the Lofs of
any valuable Author that was before them.
Now I fuppofe it will be manifeft to the
unprejudiced, who confider what has been ex-
plained
§ 6. of MUSIC 'IC 573
i] plained both of the ancient and modern Prin-
ciples and Theory of Harmonicks, that they
Shave not known more of it than we do, plainly
Ibecaufe we know all theirs ; and that we have
( improven upon their Foundation, will be as plain
* ! from the Accounts I have given of both, and
i the Comparifon I have drawn all along in ex-
'l plaining the ancient Theory ; therefore I need
jinfift no more upon this Part. The great Di£-
1 pute is about the Practice.
T o underftand the ancient Practice of Mu-
fick, we are firit to confider what the Name
j fignified with them. I have already explained
: its various Significations $ and fliewn, that in the
'imoft particular Senfe, Mufick included thefe
'Three Things, Harmony, Rythmus and Verfe :
If there needs any Thing to be added, take thefe
i few Authorities. In Plato's firft Alcibiades, So-
I crates asks what he calls that Art which teaches
ko fingy play on the Harp, and dance? and
\ makes him Anfwer, Mufick : But (inging among
them was never without Verfe. This is again
j confirmed by Plutarch, who fays, " That m
judging of the Parts of Mufick, Reafon and
Senfe muft be employed ; for thefe three
c muft always meet in our Hearing, viz, Sound,
whereby we perceive Harmony ; Time,
whereby we perceive Rythmus ; and Letters
or Syllables, by which we underftand what
Cc is faid." Therefore we reafonably conclude,
that their Muiick conftfted of Vcrfes fung by one
pr more Voices., alternately, or in Choirs; fome-
times
'i cc
cc
cc
cc
J74 -^Treatise Chap. XIV*
times with the Sound of Inftrumeilts, and fome
times by Voices only ; and whether they had!
any Mufick without Singing, fhall be again con-
fidered.
Let usnowconfider what Idea their Writers
give us of the practical Mufick : I don't fpeak
of the Effects, which fliall be examined again^
but of the practical Art. This we may expert,
if 'tis to be found at all, from the Authors who
write ex profejfo upon Mufick, and pretend to
explain it in all its Parts. I have already fhewn,
that they make the mufical Faculties (as they
call them) thefe, vip, Melopoeia, Rythmopxia^
and Poefis. For the Fir ft y to make the Com-
parifon right, I (hall confider it under thefe
Two Heads,, Melody and Symphony,, and begin
with the laft. I have obferved, in explaining
the Principles of the ancient Melopma, that it
contains nothing but what relates to the Con-
duel: of a (ingle Voice, or making what vve call
Melody: There is not the leaft Word of the Con-
cert or Harmony of Parts ; from which there
is very great Reafon to conclude, that this was'
no Part of the ancient Practice, and is altoge-
ther a modern Invention, and a noble one too;
the firft Rudiments of which I have already
faid we ow to that fame poor Friar ( as Sir
William Temple calls him ) Guido Aretinus.
But that there be no Difference about mere
Words, obferve, that the Quemon is not. Whe-
ther the Ancients ever joyned more Voices or
Inftruments together in one Symphony -, but,
whether feveral Voices were joyned, fo as each
had .
§ 6. of MUSICK. 57T
had a diflinft and proper Melody, which made
among them a Succcflion of various Concords ;
and were not in every Note Vni/bns, or at the
fame Diftance from each other, as 8ves ? which
laft will agree to the general Signification of the
Word Symphonici; yet 'tis plain, that in fuch Cafes
there is but one Song, and all the Voice* perform
the fame individual Melody ; but when the Parts
differ, not by the Tenfion of the Whole, but by
the different Relations of the fucceflfive Notes,
This is the modern Art that requires fo peculiar
a Genius, and good Judgment, in which there-
fore 'tis fo difficult to fucceed well The
ancient Harmonick Writers, in their Rules and
Explications of the Melopma, fpeak nothing of
this Art : They tell us, that the Melopma is
the Art of making Songs $ or more generally^
that it is the Ufe of all the Parts and Principles
that are the Subjects of harmonic al Contempla*
tion* Now is it at all probable, that fo confi-
derable an Ufe of thefe Principles was knows
among the Ancients^ and yet never once men-
tioned by thofe who profefled to write of Mm*
fick in all its Parts ? Shdl we think thefe con-
cealed it, becaufe they envied Pofterity fo valu-
, able an Art? Or, was it the Difficulty of explain-
ing it that made them filent ? They might at
leafc have faid there was fuch an Art; the Defi-
nition of it is eafy enough : Is it like the reil
of their Conduct to negje£t any Thing that
Blight redound in any Degree to their own Pratfe
and Glory ? Since we find no Notice of thfe
Ait
'■576 ^4 Treatise Chap. XI V*
Art tinder the Melopoeia, I think we cannot ex-
peel: it in any other Part. If any Body Should
think to find it in the Part that treats of Sy-
ftems, became that expreffes a Composition of
... feveral Things, they'll be difoppointed : For thefe
Authors have confidered Syftems only as greater
Intervals betwixt whofe Extremes other Notes
are placed, dividing them into lelfer Intervals^
In fuch Manner as a {ingle Voice may pafs a-
greeably from the one Extreme to the other*
But in diftingn ifliing Syftems they tell Us, fome
e.recrvy.<pooMiome fodtyttvot^ i. e. fome confonant
fome dijfonant : Which Names expreSfed the
Quality of thefe Syftems, viz. that of the firfr 3
the Extremes are fit to be heard together,
and the other not $ and if they were not ufed in
Confonance, may fome fay, thefe Names are
wrong applied : But tho' they Signified that
Quality, it will not prove they were ufed in Con-
fonance, at leaft in the modern Way : BeSides,
when they fpeak plainly and exprefly of their
Ufe in Succelfion or Melody, they idfe the
fame Names, to Signify their Agreement : A nd
if they were ufed in Confonance in the Manner
defcribed, why have we not at leaft fome gene-*
ral Rules to guide us in the Practice ? Or rather,
does not their Silence in this demonftrate there
was no fuch Practice ? But tho' there is nothing
to be found in thofe who have written more
fully and exprefly on Mufick, yet the Advocates
for the ancient Mufick find Demonstration
enough, they think, in fome Paffages of Authors
v that have given traniknt Defcriptions of Mufick:
But
§ 6. of MUSICK. i?7
But if thefe Paffages are capable of any other
good Senfe than they put upon them, I think the
Silence of the profeffed Writers on Miifick will
undoubtedly caft the Balance on that Side. To
do all Juftice to the Argument, I (hall produce
the principal and fulleft o^ thefe Kind of Para-
ges in their Authors Words. Ariftotle in his
Treatife concerning the World, Ttspi xocrpx^Lib,
5. anfwers that Queftion, If the Word is made
of contrary Principles, how comes it that it is
not long ago diifolved ? He fhews that the Beau-
ty and Perfection of it confifts in the admirable
Mixture and Temperament of different Things^
and among his Iliuftrations brings in Mufick thus,
Mxcrixq iis o%e?g apa, y.oa ficzpsTg, fictxpvg ts y,ca
@pa%sTg QQoyfeg pi^otSU^ h foaQSpaig Quvccig,
[a(wj obtereteffsv dpftovfav, which the Tranflators
juftly render thus, Mujica acutis & gr ambus
foniS) longifque & brembus una permixtis in
dwerfis vocibus^ unum ex Mis concent um red-*
dit^ i. e, Muftck, by a Mixture of acute and
grave, alfo of long and fhort Sounds of different
Voices^ yields one abfolute or perfect Concert,
Again^, in Lib. 6, explaining the Harmony of
the celeftial Motions, where each Orb, fays
he, has its own proper Motion, yet all tend to
one harmonious End, as they alfo proceed from
one Principle, making a Choir in the Heavens
by their Concord, and he carries on the Compa-
rison with Mufick thus : Ka9cforsp fe ev %ppu no-
pvfiocfe KOLTotpipLvrsg^ cvvs^YjXsX itag x°P°G <% v ~
ijpuv sd- ots ml yvvMxojv sv foccftipctig Quvca'g ctp-
Ttpaig ml fioLpvtipoug fim &pfiodo& tftpeXrj xepctv-
O o yvyrwy.
578 A Treatise Chap. XIV.
vJvtwv. Qiiemadmodum fit in Choro, ut aufipi-
cianti prafuli dut prjecentori^ accinat omnis
chorus^ e viris inter dum fmninifique compqfitus,
' qui diverfis ipfis vocibus^ gravibus ficilicet &
acutis concentum attemper ant. i. e. As in a
Choir ,after the Precentor the whole Choir fings,
compofed forretimes of Men and Women, who
by the different Acutenefs and Gravity of their
Voices, make one continuous Harmony.
Let Seneca appear next, Epiftle 84. K011
oides quam multorum vocibus Chorus conjiet ?
' Unus tamen ex omnibus Conus redditur^aliqua illic
acuta eft, aliqua gravis^ aliqua media, Acce-
dunt ( viris fozmina^ interponuntur tibi<e> fingu-
' lorum latent voces, omnium apparent. z.e. Don't
you fee of how many Voices the Chorus con-
iiits? yet they make but one Sound: In it fome
are acute, fome grave, and fome middle: Wo-
men are joyned with Men, and Whittles alfo
put in among them: Each fingle Voice is con-
cealed, yet the Whole is manifeft.
Cassiodorus {zys^Symphonia eft tempera-
■ fntntum fonitus grams ad acutum, vel acuti
ad gravemjnodulamen efficiens, five in voce five
in percufjione, five in flatu. i, e. Symphony is
on Adjuftment of a grave Sound to an acute, or
'an acute to a grave, making Melody.
Now the moft that can be made of thefe
Parages is y That the Ancients ufed Choirs of
feveral Voices differing in Acutenefs and Gravi-
ty j which was never denied : But the Whole
of thefe Definitions will be fully anfwered, fup-
poiing
§ 6. of MUSIC K. $79
poling they fun^ all the fame Part or Song only
in different Tenfions, as %ve in every Note*.
And from what was premifed I think there is
Reafon to believe this to be the only true Mean-
ing. %
But there are other confiderable Things to
be faid that will put this Queftion beyond all
reafonable Doubt. The Word Hemitonia {ig~
nifies more generally the Agreement of feveral
Things that make up one Whole ; but fo do
feveral Sounds in Succeffion make up one 8ong>
which is in a very proper Senfe a Compofition*
And in this Senfe we have in Plato and others
feveral Comparifons to the Harmony of Sounds
in Mufich But 'tis alfo ufed in the ftriSt Senfe
for Conformance, and fo is equivalent to the Word
Sjmphonia. Now we ihall make Ariftotle clear
his own Meaning in the Paffages adduced \ He
ufes Sjmphonia to exprefs Two Kinds of Gonfo*
nance ; the one, which he calls by the general
Name Sjmphonia, is the Confonance or Two
Voices that are in every Note unifon, and the
other, which he calls Antiphonia, of Two Voi-
ces that are in, every Note $ve: In his Pro-
blems, § 19* Prob. 16. He asks why Sjmphonia
is not as agreeable as Antiphonia ; and anfwers,
becaufe in Sjmphonia the one Voice being al-
together like or as One with the other, -they
eclipfe one another. The Sjmphoni here plain-
ly muft fignify Unifons, and he explains it elie-
where by calling them Omophoni: And that the
8<ve is the Antiphoni is plain, for it was a
common Name to %ve ; and Arijiotle himfelf
O o 1 explains
j8o /Treatise Chap. XIV.
explains the Antiphoni by the Voice of a Boy
and a Man that are as Neie and Hypatejwh.ic\i
were 2ve in Pythagoras^ Lyre. Again,. I own
he is not fpeaking here of Uiiifoti and $pe (imply
/ confidered, but as ufed in Song : And tho* in
modern Symphonies it is alfo true, that Unifon
cannot be fo frequently ufed with as good Effe£t as
8c^,yethis Meaning is plainly this^viz. that when
Two Voices fing together one Song, 'tis more
agreeable that they be %ve than anifon with one
another, in every Note: This I prove from
the iyth Probl. in which he asks why Dia-
pente and Diateffaron are never fung as the
Antiphoni 1 He anfwers, becaufe the Antipho-
ni, or Sounds of %®e, are in a Manner both the
fame and different Voices -, and by this Likenefs,
where at the fame Time each keeps its own
diitin£l Character, we are better pleafed : There-
fore he affirms, that the %ve only can be fung in
Symphony &oi itavw TJ^wdcc (jlovyj a&rcti. )
Nov/ that by this he means fucli a Symphony as
I have explained, is certain, becaufe in mo-
dern Counterpoint the qth, and efpecially
the $th are indifpenfable ,- and indeed the
$th with its Two $ds, are the Lifeof- the
Whole. Again, in Probl. 18. he asks why
why the Diapajon only is magadifed ? And an-
fwers, becaufe its Terms are the only Antipho-
ni : Now that this fignifies a Manner of Singing,
where the Sounds are in every Note 8ve to one
another, is plain from this Word magadifed,
taken from the Name of an Inftrument fiayd-
hog, in which Two Strings were always ftruck
toge-
§ 6. of MUSIC TC jit
together for one Note. Athentus makes the
Magadis the fame with the Barhiton and
JPecJis j and Horace makes the Mufe \Poly } hym-
nia the Inventor of the Barhiton, — Nee Po-
lyhymnia l^esboum refifgit tendere Barbiton.--
And from the Nature of this Inftumcnt, that it
had Two Strings to every Note, fome think it
probable the Name Polyhymnia was deduced.
Athen<sus reports from Anacreon, that the Ma-
gadis had Twenty Chords ; which is a Num-
ber fufficient to make us allow they were dou-
bled ; fo that it had in all Ten Notes : Now
anciently they had but Three Tones or Modes,
and each extended only to an %<tie. and being a
Tone afunder, required precifely Ten Chords ;
therefore Athen<eus corrects PojfidoniusfoY: fay-
ing the Twenty Chords were all diftintt Notes,
and neceffary for the Three Modes, But he
further confirms this Point by a Citation from
the Comick Poet Alexandrides^ who takes a
Comparifon from the Magadis ^ and fays, / am,
like the Magadis, about to make you imderftand
a Thing that is at the fame Time both fub lime
and low ; which proves that Two Strings were
{truck to^eth^r, and that thev were not uni/on,
He reports alfo the Opinion of the Poet Jon y
that the Magadis coniifted of Two Flutes^
which were both founded together. From all
this 'tis plain, 1 hat by magadifed, Ariftotlc
means fuch a Confonance of Sounds as to be in
every Note at the fame Diftance,and eonfequent-
ly to be without Symphony and Parts according
%o the modern Pra&icc, Athenaus reports aifq
5?x ^Treatise Chap. XIV.
of Pindar ', that he called the Mufick fung by a
Boy and a Man Magadis ; becaufe they fung
together the fame Song in Two Modes, Mr,
Per auk concludes from this, that the Strings
of the Magadis were fometimes 3d/, becaule
Ariftotle fays,the qth and $th are never maga-
di/ed .' But why may not Pindar mean that
they were at an %<ves Diftance ; for certainly
Jlriftotle ufed that Comparifon of a Boy and a
Man to exprefs an $ve ' Mr. Per auk thinks it
muft be a 3 d becaufe of the Word Mode, where-
of anciently there were but Three ,- and confirms
it by a Pailage out of Horace , Epod. 9, Sonants
miftum tibiis carmen lyra ; hac ' Dorium illis
Barbarum : By the Barharum, fays he, is to be
underflood the Lydian^ which was a Ditone
above the Dorian : But the Difficulty is, that
the Ancients reckoned the Ditone at beft a con-
tinuous Difcordj and therefore 'tis not probable
they would ufe it in fo remarkable a Manner :
But we have enough of this. The Author laft
named obferves, that the Ancients probably had
a Kind of fimple Harmony, in which Two or
Three Notes were tuned to the principal Chords
of the Key^ and accompanied the Song. This
he thinks probable from the Name of an Initru^-
ment Pandora that Athenaus mentions j which is
likely the fame with the Mandora^an Inftrument
not very long ago ufed, fays he, in which there
were Four Strings, whereof one ferved for the
Song 5 and was ftruck by a Pie tlrum or Quill tied
$9 the Forefinger; The other Three were tuned
§ 6. tf MUSICK. y.&3
fo as Two of them were an %ve, and the other
a Middle dividing the %ve into a qth and $th:
They were ftruck by the Thumb, and this re-
gulated by the Bythmus or Meafure of the Song,.
i. e. Four Strokes for every Meafure of common
Time, and Three for Triple. He thinks Horace
points out the Manner of this Inftrument in Ode
6, Lesbium fervate pedem, meique pollicis
iblum, which he thus tranilates. Take No-
tice, you who would joyn jour Voice to the
Sound of my Lyre, that the Meqf'ure of my Son^
is Sapphick, which the fir iking of my Thumb
marks out to you. This Inftrument is parallel
to our common Bagpipe.
The PatTages of Ariftotle being thus cleared, I
think Seneca and Caffiodorus may be eafiiy given
up. Seneca fpeaks of vox media, as well as acuta
and gravis \ but this can fignify nothing, but
that there might be Two %ves, one betwixt the
Men and Women ^and the flirill Tibia might be
%ve above the Women > But then the latter
Part of what he fays deftroys their Caufe .• tor
fingulorum voces latent can very well be £ai$
of fuch asfingthe fame Melody tfnifhn or OBave\.
but would by no Means be true of feverai
Voices performing a modern Symphony, where
every Part is confpicuous, with a perfect Har-
mony in the Whole. For Caffiodorus^ I think
what he fays has no Relation to Confonanc: >
and therefore I have tranllated it, An Adjujh
merit of a grave Sound to an acute, or an acute
to a grave making Melody : If it he alledg/V;
that temper amentmn may (ignifie a Mixture,! (h- < ! - } .
Oq4 v|$|
584 ^Treatise Chap. XIV.
yield it ; but then he ought to have faid, Tent'
per amentum fonit us gravis & acuti $ for what
means fonitus grams ad acutum, and again
acuti ad gravem ? But in the other Cafe this
is well enough, for hemeans>That Melody may
confift either in a Progrefs from acute to grave,
or contrarily : And then the Word Modulamen
was never applied any other way than to fuc-
ceffive Sounds. There is another PafTage which
If. Vojjius cites frorn ALU an the Platonic^
JLvfjLQwia $s s<tl SvoTv yj nXeiovw @9oyfav o%dn$j,
tied (3apvTy]Ti foatpspovlu)/ Kara, to dvro 7fjucrtg ml
xpacrig, i. e. Symphony confifis of Two or more
Sounds differing in Acutenefs and Gravity, with
the fame Cadence and Temperament: But this
rather adds another Proof that what Sympho-
nies they had were only of feveral Voices ling-?
ing the fame Melody only in a different Tone.
After fuch evident Demonftrations, I think
there needs no more to be faid to prove that
Symphonies of different Parts are a modern Im-
provement. From their reiedHng the ^ds and
6ths out of the Number of Concords, the fmall
Extent of their Syftem being only Two ObJa<ves y
and having no Tone divided but that betwixt
Mcfe and Paramefe, we might argue that they
had no different Parts : For tho' fome fimple
Compofitions of Parts, might be contrived with
thefe Principles, yet 'tis hard to think they would
lay the Foundations of that Practice, and carry
it no further ,• and much harder to believe they
would never ipeak one Word of fuch an Art
and Pra&ice 3 where they profefs to explain all
' :r i - ^ ■-" dig
§6. of MUICK. 5 8 J
the Parts of Mufick. But for the Symphonies
which we allow them to have had, you'll ask
why thefe Writers don't fpeak of them, and
why it feems fo incredible that they fhould have
had the other Kind without being ever mention-
ed, when they don't mention thefe we allow ?
The Reafon is plain, becaufe the Mufician's
Buiinefs was only to compofe the Melody , and
therefore they wanted only Rules aboift that ;
but there was no Rule required to teach how
feveral Voices might joyn in the fame Song, for
there is no Art in it : Experience taught them
that this might be done in Unifon or Otlave;
and pray what had the Writers more to fay
about it ? But the modern Symphony is a quite
different Thing, and needs much to be ex-
plained both by Rules and Examples. But 'tis
Time to make an End of this Point : I (hall
only add, That if plain Reafon needs any Au-
thority to fupport it, I can adduce many Mo-
derns of Character, who make no Doubt to fay,
That after all their Pains to know the true State
of the ancient Mufick^ they could not find the
leaft Ground to believe there was any fuch
Thing in thefe Days as Mufick in Parts, I
have named Perrault, and ftiall only add to
him Kircher and Doctor Wallis^ Authors of
great Capacity and infinite Induftry.
Our next Comparifon ftiall be of the Melo-
dy of the Ancients and Moderns; and here
comes in what's neceflary to be faid on the other
Parts of- Mufick) viz. the Rythmus and Verfe.
In order to this Cqmparifon, I fliall diftinguiih
Melody
jt6 ^Treatise Chap. XIV.
Melody into vocal and inftrumental. By the
firft I mean Mufick fet to Words, efpecially
Verfes ; and by the other Mufick compofed on-
ly for Inftruments without Singing. For the
vocal you fee by the Definition that Poetry
makes a neceffary Part of it : This was not
only of ancient Practice, but the chief, if not
their only Practice, as appears from their De-
finitions of Mufick already explain'd. 'Tis not
to be expected that I fhould make any Com-
parifon of the ancient and modern Poetry $ 'tis
enough for my Purpofe to obferve. That there
are admirable Performances in both; and if we
come fhort of them, I believe 'tis not for want
either of Genius or Application : But perhaps
we fhall be obliged to own that the Greek and
Latin Languages were better contrived for
pleafing the Ear. We are next to conlider, that
the Rythmus of their vocal Mufick was only
that of the Poetry, depending altogether on the
Verfe, and had no other Forms or Variety than
what the metrical Art afforded : This has been
already fhewn, particularly in explaining their
mufical Notes ; to which add. That under the
Head of Mutations^ thofe who confider the
JRythmus make the Changes of it no other than
from one Kind of metrum or Verfe to another,
as from Jambick to Choraick : And we may
notice too. That in the more general Senfe,
the Rythmus includes alfo their Dancings, and
€l\1 the theatrical Action. I conclude therefore
that their vocal Mufick confifted of Verfes, fet
to mufical ToneX) and fung by one or more
Voices
$ 6. of MUSIC K. j 8^
Voices in Choirs or alternately ; fometimes with
and alfo without the Accompanyment of In-
ftruments: To which we may add, from the
laft Article, That their Symphonies confifted
| only of feveral Voices performing the fame Song
I in different Tones as JJnifon and OSiave. For
inftrumental Mufick (as I have defined it) 'tis not
fo very plain that they ufed any : And if
they did, 'tis more than probable the Rythmus
was only an Imitation of the poetical Numbers,
and confifted of no other Meafures than
what were taken from the Variety and Kinds of
their Verfes ; of which they pretended a fu£
ficient Variety for expreffmg any Subject accor-
i ding to its Nature and Property : And fince the
chief Defign of their Mufick feems to have been
! to move the Heart and Paflions, they needed
no other Rythmus. I cannot indeed deny that
there are many Paffages which fairly iniinuate
I their Practice upon Inftruments without Singing,-
j! fo Athena us fays, The Synaulia was a Contefl
I of Pipes performing alternately without Jinging.
And QiiPntilian hath this Expreffion, If the
Numbers and Airs of Mufick have fuch a ~per-
tue^ how much more ought eloquent Words to
have ? That is to fay, the other has Virtue
I or Power to move us, without Reipect to the
Words. But if they had any Rythmus for in-
ftrumental Performances, which was different
from that of their poetical Meafures^ how
comes it to pafs that thofe Authors who have
been fo full in explaining the Signs by which
their Notes of Mufick were reprefented, fpeak
not
588 ^Trpatise Chap. XIV.
not a Word of the Signs of Time for Inftru-
ments ? Whatever be in this, it muft be own-
ed that Singing with Words was the moft an-
cient Practice of Mufick, and the Practice of
their more folemn and perfect Entertainments,as
appears from all the Inftances above adduc ed,to
prove the ancient Ufe and Efteem o£ Mufick 1 And
that it was the univerfal and common Practice,
even with the Vulgar, appears by the paftoral
Dialogues of the Poets, where the Conteft is
ordinarily about their Skill in Mufick^nA chief-
ly in Singing.
L e t us next confider what the prefent Pract-
ice (among Europeans at leaft) confifts of. We
have, fir ft, weal Mufick -, and this differs from
the ancient in thefe Refpe&s, viz. That the
Gonftitution of the Rythmus" is different from
that of the Verfe, fo far, that in fetting Mufick
. to Words, the Thing principally minded is, to
accommodate the long and fliort Notes to the
Syllables in fuch Manner, as the Words may be
well feparated, and the accented Syllable of
every Word fo confpicuous, that what is
fung may be diftinclly nnderftood : The Move-
ment and Meafure is alfo fuited to the different
Subjects, for which the Variety of Notes, and the
Conftitutioxis or Modes of Time explained in
Chap. 1 2. afford funicient means. Then we
differ from the Ancients in our inftrumental Ac-
companyments, which compofe Symphonies
with the Voice, fome in Uni/bth others making
a diftintt Melody \ which produces a ravifhing En-
tertainment they were not blefl with, or at leaft
with
§ 6. of MUSI CK* j 89
without which we ftiould think ours imperfect.
Then there is a delightful Mixture of pure in-
ftrumental Symphonies, performed alternately
with the Song. Ldfily^ We have Compositions
fitted altogether for Inftruments : The Defign
whereof is not fo much to move the PafTions,as
to entertain the Mind and pleafe the Fancy
with a Variety of Harmony and Rythmus ; the
principal Effect of which is to raife Delight and
Admiration. This is the plain State of the an-
cient and modern Mufeckfm refpeft ofPraclice:
But to determine which of them is moft perfect
will not perhaps be fo eafily done to fatisfie
every Body; Tho 1 We believe theirs to have
been excellent in its Kind, and to have had no-
ble Effects ,• this will not pleafe fome, unlefs we
acknowledge ours to be barbarous, and altoge-
ther ineffectual. The Effects are , indeed the
true Arguments j but how fhall we compare
thefe, when there remain no Examples of an-
cient Compofition to judge by ? fo that the De-
fenders of the ancient mufick admire a Thing
they don't know ; and in all Probability judge
not of the modern by their perfonal Acquain-
tance with it, but by their Fondnefs for their
own Notions. Thole who ftudy our Mufick^
and have well tuned Ears, can bear Witnefs
to its noble Effects : Yet perhaps it will be re-
plied. That this proceeds from a bad Tafte^ and
Jomething natural^ in applauding the be ft Thing
we know of any Kind, But let any Bcdy pro-
duce a better, and we (hall heartily applaisd it.
They bid us bring back the ancient Muficians 9
and
jpo ^Treatise Chap. XIV.
and then they'll effectually fliew us the Diffe-
rence; and we bid them learn to underftand the
modem Mujick, and believe their own Senfes:
In fhort we think we have better Reafon to de-
termine in our own Favours, from the Effects
we 9 dually iee\ than any Body can have from
a Thing they have no Experience of] and can
: pretend to know no other Way than by Report :
But we fhall confiderthe Pretences of each Par-
ty a little nearer. I have already obferved, that
the principal End the Ancients propofed in
their Mujich^ was to move the Paffions; and to
this purpofe Poetry was a neceffary Ingredient.
,We have no Difpute about the Power of poeti-
cal Compofitions to affect the Heart, and move
the Palfions, by fuch a ftrong and lively Repre-
fentation of their proper Objects, as that noble
Art is capable of : The Poetry of the Ancients
we own is admirable ; and their Verfes being
fung with harmonious Cadences and Modulati-
ons, by a clear and fweet Voice, fupported by
the agreeable Sound offome Inftrument, in fuch
Manner that the Hearer underftood every Word
that was faid, which was all delivered with a
proper Action, that is. Pronunciation and Ges-
tures fuitable to, or expreifive of the Subject, as
"we alfo fuppofe the Kind of Verfe, and the
-Modulation applied to it was; taking their vocal
Mufich in this View, we make no Doubt that
it had admirable Effects in exciting Love, Pity,
Anger, Grief, or any Thing elfe the Poet had
a Mind to : But then they muft be allowed to
'affirm,' who pretend to have the Experience of
§ 6. of MUSIC K. j 9 i
it^ That the modern Mufick taking it in the
fame Senie, has all thefe Effects. Whatever
Truth may be in it, I fhall pafs what Doctor
JVallis alledges, viz. That thefe ancient Effe&s
were mofi remarkably produced upon Rufticks,
and at a Time when Mufick was new, or a
very rare Thing : But I cannot however mifs
to obferve with him, That the Paffions are eafi-
ly wrought upon. The deliberate Reading of
a Romance well written will produce Tears,
Joy, or Indignation, if one gives his Imaginati-
ons a Loofe ,• but much more powerfully when
attended with the Things mentioned ; So that
it can't be thought fo very myfterious and won-
derful an Art to excite Paffion, as that it fhould
be quite loft. Our Poets are capable to exprefs
any moving Story in a very pathetick Manner:
Our Muficians too know how to apply a fuit-
able Modulation and Rythmus : And we have
thofe who can put the Whole in Execution ;
fo that a Heart capable of being moved will be
forced to own the wonderful Power of modem
Mufick : The Italian and Englifh Theatres
afford fuffi cient Proof of this ; fo that I believe,
were we to collect Examples of the Effects that
the acling of modem Tragedies and Operas
have produced, there would be no Reafon to
fay we had loft the Art of exciting Paffion.
But 'tis needlefs to infift on a Thing which fo
many know by their own Experience. If fome
are obftinate to affirm, That we are fill behind
the Ancients in this Art^ becaufe they have ne^
wr felt fiich Effeftsof it* I fhall ask them if
thev
59* -^Treatise Chap. XIV.
they think every Temper and Mind among the
Ancients was equally difpofed to relifti, and be
moved by the fame Things ? If Tempers dif-
fered then,why may they not now,and yet the
Art be at leaft as powerful as ever ? Again
have we not as good Reafon to believe thofe
who affirm they feel this Influence, as you who
fay you have never experienced it ? And if
you put the Matter altogether upon the Autho-
rity of others, pray, is not the Teftimony of
the Living for the one, as good as that of the
Dead for the other?
But ftill there are Wonders pretended to
have been performed by the ancient Muf/ck,
which we can produce nothing like ; fuch as
thofe amazing Tranfports of Mind, and hurrying
of Men from one Paffion to another, all on a
fudden, like the moving of a Machine,of which
we have fo many Examples in Hiftory, See
Page 495. For thefe I iliaJl anfwer. That what
we reckon incredible in them may juftly be laid
upon the Hiftorians, who frequently aggravate
Things beyond what's ftridtly true, or even their
Credulity in receiving them upon weak Grounds j
and molt of thefe Stories are delivered to us by
Writers who were not themfelves Witneffes of '
them, and had them only by Tradition and com-
mon Report. If nothing like this had ever been
juftly objected to the ancient Hiftorians, Ifhould
think my felf obliged to find another Anfwer :
But fince 'tis fo, we may be allowed to doubt
ofthefeFacls, orfufpeft at leaft that they are .
in a great Degree hyperbolical. Confider but
the
<\>
§ 6. 'of MUSIC K. it}
Circumftances of fome of them as they are told,
and if they are literally true, and can be accoun-
ted for no other Way but by the Power of
Sound, I muit own they had an Art which is
loft : For Example, the quelling of a Sedition ;
let us reprefent to our felves a furious Rabble,
envenomed with Difcontent, and enraged with
Oppreffion ; or let the Grounds of their Rebel-
lion be as imaginary as you pleafe, (till We muft
confider them as all in a Flame; fuppofe next
they are attacked by a skilful Mufi'ciah, who
addreffes them with his Pipe or Lyre,- how like-
ly is it that he {hall perfwade them by a Song
to return to their Obedience, and lay down their
Arms? Or rather how probable is it that he
may be torn to Pieces, as a folemn Mocker of
their juft Refentment ? But that I may allow fome
Foundation for fuch a Story, I {hall fuppofe
a Man of great Authority for Virtue, Wifdom
ftnd the Love of Mankind, comes to offer his
humble and affectionate Advice to fuch a Com-
pany ; I fuppofe too, he delivers it in Verfe,
and perhaps fings it to the Sound of his Lyre*,
(which feems to have been a common Way of
delivering publick Exhortations in more ancient
Times, the Mufick being ufed as a Means to
gain their Attention.) I don't think it impof-
fible that this Man may perfwade them to
Peace, by reprefenting the Danger they riin^
aggravating the Mifchief they are like to bring
upon themfelves and the Society, or alfo cor-
recting the falfe Views they may have had of
j£hijpgs t But then will any Body fay* all this
P g M
594 -^Treatise Chap. XIV.
\s the proper Effect of Mufick, unlefs Reafon-
'ng be alfo a Part of it ? And muft this be an
Example of the Perfection of the ancient Art,
and its Preference to ours ? In the fame Man-
ner may other Inftances alledged be accounted
for, fuch as Pythagoras*s diverting a young Man
from the Execution of a wicked Defign, the
Reconcilement of Two inveterate Enemies, the
curing of Clytemneflrds vicious Inclinations, (j c.
Horace's Explication of the Stories of Orpheus
and Amphion, makes it probable we ought to ex-
plain all the reft the fame Way. For the Story
of Timotheus and Alexander, as commonly re-
prefented, it is indeed a very wonderful one, but
I doubt we muft here allow fomething to the
Boldnefs or Credulity of the Hiftorian : That Ti-
motheus, by ringing to his Lyre, with moving
Gefture and Pronunciation, a well compofed
Poem of the Achievements of fome renowned
Hero, as Achilles, might awaken Alexanders
natural Paffion for warlike Glory, and make him
exprefs his Satisfaction with the Entertainment
in a remarkable Manner, is nowife incredible :
We are to confider too the Fondnefs he had for
the Iliad, which would diipofe him to be mo- ,
ved with any particular Story out of that: But
how he fhould forget himfelf fo far, as to com-
mit Violence on his beft Friend, is not fo eafily
accounted for, unlefs we fuppofe him at that.
,Time as much under the Power of Bacchus as
of the Mvfes : And that a fofter Theme fung
with equal Art, fhould pleafe a Hero who was
not
:0. of MUSIC K, ~~ 5p y
not infenfible of Venus 's Influences is rio Myfte-
ry, efpecially when his Miftrefs was in Compa-
ny : But there is nothing here above the Power
of modern Poetry and Mujick, where it meets
with a Subject the fame Way difpofed^ to be
wrought upon* To make an End of this, t
muft obferve, that the Hiftorians, by faying too
much, have given us Ground to believe very,
little* What do you think of curing a raging
Peftilence by Mujick . ? For curing the Bites of
Serpents, we cannot fo much doubt it, hnce that
of the Tarantula has been cured in Italy* But
then they have no Advantage in this Inftance :
And wemuft mind too that this Cure is not per-
formed by exquihte Art and Skill in Mujick j it
does not require a Correlli or Valentitii^ but is
performed by Strains difcovered by random
Trials without any Rule : And this will fervc
for an Anfwer to all that's alledged of the Cure
of Difeafes by the ancient Mufich
,'T i s Time to bring this Comparifon to an
End ; and after what's explained I (hall make
no Difficulty to own, that I think the State of
Mujick is much more perfe£t now than it was
among the ancient Greeks and Romans* The
Art of Mujicky and the true Science of Harmo*,
ny in Sounds is greatly improven. I have allow-*
ed their Mujick (including Poetry and the the-
atrical Aclion) to have been very moving ,• but
at the fame Time I muft fay, their Melody has
been a very fimple Thing, as their Syjtem or
Scale plainly fliews, whofe Difference from the
fiiodern, I have already explained*
I P 1 A nd
j9<5 i4 Treatise Chap. XIV.'
And the confining all their Rythmus to the po-
etical Numbers, is to me another Proof of it,
and {hews that there has been little Air in their
Muftck - 3 which by this appears to have been only
of the recitative Kind, that is, only a more
mufaal Speaking, or modulated Elocution ; the
Character of which is to come near Nature, and
be only an Improvement of the natural Accents
of Words by more pathetick or emphatical Tones-,
the Subject whereof may be either Verfe or
Profe. And as to their Instruments of Mu-
fick, for any Thing that appears certain
and plain to us, they have been very fimple.
Indeed the publick Laws in Greece gave
Check to the Improvement of the Art of Har-
mony, becaufe they forbade all Innovations in
the primitive fimple Mufick; of which there are
abundance of Teftimonies, fome whereof have
been mentioned in this Chapter, and I fhall add
"what Plato fays in his Treatife of the Laws,
viz. That they entertained not in the City the
Makers of fuch Inftruments as have many Strings,
as the Trigonus and Petlis ; but the Lyra and
Cithara they ufed, and allowed alfo fome fimple
JTifiuld in the Country* But 'tis certain, that
primitive Simplicity was altered ,- fo that from
' a very few Strings, they ufed a greater Number :
But there is mUch Uncertainty about the Ufe of
them;, as whether it was for mixing their Modes y
and the Genera, or for ftriking Two Chords
together as in the' Magadis. Since I have men^
tioned In (iriiments, I muft obferve Two Things,
Firfli That they pretend to have had Tibia of
' diffe-
§ 6. of MUSIC K. j 9 ?
different Kinds, whofe fpecifick Sounds were
excellently chofen for exprelfing different Sub-
jects. Then, there is a Defcription of the 0r~
ganumhydratilicum in Tertullian, which fome
adduce to prove how perfect their Inftruments
were. — SpebJa portent of am Archimedis Wunir
ficentiam ; organnm hydraulicum dico,tot mem-'
bra, tot partes, tot compagines, tot itinera
*vocum, tot compendia fonorum, tot commercia
modorum, tot acies tibiarum,(j una moles erunt
omnia ; where he had learnt this pompous De-
fcription of it T know not • for one can get but
a very obfcure Idea of it from VitrnviuS) even
after Kircher and Voffius's Explications. But I
hope it will not be pretended to have been more
perfect than our modem Organs : And what have
they to compare of the ftringed Kind, with our
Harpflchordsj and all the Inftruments that; are
(truck with a Bow ?
After all, if our Melody or Songs areonly e-
qualtothe Ancients, I hope the Art of Mufick is
not loft as fome pretend. But then, what an Im-
provement in the Knowledge of pure Harmony.
has been made, fince the Introdu6tion of the mo-
dern Symphonies ? Here it is, that the Mind is ra-
viflied with the Agreement of Things feemingly
contrary to one another. We have here a Kind
of Imitation of the Works of Nature, where dif-
ferent Things are wonderfully joyned in one
harmonious Unity : And as fome Things appear
at firft View the fartheft removed from Symme^
try and Order, whichirom the Courfe of Tilings
jve learn to be absolutely necetfary for thePerfecli-
P p 3 °9
598 ^Treatise Chap. XIV.
on and Beauty of the Whole j foDifcords being
artfully mixed with Concords, make a more per-
fect Compofition, which furprifes us with De-
light If the Mind is naturally pleafed with per-
ceiving of Order and Proportion, with compar-
ing feveral Things together, and difcerning in
the midft of a feeming Confufion, the moft per-
fect and exa&Difpofitionand united Agreement j
,then the modern Concerts muft undoubtedly be
allowed to be Entertainments worthy of our
Natures : And with the Harmony of the Whole
We muft confider the furprifing Variety of Air,
which the modern Conftitutions and Modes of
jTime or Rythmus afford; by which, in our in-
ftrumental Performances, theSenfe and Imaginat-
ion are fo mightily charmed. Now, this is an
'Application of Mufick to a quite different Pur-
pofe from that of moving Paffion : But is it rea-
sonable upon that Account, to call it idle and
Infignirlcant, as fome do, who I therefore fuf-
f)edt are ignorant of it ? It was certainly a noble
Ufe of Mufick to make it fubfervient to Mora-
lity and Virtue ; and if we apply it lefs that
iWay, I believe 'tis becaufe we have lefs Need
of fuch Allurements to our Duty : But whatever
be the Reafon of this, 'tis enough to theprefent
Argument, that our Mufick is at leaft not infe-
rior to the ancient inthepathetick Kind : And if
it be not a low and unworthy Thing for us to
be pleafed with Proportion and Harmony, in
which there is properly an intellectual Beauty,
then it muft be conferfed, that the modern Mu-
fick ismqre jerfefl; than the ancient. But why
§ 6. of MUSIC K. yp9
muft the moving of particular Paflions be the
only Ufe of Miifick ? If we look upon a noble
Building, or a curious Painting, we are allowed
to admire the Defign, and view all its Propor-
tions and Relation of Parts with Pleafure to
our Underftandings, without any refpeel: to the
Paflions. We muft obferve again, that there is
fcarce any Piece of Melody that has not fome
general Influence upon the Heart ; and by being
more fprightly or heavy in its Movements, will
have different Effects; tho' it is not defignedto
excite any particular Paflion,and can only befaid
in general to give Pleafure, and recreate the
Mind. But whyftiould we difpute about a Thing
which only Strangers to Mufick can ipeak ill of?
'And for the Harmony of different Parts, the De-
fenders of the ancient Mufick own it to be a va-
luable Art, by their contending for its beingan-
cient : Let me therefore again affirm, that the
Modems have wonderfully improven the Art of
Mufich It muft be acknowledged indeed, that
to judge well, and have a true Relifli of our
more elaborate and complex Mujick, or to be
fenfible of its Beauty, and taken with it,requires
a peculiar Genius, and much Experience, with-
out which it will feem only a confufed Noife ;
but I hope this is no Fault in the Thing. If
one altogether ignorant of Painting looks
upon the moll curious Piece, wherein he finds
nothing extraordinary moving to him, becaufe
the Excellency of it may ly in the Defign and
admirable Proportion and Situation of the Parts
which he takes no Notice of : Muft we there-
P p 4 fore
$oo ^Treatise Chap. XIV»
fore fay, it has nothing valuable in it, and ca-
pable to give PJeafure to a better Judge i What,
in Mufick or Painting, would feeni intricate
-and confufed, and fo give no Satisfaction to the
unskilled, will ravifti with Admiration and Der
light, one who is able to unravel all the Parts,
obferve their Relations and the united Concord
of the Whole. But now, if this be fuch a real
and valuable Improvement in Mufick, you'll ask,
How it can be thought the Ancients could be ig-
norant of it, and fatisfy themfelves with fuch a
(imple Mufick, when we confider their great
Perfection in the Sifter Arts of Poetry and Paintr
ingj and all other Sciences. I fhall anfwer this
by asking again, How it comes that the Ancients
left us any thing to invent or improve > And
how comes it that different Ages and Nations
liave Genius and Fondnefs for different Things.
The i\ncients ftudied only how to move the
Heart, to which a great many Things neceffari-
]y concurred, as Words, Tune and Action \ and
by thefe we can ftill produce the fame Effects ;
but we have alfo a new Art, whofe End is rather
to entertain the Understanding, than to move par-
ticular Pajfions. What Connection there is betwixt
their improving other Sciences and this, is not
fp plain as to make any certain Conclufion from
it. * And as to their Painting, there have been
very good Reafons alledged to prove. That they
followed the fame Tafte there as in the Mufick,
f". e. the (imple obvious Beauties, of which eve-
ry Body might judge anci be fenfible. Their End
$yas to pleafe and move the People., which is
"V'* bet-
§ <?. of MV SICK. 601
better done by the Senfes and the Heart than by
the Underftanding $ and when they found fuffr-
cient Means to accompliili this, why fhould we
wonder that they proceeded no further, efpeci-
ally when to have gone much beyond,
would likely have lofed their Defign. But, fay
you,this looks as if they had been fenfible there
were Improvements of another Kind to be made :
Suppofe it was fo, yet they might flop when,
their principal End was obtained. And Plu+
tarch fays as much, for he tells us it was not Ig-
norance that made the ancient Mulick fo fimple,
but it wasfo out of Politick : Yet he complains,
that in his own Time, the very Memory of the
ancient Modes that had been fo ufeful in the
Education of Youth, and moving the Paffions
was loft thro 1 the Innovations and luxurious Va-
riety introduced by later Mulicians ; and now,
when a full Liberty feems to have been taken,
may we not wonder that fo Jittle Improvement
was made 9 or at leaft fo little of it explained and
recorded to us by thefe who wrote of Muhck*
after fuch Innovations were fo far advanced,
I fhall end this Difpute, which is perhaps too
tedious already, with a fhort Confideration o{
what the boldeft Accufer of the modern Muftck,
Jfaac VojfiuS) fays againft it, in his Book de poe-
matum cantu &. viribus Rythmi. He obferves,
what a wonderful Power Motion has upon the
Mind, by Communication with the Body; how
we are pleafed with rythmical or regular Moti-
on ; then he obferves, that the ancient Greeks
and Latins perceiving this., took an infinite
Paitfo
6ot ^Treatise Chap. XIV;
Pains to cultivate their Language, and make it
as harmonious, eipecially in what related to the
Jiythmus> or Number, and Combination of long
and fliort Syllables, as potifible ; to this End par-
ticularly were the pedes fnetrici invented, which
are the Foundations of their Verification ; and
this he owns was the only Rythmus of their
Mufick, and fo powerful, that the whole Effect
of Mufick was afcribed to it, as appears, fays
he, by this Saying of theirs, to nciv napd iA%<n-
Koig a (uStf/lflfi And to prove the Power attri-
buted to the Rythmus, he cites feveral other
Paffages. That it gives" Life to Mufick, efpe-
cially the'pathetick, will not be denied ; and we
iee the Power of it even in plain Profe and Ora-
tory : But to make it the Whole, is perha ps at
tributing more than is due: I rather reckon the
Words and Senfe of what's fung, the principal
Ingredient ; and the other a noble Servant to
them, for railing and keeping up the Attention,'
becaufe of the natural Pleafure annexed to thefe
Senfktions. 'Tis very true, that there is a Con-
nection betwixt certain Paflions, which we call
Motions of the Mind, and certain Motions in
our Bodies ; and when by any external Motion
thefe can be imitated and excited, no doubt
we ftiall be much moved; and the Mind, by that
Influence, becomes either gay, foft, brisk or
drowfy : But how any particular Paflion can
be excited without fuch a lively Reprefentation
pf its proper Object, as only Words afford, is
not very intelligible ; at leaft this appears tome
|he moft juft p4 effectual Way, But let us
the
§ 6\ of MUSI CK. 6o 3
hear what Notion others had of this Matter,
Quintilian fays, If the Numbers of Mufick have
fuch Influence, how much more ought eloquent
Words to have ? And in all the ancient Mufick
the greater!: Care was taken, that not a Syllable
of the Words fhould be loft, for fpoiling the
Senfe, which Voffius himfelf obferves and owns.
PancirolliiS) who thinks the Art loft, afcribes
the chief Virtue of it to the Words. — Siquidem
una cum melodia Integra percipiebantur verba :
And the very Reafon he gives, that the modern
Mufick is lefs perfedt, is, that we hear Sounds
without Words, by which fays he, the ear is a
little plealed, without any Entertainment to the
Understanding : t But all this has been confidered
already. Voffius alledges the mimick Art, to
prove, that the Power of Motion was equal to
the moft eloquent Words ; but we ilialj be as
much ftraitned to believe this, as the reft of their
Wonders. Let them believe it who will, that
& Pantomime had Art to make himfelf eaiily un-
derftood without Words, by People of all Lan-
guages : And that Rqfcius the Comedian, coold
exprefs any Sentence by his Geftures, as fignirt-
cantly and varioufly, as Cicero with all his O-
ratory. Whatever this Art was, 'tis loft, and
perhaps it was fomething very furprifing j but
'tis hard to believe thefe Stories literally. How-
ever to the Thing in Hand, we are concerned
only to confider the mufical or poetical Ryth-
mus.
Voffius fays, that Rythmus which does not
icontaia and exprefs the. very Forms and Figures
of
&>4 A Treatise Chap. XIV.
of Things, can have no Effect ; and that the
ancient poetical Numbers alone are juftly con-
trived for this End. And therefore the modern
Languages and Verfe are altogether unfit for
Mujicky and we (hall never have, fays he, any
right meal Mujick, till our Poets learn to make
Verfes that are capable to be fung, that is, as
he explains it, till we new model our Langua-
ges, reftore the ancient metrical Feet, and ba-
»i{h our barbarous Rhimes. Our Verfes, fays
he, run all as it were on one Foot, without Di-
ftin&ion of Members and Parts, in which the
Beauty of Proportion is to be found j therefore
he reckons, that we have no Rythmus at all in
our Poetry j and affirms, that we mind nothing
but to have fuch a certain Number of Syllables
in a Verfe, of whatever Nature, and in whate-
ver Order. Now, what a rafti and unjufl Cri-
ticifm is this ! if it was fo in his Mother Ton-
gue, the Dutch, I know not; but I'm certain it
is otherwife in EngliJIo. 'Tis true, we don't fol-
low the metricalCompofition of the Ancients ;yet
we have fuch a Mixture of ftrong and foft, long
and fliort Syllables, as makes our Verfes flow,
rapid, fmooth, or rumbling, agreeable to the
Subjecl. Take any good Englifh Verfe, and by
a very fmall Change in the Tranfpohtion of a
Word or Syllable, any Body who has an Ear
will find, that we make a very great Matter of
the Nature and Order of the Syllables. But
why rnuft the ancient be-the only proper Metre-
for Poetry and Mufickl He fays, their Odes were
&&& as to the Jfy'thmus 7 in the fame Manner
8il
§ 6. of MUSIC K. tof
as we fcan them, every pes being a diitinct Bar
or Meafure, feparate by a diftinct Paufe ; but in
the bare Reading, that Diftinction was not ac-
curately obferved, the Verfe being read in a
more continuous Manner. Again he notices,
that after the Change of the ancient Pronunci-
ation, and the Corruption of their Language,
the Mufick decayed till it became a poor and in-
(ignificant Art. Their Odes had a regular Re-
turn of the fame Kind of Verfe ; and the fame
Quantity of Syllables in the fame Place of every
fimiar Verfe : But there's nothing, lays he, but
Confufion of Quantities in the modern Odes -, io
that to follow the natural Quantity of our SyU
lables, every Stanza will be a different Sorig, o-
therwife than in the ancient Verfes : ( He
fhouldhave minded, that every Kind of Ode was
not of this Nature; and how heroick Verfes
were fung, if this was neceffary, I cannot fee,
becaufe in them the BaBylus and Spondeus are
fometimes in one Place of the Verfe, and fome-
times in another. ) But inftead of this, he fays,
the Modems have no Regard to the natural
Quantity of the Syllables, and have introduced
an unnatural and barbarous Variety of long and
(hort Notes, which they apply without any Re-
gard to the Subject and Senfe of the Verfe, or
the natural . Pronunciation : So that nothing
can be underftood that's fung, unlefs one know*
it before j- and therefore, no wonder, lays he, that
our vocal Mufick has no Effects. Now here is in-
deed a heavy Charge, but Experience gives me
Authority to affirn* ix. to be a^fpiutely 'falfe. Wo
have
606 A Treatise Chap. XIV.
have vocal Misfick as pathetick as ever the an-
cient was. If any Singer don't pronounce in-
telligibly, that is not the Fault of the Mufick^
which is always fo contrived, as the Senfe of
the Words may be diftinclly perceived. But
this is impoffible, fays he, if we don't follow the
natural Pronunciation and Quantity, 4 which is
I think, precarioufly faid ; for was the Singing of
the ancient Odes by feparate and diftincl: Mea-
fures of metrical Feet, in which there muft fre-
quently be a Stop in the very Middle of a Word,
Was this I fay the natural Pronunciation, and
the Way to make what was fung beft under-
ftood ? Himfelf tells us, they read their Poems
otherwife. And if Practice would make that
diftihcl enough to them, will it not be as fufficient
in the other Cafe. Again, to argue from what's
{Irictly natural, will perhaps be no Advantage
to their Caufe ; for don't we know, that the
Ancients admitted the moft unnatural Pofitions of
. Words, for the fake of a numerous Stile, even
in p; ; ain Profe ; and took {till greater Liberties
in Poetry, to depart from the natural Order in
which Ideas Iy in our Mind ; far otherwife than
it is in the modern Languages, which will there-
fore be moe .eafily and readily underftood in
Singing, if pronounced diftinclly, than the anci-
ent Verfe could be, wherein the Conftruclion of
the Words was more difficult to find, becaufe of
tbeTranfpofitions. Again the Difference of long
and fhort Syllables in common Speaking,'is not ac-
curately obferved'j not even in the ancient Lan-
guages i for JSxampk) in common Speakings
who
§ 6. of MU SICK. 607
who can' diftinguifh the long and fhort Syllables
in thef e Words,y?//7j", nivis, mifit. The Senfe
of a Word generally depends upon the right Pro-
nunciation of one Syllable, or Two at moft in
very long Words ; and if thefe are made con-
fpicuous, and the Words well feparated by a
right Application of the long and fhort Notes,
as we certainly know to be done, then we fol-
low the natural Pronunciation more this Way
than the other. If 'tis replied, that fince we
pretend to a poetical Rythmus*, fuitable to dif-
ferent Subjects, why don't we follow it in our
Mufick ? I ftiall anfwer, that tho' that Ryth-
mics is more diftinguifhed in the Recitation of
Poems, yet our mufical Ryththus is accommo-
dated alfo to it j but with fuch Liberty as isne-
ceffary to make good Melody $ and even to
produce ftronger Effects than a fimple Reciting
can do ; and I would ask, for what other Rea-
fonthe Ancients fung their Poems in a Manner
different from the bare reading of them ? Still
he tells us, that we want the true Rythmus, which
can only make pathetick Miifick ; and if there
is any Thing moving in our Songs, he fays,
'tis only owing to the Words ; fo that Profe
maybe fung as well as Verfe: That the Words
ought naturally to have the greateft Influence,
has been already confidered; and I have feen
no Reafon why the ancient poetical Rythmic
fliould have the only Claim to be pathetick ;
as if they had exhaufted all the Combinations
of long and fhort Sounds, that can be moving
pr agreeable : But indeed the Queftion is a-
bout
<?o8 ^Treatise Chap. XlV.
about Matter of Fac\ therefore I fhall
appeal to Experience, and leave it; after I have
minded you, that by this Defence of the modem
.Muficky I don't -fay it is all alike good^
or that there can be no juft bbjedtion laid-a-
gainft any of our Compofltions, efpecially in the
fetting of Mufick - to Words ; I. only fay, we
have admirable Compofitions, and that the Art
of Mufick, taken in all that it is capable of, is
more perfect than it was among the old Greeks
and Romans^ at leaft for what can poflfibly be
made appear*
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