■ . ■ ' 
 
THE LIBRARY OF THE 
 
 UNIVERSITY OF 
 
 NORTH CAROLINA 
 
 V781.3 
 
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 Music Library 
 
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 be taken from the 
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Digitized by the Internet Archive 
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A N 
 
 S S A 
 
 TOWARDS A 
 
 RATIONAL SYSTEM of MUSIC 
 
 By JOHN HOLDEN. 
 
 Mufic 
 
 To which refpondent ihakes the varied foul. Thomson. 
 
 €ntereu in Stationery -$all* 
 
 GLASGOW: 
 Printed by Robert Urie for the Author. MDCCLXX, 
 

T G 
 
 His Grace WILLIAM, Duke of MONTROSE, etc 
 
 Chancellor of the University of Glasgow, 
 
 Sir ADAM FERGUSSON, of KILKERRAN, Baronet, 
 
 Lord Rector, 
 
 The Reverend Mr. JOHN CORSE, Dean of Faculties; 
 
 AND TO 
 
 Doctor WILLIAM LEECHMAJ, Principal, 
 
 And all the PROFESSORS of Glasgow College; 
 
 PATRONS of useful and polite LITERATURE; 
 
 This ESSAY is humbly infcribed, 
 
 by their moft obedient Servant, 
 
 JOHN HOLDEN. 
 
PREFACE. 
 
 r 1 y H E defign of the following Treatife is to explain, in a rational and familiar way, and to dif- 
 -* pofe, in a fyftematic order, thofe particulars with which every one ought to be acquainted, 
 who defires either to perform Mulic with propriety and fpirit, or to hear it with judgment and tafte ; 
 and therefore I have intitled it, An Essay towards a Rational System of Music. 
 
 Having hit upon feveral new obfervations, which, according to the opinions of the beft judges in 
 Mufic, whom I could confult, deferved to be communicated to the Public ; and confidering alfo, that 
 many valuable improvements, lately made in this fcience by foreign authors, and particularly by the 
 French, had not yet made their appearance, in an intelligible form, in our own language, I was ir- 
 refiftibly conftrained to attempt this talk, however unqualified for it. 
 
 The plan of this work was formed feveral years ago, and the fucceffive chapters printed off at 
 different times, as they could be got ready ; for I found a neceffity of giving fome months attention, 
 at all leifure hours, to every particular fubject, in order to bring them to that ftate, imperfect as it is, 
 in which they now appear. 
 
 I shall therefore make no other apology for the faults of the following Piece than this, which 
 feems to be the only proper one an author can make, / have done my befl : and as I have had occafion 
 to enter into fome enquiries where abfolute certainty is hardly to be expected, fo I am equally prepar- 
 ed to enjoy the approbation of the judicious, fo far as I am right; or to profit by their candid correcti- 
 ons, where I may be wrong. 
 
 *b 
 
C vi ] 
 
 O N 
 
 N 
 
 PART I. 
 
 The Rudiments of practical Mufic. 
 
 CHAP. I. 
 Of the natural Scale. 
 
 Preliminary explanations, Art. I to 6 
 Tones and femitones, where fituated, 7 to 14 
 The whole fcale confijls but of feven founds 
 
 repeated, I J to 27 
 
 A new rule for determining the places of 
 
 tones and femitones, 1 8 
 
 further remarks on the fcale, 19 
 
 Practice recommended with cautions, 20 
 Different effeils of notes, 21, 22 
 
 The fourth of the fcale variable, 23 
 
 andthefixth, 24 
 
 Thefe variations ufeful to be known, 25 
 
 and not imperceptible, 26 
 
 CHAP, II. 
 
 Application of the Scale. 
 
 Direclions to the finger, 2j, 28 
 
 A conjlani remembcrance of the key is both 
 
 neceffary and natural, 29 to 31 
 
 Of the ufe offingingfyllables, 32,33 
 
 Tunes ofajbarp fries to be examined, and 
 
 diflinguijhed from thofe of a flat feries, 
 
 34 to 36 
 
 Remarks on the foregoing chapter, 37 
 
 CHAP. III. 
 Of the modern Syftem of Mafic. 
 
 Of a femitonic fyflem denoted by letters and 
 
 other characters, 38 to 41 
 
 The twelve different fcales felecled, and 
 
 their founds denominated, 42 to 48 
 
 Remarks on the general fcheme, 49 to 51 
 The three vocal oclaves difpofed on eleven 
 
 lines, and their fpaces, 52,53 
 
 Of cliffs, and their ufe, 54, 55 
 
 Qf enlarging the compafs of mufic by leger 
 
 lines, 56 
 
 Offoarps or flats after the cliff, 57 to 59 
 A new rule for finding the place ofxs\\ and 
 
 of the key, 60 
 
 Of diftinguifnng the fliarp and flat feries 
 
 in written muff c, and when feveral parts 
 
 are fet together, 61 
 
 A practical direilionfor the ready finding 
 
 the degrees of any fcale, 62 
 
 CHAP. IV. 
 Of Time. 
 
 Of the meaning of notes, 63, 64 
 
 Kefs, 65 
 
 Augmentations, 66 
 
 Diminutions, 67 
 
 All regular mufic divided into meafures, 
 68, 69 
 Of bars, 70 
 
 Of the fubdivijions of the meafure, 7 1 
 
 Of comtnon time, 72 to 78 
 
 Of triple time, 79 to 84 
 
 Of mixed triple time, 85 to 87 
 
 Of cojnpound triple time, 88 to 91 
 
 The different values of rejls, 92 
 
 Ofmufic in f core, 93 
 
 Of accented and unaccented parts of the 
 
 meafure, 94 to 97 
 
 Of prctracled and fyncopated notes, 98, 99 
 
 and their correfpondent refts, 100 
 
 Poetry and mufic compared, 10 1 to 103 
 Hints relating to expreffton 104, 105 
 
 Of the abfolute time of mufic, 106 to 108 
 
 CHAP. V. 
 
 Mifcellaneous Explanations. 
 
 Of a double bar, 
 — a hold and clofe, 
 
 109 
 1 10 
 
THE CONTENTS. 
 
 Of a repeat, 
 
 Article 1 1 1 
 
 — Da capo and dal fegno, 112 
 
 — a diretl and volti, 1 1 3 
 
 — a tye andjlaccato 1 1 4 
 
 — Forte and Piano, 1 1 5 
 
 — a trill orjhake, 1 16 
 
 — Appoggiatura, 1 1 7 
 
 — Recitative and air, 1 18 
 
 — Symphony and fong. Prelude and in- 
 
 terlude, 119 
 
 Titles of different forts of airs, 120 
 Italian ntsords ufed to denote the time and 
 
 manner of performance, 121 
 
 Comparative degrees of 'movement ', 123 
 
 Titles of entire pieces, 1 2 4 
 
 CHAP. VI. 
 Of Harnaonical Confonances. 
 
 Definition of a chord, 1 25 
 
 Of otlaves, Jingle, double, and triple, 126 
 Of the p erf eel chord, 1 2 7 
 
 Names of compounded intervals, 128/0130 
 Of the full perfeel chord, 13 1, 133 
 
 Of diflocations and omiffions, 133 to 137 
 Inverftons, 138 
 
 Of the fiat perfeel chord, 139,140 
 
 Of the tiuo forts of perfeel chords, 141. 142 
 D'iflinclion of chords into final and medi- 
 al, 143, 144 
 Examples, 145, 146 
 Of all the perfeel chords in the fcale, 147, 
 Of the choice of chords, 149, 150 
 Of the fundamental and figured thorough- 
 bafs, 151, 152 
 Anew definition of concord and difcord, 1 5 3 
 Enumeration of concords, 154 
 Of the fundamentals of concords, 155 
 
 CHAP. VII. 
 Of Diffonances. 
 
 General remarks, 156 /o 158 
 
 Of diffonances by addition, 1 59 /a 170 
 
 Of omiffions in thefe chords, 171, 172 
 
 Of diffonances by fufpenfion, 173 /a 175 
 0/~ /^ fundamentals of diffonant chords, 
 176 /o 185 
 Of extraneous fundamentals introduced by 
 fufpenfion, 106, 187 
 
 Recapitulation, 188 
 
 CHAP. VIII. 
 Of Fundamental Progreffions. 
 
 Of cadences, 189, 190 
 
 Of the perfeel regular cadence, 191 /o 196 
 V P a Jf a g es derived from the regular ca- 
 dence, 197 /o 199 
 Of the perfeel cadence on the fourth, fifth, 
 and fxth of the fcale, and of the occa- 
 Jionally altered notes in each y 200 to 205 
 Of irregular cadences, 206/0 208 
 Of the falfe cadence, 209 
 Of the gradation on the fifth, 2 IO to 2 1 2 
 Enu?neration of fundamental paffages hi- 
 therto treated of, 213 
 Of hartnonic phrafes, 214* 215- 
 Examples, 216/02 18 
 Reafons for various omiffions, 219 
 Remarkable circumftances accounted for 
 on the foregoing principles, 220 /o 225 
 £///$<? /(?*/> o/rf third or fixth, and the ft ep 
 of a fecond, 226/0232 
 Of modulation, and the fucceffion of occafi- 
 onal new keys, 233/0243 
 
 Vll 
 
 P radical rules, 244,245 
 
 Of imitations, 246 
 
 Examples, 247,248 
 
 0/ /^ fundamental fequence offevenths, 
 and the preparation ofdifcords, 249/0254 
 Of fundamental licences, 255,256 
 
 CHAP. IX. 
 
 Of the Flat Series. 
 
 Peculiarities of the flat fries, 257, 258 
 
 Of finding the key by the marks at the cliff, 
 and contrarinuife, 259,260 
 
 Various methods of fitting mujic of a flat 
 fries, 262 
 
 Praclice recommended, <with cautions, 263, 
 264 
 
 Of the fundamental progreffions, and the 
 peculiarities of flat harmony, 265 to 272 
 
 Annotations on thefirjl leffons ofPafquali's 
 thorough-bafs, as examples of funda- 
 mental progreffions, 273/0276 
 
 CHAP. X. 
 Of Chromatic. 
 
 General idea, 277,278 
 
 Of afcending chromatic, 279 to 282 
 
 Defending chromatic, 283 to 285 
 
 Peculiarities of chromatic, 286 /o 288 
 
 CHAP. XI. 
 Of Plain Defcant. 
 
 Definitions and preparatory obfervaiions, 
 
 289/0293 
 
 Rules for the thorough-bafs 'with regard to 
 
 the paffages of chords, 294 
 
THE CONTENTS. 
 
 Licences with regard to the fundamen- 
 tal, .viz. Holding, 295 
 Svppoftion, 296 
 Substitution, 297 
 Of chords, including extreme farp or flat 
 intervals, 298 
 Of chromatic in the. lafs, 299 
 Direction to the learner, 300 
 Rules for the thorough-bafs with regard to 
 the treble, 301 
 Rules for the inner parts and examples, 
 302 to 304 
 Of pajfages derived from the fundamental 
 fequence of fevenths, 305 
 
 CHAP. XII. 
 Of Figurative Melody. 
 
 Divifon of the fubjecl, 306 
 
 Of fufpenfions and their inverfions, 307 to 
 
 Offubftitutions, and their peculiarities ,3 1 4 
 Of divifions in an upper part by confonant 
 intervals, 315, 316 
 
 Of the fame kind of divifon in thebafs, 317 
 Of divifions by paffing notes, ' , 318 
 
 — Liberties in compliance with tafle, 319 
 
 — Imitation, 320 
 
 — Greater on the K fundamental than on 
 
 any other, 32 1 
 
 Of fever al parts fet together, 322 
 
 Of the infertion of reft s, 323 
 
 Rules for diflinguifbing ejfential from paf- 
 Jing notes, 325 
 
 Of the exprejjion of figurative melody, 326 
 Concluding obfervations, 327 to 329 
 
 PART II. 
 The Theory of Mufic. 
 
 chap. 1. 
 
 Of Single Mufical Sounds. 
 
 Method of proceeding laid down, 1,2 
 
 Of the pulfes of mufical founds, 3 *<? 9 
 
 A new principle propofed, illujlrated, and 
 
 defended, 10 to 18 
 
 CHAP. H. 
 Of mufical Sounds in Succeffion. 
 
 Of the fimilarity of oilaves, 19 
 
 The rememberance of the key, 20 
 
 Of the module and its fubdimfions, 21 to 23 
 Of 'themannerof parcelling the key, 24/0 26 
 
 and the fifth of the fcale, 2 7 
 
 The whole natural defending fcale afcer- 
 
 tained and examined, 28 to 38 
 
 Of fundamental cadences, 39, 40 
 
 The variation of a comma in fome founds, 
 41, 42 
 The introduclion of 'the per -feci fourth, 43—45 
 Objections anfwered, 46, 47 
 
 CHAP. III. 
 Of Harmonical Arithmetic. 
 
 Mufical intervals and ratios compared, 
 48 to 50 
 
 Reduction of their terms, 51 
 
 Addition of intervals, 52 to 54 
 
 Subtratlion, 55 to 57 
 
 Multiplication, 58 
 
 Divifon, 5 9 
 
 P radical rule for comparing the magni- 
 tudes of intervals, 60 
 
 Procefs for finding fmall numbers which 
 fall nearly exprefs a ratio of incofti- 
 menfurables, 61 
 
 The fy ft em of modulation laid down and, 
 examined by the preceding rules, 62 to 1 ] 3 
 
 CHAP. IV. 
 Of Combined Sounds. 
 
 Of two founds generating or implying a 
 third found, 74, 75 
 
 The full perfecl chord examined in this 
 view, 76/079 
 
 Of the different arrangements, inverfions, 
 and parts of the perfecl chord, 80 to 82 
 
 The fy ft em of modulation farther explain- 
 ed, 83, 84 
 
 Of leading chords, 8 J to 87 
 
 Of the flat perfect chords 88 
 
 Of other chords improperly called perfect, 
 89, 90 
 
 Concluding remarks, 91 
 
 Of imperfections in the femitonic fyftem, 
 92 /094 
 
 Experiments propofed and related, 95 /e 98 
 
 * # * A juft fenfe of the refpeft due to 
 the Public, conftrained the Author to o- 
 verfee and correct the printing of this trea- 
 tife with more than ordinary care, fo that 
 there is not one typographical error which 
 can obfeure the fenfe, except the follow- 
 ing, which the reader is requefted to cor- 
 rect, viz. 
 Art. 207, 1. 1. for regular read irregular. 
 
PART I. 
 
 The RUDIMENTS of Practical Music 
 
 Chap. I. Of the natural Scale of M \j s i c. 
 
 ... A SUCCESSION of feveral founds, properly 
 
 n ice . 2'^ qualified to excite agreeable fenfations in the 
 hearer, is called Mufic; and to underffand mufic is to be able 
 to define what are the proper qualities of founds for this pur- 
 pofe; and alfo to judge, when founds are heard, whether they 
 have, or have not, thefe neceffary qualifications. 
 
 2. The mod: obvious difiincYions, among the founds of mu- 
 fic, are thofe of the tune or pitch of different founds, and of 
 their time or duration: and thefe two articles of tune and time, 
 as they are the mod: remarkable, fo they are alfo the moft im- 
 portant, and indeed the only things which a learner mufl endea- 
 vour to underfland. —We fhall firft attempt' to explain what 
 a beginner ought to obferve concerning the article of tune. 
 
 3. It is cuftomary, in comparing together two founds of dif- 
 ferent degrees of tune, to call the thriller found, acuter or 
 higher, and the other, graver or lower : thus a boy's voice is a- 
 cuter or higher than a man's voice, and a bafs viol is graver or 
 lower than a violin ; and when a voice or infirument moves from 
 a graver to an acuter found, we fay it afcends •, and contrari- 
 wife, the motion from an acuter to a graver found, is called, 
 defcending. * 
 
 4. In this comparifon of the tune of founds, we do not con- 
 fider their different ftrength or weaknefs; in regard to which 
 they are called loud or firong, and foft or weak ; only the par- 
 ticular tone or pitch of each, fo that the terms high and low 
 relate to the pitch of founds, but loud and foft to their ftrength. 
 
 5. In order to reprefent to the eye the different tune or pitch 
 of founds, five parallel lines are drawn, and are called a ftaffof 
 lines ; and the degrees of tune are fhewn by the fituations of cer- 
 tain characters upon thefe lines, or in the fpaces between them. 
 The character, which reprefents a higher found, has a higher 
 place, and the lower found, a lower place in the fiaff* of lines. 
 
 6. There are certain fleps or degrees of tune which conftitute 
 what is called the natural fcale of mufic : and it is very remark- 
 able, that among all the infinite variety which mufic affords, the 
 fame natural fcale is always retained, and the fame mutual rela- 
 tions, in regard to the pitch, exift among the feveral founds of 
 every tune, how various foever they may be in other refpecls. 
 
 The fucceffive founds of a tune fometimes afcend or defcend 
 by the fingle degrees or ffeps of the fcale, and fometimes by 
 larger intervals or leaps, but always fo as to pitch on fome one 
 or other of the degrees of this natural fcale. 
 
 v, when applied to the degrees of tune, has no foundation 
 
 * This remark appeared necefTary, becaufe, in truth, the propriety of the terms high and „.., 
 in the nature of found, and depends entirely upon cuflom ; and among the ancient Greeks, the graveft tones were railed high, and the acuteft low : pro- 
 bably becaufe the longeft and graved firings of their lyres had the uppermoft places. 
 
Of the Natural Scale. 
 
 Part I. 
 
 Hence we may partly conceive the great importance of a 
 thorough knowlege of this natural fca'.e ; it is indeed to mufic, 
 what the alphabet is to fpeech : for, as all the various articulati- 
 ons of fpeech are denoted by a few letters, properly ranged ; fo 
 all the different paiTages of mufic are only different arrangements 
 of a few primitive founds of certain proper degrees of pitch. 
 
 7. Thefe fteps of the natural fcale are not all of equal fize, 
 but are fome of them nearly * the doable fize of others; and 
 therefore the larger fteps are called, tones; and the fmaller, 
 femitones. 
 
 Now, the flepof a tone, not being any way vifibly diftinguifh- 
 ed from that of a femitone, in the fituation of the characters 
 of our written mufic; becaufe, here each ftep, whether greater 
 or lefs, is denoted by an equal riling or falling of the character, 
 viz. from a line to the adjacent fpace, or from a fpacc to the 
 adjacent line : we have, therefore, the more occafion to fix 
 well in our memory which of thefe, feemingly equal fteps, are 
 to be efteemed tones, and which femitones ; that we may readi- 
 ly know what degree of found is really .meant by each line and 
 fpace. 
 
 8. To determine the places of thefe tones and femitones, in 
 the natural fcale, we obferve, that in tinging the eight fuccef- 
 five notes, commonly called the afcending Gamut, as reprefent- 
 ed, PL I. Exam. I. the voice, as to the pitch of the firft note, 
 is quite undetermined, and ready to fall in with any mufical 
 found, which may be propofed by any other voice or inftru- 
 ment, provided it be not above or below its compafs ; but after 
 the tone or pitch of the firft note is determined, it fo influences 
 and governs all the following notes, that each of thefe muft 
 bear its proper relation to the firft, or elfe the fucceffion will be 
 harfh and (hocking inftead of mufica!. 
 
 This firft note of the Gamut, which thus governs all the reft, 
 is called the Key note ; and the following notes of the afcend- 
 ing fcale, are called, fecond, third, fourth, fifth, fixth, fe- 
 venth, and fo on, as fhewn by the letter K, and the figures 
 fubjoined to each, in Exam. 1. f 
 
 9. Now, let the firft four notes be fung over, applying the 
 fame fyllable, fuppofe la, to each note ; a very little attention 
 will ferve to difcover, that the voice, in pafTing from the third 
 to the fourth note, does not take fo large a ftep, as it does from 
 the key to the fecond, or from the fecond to the third : there is 
 fomething remarkably grave in the fourth note, which cannot 
 efcape the notice of the mod unfkilful hearer, provided he 
 hath a genius for mufic, and be previoufly warned of fuch a 
 property; although, without due attention to this and other 
 fuch like peculiarities, a perfon may fingall his life, and yet ne- 
 ver conceive the leaft idea of the nature of the fcale ; much in 
 fame manner, as a perfon may ufe his faculty of fpeech, with- 
 out any knowlege of reading and writing. 
 
 The fteps from the key to the fecond, and from the fecond 
 to the third, are tones ; and that from the third to the fourth, 
 is a femitone. 
 
 1 o. Let the whole eight notes of the afcending Gamut, Ex. I. 
 be now fung, and it will be eafy to obferve, that the voice 
 takes a pretty large ftep from the fourth to the fifth ; and that 
 the fixth, feventh, and eighth notes follow the fifth in the fame 
 manner, and with the fame air, in the latter half of the Gamut, 
 as the fecond, third, and fourth, followed the key in the form- 
 er half: fo that the Gamut naturally divides into two equal 
 and fimilar half-gamuts: 
 
 The firft containing, the key, fecond, third and fourth ; 
 
 The latter containing, the fifth, fixth, feventh and eighth : 
 
 * We fay nearly, becaufe, on fearching further into the nature of mufic, it is found, that the femitone is not the exact half of a tone; but in common 
 practice, it is fuppofed to be fo, without caufing any fenfible error. 
 
 | In this Example, there are certain characters at the beginning of the ftafFof lines, which the reader need not take any notice of at prefent; their mean- 
 ing and ufe will be explained in a proper place. 
 
Chap. I. 
 
 And as the femitone lay betwixt th 
 
 firft half, (Art. 9.) fo the femitone lies betwixt the feventh and 
 
 eighth, in the latter half; and ail the other degrees are tones. 
 
 The ftep of a femitone is diftinguifhed in Exam. I. by a 
 fmall s over the notes. 
 
 11. As the fituation of thefe tones and femitones cannot be 
 
 Tones and Semitones, -where fitualed. g 
 
 third and fourth, in the fcale, as (hewn, Exam. V. where it may be obferved that mi, 
 which is appropriated to the feventh note, is the only fyllable 
 which occurs but once ; fa anfwers to the key, the fourth and 
 eighth; fol is the fecond and fifth, and la the third and fixth; 
 which is a great inconvenience, becaufe of the double meaning 
 of our fyllables ; however, it is of life to remember this difpofi- 
 
 too ftrongly imprinted on the memory, we fhall here defcribe tion of the fyllables, for underfianding the authors who have 
 feveral different ways of conceiving the fteps of the fcale, fome wrote on the fubject of mufic in our own language. 
 
 of which may probably fuit one leader, and others another. 
 
 The fteps of the afcending Gamut are firft two fucceffive 
 tones, then a femitone ; then three fucceffive tones, and then 
 another femitone. Two fucceffive tones are called a ditone; 
 and three fucceffive tones, a tritone: fo that the afcending fcale 
 is firft a ditone, and a femitone ; then a tritone, and a femitone. 
 See Exam. II. 
 
 To affift the memory in this matter, our authors obferve, 
 that in afcending above mi, we have, fa, fol, la, twice over ; and 
 in defcending below mi, we have, la, fol, fa, twice repeated, and 
 then mi comes again. See Exam. VI. — and that the femitones 
 always lie below the fyllable fa. 
 
 r 5. Thefe methods of naming the degrees of the fcale, by 
 finging fyllables, lead us to obferve another property of founds, 
 
 Each half of the Gamut, as defcribed, Art, 10. is called viz. that as the key, and the eighth, are both called by one name, 
 
 a tetrachord ; and in this way, the Gamut is compofed of two 
 disjoined tetrachords : the ftep from the fourth to the fifth, be- 
 ing from the uppermoft note of one tetrachord, to the loweft 
 of the other, .is called, the diazcuclic tone, or tone of disjuncti- 
 on. See Exam. IIL 
 
 13. The Italians and French apply the finging fyllables, ut, 
 re, mi, fa, fol, la,fi, ut *, to the eight notes of the Gamut, (fee 
 Exam. IV.) with great propriety ; for, by this means, the fteps 
 from the third to the fourth, and from. the feventh to the eighth, 
 being from mi x.ofa, and from fi tq ut, are naturally fung fe- 
 mitones; becaufe the fyllables ml and fi, partly oblige the fing- 
 er to keep their founds more acute, while fa is a more open 
 fyllable, and requires a graver pronunciation, fuitable to the 
 found of the fourth note, Art. 9. 
 
 14. Our Britifh rnuficians me no more than four of thefe fyl- 
 lables, viz. fa, fol, la, mi, which they apply to the notes of the 
 
 fo they are in reality, both of the fame nature, and produce the 
 fame effect, on the hearer; fo that, when they are heard together, 
 they feem to be both one and the fame found. For inftance, 
 fuppofea man and a boy were to begin to fing the Gamut toge- 
 ther, the boy would naturally give the fame found to the firft 
 note, which the man arrives.at in the eighth or laft note; fo that 
 the boy's loweft note would be the fame with the man's high- 
 eft; and yet .the two would agree together fo perfectly, that a 
 hearer would fcarce be able todiftiuguifh any difference ; where- 
 as, if the boy were to begin his Gamut with any other found 
 but this eighth, thedifagreement would be very perceivable, and 
 in many cafes very offenfive. For another inftance, fuppofeone 
 man to fing over the afcending Gamut, and the very inftant, 
 when he arrives at the eighth, or uppermoft found, another man 
 begins at the key, or loweft found ; the firft and laft founds 
 of the Gamut thus heard together, will feem but as one. 
 
 * Inftead of the f) liable ut, they fubftitute do in practice, being more eafy to pronounce: but as all the theorifts retain the fyllable 
 kept it here ; as one chief uie of this article is to enable the reader to underfland what the foreign authors mean by thefe fvllables.. 
 
4 Of the Natural Scale. Part I. 
 
 This interval of found is called an octave ; becaufe the upper is it : and thus, how far foever we extend the fcale of mufic, both 
 upon the eighth degree of the fcale, when the lower is on the upward and downward, it will ftill be but a number of fingle 
 
 firft, or key. 
 
 16. Now, when a man and a boy proceed to fi rig the whole eight 
 notes of the afcending Gamut, the boy's voice begins, as before 
 mentioned, an oftave above the man's voice ; and afcending all a- 
 long by the fame fteps, keeps always the fame interval of di- 
 ftance between them, and comes at laft to a found, as much a- 
 bove the man's laft note, as this laft is above the man's firft. 
 The boy's voice is ftill an octave above the man's, and they 
 perfectly agree together throughout the whole fcale. 
 
 We infer from hence, that if a man's voice afcend above the 
 8th, to a yth, 10th, 1 1 th and i 2th found, etc. he muft juft 
 make life of the fame fteps of the fcale over again, becaufe the 
 8th found is in all refpe&s like the former key; to which the 
 9th is a fecond, the i oth a third, the 1 1 th a fourth, the I 2th a 
 fifth, etc. that is, when we afcend above the eighth, we come into 
 the boy's Gamut, which is, in all refpefts, like the man's, but 
 an oclave higher. Thus alfo, if the boy were to defcend below 
 his firft, or key note, he would come into the man's Gamut, 
 and muft therefore, next below the key, take the 7th, then the 
 6th, then the 5th, etc. each an oftave lower. 
 
 1 7. This being the cafe, we have not, in the whole compafs 
 of mufic, any founds effentially different from thofe which exift 
 within the compafs of one oftave, which, as we have already 
 feehy are naturally no more than feven : and the learner's bu- 
 finefs, in regard to the tune of founds, is brought within a 
 much narrower compafs than he probably may have expected. 
 Let him obferve and remember, that whatever found be confti- 
 
 Gamuts, each confifting of eight founds, at the very diftances a- 
 bove defcribed ; and the whole connected together, by making 
 the eighth found of each Gamut to coincide with the firft of that 
 which is above it. 
 
 If a long train of notes be placed upon lines and fpaces, as in 
 Exam. VI. the learner will, as yet, be at a lofs to know which 
 of them is to be efteemed the key, or firft note of his Gamut ; 
 as this depends upon the characters, and marks prefixed to each 
 example ; and their meanings are not 10 be underftood, till fome 
 other particulars have been explained : but, fuppefing the place 
 of the key note to be known ; which is, in the former examples, 
 upon the fecond line, and here on the third fpace * ; the ap- 
 plication of the K, and the numbers 2, 3, 4, etc. or, of the 
 Britilh finging fyllables, fa, fol, la, etc. or of the Italian fylla- 
 bles, ut, re, mi, fa, etc. will be very eafy, if what is above faid 
 be w r ell underftood. 
 
 This example is extended both upward and downward, be- 
 yond the compafs of the ufual ftaff of five lines, upon fmall ad- 
 ditional lines, which are occafionally drawn for this purpofe, 
 and are called, leger lines. It will eafily be feen, that inftead 
 of the number eight, as in the former examples, here comes a 
 repetition of the key, agreeable to whatisobierved above. The 
 Britiili fa, and the Italian tit, always correfpond with the key, 
 and the other fyllables follow in order; and the places of the fe- 
 mitones, marked above with s, are according to the preceding 
 rules. 
 
 18. Having thus eftablifhed the connexion of feveral Gamuts, 
 
 tuted a key, it is always to be confidered as the lowed note of in order to extend the fcale of mufic at pleafure, we may here 
 one Gamut, and the higheft note of another ; it is fomething add another rule for finding the places of the femitones, when 
 like a knot which connects the founds abo\x with thofe below the place of the key is known, which feems to be more fimple, 
 
 * In counting the lines and fpaces of a flafF for mufic, we always begin at the bottom line, and call it the firft; and the fpace next above it, is called the 
 firft fpace: and thus, the uppcrmoft line is the fifth. This cuftom, fo contrary to that of counting lines, upon other occafions, has probably been introdu- 
 ced in conformity to the manner of counting the degrees of the fcale, which, as we have already feen, are always numbered upwards. 
 
Chap. I. 
 
 Various Ways of conceiving 
 
 and more eafily remembered than any of the foregoing, viz, pb- 
 ferve the place of the key and of the third ; The femitones are ah 
 luays the next below the key, and the next above the third ; and 
 all the other degrees are tones. 
 
 19. It will be ufeful to obferve here, that as the Gamut Was; 
 called two disjunct tetrachords, Art. 12.; fo the upper tetrachord 
 of each Gamut is connected with the lower tetrachord of that 
 which lies above, by means of the key-note which is common 
 to both; and thus the two half Gamuts, which lie on each fide 
 of the key, are conjunct tetrachords.; and the fcale of mufic, ex- 
 tended at pleafure, may be conceived as confifting of tetrachords 
 disjunct and conjunct, alternately; the conjunctions being at 
 every key note, and the disjunctions between every fourth and 
 fifth. 
 
 For the better fixing of thefe ideas, it may alfo be remarked, 
 that the disjunctive or diazeuElic tone is always the loweft ftep 
 of the tritone, Art. 1 1. and that in the Italian fyllables it is al- 
 ways from fa lofol, Art. 13.- The ftep horn fa tofol in our 
 Britifh fyllables, Art. 14. is ambiguous *'. 
 
 We have defignedly dwelt the longer on this explanation of 
 the fcale of mafic, both becaufe it is the foundation on which 
 
 the Scale. Practice necefjary. e 
 
 all our knowlege of the proper tune of founds mult be built; 
 and alfo, becaufe few or none of the authors who have conde- 
 scended to write introductory treatifes on mufic, feem to have 
 fufficiently infifted on this matter, 
 
 . 20. Befides the reading and retaining what is pafi, we have 
 now another talk to impofe on the learner: it is not enough to 
 know fpecul^tively what name belongs to each note,, and where 
 the tones and femitones lie; but he mult alfo know practically 
 when he hears the founds themfelves, whether they be juftly 
 performed or not. Speculations here are of no ufe, only as far 
 as they can be put in execution.- , 
 
 For this purpofe, he mult pracYife. the afcending and defcend- 
 ingGamut, as represented, PI. I. Exam. VII. till the proper found 
 of each note become quite familiar; and, if a mere beginner, he- 
 ought not to trtiit altogether to his own ear and judgment, but 
 feek all opportunities of the accompanyment of fuch as are more 
 expert; for it is a well known fact, that falfe and ill-tuned 
 notes, by frequent ufe, will become tolerable; thus, by frequent- 
 ly hearing an .ill-tuned ring of bells, or a falfe flopped violin,, 
 or fome other mufical inftrument iiijudicioufly managed, ma- 
 ny, pofiefled of good natural talents for mufic,. have acquir- 
 
 * There is no doubt, but when our fyllables were firft reduced to four, it was done with an eye to the confutation of the fcale by tetrachords : though 
 atccording to the above-defcribed pofitiori of the tetrachords, we find no manner of correfpondence between them and our fyllables. 
 
 In defence of the arrangement of our fyllables, it muft be obferved, that the tetrachord of the Greeks, from whom we borrow the name, was a different 
 fyftem of founds froTn what is above defci'ibed : theks was firft a femitone, then two tones; whereas ours is firft two tones, and then a femitone : fo' that 
 in applying the tetrachords of the Greeks to our fcale, we muft begin on our third and feventh notes ; and fuppofe the conjunctions to be at every third, and- 
 the disjunctions between every fixth and feventh; as will eafily appear by infpecting the fcale, Exam. VI. 
 
 In this view we'fhall find the tetrachord to be ini,'fa,j'alja, conftantly; only at the conjunctions, when the »ii of one tetrachord coincides with the la of 
 another; the mi is fupprefled, and la is kept in its place : and thus the diazeudtic tone is from la to rhi. ■ 
 
 Had not the nature, or perhaps more properly, the fpecies of our modern mufic, compelled us to acknowlege the other divifion of the tetrachords much 
 more obvious; we fliould gladly have retained that which at once both correfponds, with the fyftcms of the ancients, and with cur prefent iinging fyllables. 
 
 How far the fpecies of the ancient mufic might tend to familiarize certain pafi'ages, which feem vaftly unnatural to us at prefenf, is liard to fay. This we 
 know, that we cannot, without confiderable precaution, begin to fing with any note, except the key ; neither can we finifh with any other: whereas the 
 Greeks feem to have begun, where their tetrachord begins, viz. on the feventh. 
 
 If it be. altogether owing to cuftom, that we adopt a different fcale from that of the ancients ; that cuftom is, to us, as inviolable as natural inflinct; and 
 therefore ought to be carefully attended to. 
 
 The greateft of modern theorifts have found the fame necefhty of relinquifhing the ancient tetrachord. See M. D'Alemoert's El'emens ds Mufiqtte, Part I. 
 Chap, v, and vi. 
 
Of the Natural Scale. 
 
 ed bad habits of Tinging falfe, which have not been caft off with- 
 out the greateft difficulty. 
 
 The moft proper time to guard againft fuch unnatural habits 
 is at our firft beginning ; and one may almoft venture to aflert, 
 that he who habitually fings the Gamut perfectly true, will ne- 
 ver after fing any thing remarkably falfe. 
 
 A well tuned voice is, doubtlefs, the beft accompanyment for 
 a learner of vocal mufic, where fuch can be procured; and it 
 can feldom happen, that any who would learn to fing, can be 
 at a lofs for fuch affiftance: yet the learner need not confine 
 himfelf in this refpect, but may "fing along with any inftru- 
 ment without fcruple, provided it be in a judicious hand. 
 
 2i. Another caution, in regard to this practice, may be no 
 lefs neceffary ; that the learner do not miftake the main defign 
 for which it is recommended at this time. It is not fo much 
 for the fake of acquiring a habit of performing the notes juftly ; 
 but chiefly, that he may learn, as it were, to feel the difference 
 between the ftep of a tone and that of a femitone; and to per- 
 ceive the different natures and effects of the different notes of the 
 fcale, according to their feveral relations, as 2d, 3d, or 4th, etc. 
 to the key. He muft not only learn to fing or play the Gamut, 
 but he muft learn to know, in fome meafure, how he does it, 
 and what it is when done. 
 
 There is fomething peculiar in every note, which well deferves 
 our regard. The key note is remarkably bold and command- 
 ing ; the third and feventh have fomethingfnpplicativein them : 
 and it is on one of thefe tones which the beggar chiefly dwells, 
 if he ufe any tone at all * : the fixth is a kind of plamuve found ; 
 
 Part. I. 
 
 the fourth, as obferved before, is grave and folemn; the fifth 
 partakes of the nature of the key, and the fecond is not unlike 
 to the fixth. It muft be obferved, that thefe properties are not 
 infeparable from the founds ; but that the fame found will have 
 different effects, according to the different manners in which it 
 is introduced : this will be better underftood afterwards: we 
 fpeak here only of the moft ufual properties of thefe founds, 
 which depend on the perception of their feveral relations to the 
 key note +. 
 
 22. It is obfervable, that the fame peculiar property of found 
 always correfponds with the fame fituation in every tetrachord 
 of the fcale, as defcribed, Art. 1 2, and 19. The lowed note 
 of every tetrachord is bold and commanding; the next above 
 is plaintive ; the third of each tetrachord is fupplicative ; 
 and the fourth, or highefl note, is grave, except at the con- 
 junctions, when it coincides with the loweft of the next tetra- 
 chord : in which cafes, the found is rather conceived as the 
 loweft of the tetrachord above, than as the higheft of that below. 
 
 From this fimilarity of effect of the notes in every tetrachord, 
 arifes that fimilarity of air in each half-gamut, defcribed, Art. 
 10.; which is thelolecaufe of our adopting this divifion of the 
 tetrachords, preferable to any other. See the note to Art. 19. 
 
 23. There is yet fomething further to be obferved in regard 
 to the fourth note, which is not quite fo grave a tone in the 
 afcending, as in the defcending Gamut : this, the more expert 
 mufician will readily perceive : and he who cannot at prefent, for 
 want of practice, enter into the examination of fuch minute 
 differences, will yet arrive at the knowlege of them much foon- 
 
 * The difference between a mufical tone and natural fpeech, is very obvious. The tone is a found continuing for fome time on the fame degree of 
 pitch; while the founds of natural fpeech are perpetually varying; and, as it were, fliding either upwards or downwards during the pronunciation of 
 every fyllable. 
 
 f It is plain, that if we were only to confider mufical founds fingly, without any regard to their relations to, or dependencies upon each other; no fuch 
 properties as thefe could be attributed to any one found more than another : we might indeed afcribe different properties to the founds of different inftru- 
 ments; thus, the trumpet is bold, the violin is chearful, the baffoon is folemn, etc. but then, in this view, all the founds of the fame inftrument are of 
 one fort. 
 
Chap. I. Cautions in regard to PracJice. Different Effects of the Notes. The fourth variable, and the fixth. 7 
 
 er, by having them previously pointed out to him, .at his firft 
 fetting forward. 
 
 It is for this reafon we diftinguifh the defcending fourth, by 
 the name of a grave fourth, and mark it with a grave ac- 
 cent (\) above the note in Exam. VII. j 1 , 
 
 The comparifon of the two fourths, as they occur in Ex. VII. 
 is pretty difficult, becaufe of the intervening notes : but if 
 the firft half of the afcending Gamut, and the latter half of the 
 defcending Gamut be joined together, and fung, as reprefented, 
 Exam. VIII.; the difference of the two fourths, when thus 
 brought together, will be more perceivable; and the hearer will 
 eafily obferve, that the fucceffion is more mufical, when the latter 
 fourth is made diftinguifhably graver than the former. 
 
 The difference here fpoke of will be beft' perceived, when the 
 fucceffion of founds is neither very quick nor very flow ; and for a 
 direction in this refpedt, let the names K. 2. 3. 4. 4. 3.^2. K. 
 be applied to the notes, as fubjoined in Exam. VIII.,; giving 
 each note as much time as is neceffary for a diftinct,. but not 
 rapid, pronunciation of the word ; and making each of them as 
 nearly of an equal length as poffible. 
 
 Thus we have in reality two forts of fourths ; although this 
 diftin&ion has, I believe, efcaped the notice of all former writ- 
 ers on mufic: it is neverthelefs a faft confirmed by the experi- 
 ence of the beft performers ; and a circumftance not' only of 
 great importance in itfelf, but which alfo leads to the knowlege 
 of feveral other ufeful, and hitherto intricate particulars. 
 
 24. All the other notes of the fcale are the very fame both in 
 afcending and defcending; yet wc have a reafon for diftinguifh- 
 ing the fixths of the Gamut by the name of acute fixths ; as 
 fhewn by the acute accent (/) placed over them, in Ex. VII. 
 becaufe, in other kinds of fucceffions, there frequently occurs a- 
 nother fort of fixth, which is diftinguifhably graver than thefe ; 
 and yet this other fixth cannot fo properly be called grave, as 
 will be fhewn hereafter. 
 
 This difference between the fixths in different paffages, al- 
 though in reality, not fo perceivable as that of the fourths a- 
 bove-mentioned, is obferved by M. D'Alembert, in his Elemens 
 de Mufiqiie, Art. 63. and note (0), and by fome other modern 
 authors of the firft rank; although neither the epithets acute 
 and grave, nor any others equivalent, have been currently made 
 ufe of for diftinction, nor any new character admitted into our 
 written mufic, for that purpofe. 
 
 We have made choice of thefe accents to fupply this defect; 
 bothbecaufe their meanings are moft univerfally known; and 
 alfo, -becaufe they can be very eafily added with a pen, in any 
 pieces of mufic already printed ; and as eafily fupplied in cut 
 plates of mufic, by all fuch as judge them ufeful. 
 
 It is unqueftionably an advantage, to have every difference 
 which really exifts among the founds of a tune, fomeway vifibly 
 reprefented in our written mufic; and without fome fuch di- 
 ftinguifhing marks as thefe, the proper founds of the fourth 
 and fixth notes would often be dubious, and always depend 
 on the tafte of the performer; whereas thefe marks take away 
 all ambiguity, and bring the variable founds of the fourth and 
 fixth notes under as certain rules, as the other invariable notes 
 of the fcale. 
 
 25. .All the foregoing remarks and directions concern every 
 inflrumental performer juft as much as the vocal ; although 
 fome of them may feem particularly addreffed to the latter only. 
 It is abfolutely neceffary, that every mufician be well verfed in the 
 nature of the fcale, whether the founds he ufes be under the im- 
 mediate direction of his ear and judgment, as with the voice and 
 violin; or whether the founds be partly fixed, and yet capable 
 of fmall occafional variations, as on the German flute; or whe- 
 ther they be, for the time, quite fixed and immutable, as on all 
 key'd inftruments. 
 
 A great part of our enjoyment of mufic is owing to the per- 
 ception of certain due relations among the feveral fucceffive 
 
Of the Natural Scale. 
 
 8 
 
 founds, when compared with one determinate found as a key. 
 Now, a performer, above all others, mould be a complete judge 
 of the juftnefs of thefe relations, in order to relifh, and enter 
 into the true/pint of his piece ; without which, he ftands but 
 little chance of executing with proper tafte. 
 
 The performers on key'd inftruments feem to have the leaft 
 occafion for obferving the fmall differences incident to the fourth 
 and fixth notes, as defcribed in the two preceding articles t be- 
 caufe, although ever fo well known, they cannot be executed 
 on thefe inftruments: but then, if the performer have lef& to 
 do with thefe fmall occafional variations, the tuner has fo much 
 the more; for, in truth, thefe inftruments ought, as near as 
 poffible, to be fo tuned, as that every one of the fixt founds 
 may be a kind ot a mean between the two different natural 
 founds, which it may occafionally be put to reprefent. 
 
 •This method of adjufting the fixed founds of an organ or 
 harpfichord, fo as to anfwer equally well in all kinds of paifages, 
 is called temperament ; of which more particularly in a proper 
 place. ' 
 
 26. We ought not to conclude this chapter, without ac- 
 
 Part I. 
 
 quainting the reader, that although the difference between the 
 ftep of a tone and that of a femitone be eafy to feel ; and the dif- 
 ferent effects of the feveral notes of the Gamut, not very hard 
 to perceive f> yet to difcover, in all paffages, thefe fmall alterati- 
 ons to which the fourth and fixth notes are liable, is one of the 
 moft critical matters in the whole practice of mufic, and what 
 no beginner ought to expect, let his natural talents be ever 
 fo good, without a great deal of practice. It is by practice 
 only, that our natural faculties of retaining and comparing mu- 
 fical founds are to be improved and perfected; and therefore, 
 fuch as cannot immediately difcover the difference of the two 
 fuccefllve fourths, as defcribed, Art. 23. ought not, on that ac- 
 count, either to difpute the reality of the fact, or to doubt of 
 the fufficiency of their own talents for mufic. 
 
 If thefe differences had been very obvious, the overfight 
 of all the writers on mufic, from theearlieft antiquity, down to 
 the prefentage, would have been inexcufable; fince none of them 
 have given us any diftinct account of thefe occafional varieties: 
 however, they are far from being imperceptible, when a prac- 
 titioner partly knows what they are, and where to expect' them. 
 
 Chap. II. Application of the N a t u . r a l Scale. 
 
 . ., TT7HEN by often practifing the Gamut, as fcend and defcend by fingle ffeps, he may occafionally do it by 
 
 /• YV recommended above, the proper found of larger leaps ; and thus become able to give the proper founds to 
 
 each note is become familiar to the learner, it will be very eafy any mufical * fucceffion of notes which may be propofed. 
 to vary their order; and inftead of being confined always to a- If it be required, immediately after the key, to found the 
 
 * A fucceffion of notes, propofed at random, might chance to be very difficult, even for the mod expert, to execute; fo that a learner need not be 
 difcouraged, on finding it almoft impoffible to perform fome leaps, which are but too frequently found in initial leflbns : he may, on thefe occafions, 
 reft allured, that fuch leaps, as are very hard to execute, have but little to do in mufic : becaufe here, in general, whatever is eafy, is agreeable; and what- 
 ever is difficult, is harfh. 
 
 Thefe impracticable combinations of founds, in mufic, are fomething like clutters of confonants, without a proper intermixture of vowels, in writing; 
 which no body can pronounce, and therefore do not at all belong to fpeech. 
 
Chap. II. Direclions to the Singer, 
 
 third, dr fourth, or fifth, or fixth above; the directions ufu- 
 ally given to a learner of vocal mufic are ; firft, to run up the 
 fteps of the fcale, from the key, to the note required, as quick- 
 ly as poffible, and afterwards to join the firft and laft notes to- 
 gether, omitting the intermediate fteps. The fame directions 
 are given for learning to execute the defcending leaps, from the 
 key to the fixth, or fifth, or fourth, or third, or fecond below. 
 
 Again, if it be propofed, immediately after the fecond, to 
 take the fourth, or fifth, or fixth ; the infertion of the interme- 
 diate fteps of the fcale, in as fhort notes as poffible, is ftill di- 
 rected; until the learner be fo far advanced in the command of 
 his voice, and fo habitually acquainted with the peculiar air 
 and effect of each leap, that he has no more occafion to touch 
 the intermediate fteps at all. 
 
 By this method, a perfon may learn to perform all leaps, from 
 any one note of the fcale, to any other ; and it is often of no 
 fmall ufe, even to the more expert, in difficult or unufual paf- 
 fages. 
 
 28. If the peculiar effecl: of each note of the fcale, as menti- 
 oned, Art. 21. be carefully attended to, it will greatly affift 
 the finger in the attainment of this neceffary qualification; and 
 prevent the troublefome, and oftentimes difagreeable infertion 
 of the intermediate notes; which frequently cannot be heard 
 without altering, and perhaps fpoiling, the genuine air of the 
 tune. 
 
 Thus, for inftance, when the fourth is required to be fung, 
 the learner, who is acquainted with the peculiar grave tone of 
 the fourth, will be able immediately to give its proper found, 
 without regarding how many fteps of the fcale are between it, 
 and the former note : thus alfo, the fupplicative third and fe- 
 venth ; and the plaintive fecond and fixth, may be kept in rea- 
 dinefs to be inferted, whenever they are required. 
 
 "The Key always to be kept in Mind. 9 
 
 In this way, we are fuppofed to keep the key note conftant- 
 ly in view, during the whole courfe of a tune ; and to confider 
 all the other notes of the fcale, chiefly with regard to their 
 feveral relations to the key 5 and that this is the beft and moft 
 natural way, I might venture to alien from the uncommon progrefs 
 fome beginners have made, by being taught after this method; 
 but more pofitively from the following confederations, which 
 are obvious to all readers who have any tafte for mufic. 
 
 29. The common Pfalm tunes of the church are more uni- 
 rerfally known than any other pieces of mufic; and therefore, 
 we Jfhall make ufe of thefe for examples, whenever they will 
 ferve the purpofe. 
 
 Let a perfon fing, or play over the well known Pfalm tune, 
 called York or Stilt, as reprefented, Exam. IX. and if he fings, 
 let him apply the fyllable la to each note, rather than the words 
 of any verfe ; the hearer will find, all along, a fort of expecta- 
 tion of fomething ftill to follow, until the performer arrives at 
 the laft note of the fourth line ; at which note, the expectation 
 ceafes, and the hearer is perfectly fatisfied with a conclufion on 
 that note : whereas, if the performer were to ftop at any other 
 place of the tune, fuch an abrupt ending would be quite unfa- 
 tisfactory, and a plain difappointment to the hearer. * 
 
 This natural expectation, which never ceafes till fome pro- 
 per fucceffion of notes occur, leading to, and terminating upon 
 one certain found, is not peculiar to a few tunes only, but is 
 common to all : we have chofen York tune, as an inftance, only 
 on acconnt of its eafinefs : and here let it be obferved, that if a- 
 ny reader be not already acquainted with this tune ; at leaft, fo 
 as to fing or play it by the ear, as it is ufually called, he muft in- 
 evitably learn to do this, before he can properly proceed any 
 further. 
 
 Thefe fucceffions of notes, which properly lead to the final 
 
 * "We have, in this Example, directed the finger to ufe the unmeaning fyllable la, rather than any connected words ; becaufe, the natural expectation, 
 intended to be exemplified, is often kept up by the uniinifhed fenfe of the words fung; when otherwife, the mufic might admit of a final clofe. This ex- 
 pectation has alfo a dependence on the time and mtafure of the mufic, as will be fhewn when we come to ipeafc of thefe particulars. 
 
^Application of the Scale. 
 
 found, are called regular cadences ; and will be more particular- 
 ly defcribed hereafter; and this final found, with which we are 
 obliged to conclude, becaufe no other will fully fatisfy the hear- 
 er, is always the key note of the tune ; and in all tunes of the 
 fame fort * with that we have here ufed, the key of the tune is 
 alfo the key of the natural fcale, as explained in the former 
 Chapter ; and all the other notes are feconds, thirds, fourths, etc. 
 to the key. 
 
 Now it muft be acknowleged, that the hearer, who fo con- 
 flantly expects the occurrence of this key note at the conclufi- 
 on, muft, all along, keep the idea of it in his mind ; for if ever 
 he were fuppofed to have totally forgot it, what then could 
 hinder him from being as fully fatisfied with a final clofe, upon 
 fome other note ? 
 
 It is plain then, that in the practice of mufic, the key note 
 is conftantly kept in mind; and all other notes which are ad- 
 mitted, are fomeway compared with the key. Thefecomparifons, 
 and the confequent perceptions, are, indeed, the very effence of 
 mufic. It is impoffible for us to hear two different founds, ei- 
 ther together, or in fucceffion, without attempting to make 
 fome comparifon, either between one and the other, or between 
 each of them, and fome third found, with which our mind 
 may previoufly be poffelfed, and which we regard as a key note. 
 
 30. As we neceflarily conclude every piece of mufic with the 
 key note, fo we naturally begin with the fame : this is the cafe 
 in York tune, and moft part of the other Pfalm tunes in com- 
 mon ufe. If we fometimes begin with another note of the 
 fcale, it muft be one of thofe which are more nearly related to 
 the key, viz. the third or the fifth f; and the initial paffages 
 muft be fuch, as will fufficiently point out, to an experienced 
 
 Part I. 
 
 hearer, the real key of the following piece. This neceflity of 
 having the key plainly pointed out, at the beginning of every 
 piece, may be accounted for, by obferving that fo long as a hearer 
 remains in fuipenfe, which note to fix upon for a key, he has 
 no enjoyment of the mufic; and if he were to pitch upon a 
 wrong one, it would be ftiil worfe ; for the fucceffive founds, 
 when compared with a different key from that which the com- 
 pofer intended, would either produce jarring and offeniive fen- 
 fations, the very reverfe of mufic ; or they would, at beft, pro- 
 duce fuch as were altogether foreign to the true air and defign 
 of the piece. 
 
 For an inftance of this, fuppofe a man to come within hear- 
 ing of the fineft and moft connected piece of mufic, at the mid- 
 dle of fome ftrain ; fuch a man, not having heard the beginning, 
 and confequently, not knowing what note to pitch upon for a 
 key, would not know what to make of it; it would, for a little 
 time, feem to him an unmeaning, unconnected jargon of founds; 
 till fome pafiage occurred which fhould point out to him the 
 proper key of the whole ; and the more he was practifed in 
 mufic, he would the fooner be able to determine what was the 
 initial key, and would be the final, from the fpecies of the 
 middle paffages. An unfkilful hearer, in fuch a cafe, might 
 probably never be able toafcertain the proper key, till the final 
 clofe; much like a fchool-boy, who enters, when the mafteris 
 in the middle of one of Euclid's demonftrations ; and can nei- 
 ther difcover the connexion, nor the defign of the reafoning, 
 for want of knowing what is the principal propofition intended 
 to be demonftrated. 
 
 3 1 . It is the uninterrupted rememberance of one certain found, 
 as a principal key, which connects together all the fucceffive 
 
 * There is another fort of tunes which conclude upon the fixth, or fecond of the natural fcale; or rather, which afiume a different feties made up, by 
 piece-meals, out of two natural fcales, fo fituated, that the fifth of one fcale, is the key of the other ; and therefore, the fixth of one fcale, is the fame 
 found as the fecond of the other; and this found, partly fixth, and partly fecond, is to fuch tunes a final, or key note; preferable to the real key of either 
 of the two natural fcales, when thus intermixed. 
 
 f This nearer affinity between fome certain notes of the fcale, will be particularly explained, under the head of harmonica! confonances. 
 
Chap. II. *d conftant Rememberance of the Key 
 
 parts of one piece of mufic. When a perfon has fixed his at- 
 tention on one found as a key note, if other founds accidentally 
 intrude upon his ears, which belong to a different key, they have 
 often a moft difagreeable effect ; although thefe other founds, 
 when compared with their proper key be ever fo agreeable ; yet, 
 in cotnparifon with his key, they muff be in fome degree extra- 
 neous, and they may be quite incompatible * : and before he 
 can hear fuch incompatible founds, with fatisfaction, he muff 
 quit his own former key, and turn his attention altogether on 
 that to which they properly belong : this is what we uiually ex- 
 prefs by a man's being put out of his tune. 
 
 He that will refolutely go on with his own tune, when fuch 
 founds interfere, finds himfelf under a neceffity of confining his 
 attention clofely to his own notes ; and, as it were, fhutting his 
 ears to thefe incompatible founds. The Dutch concert f is a 
 moft admirable leffon for this purpofe. 
 
 The faculty of retaining theimpreiTions of mufical founds, for 
 fome confiderable time after they ceafe to be heard, is purely 
 natural, and requires no improved abilities at all. The plough- 
 driver who quits his tune abruptly, whenever occafion re- 
 quires ; and refumes it again, juft where he broke off. even af- 
 ter an interruption of feveral minutes, is a notable inftance of 
 this. It feems as natural to remember the tone of founds which 
 we have lately heard, as the (hape or colour of objects which 
 we have lately feen : but to turn our attention readily from one 
 key note, and the founds dependent upon it, to another dif- 
 ferent key, which introduces with it a different fett of founds, is 
 
 , proved to be both neceffary and natural. I L 
 
 an attainment due to practice alone, and a kind of force upon 
 nature. 
 
 32. From thefe, and numberlefs other inftances, which will 
 daily occur to the diligent obferver, it will plainly appear, that 
 a conftant rememberance of the key note is both neceffary for 
 the purpofes of mufic, and natural to the human mind ; from 
 hence we infer, that whoever practices mufic, with a view to 
 improve his judgment, concerning the propriety and juftnefsof 
 each fucceffive found, ought principally to regard the relation 
 of every one of them to the key ; he ought never to ufe a found 
 without confidering it as fuch a degree of the natural fcale ; 
 and for this purpofe, if he apply feven different finging fyllables, 
 to the feven degrees of the Gamut, fuch as thofe of the Italians, 
 defcribed, Art. 13. ; he ought to regard his fyllables as a kind 
 of technical terms, which, in plain Englifh, mean no more than 
 the key, the fecond, the third, etc. of the natural fcale, refpec- 
 tively ; and herein confifts the advantage of thefe fyllables, that 
 he who confiders them in this manner, adopts into his language 
 a fett of (hort, fluent monofyllables, which can eafily be applied 
 to the founds of a tune; and which, to him, fufficiently exprefs 
 the feveral degrees of the natural fcale, and confequently the 
 proper effect of every note he ufes. 
 
 It is very certain, that by conftantly ufing fuch fyllables, and 
 confidering them in the manner above directed, a finger will, in 
 time, affociate the idea of each found in the fcale, with its pro- 
 per fyllable, fo that he will habitually give fa the found of a 
 fourth, fol that of a fifth, la that of a fixth, etc ; and will plainly 
 
 * We call every different key, in fime 'degree extraneous, or foreign ; but yet fome are much more fo than others. There is fuch an affinity hetween the 
 key and its fifth or fourth, that we can for a little while turn our attention upon either of thefe as an occafional new tey, without entirely relinquiihing our 
 principal key ; but all other natural fcales, except thefe dependent upon the fifth or fourth, are incompatible with the principal fcale. But of this, more hereafter. 
 
 f So called, in burlefque ; when a company of muficians fall to, all at once, each finging or playing a different tune on a different key, and in a different 
 inood of time, as his own choice happens tc lead him ; and he that can hold out his tune longetr, is efteemed the beft performer. 
 
 It is a droll enough talk, efpecially for a fing;r, to keep his own tune, and at the famctime to be upon the watch, fo as to be able to point out any other 
 of the company, when he goes wrong. 
 
1 2 lApplicatiofi of the Scale 
 
 perceive an error, when any fyllable is not applied to its own 
 
 Part I. 
 
 found : but we muft own that it requires a long time, and a care- 
 ful application, to arrive at this degree of proficiency ; and after 
 we have got thus far, we may ftill be at a lofs, when we would 
 apply the words of a verfe to a tune, at firft fight : fuch a pro- 
 ficient may be able lofolfa, as it is called, but yet not able to 
 fing, at fight. 
 
 33. Confidering that we have occafion, in finging, to apply 
 all forts of fyllables to every found; and in inftrumental mufic, 
 we have nothing to do with articulate fyllables at all; it feems 
 not worth the while to be at fo much pains to afTociate every 
 found with a particular fyllable; and ifitbean unneceffary trou- 
 ble to learn fol-fa-ing in the Italian method, the Britifh me- 
 thod, defcribed, Art. 14. is ftill more difficult, and lefs to the 
 purpofe; for which reafon, our moft difcerning authors have, 
 long ago, juftly reprefented_/j/-_/k-z7zg-, as a very laborious and 
 ufelefs attainment. 
 
 Thefe are fome of the reafons for which we would advife e- 
 very learner of mufic, whether vocal or inftrumental, not to 
 beftow much time in aflbciatingunmeaning fyllables with founds, 
 which are already fuffkiently diftinguifhable by their effects, as 
 fhewn, Art. 21.; and whofe fituations in the natural fcale are 
 better and more familiarly exprefled, as we have hitherto done, 
 by the ordinal numbers : let him rather turn his attention on 
 the very founds themfelves ; and if he have formerly made him- 
 felf perfect in the Gamut, as recommended in the laft chapter; 
 Jet his next ftudy be, whenever he either performs, or hears a 
 piece of mufic; firft, to afcertain its proper key, which he 
 may always do by attending to the characteriftics of the key 
 already given, Art. 29. and then to determine what degree of 
 the natural fcale correfponds with each fucceffive found. 
 
 Thus, by obferving that the firft found of York tune is the 
 fame as the laft, the learner may conclude it to be the key note 
 of the fcale j which being determined, he may, if neceffary, 
 
 run over all the fteps of the Gamut pretty quickly, both up- 
 ward and downward, for the better fixing in his mind the idea 
 of each found, before he proceed to the tune. This done, he 
 will eafily perceive that the fecond note of the tune is the fame 
 found as the third of the Gamut; and that the third note of the 
 tune is the fifth note of the Gamut : fo that the three firft notes 
 of York tune are, the key, the third, and the fifth of the 
 fcale. 
 
 Proceeding in the fame manner through the whole tune, he 
 will find each note correfponding with that degree of the Ga- 
 mut, which is fptcificd by the K, or the figure fubjoined, in 
 Exam. IX. : And if he fhould happen to be in any doubt con- 
 cerning the real degree of any note, he may eafily have recourfe 
 to the key, which, at any event, he will remember; and by 
 running either upward or downward from the key, till he comes 
 to the fame found, he will infallibly difcover what it is. 
 
 34. In this manner, we fhall fuppofe the learner to examine 
 every plain eafy tune, with which he is acquainted, it it be of 
 the fame fort with this ; but it muft be obferved, that there is 
 another fort of tunes, which, as hinted in the note to Art. 29. 
 have a different fcale, and are called tunes of a mixed or flat fe- 
 ries ; in contra-diftinction to the fort we now propofe to be ex- 
 amined, which are called tunes of a natural or (harp feries., 
 
 The full explanation of the mixed feries muft be delayed till 
 the affinity between the key and its fifth or fourth, mentioned 
 in the note to Art. 31. be more fully handled; only it becomes 
 neceffary here to give fome directions, whereby the learner may 
 know whether any tune, with which he is acquainted, be of 
 the one or the other fort; fo that he may not be baffled in at- 
 tempting to apply the natural fcale to the wrong fort of tunes. 
 
 The moft palpable difference between thefe two forts of tunes 
 is in the third note; which, as we have already feen, is a kind 
 of fupplicative found, and only a femitone below the fourth in 
 the natural fcale; whereas, the third of the flat fcale is a 
 
Chap. II. T'unes ofajbarp Series, how to be examined, and how diflingu'ifbed from thofe of a flat Series. 
 
 remarkably grave found, much like the fourth of the natural 
 fcale, and is only a femitone above the fecond. 
 
 The third of the natural fcale is called a major ox greater third, 
 and is two tones above the key; and the third of the flat fcale 
 is called a minor or lefs third, being only one tone and a femi- 
 tone above the key. 
 
 Thefe two forts of thirds are alfo fometimes diftinguifhed by 
 the epithets fiarp and fat, rather improperly, as will appear 
 when we come to (hew what are the received meanings of thefe 
 epithets among muficians : however, our bufinefs is not fo 
 much to afcertain the propriety of words, as of founds. 
 
 35. The direction given, Art. 29. for afcertaining the key 
 note, is equally good in both forts of tunes. Now, we (hall fup- 
 pofe that the learner, after he has found the proper key of any 
 tune, proceeds to try whether it be natural for him to take the 
 greater, or the lefs third above the key, while the rememberance 
 of that tune lafts ; and in this trial he cannot be deceived ; be- 
 caufe it is very difficult to turn our attention from one fort of a 
 fcale to the other, while we retain the fame key note ; therefore, 
 if he find it natural to take the lefs third above the key, imme- 
 diately after he has either played, or fung, or heard any par- 
 ticular tune, or even when he has been only thinking of it, he 
 may fafely conclude that it belongs to the flat feries; and there- 
 fore may, for the prefent, defer all further examination of fuch 
 a tune. 
 
 For one infiance of the change of attention neceflary to be 
 made between the fharp and flat feries, we fhall fuppofe the 
 reader to be acquainted with the old Pfalm tune, called Martyrs, 
 reprefented Exam. X. which is of the flat fort. Let him firft 
 fing, or play this tune over, and then immediately try to begin 
 York tune in the fame found with which hefinifhed this other; 
 he will find that in (lead of the lefs third, which is the fecond 
 note of Martyrs tune, he mule now apply a quite different 
 found, viz. a greater third, in order to come into the true air 
 
 *3 
 
 of York tune; and that before he can properly found fuch a 
 greater third, the rememberance of the lefs third, and indeed 
 of every other note, which he had formerly applied in Martyrs 
 tune, muft be entirely difregarded, and quite new ideas fubfti- 
 tuted in their ftead. The truth is, thefe two different fcales are 
 derived from quite different ways of conceiving the key note, 
 The key of the flat feries is confidered as a plaintive found, 
 much like the fixth of the natural fcale ; and can never, to our 
 imagination, affume the boldnefs neceflary to the key of a fharp 
 feries, Art. 21. until every idea of its effect, as key of a flat 
 feries, be entirely fet afide. 
 
 To make this change of attention flill more perceivable, let 
 the learner, after he has begun either of thefe tunes, and found- 
 ed only two or three of the firft notes, flop fhort there, and 
 attempt to begin again on the fame key as before, but with the 
 other tune: this experiment he may repeat feveral times, be- 
 ginning the one and the other alternately; and thus by com- 
 paring the beginning of T01 k tune, with the beginning of Mar- 
 tyrs, he may, in a few minutes, acquire a more diflin£t idea of 
 the difference between a greater and a lefs third, than can be 
 conveyed to him by all the words in the world, without fome 
 fuch trial. 
 
 It is needlefs to dwell any longer on the difference between 
 thefe two forts of fcales ; as almoft every one who has a turn 
 for mufic, will perceive it at the firft glance of thought: and 
 fuch as cannot partly find it out, by comparing the beginnings 
 of Tork and Martyrs together, as directed above, need not ex- 
 pect to receive much intelligence from any thing we can fay on 
 the fcience of mufic; at leaft, not till they have made further 
 progrefs in the art, by the affiftance and inftrucYions of a 
 roafter. 
 
 36. It is impoffible here to point out any particular tunes 
 which will be the moft proper for every reader to examine ; 
 becaufe, thofe only are the moft proper for this purpofe, which 
 
Application of the Scale. 
 
 Part I. 
 
 the reader can already fing or play by the ear ; and of this fort 
 every different reader will have a different collection. Among 
 the common Pfalm tunes, the Old C. Pfalm, French, London 
 New, St. David's, St. Ann's, Stroudwater, St. Matthew's, etc. 
 are all tunes of the natural, or (harp feries ; and therefore may be 
 examined at this time: and Dundee or Windfor, St. Mary's, 
 Walfal, Bangor, Crowle, Norwich, St. Neot's, St, Ive's, Zi- 
 on, etc. are all tunes of the mixed, or flat feries; and, therefore, 
 mud be omitted at prefent. 
 
 The compafs of fome tunes lies wholly or chiefly above the 
 final key ; fuch as, French, London New, and St. David's : 
 others are for the mod part below the final key ; fueh as, St. Ann's : 
 and others extend both above and below the key ; fuch as, the 
 Old C. Greenock, St. Paul's, Stroudwater, and St. Matthew's. 
 This laft mentioned fort of tunes have been fometimes called 
 plagal, to diftinguiih them from the other, which are denomi- 
 nated authentic ; but this diftinction feems to be of very little 
 ufe, in the prefent flate of mufic. 
 
 37. In impofing upon the learner this tafk of confidering 
 and examining the tunes with which he is acquainted, we 
 are departing from the common courfe of introductory treatifes 
 on this fubject. The learner is generally left to the quiet en- 
 joyment of every ftrain which he can already perform, and is 
 hurried on to a parcel of initial leffons, which he neither knows 
 how to perform, nor can the dumb leaves of a book lend him 
 the leaft affiftance. It is plain, that a book can neither fing, 
 nor fpeak ; it can only refer the reader to certain words 
 which he has formerly afTociated with their reprefentative let- 
 ters ; or to certain founds, which he has afTociated with their 
 proper characters and names. Therefore we prefume, that 
 
 as a perfon muft be able to pronounce a great many words be- 
 fore he begins to learn the alphabet; fohe fhould be acquaint- 
 ed with feveral tunes, before he begins to ftudy the application 
 of the fcale, which is the alphabet of" mufic ; and as we fhould 
 fir ft learn to fpell and read our mother-tongue ; (o we fhould firft 
 learn to apply the fcale, if I may ufe the exprefflon, to our 
 nurjis tunes. 
 
 The courfe we propofe to the learner is briefly this ; firft, 
 to determine the proper key of any tune, with which he is ac- 
 quainted, by the directions given Art. 29.; then to try whe- 
 ther it be of the fharp or flat feries, by the rule, Art. 35.: if it 
 happens to be of the flat fort, he may defer all further exami- 
 nation for the prefent; but if it be of the fharp feries, let him 
 proceed according to the method prefcribed, Art. 33. to afcer- 
 tain the degree of each fucceffive found, through the whole 
 tune : and if thefe enquiries fhould prove a little abftrufe and 
 difficult at firft, he may affure himfelf that every fucceffive trial 
 will render them more and more eafy ; until the proper degrees 
 of every fticceffion of founds which he hears, become even as 
 obvious to his imagination, as the letters which fpell any word. 
 
 This is the perfection he ought to afpire at, and thefe are the 
 readieft and moft effectual fteps, we can propofe to be taken 
 for the attainment of it . 
 
 This will ferve to give an idea of what we call the practical 
 application of the fcale; in which, the learner's progrefs muft 
 depend altogether on his own diligence in examining, and na- 
 tural ability in judging of mufical founds. — We come next to 
 fhew how thefe degrees are reprefented by the lines and fpaces 
 of our written mufic, which will afford fufficient matter for 
 another chapter. 
 
Chap. III. 
 
 The Semi tonic Sy/iem defcribed, and its fixt Sounds denoted by Letter. 
 
 ijr 
 
 Ch 
 
 III. Of the Modern System o/Music, 
 
 . ft ^TT^HE natural fcale, of which we have hither- 
 Ar ic e 3 . J^ to treated, has always been, and will always 
 be the fame in all ages and countries i it feems to have been one 
 of thofe laws which the great Author of nature prefcribed to 
 bimfelf, in the formation of the human mind, that fuch certain 
 degrees of found mould, conltitute mufic: but the names by 
 which thefe degrees have been diftinguifhed, and the ways of 
 reprefenting them in writing, being things quite arbitrary, have 
 undergone many alterations; infomuch, that a whole volume 
 would fcarce fuffice to give art account of all the different fyf- 
 tems, which are to be found among the authors of different ages. 
 
 Happily for us, of the prefent age, we have one eftabliftied 
 fyftem, which is received by all the muficians in Europe; and 
 therefore, we may fafely confine ourfelves to the explanation of 
 this modern fyftem, leaving all thofe of the antients, as matters 
 of more curiofity than ufe. 
 
 39. We fuppofe all thofe intervals in the natural fcale, which 
 we have hitherto called tones, capable to be fubdivided into 
 two femitones each, by inferting an intermediate found between 
 them: and thus, inftead of five tones, and two femitones, 
 which, as formerly feen, are the natural intervals between any 
 one found and its octave, we have always twelve femitones. 
 
 By this means we conceive a fyftem of founds, indefinitely 
 proceeding both upward and downward, by fteps which we call 
 femitones each; and we fuppofe, that if any one of the founds 
 of this fyftem be conftituted the key of a natural fcale, we (hall.' 
 always find, among the other founds of the fyftem, all the o- 
 ther degrees of the fcale, fufficiently near for practice; by o- 
 m-itting one found, and taking the next, when a ftep of a tone 
 
 is naturally required ; and by taking the two contiguous founds 
 for the natural femitone, when it occurs. 
 
 40. Thus we have a femitonic fyftem of founds, which we 
 fuppofe fixed and immutable; yet fo regulated, as that any one 
 of thefe fixed founds may be the key of a natural fcale. For 
 denoting thefe fixed foiinds, we make ufe of the feven firft let- 
 ters of the alphabet, in this manner, viz. 
 
 T*A*BC*D*EF*G*a*bc*d*ef*g*a*bc*d*ef*g. 
 
 The loweft found is reprefented by the Greek T on the left 
 hand ; the next found, which is a femitone above T, is repre- 
 fented by an afterifm *, the next femitone is A, the next is an 
 afterifm, the next is B, and the next C ; fo that the letters 
 B and C are placed contiguous to each other. In the fame 
 manner, the letters E and F are contiguous ; and all the other 
 have an intermediate found between them, reprefented by the *. 
 
 Thus, the feven letters, with five afterifms interfperfed, re- 
 prefent twelve degrees, or an octave, of this femitonic fyftem, 
 Art; 39.: fo that, when we come to a repetition of the fame 
 letter over again, we come alfo to the fame found, an octave 
 higher, Art. 1 5; and 16. ; and for diftinguifhing thefe founds 
 in different octaves, we have put the loweft octave in capitals; 
 the middle octave in Roman fmall ; and the uppermoft in Italic 
 fmall, in the abo've fcheme. 
 
 41. Thefe three octaves comprehend the whole compafs or- 
 dinarily affigned to the human voice. For inftance, T repre- 
 fents the graveft found, which a man of a tolerable voice can 
 clearly form ; and g is the acuteft found, which a Woman's 
 voice is ordinarily fuppofed to reach: and, although the fcale 
 
Of the Modern Syjiem of Mufic. 
 
 16 
 
 for fome inftruments extends much farther, both upward and 
 downward, yet thefe three, which we may call vocal octaves, 
 are all for which we affign lines and fpaces in our written mu- 
 fic : whatever exceeds this compafs, is put upon leger lines, 
 Art. 17, 
 
 42. According to the above difpofition, it will eafily be ob- 
 ferved, that if C be conftituted the key of a natural fcale, all 
 the degrees of the fcale will coincide with the letters C. D. EF.G. 
 
 a. b c. etc. becaufe, in this cafe, the femitones, which are na- 
 turally the next above the third, and the next below the key, 
 Art. ,8. are from E to F, and from b to c; which agrees ex- 
 actly with the difpofition of the letters in our fyftem; fo that, 
 in the key of C, we have no bufinefs with any of the founds 
 marked with the *. 
 
 43. The cafe is different if we make any other found a key; 
 for inftance, G. The rule requires, that the femitones in the 
 key of G, be above b, and below G; now, above b, we have 
 c, which is properly a femitone; but below G, we have F, a 
 whole tone: therefore, inftead of this F, we muff ufe the note 
 at the *, next below G, which is a femitone higher than F, 
 and which we therefore call F fharp ; and fuch elevation of 
 a note, a femitone above its proper found, is fhewn by affixing 
 this character $, called a fharp, to its denominative letter. 
 
 44. If F be conftituted a key, the femitone from E to F, be- 
 low the key, is right fituated ; but above a, the third, we have 
 
 b, a whole tone, inftead of a femitone ; fo that, in the place 
 of this b, we muft take the note at the *, next below b, which 
 being -a femitone lower than b, is called b flat ; and this depref- 
 fionofanote, a femitone below its proper found, is denoted 
 by affixing this character \), called a flat, to its denominative 
 letter. 
 
 45. The proper found of any letter is called natural, to dif- 
 tinguifh it from that of the fame letter, when elevated a femi- 
 tone by an occafional fharp, or depreffed the fame quantity by 
 
 Part I. 
 
 aflat; both which altered founds are denominated fictitious: 
 according to this diftinction, all the notes of the fyftem 
 marked with the * are fictitious ; although they may fome- 
 times be called the fharp of the letter below ; and at other times, 
 the fiat of the letter above: thus the *' between G and a. is 
 fometimes called G fharp, and at other times, a, flat ; and fo 
 ot the reft. 
 
 46. To determine what name properly belongs to every fic- 
 titious note, which, in any particular fcale, we find a neceffity 
 of ufing, it may be obferved, that for every fictitious note, which 
 we take in, we muft of courfe leave out a natural note, either 
 immediately above or below ; and therefore, we allot to fuch 
 fictitious note, the name of the natural note, whofe place it fup- 
 plies, with the proper addition of fharp or flat, according to its 
 fituation above or below the natural. 
 
 By this means the feven letters, which denominate the natu- 
 ral notes of the fyftem, come to be all of them ufed, in every 
 particular fcale, though differently affected with occafional (harps 
 or flats. — The following general fcheme will fufficiently illuf- 
 trate this. 
 
 47. For the better underftanding of this fcheme, we mayob- 
 ferve, Firft,That there can be no more than twelve different fcales, 
 anfwering to the twelve femitones in an octave, any one of which 
 may be conftituted a key. Secondly, That in the order thefe 
 different fcales are here ranged, every fubfequent fcale has either 
 one fharp more, or one flat fewer, than that immediately pre- 
 ceding. Thirdly, That the fourth of every preceding fcale, by 
 adding a fharp to it, or by taking away a flat from it, always be- 
 comes a feventh in the fubfequent fcale. Fourthly, That for 
 rendering the comparifon between each of thefe fcales and the 
 general fyftem more obvious, we have carefully placed each note 
 with its proper name, exactly under the correfportdent note in 
 the general fyftem. Fifthly, That we have begun each of the 
 particular fcales with its key note, and terminated with the oc- 
 
Chap. III. The Sounds of each particular Scale felecled, and denominated by Letters and other Marks. 17 
 
 tave of the key, in order that the analogy between each of 
 thefe progreflions and the natural fcale, formerly defcribed, 
 may be more diftin&ly feen. Neverthelefs, each of them may 
 be continued at pleafure, either upward or downward, by a re- 
 
 ed into that of D flat, in the fubfequent line ; being the very 
 fame founds, but differently named, viz. having five flats and 
 two natural notes ; and thus, when we are come to fix fliarps 
 as inN° 7. we take five flats for the next fcale, rather than fe- 
 
 petition of the lame letters, in the fame order. Sixthly, That ven fliarps ; and fo proceed to diminifli the number of flats 
 
 when the number of fharps amounts to feven, all the natural for each fubfequent fcale, until none of them remain, and 
 
 notes are a femitone elevated, being each of them afFe&ed by a we arrive again at the fame natural fcale of C, with which we 
 
 fliarp ; and in this cafe, the fcale of C lharp, N° 8. i9 convert- began. 
 
 48. Scheme of the Formation of the Twelve particular Scales from the General System. 
 
 Genera] 
 
 Syftem. 
 
 N°i. Sharp 
 
 Series of C 
 
 2. 
 
 ofG 
 
 3- 
 
 ofD 
 
 4- 
 
 of A 
 
 5- 
 
 ofE 
 
 6. 
 
 ofB 
 
 7- 
 
 ofF^ 
 
 "•I 
 
 ofC& 
 or of D\) 
 
 9- 
 
 of A\) 
 
 10. 
 
 ofE[, 
 
 11. 
 
 ofB[, 
 
 12. 
 
 ofF 
 
 The fame as at firft, C 
 
 * D 
 
 . D 
 
 . D 
 
 . D 
 
 
 E b . 
 E b . 
 
 Eb • 
 E b . 
 
 * G 
 
 . G 
 
 F*G 
 F^G 
 
 F* • G^a 
 
 . F* 
 E&F& 
 F G\, 
 
 F 
 F 
 
 F 
 
 F 
 
 F 
 
 G*. 
 
 Ga[, 
 Gaj, 
 G . 
 G . 
 G . 
 
 b c 
 b c 
 
 e f 
 
 g * a, etc. 
 all natural. 
 F fliarp. 
 F and C fliarp. 
 
 F C and G fliarp. 
 
 b . c% . d$ e . . F C G and D fliarp. 
 :b FCGDandA fliarp. 
 
 c%d 
 
 a&b . c% 
 
 FCGDA and E fliarp. 
 
 bb 
 
 : c% all fliarp. 
 
 d b . . . 
 
 e\> • 
 
 . c 
 
 b c 
 
 BEADandGflat 
 B E A and D flat 
 B E and A flat. 
 B and E flat. 
 B flat, 
 all natural. 
 
 } 
 
18 
 
 Of the Modem 
 
 49. It will eafily appear, that, as in beginning from the top 
 of this fcheme, every fubfequent fcale differs from the preced- 
 ing only in one note ; which, being always a fourth in the pre- 
 ceding fcale, is elevated a femitone, and becomes a feventh 
 in that which follows ; fo if we begin at the bottom, and pro- 
 ceed upward, we (hall find every fcale differing from the next 
 below; in that the feventh of the lower fcale, by being de- 
 preffed a femitone, becomes a fourth in the upper: and thus it 
 mull always be, that the fharpened fourth becomes a feventh, 
 and the flattened feventh becomes a fourth, of a new fcale. 
 
 50. The fourth and feventh are the only notes liable to be dif- 
 placed, in paffing from one key to another : this remark belongs 
 more properly to another place ; only it feemed not amifs to 
 take notice of it here, that the reader may be the better recon- 
 ciled with the feeming irregularity in the arrangement of the fe- 
 veral fcales above. The truth is, the fourth and feventh are 
 the two terms of the tritone, Art. 11.; and by.elevating the 
 lower or depreffmg 1 ' the upper term of the tritone, we make it a 
 ditone and a femitone; and at the fame time, what was before 
 a ditone and femitone, beqomes a tritone; fo that the proper 
 degrees of the natural fcale flill exift, but the key note is chang- 
 ed : whereas, if we were to fharpen any other note except the 
 fourth, or flatten any other except the feventh, fuch a diiloca- 
 tion would fpoil the order of the founds, and render them not 
 confiftent with any natural fcale. 
 
 Take any of the fcales, in Art. 48. at pleafure; fuppofe that 
 of C, which is there the firft and laft, viz. 
 
 . C . D . E F 
 
 . b 
 
 it will plainly appear that if anyone of thefe founds, viz. C, or D, 
 or F, orG, or a, were flattened, and all the reft continued as they 
 are, there would be an undivided interval of three femitones, 
 called a trihemitone, between the flat note and the next above ; 
 but there is no fuch undivided interval in the natural fcale. In 
 
 Syftem of Mufic. Part I. 
 
 the fame manner, if any one of thefe, viz. D, or E, or G, or a, 
 orb, were made fharp, and all the reft remaining unaltered; 
 the fame undivided and unnatural interval of three femitones 
 would be produced, between the fharp note and the next ber 
 low ; fo that it is evident no one note can be flattened, except 
 E or b, nor can any one be fharpened, except C or F, without 
 producing a trihemitone. 
 
 Now, if C were made fharp, or E made flat, we fnould have 
 four fucceffive tones together, viz. from F to C$, or from E[> 
 to b, which is inconfiftent with the fteps of the natural fcale ; 
 therefore, none but F can be fharpened, and this produces 
 the fcale of G, as in N° 2.; and none but b, can be flattened, 
 and this produces the fcale of F, as in N° 12. In the fame 
 manner it will appear, that the fharpened fourth of every fcale, 
 Art. 48. produces the fcale next below, and the flattened fe- 
 venth produces that next above-; and that thefe are the only 
 fmgle alterations which. can be made,. without fpoiling the natu- 
 ral order of the founds. 
 
 ' 51. There is yet fomething to be obferved in regard to the 
 feventh fcale in. Art. 48. which has fix fharps, viz. F, C, G, D, 
 A, and E; and of which Y% is the key. In this pofition, t% 
 takes the place off natural ; and this cannot be avoided, with- 
 out tranfgreffing the rule given, Art. 46. viz. that all the 
 feven letters muff be ufed in naming the founds of each particu- 
 lar fcale ; and this rule is of the utmoft importance, becaufe, 
 as we fhallfhortly fee, 'each letter has its proper line or fpace 
 allotted in our written. mufic ; at the beginning of which, we 
 prefix the proper mark, to fhew if fuch letter be affected with 
 either fharp or flat, in the particular fcale we are then making 
 life of: fo that if one letter were to occur twice over, different- 
 ly affected, and the next letter were quite left out, we mould 
 have one double-meaning line or fpace, which muft occafionally 
 denote two different founds, and an adjacent fpace or line, with- 
 out any meaning at all, as having no found appropriated to it. 
 
Remarks on the foregoing Scheme. Of naming Lines and Spaces by Letters. 
 
 Chap. III. 
 
 Thus if we were to ufe f, inftead of e% in the feventh fcale, 
 Art. 48. wefhould have the letter f, twice over, viz. firft natu- 
 ral, and then fharp, and the letter e, would be quite excluded ; 
 and thus the line or fpace allotted to e, would have no meaning, 
 while that of f, would have more than could conveniently be ex- ' 
 pre fled. 
 
 For the fame reafons v/e alfo take B$: in the place of C natu- 
 ral, in the eighth fcale, Art. 48. when all the {even letters are 
 fharp ; but this fcale, as already (hewn. Art. 47. is better ex- 
 prefled with five flats, in the fubfequent line. 
 
 52. We now proceed to fhew how thefe feveral fcalesare ex- 
 preffed by the lines and fpaces of our written mufic ; and here 
 as obferved, Art. 41. we have only three octaves, called vocal 
 octaves, to provide for ; we therefore take eleven lines, which, 
 with their intermediate fpaces, are fufficient for this purpofe. 
 
 Upon thefe lines and fpaces v/e fuppofe the letters to be rang- 
 ed in order, as fhewn below ; where obferve, that the whole 
 compafs of three octaves, being much more than any fingle 
 voice can perform; we therefore aflign the five loweft lines, 
 with their fpaces, to the graved: of men's voices, which we call 
 Bafs ; and thefe five loweft lines are called the Bafs ftaff: and 
 the five uppermoft lines we appropriate to women's voices, 
 which we call Treble; and thefe we call the Treble flaff. Thus 
 the lowed octave, denoted by capital letters, belongs properly to 
 the Bafs; and the higheft octave, denoted by Italic fmall letters, 
 belongs to the Treble. 
 
 Between the Bafs and the Treble ftafF, there is one interme- 
 diate line, which exprefies the c of the middle octave, viz. of 
 that octave which we denote by Roman fmall letters, and which 
 may be called the Tenor octave. To this intermediate line, 
 we therefore fometimes add the two loweft lines of the Treble 
 ftaff, and the two uppermod of the Bafs ftafF; and fometimes 
 more of the Bafs lines, and fewer of the Treble ; or more of the 
 Treble, and fewer of the Bafs, fo as always to make up five lines 
 
 19 
 
 in whole; which we allot to the middle voices, under the name 
 of Tenor, Medius or Contra. 
 
 53. Scheme of the Difpofition of the Letters upon Lines and Spaces. 
 
 2* 1 
 
 The Treble, or G cliff. 
 
 I 
 
 The Tenor, or C 
 
 ciiff. H — c 
 
 g 
 
 ej 
 
 The 
 
 3, orF 
 
 ciiff Q; 
 
 -D S^ 
 pa 
 
 54. Now, in order to fhew what letter belongs to each line 
 and fpace of our written mufic; and alfo, whether fuch letter 
 ought to be underdood as in the Bafs, or Tenor, or Treble oc- 
 tave ; we prefix to every ftaff of lines a certain mark, called a 
 cliff; of which we have three forts, as reprefented above, viz. 
 
 iji, The Bafs cliff is a kind of inverted C, with two points af- 
 
Of the Modern Syfte?n of Muftc. 
 
 Part. I. 
 
 terit ; and this fhews that the line on which it is placed, i. e. the 
 line between the two points, is F, of the Bafs ottave. It is gene- 
 rally fet upon the fourth line of the Bafs ftaff, as in the pre- 
 ceding fcheme. 
 
 idly, The Tenor cliff is two upright and two tranfverfe 
 ftrokes ; and fhews that the line between the tranfverfe flrokes is 
 c, of the Tenor octave. This cliff is occafionally fet upon any 
 of the five lines, except the uppermoft : wherever it ftands, the 
 lines above it may be conceived as belonging to the Treble ftaff, 
 and the lines below to the Bafs. 
 
 3 dly, The Treble cliff partly refembles a carelefs G ; and 
 (hews that the line on which it ftands is the g, immediately be- 
 low the treble o&ave. This cliff is always fet, as above, on the 
 fecond line of the Treble ftaff 
 
 55. The name of the cliff line being thus determined, the 
 names of all the other lines and fpaces follow of courfe, viz. by 
 counting upward from the cliff, in the alphabetical order of the 
 letters ; obferving after G, to begin again with A ; and by count- 
 ing downward from the cliff, in the fame order reverfed *. 
 
 The learner ought to praftife this method of naming the lines 
 and fpaces by letters, till it become quite familiar to him ; efpe- 
 cially in the Bafs and Treble ftaffs; not only, becaufe thefe two 
 are moft commonly ufed, but alio, becaufe the Tenor ftaff, as 
 fhewn above, is made up by joining fome part of each of thefe 
 to the intermediate, or Tenor cliff line; fo that the Tenor ftaff 
 has nothing properly its own, but the cliff line ; whatever is a- 
 bove may be conceived as borrowed from the Treble; and what- 
 ever is below, from the Bafs. 
 
 * Though any of the feven letters, placed at the beginning of a ftaff, might have anfwered the purpofe of a cliff, to determine the places of all the reft; yet 
 the firft inventor probably made choice of thefe three letters, C, G, and F, rather than any other ; becaufe, as has been already fhewn, Art. 41. tlfeq. 
 thefe are the letters which can be conftituted keys, with fewer diilocations of the natural founds of the fyftem than any other; fo that they may, properly 
 enough, be called keys, or cliffs, by way of eminence, even in our modern fyftem. 
 
 Among the antients, thefe three letters were dill more diftinguilhable. Gui'do Arctinus, to whom the invention of lines, marked with cliffs, is generally 
 attributed, had, indeed, no other keys but thefe ; neither were the fcales, even of thofe, quite entire in his fyftem, according to our modern form; for he 
 admitted none of our fictitious notes, except bty ; whereas the modern fcale of G requires F fharp, for its feventh note, which was not in Guido's fyftem. 
 
 The truth is, the ancients did not confider dll the feven founds of the Gamut, as depending immediately on the principal key ; but fome of them, in cer- 
 tain paffages, were fuppofed to depend either upon the fifth or fourth, as an occafional new key, [See the firft note to Art. 31.] and the feventh note in 
 particular was always referred to one of thefe; fo that the fcale of any found was fuppofed to be, at the moft, only fix notes, called a hexachord ; to which 
 Guido appropriated the fix fyllables, ut, re, mi, fa, fo], la, beginning firft from r, then from C, then from F, then from G, etc. as here reprefented. 
 
 B C 
 
 . D 
 
 . E F 
 
 . G 
 
 . a & b c 
 
 . d . 
 
 e, 
 
 mi fa 
 
 . fol . 
 
 , la . 
 
 
 
 
 
 ut 
 
 . re . 
 
 mi fa 
 
 . fol 
 
 . la . . . 
 
 
 
 
 
 ut 
 
 re , 
 
 . mi fa . fol , 
 
 . la 
 
 
 
 
 
 ut . 
 
 re . mi fa . 
 ut . 
 
 fol 
 re . 
 
 . la 
 
 etc. 
 
 We fhall have occafion to confider this fcheme hereafter; it may ferve at prefent to fliew Guido's reafon for placing fo particular a regard on thofe letters. 
 It will alfo ferve to account for thofe, feemingly, barbarous names which the muficians ftill ufe, viz. Gamut, A re, B mi, Cfa ut, Dfol re, etc. : as thefe names 
 will appear to be taken exa&ly from Guido's fyftem ; by annexing to each letter all the fyllables by which it may occafionally be cxpreffed, and which, in 
 this fcheme, fall directly below the letter. 
 
Chap. III. 
 
 Of Cliff's, and the Sharps or Flats placed after them. 
 
 56. Thofe founds which afcend ftlll higher than the Treble 
 octave, and which are therefore placed upon leger lines above 
 the Treble ftaff, are denominated by the fame letters, in the 
 fame order, with the addition of the words in alt. ; thus, the 
 fpace above the five lines being called G, the firft leger line a- 
 bove is A in alt. the fpace above that line is B in alt. the fe- 
 cond leger line is C in alt. and fo on. 
 
 On the contrary part, the notes below the Bafs octave, which 
 are placed on leger lines below the Bafs ftaff, are called double 
 Bafs, and expreffed by doubling the denominative letter ; thus, 
 T being the lowed: line of the Bafs ftafF, the fpace below is cal- 
 led double F, and denoted thus F F ; the firft leger line is E E ; 
 the fpace below that is D D ; the fecond leger line is C C ; and 
 fo on. 
 
 57. When any letter ought to be either fharp or flat, in the 
 particular fcale, to which we propofe to adapt our tune, we 
 place the refpective character upon the line or fpace belonging 
 to fuch letter, immediately after the cliff; and we look upon 
 the (harps or flats, which thus ftand at the beginning, as .in- 
 fluencing all the notes, which ftand on the fame lines or fpaces, 
 throughout the whole tune. 
 
 For inftance, if we would write a tune of a fharp feries, for 
 a Treble voice or inftrument, upon the key of G ; we begin 
 with placing the Treble cliff upon the fecond line ; then we 
 confider that if G, which here is the cliff line, be intended the 
 key of the tune, we muft fharpen F, agreeable to the fecond 
 fcale, Art. 48. ; we therefore place the fharp character upon 
 the fpace below the cliff line, which is the place of F ; and be- 
 caufe the fame letter F occurs again, on the uppermoft line, we 
 may alfo place another fharp there, or we may omit this other 
 fharp at pleafure ; it being a conftant rule, that if the fame 
 letter occur twice in one ftaff of lines, and one of its places be 
 marked with either fharp or flat, the other is alfo underftood to 
 be affected in the fame manner, whether the mark be there or not. 
 
 See Plate I. Exam. I. at the beginning: alfo Exam. VII. 
 In thefe the fharp is placed upon both the upper and the lower 
 F, immediately after the Treble cliff. See alfo Exam. IV. and 
 IX. in which only the lower F is marked fharp, the other F is 
 underftood to be the fame, though not marked. 
 
 The key of the natural fcale in each of thefe Examples is G, 
 as we have all along fuppofed it to be; the reader will now un- 
 derftand that fo it muft be, whenever F is made fharp. 
 
 In Exam. VI. there is neither fharp nor flat after the cliff, 
 and therefore, all the letters are natural ; from whence we con- 
 clude that C is the key of the natural fcale, Art. 42. and by 
 counting upward fromG, the cliff line, we find the third fpace, 
 viz. the fpace above the middle line belongs to C ; therefore, we 
 conclude this to be the place of the key. See Art. 1 7. 
 
 In Exam. X. we have two flats, viz. B and E, after the cliff, 
 which correfponds with the eleventh fcale of Art. 48.; and 
 therefore, B[? ought to be the key of the natural fcale; but yet 
 this tune, being one of the mixed or flat feries, has its final or 
 key note upon G, which, as will eafily be feen, is the flxth of 
 the natural fcale. See Art. 35. and the fecond note to Art. 29. 
 
 58. In the fame manner, we may always determine the key 
 of a fharp feries, by the fharps or flats which follow the cliff: 
 and this we may do, by obferving only the number of them, al- 
 though they fhould happen to be a little mifplaced ; for when- 
 ever we fee two fharps, we may conclude that they are, or 
 ought to be, upon F and C, and that D is the key of the fharp 
 feries ; agreeable to the third fcale, Art. 48. Three fharps 
 muft be F, C, and G, with A as key of the fharp feries, as in 
 the fourth fcale; and fo on. 
 
 In like manner we may conclude, when we fee one flat, that 
 its place is B, having F for a key. Three flats are B, E, and 
 A, with B[; asa key; and fo of the reft: obferving always 
 that if the fame letter occur twice in the compafs of the ftaff, 
 and both its places be marked fharp or flat, yet thefe two 
 
Of the Modern Syfteiri- of Muftc. 
 
 Part I. 
 
 marks are to be accounted only as one, agreeable to the pre- 
 ceding Article. 
 
 On the contrary, if we would write a tune of a {harp feries, 
 and intend to make fuch a certain note the key, we may al- 
 ways, by infpecYing the Scheme, Art. 48. find out the particu- 
 lar fcale of our intended key, and the number and pofition of 
 the {harps or fiats, necefTary to be prefixed. 
 
 59. From what is already faid, it will plainly appear, that 
 every perfon, who would make any progrefs in the knowlege of 
 our written mnfic, muft be familiarly acquainted with the two 
 preceding Schemes, Art. 48. and 53.; and it would be well 
 if every reader, who finds himfelf not able fufficiently to re- 
 tain them in his memory, would be at the pains to copy them 
 fair over with his own hands; for I can fafely affirm, from ex- 
 perience, that by once actually copying fuch fchemes as thefe, 
 he- will acquire a more complete idea of them, than by forty 
 times looking tranfitorily over them. 
 
 60. There can be no occafion for any more than once copy- 
 ing over the Scheme, Art. 53. for imprinting it fufficiently 
 on the memory of a learner; but that in Art. 48. being more 
 complex, the learner may ftill find a difficulty in retaining it fo 
 completely in mind, as to be able, without the book, to apply 
 it on any occafion: therefore the muficians find advantage in 
 fubftituting a kind cf technical memory , for this purpofe. Of 
 this fort are the rules for finding mi, which are to be found in 
 different forms, among our authors; the following feems to be 
 more comprehenfive and eafier to retain, than any I have feen ; 
 
 Name thus your {harps, F, C, G, D, A, E : 
 Your fiats call, B, E, A, D, G, and C : 
 Of fliarps the laft ; of flats the following letter's mi. 
 
 The meaning is this ; when we have {harps after the cliff, whe- 
 ther there be more or fewer of them, F is to be accounted the 
 firfl; C, the fecond ; G, the third, and fo on ; according to the 
 
 order in which they fucceffively enter, in defcending from the 
 top of the Scheme, Art. 48.; and when we have named all our 
 {harps in this order, the laft named letter is always the feventh 
 of the fcale, or the mi of our Britifh fyllables, Art. 14. When 
 we have flats after the cliff, B is to be accounted the firfl: ; E, 
 the fecond ; A, the third, and fo on ; according to the order 
 in which they fucceffively enter, in afcending from the bottom 1 
 of the Scheme, Art. 48. ; and when we have named all our flats 
 in this order, the following letter, or that which would come 
 next in courfe to be flattened, is the feventh of the prefent fcale, 
 i. e. mi. This will plainly appear to be the cafe by infpecYing 
 the Scheme itfelf, with the third obfervation in Art. 47.: be- 
 caufe, in adding {harps, the laft added {harp always becomes 
 the prefent feventh ; and in taking away flats, the letter laft re- 
 ftored to its natural place is the prefent. feventh: which letter is 
 always the next following, in the above order of flat letters. 
 
 Having thus found the place of mi, which is the feventh of 
 the fcale; the femitone next above mi, is to be accounted the 
 key. A few examples will make all this abundantly plain. 
 
 For inftance, one Angle {harp is always ¥%, and here F^: be- 
 ing laft, as well as firft, is mi, and G, which is a femitone high- 
 er, is the key. 
 
 Two {harps are F$, and C$ : here C$ is mi, and D, a femi- 
 tone higher is the key. 
 
 Three {harps' are F-$, C$, and G$; here G% is the mi, and 
 A is the key : and fo of the reft, until we come to the fix {harps, 
 of which the laft is E^< ; and this being the fame note with F 
 natural , we therefore ftep a femitone higher to F$, which is 
 then the key; as in the feventh fcale, Art. 48. 
 
 Again, one fingle flat is B\), and here E, which is the next 
 following letter, is mi; and F, the femitone next above E, is 
 the key. Two flats are B\) and E|>; here A is mi; and B[j, a 
 femitone higher, is the key. Three flats are B|>, E\>, and Ajj ; 
 here D is mi, and E[> is the key ; and fo of the reft, until we 
 
Chap. III. Practical Rules for finding the Key, 
 
 come to five flats, of which the Lift is G\j, and the follow- 
 ing letter is C, which is therefore mi, and D[j, the femitone 
 next higher ; is the key ; as in the eighth fcale of Art. 48. 
 
 It may be obferved that C, which is the laft letter of the or- 
 der for flats, is never itfelf marked flat at the cliff; neither is B 
 ever marked fharp. We admit no more than fix (harps, or five 
 flats, at the cliff. 
 
 61. Having hitherto fuppofed the learner to apply his atten- 
 tion to tunes of the fharp feries only, without meddling with 
 thofe of the flat feries; it becomes neceftary here to obferve, 
 that after afcertaining the key of the natural fcale, of any written 
 mufic, by the preceding rules ; if he be not already acquainted 
 with the air of it, he muft look to the final note ; and if he 
 find that it ends upon the key of the natural fcale, he may fafely 
 conclude, that it is of the fharp fort; but if it ends upon the 
 fixth or fecond, he may conclude that it is of the flat fort. He 
 muft obferve alio, that when feveral different flafFs are placed 
 below each other, and coupled together by a large bracket, at 
 the beginning, it fignifies, that while one voice or inflrument 
 goes on with the notes of one of the ftaffs, other voices or in- 
 struments are to accompany with the notes of the other ftaffs ; 
 each in the proper degrees of found, asfhewn by the cliffs pre- 
 fixed to the feveral ftaffs ; fo that one, two, three, four, or 
 more different tunes' are to be carried on together; and the 
 joint effect, of thefe compounded tunes, when properly per- 
 formed, is called harmony ; in contradiftinction to the effect of 
 one Angle tune, which is called melody. 
 
 In regard to this fort of compofitions, we (hall only ob- 
 ferve for the prefent, that each tune, in refpecl: of the whole, is 
 called a part ; and that whatever number of parts be affigned to 
 Treble, or to Tenor voices, or inftruments, there muft always 
 be one Bafs part, which is a kind of fundamental to all the reft ; 
 and which is fuppofed to attract the hearer's attention, at the 
 
 by the Marks at the Cliff: and vice verfa. 23 
 
 ends of feveral ftrains, or portions of the tune ; and more par- 
 cularly at the final concluflon. 
 
 Thefe parts are known by their cliffs, as fhewn Art. 54. and 
 alfo by their fituation under each other: the Bafs is always put 
 loweft, the Trebles higheft, and the Tenors, if any be. are put in 
 the middle between them :' and for afcertaining the key note of 
 one of thefe compofitions in feveral parts, we muft always look 
 to the laft note of the Bafs; becaufe, the hearer's attention be- 
 ing here chiefly upon the Bafs, it is the only part which muft 
 neceffarily conclude v.iih the key. 
 
 The upper parts are often allowed to end upon the third or 
 fifth of the fcale ; fee the note to Art. 30. 5 and yet they muft 
 all have the fame key as the Bafs ; or otherwife they could not 
 be heard together, with any fatisfaction. This does not contra- 
 dict what was faid, Art. 29.: for if one of thefe upper parts, 
 which ends upon the third or fifth, were performed alone, fuch 
 an ending would not be fully fatisfaftory ; but would feem to 
 require fomething more to follow, contrary to the charadterif- 
 tics there given of the key note. 
 
 62. After afcertaining the place of the key note, of any writ- 
 ten tune, the places of all the other degrees of the fcale will be 
 known of courfe ; and it is worth a learner's while to obferve, 
 that if the key note be upon a line, the third, fifth, and feventh 
 above will alfo be upon lines; and the fecond, fourth, fixth, 
 and octave above, upon fpaces : on the contrary, if the key 
 note be upon a fpace, the third, fifth, and feventh above will 
 alfo be upon fpaces; and the fecond, fourth, fixth, and octave, 
 upon lines. This remark has been found of confiderable ufe, 
 especially in learning to ling, at firft fight, by the notes ; be- 
 caufe the finger has nothing elfe to direct his choice of notes, 
 but the knowlege of their feveral lelations to the key, and their 
 peculiar effects, as defcribed, Arf. 21.; fo that every remark 
 which tends to make thefe relations more obvious, is a valuable 
 
24 Of 'Time. 
 
 acquifition to the finger in particular, and indeed to every per- 
 former in general; for although feveral inftruments be managed, 
 in a mechanical way, by the lingers, yet the performer's ear and 
 judgment muft affift him to relith, and enter into the true fpirit 
 of his piece, as obferved, Art. 25. 
 
 We come next to explain what belongs to the article of Time ; 
 
 Part. T 
 
 this being indifpenfably neceffary to be underftood, before a per- 
 fon be qualified to attempt the performance of mufic, by help 
 of the notes only ; for it is to no purpofe that we know how to 
 give each note its proper found, unlefs we alfo know how long 
 time each found ought to be continued. 
 
 Ch a^. IV. Of T 1 m e, and the Charatlers relating thereto. 
 
 • ' \l t\ A ^ t ^ ie different degrees of tune or pitch are re- 
 3. j-^ p re f en ted by the fitnations of certain charac- 
 ters, upon different lines or fpaces, with proper cliffs, and other 
 marks prefixed, in the manner already explained ; fo the diffe- 
 rent proportions of time, or duration of the founds of mufic are 
 fhewn by feveral forts of characters, which we make ufe of, for 
 that purpofe; their names, forms, and meanings, are to befeen 
 in Plate II. Exam. XL 
 
 64. The femibreve, which is a round note, expreffes the 
 
 longeft duration of time, among our modern characters ; and. 
 may be efteemed as a kind of meafure which regulates all the 
 reft: the minim is the fame round note, with a ftem added to 
 it, and denotes half the time of a femibreve: the crotchet is a 
 black head with a ftem; and is half a minim, or a fourth part 
 of a femibreve : the quaver has a ftroke a-crofs the ftem, or elfe 
 the extremity of the ftem turned back again in the form of a 
 hook, and is half a crotchet *'. The femiquaver has two ftrokes 
 a-crofs the ftem, or elfe a kind of double hook ; and is, as its 
 
 * There is an impropriety in calling the ftraight-ftemmed note a crotchet ; which, though it may be but of little importance to the mufician, is ftill lefs 
 honor to the linguift, who fiift adopted this name into our language. The word feems to come directly from the French crochet, which fignifies a hook, or 
 pick-lock; and therefore belongs properly to the note we call a quaver ; and, which makes the nu'f-application ftill more inexcusable, the French muficians 
 have all along called their hooked note by the name of creche. 
 
 The names femibreve and minim are not very proper at prefent, but may be difpenfed with, if we confider the firfl: rife of them. For fome ages, after 
 the invention of lines and cliffs, they ufed nothing but three forts of fquare characters, of different fizes, which they placed in various manners, to denote 
 the proportions of the time of founds; thefe characters were called by the Italians, majjima, longa, and breve ; meaning the largeit, the long, and the fliort, 
 refpectively. Afterwards came the femibreve into ufe, which was at firft of a diamond form, but is now made round ; then followed the minima, or fhorteft 
 note ; and hence arofe the two names, femibreve and minim, which we dill retain. 
 
 The fhorter notes are of later date, and feem to have been taken in one after another, as the longer were caft off: however, it it is more than probable, 
 that in former times, the maffima, or large, was fung not much longer than our femibreve is at prefent, in church mufic ; fo that the alteration is rather in the 
 forms and names of the feveral notes, than in the real length of the founds of mufic. 
 
 We here fubjoin the French, and Italian names of the notes 
 
 Quaver, 
 
 English. 
 
 Semibreve, 
 
 Minim, 
 
 Crotchet, 
 
 French. 
 
 Ronde, 
 
 Blanche, 
 
 Noire, 
 
 Italian. 
 
 Semibreve, 
 
 Miiiima, 
 
 Scmimbrima, 
 
 Examp. XI. 
 
 Semiquaver, Demi-quaver. 
 
 Croche, Double Croche, Triple Crochc. 
 
 Chroma, or Fufa, Semichroma, Bif -chroma. 
 
Plate 1?* 
 
 Exam. 1. 
 
 * 
 
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 Ex.5i 
 
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 5E=oE=5r^ 
 
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 Uf. 7\?. mi 
 Ex. 6 * 
 
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 7. K. 2. 3. 4. 5. 6. 7. K. 2. 3. 4. 5. 6. 7. K,&c 
 
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 Jfa/ia?iSy/£ji~. uf. re. -w/. /«. /?/. yfe. ./£'. uf. re. zvt. /a. /&/. (a. /c. 
 
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 l^^^a^^^a^^ ^ ^j ^ a i 
 
Qhap.IY. Of 'Notes, Re/Is, Augmentations , and^Diminutions. The 'Divifion of all regular Mufic into Meafures. 25 
 
 to be performed in the time of one minim : three quavers, 
 figured over with a three, are equal to one crotchet ; and fo on. 
 See Exam. XIII. where the figured notes are placed above, and 
 
 name imports, the half of a quaver. Demi-femiquavers, fome- 
 times called, by contraction, demi-quavers, are joined by three 
 crofs ftrokes, and are each the half of a femiquaver: thefe arc 
 the fhorteft notes which ordinarily occur in modern mufic ; four 
 of them are equal, to one quaver; eight, to one crotchet; fix- 
 teen, to one minim; and thirty two, to one femibreve. 
 
 6$. After each of thefe notes, in Exam. XI. is placed ano- 
 ther character, which is called a Reft, or Paufe ; and is ufed 
 to denote a filence, or cefiation of found, equal in length to the 
 value of the, note which there precedes it, and from whence it 
 takes its name : thus, the black fquare character which projects 
 downward from any line, is called a femib.reve reft * ; the fame 
 character projecting upward from any line, is a minim reft; 
 the crotchet reft is like a tenter-hook, turned towards the right- 
 hand ; the quav;er reft is the fame, turned towards the left-hand; 
 the femiqtiayec reft is a double hook ; and the demi-femiquaver 
 reft, a triple hook, turned towards the left hand. 
 
 66. A dpjt or point put after any of thefe notes augments its 
 value, and makes it juft one half longer: thus, a pointed femi- 
 breve is equal to a femibreve and a minim, or three minims : 
 a pointed minim is equal to a minim and a crotchet, or three 
 crotchets : a, pointed crotchet is equal to three quavers, and fo 
 on. See Exam. XII. where the pointed notes are placed above, 
 and the equivalent plain notes below. 
 
 6.7. The figure 3, placed over three equal notes of any kind, 
 fhews that thofe three notes are diminished in value, fo as to, be 
 all performed in the fame time which two of them would other- 
 wife require :. thus, three crotchets, figured over with a 3, are 
 
 the equivalent plain notes below. 
 
 It may beobferved from this, and the two precedingexamples, 
 that if we pitch upon any plain note, except the longeft or the 
 fhorteft, we can always exprefs its double, or its half; and its triple, 
 or its third part, by the above methods of notation : for the 
 next longer plain note is always double, and the next fhorter is 
 half; the double note, with the addition of a point, becomes 
 triple; and the half, by figuring it over with a 3, becomes a 
 thhd part. 
 
 68. To explain the proper method of applying thefe charac- 
 ters, we obferve, that although the fucceffive founds of mufic 
 are often vaftly unequal, in refpcct to their length, or duration; 
 yet ftill there are certain due proportions of time which muft be 
 adhered to, with the greateft care and exactnefs, or otherwife 
 the mufic will appear wild and incoherent : and in order to dif- 
 cufs this matter fully, we muft firft take notice of the natural 
 divifion of all regular mufic into certain equal-timed parcels, 
 called Meafures. 
 
 69. This meafure is, in regard to the time of mufic, fome- 
 thing like the key note, in regard to the tune or pitch of founds ; 
 for as every degree of found muft bear its proper relation to one 
 eftabliihed key note, fo the length of every found muft bear a 
 due proportion to one eftabliihed meafure, which runs through 
 the whole piece f . This equal-timed meafure fometimes com- 
 prehends more, and fometimes fewer different founds, according 
 
 * The femibreve reft is alfo ufed to denote the reft of a bar, even though the bar fhould contain more or lefs than 
 fliew : there were alfo other characters formerly in ufe, which denoted the reft of two, or four, or eight bars, or m 
 and a more intelligible method of exprefling fuch long filences is fubftituted in their ftead. 
 
 The Italians name the refts in the fame manner we do, viz. [unfa di femibreve, pat/fa di minima, paufa di femiminir, 
 concife names which are well worth our notice : they call the femibreve reft, {imply uhc paufe ; the minim reft, demi-paufe 
 a figh ; the quaver reft, demi-foupir ; and the femiquaver reft, quart defoupi, 
 
 a femibreve, as we fliall hereafter 
 >re ; but they are now laid afide, 
 
 etc. ; but the French have more- 
 die crotchet reft, foapir, which is 
 
 f There is a moft remarkable analogy between the choice of the mind i 
 yet been taken notice of; and which we fliall endeavour to purfue to ud\ 
 
 egard to the time, and its choice in 
 ntage, when we come to {peak of the theory. 
 
 G 
 
 to the tune of founds, which has never 
 
2 6 Of 'Time. Part. I. 
 
 as the founds are longer or fhorter ; but ftill the length of all 
 the notes, taken together, ought naturally to be the fame, in eve- 
 ry meafure throughout a whole tune. 
 
 70. In written mufic thefe meafures are feparated by fmall 
 perpendicular lines drawn acrofs the ftaff, which we call Bars; 
 and although thefe lines be properly fo called, yet we often ufe 
 the fame word to exprefs the notes comprehended between 
 two bars, i. e. the meafure of the mufic ; fo that a meafure and 
 a bar are often fynonymous terms. 
 
 71. Befides this neceflary diftribution of the fucceffive founds 
 of mufic into certain equal-timed parcels, called meafures, or 
 bars ; it is alfo requifite that each meafure be conceived as di- 
 vided and fub-divided into fmaller equal parts : and on exami- 
 nation of the feveral kinds of divifions and fub-divifions, which 
 are admitted in the meafure of mufic, it is very remarkable, that 
 we never find any other parts, but fuch as arife either from bi- 
 fedtion or trifedtion ; that is, fuch as are either halves, or third 
 parts of the whole meafure, or of fomeof its parts. Thus the di- 
 vifion into four, or eight, or fixteen parts, arifes from a conti- 
 nual bifedtion : the divifion into five equal parts is not admitted : 
 a fixth part may be either the half of a third part, or the third 
 part of a half: a feventh part is never ufed : a ninth partis 
 the third, of a third part ; a tenth part is rejected on account of 
 its implying a divifion by five, which is not admitted ; tor the 
 tenth part would either be the half of a fifth, or the fifth part of 
 a half : an eleventh pait is impracticable: a twelfth part may 
 be either the half of a fixth, or the third part of a fourth. 
 
 Thus we might proceed to afcertain the other practicable and 
 impracticable divifions of the meafure in mufic, which have their 
 parts ftill fmaller and more numerous than thofe above-mention- 
 
 * This latter mood is called by the Italians, tempo alia breve; in contra-diitincTiion to the former, which is called tempo alia femihreve ; which terms feem 
 to intimate that the quantity of two femibreves, or a breve, molt properly belongs to the mood of the barred C: but although, in this mood,_ the notes 
 ought to be made fnorter than in the other, yet they are not fo much fhortened as to bring the whole breve into the time which a iemibreve requires, in the 
 other mood ; nor indeed is there any fixt rule for the comparative proportions of the time of a bar, in different moods. 
 
 ed ; the fame rule ftill takes place, viz. that the number of e- 
 qual parts which we imagine the meafure to contain, muff be a 
 number fome way produced, by repeated multiplication, from 
 one, or both of the fmall factors, 2 and 3. But we need not to 
 purfue this fpeculation any further at prefent. 
 
 7 2. If we conceive the entire meafure to be continually bifect- 
 ed, which is the eafieft and moft natural fub-divifion, fuch a 
 partition conl'.itutes the fpecies of time, called double or com- 
 mon time; which we mark, in written mufic, when flow, by a 
 large C after the cliff; when quicker, by the fame C, with a 
 bar drawn down through it. See Exam. XIV. and XV. 
 
 Thefe marks for denoting the fpecies of the time, are called 
 Moods. In the firft of the above-mentioned moods, which is 
 a plain C, every bar contains the quantity of a femibreve; in 
 the other, which is a barred C, we fometimes find the quantity 
 of two femibreves ; and fometimes only one in every bar *. 
 
 73. For acquiring a proper idea of the natural fub-divifions 
 of the meafure in common time, the following obfervations 
 will be found of great ufe to a learner. Upon applying a watch 
 to our ear, and Mening to its beats or pulfes, we always find 
 them proceeding by pairs, i. e. by two and two together ; which 
 is owing to the pulfes being alternately a little ftronger and 
 weaker, even in the very beft of watches ; and they are often much 
 more fo, in the common fort. Thefe pulfes we can moft eafily 
 count, 1,2, 1,2, 1,2, and fo on; and in this way of count- 
 ing, each fingle pulfe may properly enough reprefent the time 
 of a femiquaver. We can alfo, with a very little attention, place 
 our regard on the alternate ftronger pulfes, and difregard the 
 weakerones.foas to apply the fame way of counting, i;2; i;2; i;2; 
 etc. in a flower manner, to the fucceffive pairs of fingle pulfes ; 
 
Chap. IV. 
 
 Of the Sub-divifwns of the Meafure ; and fir ft of Common Time. 
 
 confidering each pair, for the time, as conftituting one pulfe of 
 a flower fort, anfwering to the time of a quaver. We can pro- 
 ceed yet further, in the fame manner, to regard two of thefe 
 pairs, i. e. four Angle pulfes as conftituting one pulfe of a ftill 
 flower fort, anfwering to the time of a crotchet ; and thefe par- 
 cels, of two pairs in each, we can alfo count, 1:2:1:2:1:2: etc. 
 or we can count them, 1! 2: 3: 4: i: 2: 3: 4: and fo on; which 
 laft method reprefents exactly the divifiori of the common time 
 meafure into four crotchets, and the fub-divifion of each crot- 
 chet into four femiquavers. 
 
 This being a moft ufeful, and, as far as I can learn, a new 
 method*; we here fubjoin a reprefentation of the fucceffive fingle 
 pulfes, by a row of frriall perpendicular lines; alternately longer 
 and fhorter, to denote the ftronger and weaker, or, as we may 
 call them, the naturally accented, and unaccented pulfes; under 
 which are placed figures to illuftrate the method of counting, 
 as above recommended, by 4 in a parcel. 
 
 I I I I I I I 
 
 I I I I 1 I I 
 
 123412 etc. 
 The fpace of time, from 1 to 2, from 2 to 3, etc. in this way, 
 is almofta fecbnd of a common houfe-clock; and the time from 
 1 to 1 again, is to be efteemed the entire meafure, or bar. It 
 matters not though the real length of the meafure ought to be 
 made fomething more, in Church mufic ; and often much lefs, 
 in Opera mufic: the learner, who by practice in counting the 
 pulfes of a watch, after this manner, has rendered this fort of 
 fub-divifion familiar to his mind, will find no difficulty in ap- 
 plying it to any meafure, whether longer or fhorter. 
 
 * Several authors advife the counting of the fingle beats of a houfe-clock for crotchets-, but a watch is vaftly preferable; becaufe the quick pulfes of a 
 watch, taken by 4 and 4 together, give an idea of the natural fub-divifion of the crotchet; and this fub-divifion is as effential to the common time moods, 
 as the primary divifion itfelf. The watch is alfo preferable, becaufe a learner will have it ready, in any place. 
 
 27 
 
 74. The cuftom of maintaining fome uniform motion, either 
 with the hand or foot, to denote the proper divifion of a tune 
 into meafures, and the fub-divifion of each meafure into certain 
 parts ; which is called beating time, is very advantageous to a 
 learner; and abfolutely necefTary for the moft expert performers, 
 when a number join together, in the parts of a harmonic com- 
 pofition : with this difference, that the learner ought to begin 
 with quicker, and more frequent motions ; becaufe, it re- 
 quires practice to judge of the equality, when the motions in 
 beating time are very flow; fo that in fuch moods as the expert 
 mufician needs only one motion for a whole bar, the begin- 
 ner ought toufe two, or, perhaps, four motions in the fame time. 
 
 75. It is beft to begin to beat flow common time with four mo- 
 tions in a bar, thus ; let the learner fit with a table or defk before 
 him, and applying a watch to his ear with the left hand ; let him 
 count its pulfes by 4 and 4 together, as directed Art. 73.: let 
 the right-hand be put down at 1 , the fingers raifed a little, with- 
 out moving the wrift from the table, at 2 ; the hand raifed to- 
 wards the left breaft, at 3 ; and from thence towards the right 
 ear, at 4 ; then down again at 1 ; and fo on. 
 
 This method muft be followed till he can f^rform thefe four 
 motions, in equal time, without the help of .he watch; which 
 will coft fome attention and practice ; but it will, in the iffue, 
 abundantly requite all the pains which may be bellowed up- 
 on it. 
 
 76. Thofe who apply themfelves to the practice of inftrumen- 
 tal mufic, in which the hands are to be otherwife employed, 
 may, inftead of the four motions of the hand, as above, fubfti- 
 tute correfpondent motions of the foot, fuch as thefe ; viz. while 
 the heel continues conftantly on the ground, let the toe be raif- 
 
•>3 
 
 cd a little, and fo kept in readinefs, to be put down exactly 
 at i ; railed again at 2; put down again at 3 ; and raifed at 4 ; 
 and fo on : and in order to attain the habit of performing thefe 
 alternate motions in equal time, fuch as have not the opportu- 
 nity of counting a watch, or a houfe-clock, will find it ufeful 
 to ftand and move the whole body, inclining a little for- 
 ward, and back again, as the toe is put down and raifed % but, 
 if this is done, it muftbe with Ipecia! caution, that the motion 
 of the body be no longer ufed, than till the learner can keep e- 
 qual time without it ; which he will eafily experience, by try- 
 ing bow he fucceeds, when fitting. There is nothing more 
 ridiculous than to fee a man's body vibrating like a pendulum, 
 all the while he fings or plays. 
 
 77. When the manner of beating ilow common time, by four 
 motions in a bar, is become pretty familiar to the learner, be 
 ought to proceed to the quicker mood of the barred C ; in which 
 it is proper to begin with only two motions in a bar; and in- 
 deed, after a little practice, even in the fioweft mood., the mo- 
 tions may be reduced to two. in a bar; viz. a putting down 
 of the hand, or toe, at the beginning of the bar, and c raifing it 
 at the middle ; tlem down again at the next beginning, and 
 fo on. r f 
 
 78 There are two other moods of common time, frequently 
 ufed in Opera mufic, which are ftill quicker than the barred C ; 
 viz. one marked with the fame barred C, but inverted, which 
 is called the retorted mood ; and another marked with a 2 above, 
 and a 4 below : fee Exam. XVI. and XVII. The retorted mood 
 contains the quantity of a femibreve in each bar; but then the 
 length of time due to each note, is only about half of what be- 
 longs to them in the flow mood ; fo that the motions are about 
 the fame quicknefs, for two in a bar in this mood, as for four 
 in a bar in the flow mood. The other, and ftill quicker mood, 
 marked with 2 and 4, placed in the form or a fraction, contains 
 only the quantity of two crotchets, or a minim in each bar ; 
 
 Of Time. Part I. 
 
 and here the motions are abont the fame quicknefs for 2 in 
 a bar, as they would be for 4 in a bar, in the mood of the bar- 
 red C ; and as it is moft proper to ufe only two motions in a 
 bar, in the mood of the barred C ; fo here it is beft to ufe only 
 one, viz. to drop the toe at the beginning of one bar, and to 
 raife it at the beginning of the next, and fo on. 
 
 79 If the entire meafure be divided into three equal parts, 
 and yet each of thefe parts fub divided into 2 and 4, etc. like 
 the parts of common, time ; this fort of divifion is called fimple 
 Triple time: for which, if the movement be flow, we take a pointed 
 femibreve, or three minims, to a bar ; and mark the mood with 
 3 above, and 2 below : if the movement be quicker, we take a 
 pointed minim, or three crochets to a bar ; and mark the mood 
 with 3 above and 4 below: fee Exam XVIII. and XIX. 
 
 80. In order to get the proper idea of this fort of time, the 
 counting of a watch by 4 and 4 fingle pulfes together, will ftill 
 be ufeful; but then we muft only proceed to the number 3, 
 and then begin with one again; inftead of going on to 4, as in 
 Art. 73. See a reprefentation of this way of counting, deiign- 
 ed after the fame manner, as that of the above-cited Article. 
 
 I . I I I I M 
 
 I I I M I I I I 
 
 etc. 
 
 3 etc. 
 
 Here, as before, the meafure is fuppofed always to begin at 1 : 
 and in the flower mood of three minims, each fingle pulie re- 
 presents a quaver, and the fingle pair reprefents a crotchet; but 
 in the quicker mood of three crotchets, each fingle pulie repre- 
 fents a femiquaver; and the fingle pair, a quaver. 
 
 8r. For learning to beat this fort of time, let a watch be ap- 
 plied to the ear, as formerly, and its pulfes counted in the man- 
 ner reprefented above; put down the right-hand at 1, raife the 
 fingers a little, without moving the wrift from the table, at 2; 
 raife the hand towards the right ear, at 3 : then down again, 
 
Chap. IV. Directions for beating Common 'time. Of the Moods, Names and Ways of 'beating'TripleT'ime. 29 
 
 at 1, and fo on. If the foot is made ufe of, Jet the toe be put 
 down, and prefTed a little to the ground, at 1 ; withdraw the 
 prefTure, but let the toe remain on the ground, at 2; raife the 
 toe, at 3; prefs it down again, at 1 ; and foon. 
 
 Here alfo, as before, in Art. 74, we have occafion to recom- 
 mend the frequent practice of one, or both, of thefe methods 
 of beating triple time, with three motions in a bar; until the 
 learner can perform the motions, in equal time, without the 
 help of a watch : after which, he will find but little difficulty 
 in conforming to any meafure, whether longer or fhorter ; by 
 making the motions refpectively flower, or quicker. 
 
 A houfe-clock may ferve, in default of a watch, by counting 
 its fingle beats, 1:2:3: 1 : 2: 3: etc. ; fo that here, every fingle 
 beat of a clock flands in flead of four pulfes of a watch; and 
 the length of each meafure is juft three feconds. 
 
 82. There are alfo two other moods of fimple triple time, 
 which exprefs quicker movements, and are often ufed in Opera 
 mufic ; viz. one marked with 3 above and 8 below ; in which 
 the meafure contains only the quantity of a pointed crotchet, 
 or three quavers ; fee Exam. XX. : and another marked with 
 3 above and 1 6 below, in which the meafure is only a pointed 
 quaver, or three femiquavers. 
 
 The eafieft way of conceiving the meaning of thefe, and all 
 other moods, which are marked by figures, placed in the form 
 of a fraction, is, to fuppofe them a real fraction, refpefting a 
 femibreve as the integer ; and thus, the lower figure, or de- 
 nominator, fhews the number of parts into which the femibreve 
 is divided; and is always either 2, 4, 8, or 16, anfwering to 
 minims, crotchets, quaver9, or femiquavers, refpeftively ; and 
 the upper figure, or numerator, fhews how many of thofe parts 
 conOitute one meafure or bar; and in all the moods of fimple 
 triple time, the upper figure is always 3 ; becaufe it is the cha- 
 rafterifiic of this fort of time, that three equal notes go to one 
 meafure. 
 
 83. There are feveral ways of naming thefe figured moods, 
 as fhewn in the following table. 
 
 Moods. 
 
 Names. 
 
 3 C Three to two, three halves, 
 
 2 £Slow triple, or triple of minims. 
 
 3 ( Three to four, three fourths, 
 
 4 I Common triple, or triple of crotchets. 
 
 3 C Three to eight, three eighths, 
 8 £ Quick triple, or triple of quavers. 
 
 3 ("Three to fixteen, three fixteenths, 
 1 6 £ Very quick triple, or triple of femiquavers. 
 
 Of thefe four feveral names of each mood, the laft, which 
 expreffes both the fort of time, and the conftituent note, feems 
 to be the moft proper, and moft intelligible. We fhall there- 
 fore hereafter name the moods in this manner. 
 
 84. The method of beating time, defcribed, Art. 81. which 
 is called three in a bar, agrees very well with the triple of mi- 
 nims, and with the triple of crotchets in Church mufic; but 
 in the triple of quavers or femiquavers, and often in the triple 
 of crotchets in Opera mufic, the fucceffion is too quick to ad- 
 mit of three motions in a bar; and in thefe cafes it is moll conve- 
 nient to ufe only one motion for a bar, putting down the hand, 
 or toe, at the beginning of one bar, and raifing it at the begin- 
 ning of the next bar, and fo on : and in this way the triple di- 
 vifion of the meafure is fupplied by the imagination, while the 
 motions determine nothing but the length of each entire mea- 
 fure. See Exam. XIX, and XX. 
 
 In the very quick triple of femiquavers, we may take fiill 
 greater liberty, and include two, or even four bars, under one 
 motion: but fiill the imagination muft fupply all the divifions 
 
 II 
 
3 o • i Oftt. 
 
 and fubdivifions, which are left undetermined by the motions 
 of beating time. 
 
 85. As we fometimes join together two, or four, quick tri- 
 ple bars, under one motion, in beating time ; fo the compofers 
 often join together two, or -four, quick triple meafures, in one 
 bar ; and from hence arife the forts of time called, in general, 
 mixt triples; which are not fo much a different fpecies of time, 
 as a different manner of barring the fimple triples. 
 
 Thus the triple of crotchets, Exam. XIX. by omitting one 
 bar, and inferring the next, and fo on, alternately, becomes a 
 fextuple, or a double triple, of crotchets, as in Exam. XXI. ; 
 and is therefore marked with 6 above : the triple of quavers, 
 Exam. XX by proceeding in the fame manner, becomes a fex- 
 tuple of quavers, as in Exam. XXII. 
 
 • The joining of four quick triple meafures in one bar, con- 
 ftitutes the dodecuple, or quadruple triple moods, as in Exam. 
 XXIII, and XXIV; which are marked with 12 above : and it 
 is evident, that any of thefe mixt moods might be brought into 
 the form of fimple triples, by inferring the omitted bars, in their 
 proper places. 
 
 86. In regard to the time of thefe mixed triples, it may ge- 
 nerally be efreemed fomething quicker than that of the fimple 
 triples, of the fame denomination; and the dodecuples, fome- 
 thing quicker than the fextuples. It is for this reafon that the 
 fextuple of crotchets, Exam. XXI. is direfted to be beat by 
 two in a bar, moderate ; when the triple of crotchets, Ex. XIX. 
 is made one in a bar, JIow '.- in the fame manner the dodecu- 
 ple of quavers, Exam. • XXIII. is called two in a bar, moderate ; 
 and the fextuple of quavers, Exam. XXII. is oneinabar,y7cw; 
 and fo of the reft. 
 
 e. . Part. I. 
 
 87. We (hall here only add the following table of mixed tri- 
 ples, which are in ufe among modern authors. 
 
 Sextuples, or Double Triples. 
 
 I Of crotchets, marked, 
 
 4 
 
 Sextuple <J Of quavers, „ 
 
 I Of femiquavers, - 
 
 Dodecuples, or Quadruple Triples. 
 
 IOf crotchets, marked, I2 
 
 I2 4 
 Dodecuple <^ Of quavers, Q 
 
 1 i 
 
 I Of femiquavers, , * 
 
 88. The only fpecies of time which now remains to be con- 
 fidered, arifes from conceiving the three primary divifions of 
 fimple triple time, to be each of them again fub-divided into 
 three other fmaller parts ; by which means the meafure becomes 
 of nine equal parts ; and this is called nonuple, or compound 
 triple time ; and diftinguifhed by the figure 9, placed above, in 
 the mood. See Exam. XXV, XXVI, and XXVII. 
 
 8g. If the learner have attained a tolerable idea of the mixed 
 triples laft fpoke of, he will meet with no difficulty in thefe. In 
 the nonuple of crotchets, Exam. XXV. the time muft be beat 
 
 ■ * Mi BrcfTard, in h 
 . ■ ' ■ kcifple of mini 
 op far, the firit rife of thefe m 
 never taken place. 
 
 DiFliiiiaire de Mufiquc, under the word tripohi, takes fome pains to propofe four other flower forts of mixed triples; viz. a fextuple 
 
 s, and a iextuple and dodecuple of femibreves : but he feems not to have confidered that the quicknefs of movement gave occafi- 
 
 bars, as fhewn Art 85.: and it is for this reafon, m oft probably, that his propofal has 
 
 loods, by omitting fome of tl 
 
Chap. IV. 
 
 Of mixt and corn-pound triple Time. 
 
 by three motions in a bar, juft the lame as in the flow triple 
 moods; but then the imagination muft fub-divide each motion 
 into three equal fmaller parts, in the fame manner as it is done 
 in the quick, or mixed, triple moods. 
 
 90. If any difficulty fhould remain, the counting of a watch 
 may ffcill be of fervice, with the following directions. Apply a 
 watch to your ear, andobferve If s ftronger, or accented pulfes ; 
 which will immediately lead you to count the pulfes, by pairs; 
 but inftead of proceeding in the natural order of the numbers, 
 i; 2; 3; etc. let each number be named three times over, be- 
 fore you go on to the next ; and after the number 3, begin a- 
 gain with 1 ; thus, 1; i; i; 2; 2; 2; 3; 3; 3; i; 1; i; etc. When 
 this method is become familiar, you may drop the repeti- 
 tions, and imagine the number, once named, to extend thro' 
 the time of three pairs, or fix fingle pulfes, according to the 
 following reprefentation, which is to be conceived in the very 
 fame manner as thofe formerly given. 
 
 I I I i I I I I I . I I I I I 
 
 The different Values ofRefls on certain Occafwns. 3 I 
 
 See now the following table of the nonuple moods, pre- 
 fently in ufe... 
 
 I 1 ! 
 
 etc. 
 
 etc. 
 
 and in the 
 
 The beginning of the meafnre is always 
 nonuple of crotchets, each fingle pulfe reprefents a quaver, and 
 the fingle pair a crotchet ; and here the meafure has three mo- 
 tions, like flow triple time. 
 
 91. The quicker nonuple moods of quavers and femiquavers 
 feldom admit of more than one motion, in the whole meafure ; 
 and as in the flow nonuple above reprei en fed, the imagination 
 has to fupply the triple fub-divifion of the time of each motion; 
 fo here, the imagination muft fuppiy both the primary triple di- 
 vifion of the meafure, and alfo the triple fub-divifion of each 
 part, in order to form the idea of nonuple time; when the 
 Motions of beating determine nothing but the length of the en- 
 tire meafure, See Exam. XXVI. and XXVII. 
 
 I Of crotchets, marked, 
 
 Nonuple <j Of quavers, -, 
 
 I Offemiouavers, ",, 
 
 92. There a 
 to the values of 
 
 re feveral particulars to be obferved, in .regard 
 the refls, or marks of filence, in each of thefe 
 triple moods. 
 
 Firft, The character commonly called a femibreve reft, as in 
 Art. 65. expreffes the time of a complete bar in all moods ; and 
 ought never to be ufed but for that purpofe. Thus, if inftead 
 of the fecond note m Exam. XVili. which is a femibreve, we 
 had occafion to fubfiitute an equal time of filence, we muft not 
 ufe a femibreve reft, but rather two minim refts ; becaufe here 
 the paufe muft not be a whole bar, as the femibreve reft would inti- 
 mate, but muft admit the following minim to complete the bar. 
 
 Secondly, . The minim reft f lands for the time of half a bar, 
 in all the mixt triple moods, except the dolecuple of crotchets'; 
 although the half b?x is never juft the time of a minim, being 
 always equal to fome one, or two pointed notes. See Exam. 
 XXI. to XXIV. where the minim reft is put to nil up half a 
 bar, at the end of each Example, ' 
 
 Thirdly, To denote the filence of one fourth of a bar, in the 
 dodecuple moods, or of one third of a bar, in the nonuple moods ; 
 which is, in each cafe, the CJbnee of one of the primary divifi-. 
 ons of the bar, we need only to place one fingle reft, whofe 
 ordinary value is but two thirds of the divifion ; and this is the 
 mod iifual method in the nonuple moods. See Exam XXV. 
 to XXVIL where the reft which is put at the end of each Ex- 
 
3 2 
 
 ample reprefents the complete divifion, or third part 
 bar: and here, if it were required to denote a filence equal on- 
 ly to two thirds of the divifion, we muff not ufe the lame cha- 
 racter, although this is its ordinary value, but rather place two 
 of the next fhorter fort, in its ftead. Thus if the fecond note of 
 Exam. XXVI. which is a crotchet, were to be converted into 
 an equal time of filence, we muff not ufe a crotchet reft, be- 
 caufe that character, here, reprefents the whole divifion, confift- 
 ing of three quavers, as at the end of the example ; but 
 we muft rather place two quaver refts, to prevent the am- 
 biguity which would arife from ufing the fame character, with 
 two different meanings. In the fame manner the filence of a 
 minim, in the nonuple of crotchets, muft be expreffed by two 
 crotchet refts ; and the filence of a quaver, in the nonuple of 
 femiquavers, muft be expreffed by two femiquaver refts. 
 
 Fourthly, As the minim reft denotes one third part of a bar 
 in the nonuple of crotchets, fo, by the fame rule, it expreffes 
 one fourth part of a bar in the dodecuple of crotchets; and 
 hence arifes a difference between the meaning of a minim reft, in 
 this mood, and its meaning in all the other mixt triple moods, 
 as mentioned under the fecond head. The minim reft, here, 
 reprefents only one fourth part of the bar, and to exprefs the 
 filence of half a bar, we are obliged to ufe two minim refts. 
 
 Fifthly, Upon the whole it appears, that the character we ge- 
 nerally call a femibreve reft, ought rather to be termed a bar 
 reft ; and that the minim reft may fometimes be more properly 
 called a half-bar reft ; and that if any change be occafionally 
 fuppofed in the value of the minim reft, when taken as fuch, 
 or of any other of the fmaller refts, fuch change is always an 
 augmentation of one half more than their ordinary value, in 
 order to make them equal to the principal divifion of the bar, in 
 fome of the mixed or compound triple moods; and correfponds 
 exactly with the augmentation of notes, by a point, Art. 66. 
 
 If fuch points of augmentation had been admitted after the 
 
 Of Time. Part I. 
 
 of the refts, the comparative values of the refts might have been kept 
 as immutably as thofe of the notes. 
 ■ However, the method prefently received is neither hard to 
 comprehend, nor burdenfome to the memory. It muft be ob- 
 ferved alfo, before we difmifs this fubject, that thefe occafional 
 augmentations are not always left to the judgment of the per- 
 former ; but that the moft correct tranfcribers generally de- 
 note the quarter-bar reft, in the dodecuple of quavers, by 
 two characters, the firft reprefenting a crotchet, and the other 
 a quaver reft. In like manner, the quaver and femiquaver refts 
 are joined together, for a quarter-bar, in the dodecuple of fe- 
 miquavers. See the laft bars of Exam. XXIII. and XXIV. 
 
 93. When mufic confifts of feveral parts, we obferved for- 
 merly, Art. 61. that all the parts which go together muff have 
 the fame key note; we may here add, that they muft likewife 
 all have the fame fpecies of time, and that the fuccefhve mea- 
 fures of all the parts muft begin exaclly at the fame inftant ; fo 
 that the motions of beating time, which anfwer to any one 
 part, muft equally anfwer to all the reft. When thefe 
 parts are carried forward together, in feveral ftaffs, properly 
 difpofed, one below another, as hinted in the above-mentioned 
 Article, the bars which divide the meafures of the feveral parts, 
 ought to fall direelly under each other ; and this is called mil- 
 fie in /core. 
 
 It often happens, that fome of thefe parts are to reft, or be 
 filent, for feveral bars together; in which cafe, if the mufic be 
 wrote in fcore, the bars are drawn in the ftaves belonging to 
 the filent parts, as well as in the other, and every filent bar is 
 marked with a bar reft : but if the different parts are written 
 out in feparate books, the filence of a number of bars is expref- 
 fed by placing one fingle bar reft, with a figure over it, to fhew 
 how many of fuch filent bars is thereby meant. 
 
 94. In the performance of mufic, there is a certain emphafis.. 
 or accent laid on the beginning of every meafure, which plainly 
 
Plate 2^Ex. XI. , 
 
 jSIames and Values. Semibreve,!. Mininl -A- Crotchet -j- 
 
 Notes and Reft *-. 
 
 -© 
 
 Quaver-^- Semiquaver-^- Demisemiquave 
 
 m, i i r j j i H^m m m 
 
 •i 
 
 Ex. XII 
 Q. 
 
 ,1. ■ J . . J^ 
 
 ExXIII, 
 
 I 
 
 •it, it ,t. 
 
 
 S3 
 
 a 
 
 K 
 
 EE 
 
 ode 
 
 Ex.XIV. four in a Bar.moderate 
 
 r^ r r r \ i \ 
 
 i 
 
 ^ 
 
 pp p p pp p p 
 
 ^ 
 
 Bffl 
 
 1234 1234 1234 1 2 34 1234 1234 1234 12 34 
 
 Ex. XV. two in a Bar, slow. Ex.XVI.two in a Bar, moderate. Ex. XVII. One in a Bar,slow. 
 
 #_ #/» P 
 
 1 j II 
 
 €t— O 
 
 jEJDEg 
 
 £>— Q 
 
 S 
 
 tw^ 
 
 12 12 12 
 Ex.XVIII three in a Bar, moderate. 
 
 2 12 1 
 
 Ex. XIX. One in a Bar, slow. 
 
 O— q — q=cp 
 
 rr^: 
 
 Q — p 
 
 cri?' 
 
 BZ 
 
 12 312312312 3 123 1 2 1 2 I 
 
 Ex.XX.One in a Bar, moderate. Ex.XXI.two in a Bar, moderate. Ex.XXII.One in a Bar, slow. 
 
 m 
 
 p-p 
 
 tm 
 
 w? 
 
 q-p 
 
 ffcrcrh-isrTcia: 
 
 p p pp pp 
 
 ess: p 
 
 1112 1 
 
 Ex.XXIII.two in a Bar, moderate. 
 
 12 12 1 
 
 Ex. XXIV One in a Bar, slow. 
 
 1 2 1 
 
 Ex. XXVI three in a Bar, moderate. 
 
 L2bt 
 
 WFPP 
 
 ^f-' i 
 
 ifff^ fi B Bl fr 
 
 J 
 
 1 2 1 
 
 Ex. XXVI. One in a Bar, slow. 
 
 2 12 
 
 Ex XXVII One in-, a Bar, moderate. 
 
 2 3 
 
 Jfr-9- 
 
 ^-HH^m£&^ 
 
 p m^g 
 
 16- 
 
 K V* m 
 
 
Chap. IV. 
 
 diftinguifhes one fpecies of time from another; fo that a hearer 
 is naturally led to diftribute a tune into its proper meafures, 
 though he fhould take no notice of the manner of beating time; 
 nay, though he fhould know nothing at all of the rules relat- 
 ing to the time of mufic. We are naturally pleafed with equal- 
 timed meafures, divided and fub-divided in certain eligible man- 
 ners ; but it is a work of time and fpeculation to afcertain what 
 thofe divifions are, which give us pleafure. 
 
 The emphafis always falls upon the number i, in the me- 
 thods of counting a watch, or clock ; and accompanies the put- 
 ting down of the hand, or toe, in beating time, as directed a- 
 bove. 
 
 95. There is no occafion to make the beginning, or empha- 
 tical part, of the meafure, always ftronger, or louder than the 
 reft, though it is fometimes beft to do fo ; for, it is not fo much 
 the fuperior loudnefs of the found, as the fuperior regard which 
 a hearer is led to bellow upon it, that diftinguifhes one part 
 of the meafure from another. This is a truth of great impor- 
 tance, as will hereafter appear, and deferves to be well fixed 
 in mind, before we proceed. For illuftration of this, it may 
 be obferved, in the method of counting a watch, reprefented, 
 Art. 72- that although the alternate ftronger pulfes, reprefent- 
 ed by the longer lines, be undoubtedly all equal, yet when we 
 count one, and pafs over the next, and count the next, and 
 pafs over the next, and fo on ; we imagine the pulfes which 
 we count, to be really ftronger than the intermediate ones, which 
 we pafs over. The fuperior regard which we beftow on the 
 counted pulfes is, here, the fole caufe of thefe imaginary ac- 
 cents. 
 
 Again, in the method of counting a watch, reprefented, Art. 90. 
 the pulfe which we count acquires a greater emphafis, than 
 either of the two following equal ones, which we pafs over. 
 
 If we extend thefe obfervations ftill further, we fhall find, 
 that in all the methods of counting a watch, the pulfe which 
 
 Of the accented, and unaccented Parts of the Meafure. 
 
 33 
 
 anfwers to i, that is, the pulfe which begins the meafure, 
 feems ftronger than any of the others ; efpecially after we have 
 continued the fame method of counting, for feveral times over. 
 We {hall find, alfo, that the pulfe which anfwers to the num- 
 ber 3, in common time, will leem ftronger than either of thofe 
 which anfwer to 2, or 4 ; and even in triple time, the fame 
 number 3, is more emphatical than 2. 
 
 The cafe is juft the fame with the fucceffive founds of mufic, 
 becaufe we naturally conceive thefe founds, as pofieffing fuch. 
 and fuch parts of the eftablifhed meafure of the piece. The 
 regard, paid to the beginning of every meafure, renders the 
 note, which pofiefies that part, more remarkable, and therefore, 
 more emphatical, than any of thofe which occupy the other 
 parts of the meafure ; the beginning of the third divifion of the 
 meafure, is alfo accented, in a fmaller degree, both in common 
 and triple time; and the beginning of every primary divifion of 
 the meafure is more emphatical than that of any of the fecondary 
 fub-divifions. All thefe remarks are direct confequences from 
 the foregoing obfervations. 
 
 96. From hence, it appears, that the ideas of the length of 
 the whole meafure, and of its proper divifions and fub-divifions, 
 are as conflantly kept in mind, through the courfe of a tune, as 
 the key note was obferved to be, in Chap. II. : and that the 
 conftant rememberance of the meafure and its divifions regu- 
 lates the time, and accent, of each found which we ufe; much, 
 in the fame manner as the key note determines the tune or 
 pitch of all thofe which follow it. 
 
 97-j To preferve the idea of the proper divifion of the mea- 
 fure, it is beft for a beginner to divide all the long notes which 
 occur into parts, anfwer^ng to the parts of the meafure which 
 it occupies, by making feveral d\[Vm&: Jwells in the found ; and 
 this may be very eafily done with the voice or violin, or any 
 wind inftrument; but is impracticable in the pulfatile kind, 
 fuch as, the harpfichord, lute, harp, guitar, etc. ; becaufe the 
 
34 
 
 Of time. 
 
 Part. I. 
 
 founds of thefe inftruments muft inevitably decreafe from ntft The fyncopated note often happens to be prolonged from the 
 
 to laft, and are not capable of any fwells, without repeating the end of one meafure to the beginning of another; in which cafe, 
 
 flroke: and hence we may conclude that it is much more diffi- if it be a femibreve or a minim, and one half of it belonging to 
 
 cult to keep true time, with fuch inftruments, except the voice each meafure, the bar may be drawn down through the middle 
 
 be accuftomed to go along with them. of the note; or it may be reprefented by two fhorter notes, 
 
 For an inftance of this way of dividing long continued founds, with the bar between them, and an arch drawn over the whole ; 
 
 let a learner fing the beginning of Exam. XIV. which is in 
 flow common time, and directed to be beat with 4 motions in a 
 bar; applying the fyllable la to each note. 
 
 Here the firft note, being itfelf a complete bar, muft employ 
 all the four motions ; let it therefore have four diftinft and e- 
 qual fwells, anfwering to the four motions, or parts of the bar, 
 thus, la — a — a — a. 
 
 The fecond and third notes, by the fame rule, are to have 
 two fwells each, thus, la — a, la — a. 
 
 The four next notes, which conftitute the third bar, are juft 
 equal to the four divifions of the meafure; and therefore are 
 fnng, fimply, la, la, la, la. It is to be obferved, that thefe 
 fwells are of no ufe, except when fome primary divifion of the 
 meafure begins within the time of one continued note. 
 
 98. All notes, which, by being longer than the primary di- 
 vifion of the meafure, muft necefTarily belong to two, or more, 
 
 which (hews that they are be confidered as one note : fee the 
 letter M, in Plate III. Exam. XXVIII.; where the notes in the 
 lower ftaffare defigned as a kind of explanation of thofe in the 
 upper ftafF. 
 
 Sometimes a note, at the end of one meafure, is carried on 
 to the beginning of the next meafure, by means of a point of 
 augmentation, in which cafe the bar is drawn between the note 
 and the point: fee letter N in Exam. XXVIII. where the point- 
 ed note in the upper ftafF is explained below. 
 
 The notes at H, I, K, in this Example, are to be efteemed 
 protracted notes ; and thofe at L, M, N, O, P, Q^ are fynco- 
 pated. 
 
 99. If it be convenient to divide the protracted note into di- 
 ftindt fwells, for preferving the proper idea of the time, as di- 
 rected Art. 97. ; it is much more necefTary, to divide the fyn- 
 copated note in this manner, by making a perceivable fwell 
 
 divifions, may be diftinguifhed into protracled and fyncopated when the accented part of the meafure begins ; and this is the 
 
 notes. Such as begin with a part of the meafure which has an 
 equal, or a greater emphalis than any of the fucceeding parts 
 which they are continued upon, are called protracted or length- 
 ened notes; but fuch as begin with a lefs emphatical part, and 
 are afterwards continued upon a more emphatical part, are cal- 
 
 reafon why we generally find it beft to ftrike thefe notes twice, 
 
 in playing the harpfichord, or any other pulfatile inftrument, 
 
 after the manner reprefented in the lower ftafF of Exam. XXVIII. 
 
 100. When an equal time of filence is fubftituted inilead of 
 
 fyncopated note, we ufe two characters reprefenting the two 
 
 led fyncopated notes : or, in other words, the protracted note parts of the note, as it ought to be divided by the fwell. Thus, 
 is moft accented at the beginning, and the fyncopated note is if the note at L, were to be converted into a filence, we muft 
 910ft accented fome time after the beginning*. place two crotchet refts, rather than a minim reft. In like 
 
 •■The word fyncopated takes is rife froffi a Greek verb, synkopto, which iignifies to beat or ftrike, and probably has been appropriated to thefe forts 
 of notes, becaule it generally happens, that we are to make one of the motions of beating time, during the continuance of the note. 
 
Chap. IV. Of protracted and fyncopated Notes. Poetry 
 
 manner, any of the notes at O, P, Q^ muft be reprefented by 
 two quaver refts, rather than a crotchet reft. 
 
 101. The divifion of mufic into equal timed meafures, an- 
 fwers exactly to the divifion of poetry \n\ofeet ; and when mu- 
 fic is adapted to poetry, thefe divifions, moft naturally, coin- 
 cide with each other; fo that he who can /can the verfes, may 
 immmediately difcover the meafure of the fong. 
 
 It muft be acknowleged, that this order is pretty often inter- 
 rupted, efpecially in the works of the more eminent compof- 
 ers; and more, or lefs, than one foot of the poetry, allotted to 
 one meafure of the mufic : but then, fuch pafiages are, in 
 fome degree, (trained and unnatural ; and are introduced for 
 variety, or for heightening the expreflion of fome paffion, etc. 
 and ought to be ufed with great caution and {kill. 
 
 The moft natural and eafy paffages are expreffive of a calm 
 unruffled temper of mind ; but when any violent emotion is 
 fuppofed to take place, the ft riff, rules both of tune and time, 
 in mufic, may, and ought to be partly fet afide. 
 
 1 02. Our attention is ftill more lhble to be diverted from 
 obferving the ftrict rules of time, by the fenfe of the words, in 
 poetry; but fo far as we may be fuppofed at liberty to regard 
 the time, of poetry, we ftiall find that the very fame rules take 
 place here, as in mufic, viz. the fucceflive feet of a verfe, 
 moft naturally, require each an equal time of pronunciation ; 
 the firft fyllable of every foot is accented ; and every foot is, in 
 imagination, divided either into three, or into four, equal parts. 
 The two firft of thefe particulars will plainly appear to all, who 
 are, in the leaft degree, accuftomed to the reading of poetry; 
 and the laft particular, though not quite fo obvious, will be 
 found equally true, on a more careful examination. 
 
 103. Befides the diftribution of mufic into equal meafures, 
 it is alfo neceflary to go yet further, and to imagine fome num- 
 bers of fuch meafures, as conftitutiog certain phrafes, or Jlrains, 
 of a tune. Thefe phrafes may, very aptly, be compared with 
 
 and Mufic compared. Hints relating to Expreflon. gy 
 
 verfes, in poetry : for, as there can be no poetry, without a 
 proper intermixture of cadences, at the ends of the lines, fo 
 there can be no mufic, without fome kind of partition into 
 phrafes. 
 
 Thefe phrafes contain more, or fewer meafures, as verfes 
 confift of more or fewer feet ; but both muft always end with 
 an accented part of the meafure. 
 
 When the fucceflive phrafes in mufic are of unequal lengths, 
 it refembles that kind of free, unconfined poetry, which is 
 commonly called Pindaric: and, as this fort of compofition is 
 the moft capable of variety of expreflion ; fo, the greateft ma- 
 ilers, both in poetry and mufic, often make ufe of it. 
 
 1 04. A lively expreflion of the feveral fentiments and paflions, 
 is undoubtedly the perfection of mufic, as well as of poetry and 
 painting. There are numberlefs different modifications of 
 founds, which a fkilful compofer may avail himfelf of, for this 
 purpofe ; fuch as the different qualities of loud and foft, of 
 hoarfe or rough, and clear or fmooth founds : the various de- 
 grees of gravity and acutenels, in the pitch of the whole piece ; 
 the different effects of certain degrees of the fcale, and of certain 
 fucceflions in the melody of fingle parts, as well as of confonan- 
 ces, in the harmony of compounded parts; befides feveral other 
 circumftances in the manner oi performance, fuch as the diftinct, 
 oxjlepping, and the indiftinct, or fliding manner ; the keeping 
 one uniform equality of loudnefs, and the occafional fwelling 
 or foftening of the founds, etc.; and, among the reft, the dif- 
 ferent moods of time, have no fmall (hare in contributing to the 
 expreflion of mufic. Thefe come in cour'e to be fpoke of, be- 
 fore we conclude this Chapter. 
 
 105. The particular manners, and modulations of the voice, 
 which, naturally, or by the cuftomof a particular country, habitu- 
 ally accompany fuch emotions of the mind, in common fpeech, 
 are the fureft guides to expreflion in mufic. From hence we 
 conclude, in general, that flow or quick movements of mufic 
 
3 6 
 
 ought to be introduced, according as the fentiment, intended 
 to be expreffed, would require a flow or quick delivery, in the 
 way of fpeaking : and of this it is very eafy to judge. For in- 
 ftance, forrow, humility, and reverence, require a flow move- 
 ment, with gentle, eafy inflexions of the voice; but joy, thankf- 
 giving, and triumph, ought to be diftinguifhed by a quicker 
 movement, with bolder inflexions, and more diflant leaps, 
 from one found to another. A moderate movement, with fre- 
 quent fwells, andfoftenings,is expreffive of tendernefs andcom- 
 paffion ; a quicker, more, uniform, and ftrongly accented move- 
 ment, exprefles refolution and fortitude. Anger is generally 
 quick, loud, and unconnected; hope and expectation, more 
 moderate, foft, and eafy, and fo of others. 
 
 106. The different forts of time have, in fome degree, each 
 their peculiar character. Common time is naturally more grave 
 and folemn; triple time, more chearful and airy. And for this 
 reafon, it is generally agreed, that every mood of triple time 
 ought to be performed fomething quicker, than the correfpon- 
 dent mood of common time ; for inftance, the meafure in 
 the flow triple of minims, ought to be made fhorter than the 
 meafure in the flow common time, marked with a plain C ; 
 and the meafure, in the triple of crotchets, fhould be fhorter 
 than the meafure, in the mood of the barred C; and fo on. 
 
 After all, it muft be acknowleged, that the abfolute time 
 which ought to be allowed to different pieces, is the moft 
 undetermined matter, that we meet with, in the whole fci- 
 ence of mufic. There is one infuperable difficulty, which 
 fruftrates ail attempts towards regulating this particular, viz. 
 the different humours and taftes of different perfons ; which 
 are fo various, that one perfon fhall think a tune much too 
 quick, for the intended expreffion, while another thinks it not 
 quick enough. 
 
 1 o/. If we proceed upon thefe principles, which feem moft 
 reafonable, that thofe who have a brifker flow of fpirits, a more 
 
 Of Time. Part I. 
 
 ready conception, and a quicker fucceffion of ideas, require 
 quicker mufic, for the fame expreffion, and vice verfa; we 
 may conclude, in regard to Church mufic, that the fame Pfalm 
 ought to be fung quicker, when the congregation confifts moftly 
 of young people; and flower, when the greater part are old : 
 quicker, in general, in a town, than in a country church; 
 quicker, in places where mufic is more generally praftifed; and 
 flower, where it is lefs in ufe : quicker, when only one Angle 
 part is fung, and flower, as the parts are more numerous ; be- 
 caufe the ideas of fingle founds are much more readily conceiv- 
 ed, than thofe of feveral founds, joined together in harmony : 
 quicker, when ihe voices are few and weak, and flower, when 
 the choir is numerous and flrong ; becaufe nothing can be quite 
 agreeable to the hearers, which feems laborious to the perform- 
 ers. Many other fuch like diflinclions, according to the various 
 circumftances, both of performers and hearers, will occur to 
 the coniiderate reader, from the fame principles. Thefe obfer- 
 vations may, with equal propriety, be extended to Opera mufic. 
 
 The Italians, whofe compofitions are juftly efleemed the 
 flandards of true tafte in mufic, do not reflricT: themfelves alto- 
 gether to the diftiniftions of flow and quick, by the feveral 
 moods, as above defcribed ; but rather make ufe of certain words, 
 placed at the beginning of the piece, and elfewhere, asoccafion 
 requires ; which ferve to direct the performer, not only in re- 
 gard to the time, but alfo the particular expreffion, and manner 
 of performance, as we fhall fhew in the next Chapter. 
 
 108. We fhall conclude this Chapter, with obferving, that 
 the writers on Church mufic feem to be pretty well agreed, 
 that the time of a fecond, may ferve, at a medium, for the 
 length of a crochet, in Pfalm tunes, in the triple of crotchets, 
 and in the mood of the barred C; and that the minim, in 
 the triple of minims, ought to be made nearly equal to the 
 crotchet, in the mood of the plain C ; and that either of thefe 
 two ought to be longer than the fecond of a clock. 
 
Chap. V. 
 
 Of- a double Bar, a Hold, a Clofe, and a Repeat. 
 
 37 
 
 Chap. V. Miscellaneous Explanations. 
 
 J f l ioo nr^HERE are Several other characters and 
 "' JL terms ufed in mufic, which have not yet 
 come under confideration. We fliall, in this Chapter, explain 
 fuch of them as are mod frequently met with, and moft necef- 
 fary to be underftood. 
 
 A double bar is ufually put at the end of every line, in Pfalm 
 tunes; but in other kinds of mufic the double bars are not fo 
 frequent, and are ufed only at the ends of fome larger fections, 
 or parts of the tune. 
 
 When the laft meafure, or bar, of the part is complete, the 
 double bar alfo fupplies the place of a fingle bar, for dividing 
 the time, as mentioned, Art. 70. et feq. : but the double bar 
 often takes place when the meafure is not complete, in which 
 cafe, the remainder of the meafure follows immediately after, 
 and the double bar fhews only the end of the fecYion or part, but 
 is of no fignifkation in refpedl: to the divifion of the time. 
 
 It muft be obferved, that although the double bar, when it 
 occurs, makes no change in the progreffion of the meafure and 
 its divifions, as denoted by the fingle bars ; yet it generally im- 
 plies a paufe in the performance; but then it is a different kind 
 of a paufe from thofe which we call refts, as explained in the 
 laft Chapter : the meafure is conceived as regularly going on 
 during the time of the reft; but the double bar implies a kind 
 of unlimited, or difcretionary paufe, not only in the progreffi- 
 on of founds, but alfo in the meafure itfelf : or, in other words, 
 the reft is a meafured filence, but the paufe, at the conclusion 
 of a part, is unmeafured ; and may be made longer or fhorter, 
 at pleafure. 
 
 no. The fame kind of unmeafured paufe or fufpenfion is 
 
 often introduced in the middle of a piece, to give the finger an 
 opportunity of dwelling upon fome particular word, beyond its 
 due time, for heightening the expreffion, or for other reafons, 
 at the difcretion of the compofer, and is called a hold. 
 
 The hold is denoted by a fmall arch, with a point in the mid- 
 dle, placed over or under the note which is intended to be pro- 
 longed, thus t or thus o. 
 
 The fame character fometimes fhews that the note, to which 
 it is affixed, is to be the laft, or clofing note of the piece ; and 
 is then called a clofe. This becomes neceffary when a piece of 
 mufic is to be finiffied with a repetition of fome of the firft or 
 middle parts. 
 
 The double ufe of this character, viz. as a hold, and as a 
 clofe, caufes no ambiguity : for it is plain, as was obferved in 
 the laft article, that the clofe always implies a hold ; and when- 
 ever this mark is ufed, fimply as a hold, the unfinifhed expec- 
 tation of the ear infallibly fhews, that it cannot, in that fituati- 
 on, be confidered as a final clofe. 
 
 in. The repetition of fome certain part of a piece of mufic 
 is marked by points affixed to the double bars, or by a pointed .S* 
 placed over the beginning, or end, of the part to be repeated ; 
 when it happens not to begin, or not to end, at a double bar. 
 
 Exam. XXIX. Plate III. reprefents feveral varieties of repe- 
 titions, which will fully fuffice for the learner's information in 
 this matter. It is to be obferved, that each letter, which ftands 
 in the If aff of lines, reprefents a portion of mufic, which may 
 be, and generally is, of a confiderable length, comprehending 
 feveral fingle bars, or meafures, although the letters are croud- 
 ed near together, and no fingle bars placed in this Exam- 
 
3 8 
 
 MifeUaneous Explanations. 
 
 pie; which is done for the fake of including all the varieties in 
 one ftaff of lines. 
 
 The portions H, I, K, are each to be repeated, becaufe of 
 the points at the two firft double bars : the pointed .S', be- 
 tween K and L, fhews that the repetition extends only to the 
 part K, fo that we muft not proceed to the part L, till after K 
 has been repeated. The parts L and M are not to be repeated, 
 becaufe the double bar which feparates them, is not pointed; 
 but the part N muft be repeated, becaufe of the ,S- at the be- 
 ginning, and the points preceding the double bar. O is not re- 
 peated, but at the beginning of P we have the mark for a repe- 
 tition, which extends over one double bar, not pointed, and to 
 the next double bar ; fo that, after P and QJiave been once 
 gone through, we muft return to P again, and fo go forward. 
 The part R is to be once performed, S is to be repeated becaufe 
 of the marks, and T is not to be repeated. See the order of 
 the parts fubjoined to the Example. 
 
 At the conclufion of fome portions of mufic, we often find 
 two parcels of notes, tied together by arches over, or under 
 each ; with the figure I , in the former arch, and 2, in the latter ; 
 this always intimates a repeat, and fhews that, for the firft time, 
 we are to take the notes marked, 1 ; and when we come to the 
 fame place, a fecond time, we are to pafs over the notes at 1, 
 and take thofe marked, 1, 
 
 112. Da capo or D. C. at the end of a piece fignifies that 
 we are to begin again at the head and conclude with the firft 
 part. Dal fegno fignifies that we are to begin again at fuch a 
 certain mark, as at a pointed .S", or any other mark, at the 
 tranfcriber's pleafure; but it is beft to put the fame mark, in- 
 tended to be referred to, over, or after the words daV fegno, 
 to prevent miftakes ; when feveral marks have occurred in the 
 preceding piece. 
 
 In thefe pieces the clofing note ought to be diftinguifhed, as 
 mentioned, Art. 1 1 o. 
 
 Part I. 
 
 1 1 3. A kind of fmall W put upon any line or fpace is called a 
 diretl or index. This is often put at the end of a ftaff of lines, 
 to fhew the place of the next note in the following ftaff, and 
 efpecially when the ftrain is incomplete. 
 
 Volti at the foot of a page fignifies, turn over ; Volti fubito, 
 is, turn over quickly, The tranfcribers of inftrumental mufic 
 ought to avoid as much as poffible, the breaking of a connect- 
 ed ftrain, at the foot of a page, becaufe of the inconvenience, 
 and lofs of time, in turning the leaf. 
 
 114. An arch o\ r er or under feveral notes, is called a tye or 
 Jlur ; and fhews that all the included notes belong to one fyl- 
 
 lable, in vocal mufic. The fame mark, in inftrumental mufic, 
 fhews that the found muft be made to flide from one note to a- 
 nother, without any breaking off, or interruption; foastoi- 
 mitate the voice, as nearly as the nature of the inftrument will 
 admit. How this and other fuch like effects are to be produced 
 on any particular inftrument, may be learned from the inftruc- 
 tion books which are compofed exprefsly for that inftrument ; 
 and ftill better from the directions and examples of experienced 
 mafters; but does not belong to this place: as we propofe on- 
 ly to give the general Rudiments of the Art, which in fome 
 meafure concern all forts of performers. We fhall only obferve, 
 that a great part of the merit of all inftrumental performance 
 confifts in a ftriking imitation of the natural inflexions of the 
 human voice; and it is on this account that we often fay, a ma- 
 fterly performer makes his inQirumexW. fpeak ; and thus, it was 
 faid of the late inimitable Sig': Tartini, that he made the violin 
 fing, rather than play. 
 
 There is indeed no other inftrument as capable of this imita- 
 tion as the violin ; but then all others may fafely be pronounced 
 defective, in fo far as they fall fliort of it. 
 
 This may ferve to apologize for the frequent addreffes we 
 have hitherto made, and fhall hereafter make, to the finger in 
 particular; becaufe the human voice is the true ftandard of all 
 
Of fever al Marks and Terms relating to particular Pafages, and to larger Sections. 
 
 Chap. V. 
 
 mufic; and he who knows how a paffage (hould be fung, knows 
 alfo how it fhould found, when played ; and wants only to be- 
 come fo far a mafter of his inftrument, as to be able to reduce 
 his knowlege to practice. 
 
 The flaccato, which is marked by fmall perpendicular dafhes 
 over each note, is juft the reverfe of the flur; and fhews that 
 the notes muft be feparated more diftinctly than ordinary. 
 
 115. Forte, fometimes wrote, by contraction, /*>. fignifies 
 ftrong or loud. Piano, or P°. fignifies weak or foft. The 
 fuperlatives Fortiffimo and PianiJJimo are fometimes ufed, to de- 
 note extreme loudnefs and foftnefs, refpectively. Moderato, 
 expreffes the mean between thefe. 
 
 116. The {hake or trill is marked by tr, put over a plain 
 note : this is by far the fineft grace in mufic, and generally 
 cofts the" performer the molt pains to acquire it. It may, with 
 fufficient propriety, be called a quick, alternate repercuffion of 
 two founds, whereof the lower is reprefented by the note itfelf, 
 and the upper is the next fuperior degree of the fcale. The 
 finger, who would attain this capital qualification of executing 
 a Ihake, muft begin with founding two contiguous degrees of 
 the fcale alternately, and Jlovjly ; increafing the quicknefs of 
 fucceflion, as the voice grows more flexible : he muft be parti- 
 cularly careful to let each found be clearly and diftinctly heard, 
 every time he recurs upon it ; left, by too much eagernefs to 
 arrive at a quick execution, he run the hazard of acquiring an 
 indiftinct and confufed one ; which is more properly Stammer- 
 ing, than finging. 
 
 Sometimes the note next below that marked with the trill is 
 inferted, juft before the end of the (hake, as fhewn, Ex. XXX. 
 and then it is called a turned (hake. At other times, none but 
 the two principal notes are ufed, as in Exam. XXXI. and then 
 it is called a plain (hake. Sometimes the make is not begun till 
 one half of the note be fung plain. Thefe and other fuch va- 
 rieties are generally left to the performer's choice, being all in- 
 
 39 
 
 difcriminately marked in the fame manner, except in leffons 
 intended only for one particular kind of inftrument; where 
 the feveral graces, proper for that inftrument, are fpecified by 
 feveral marks, eftablifhed for that purpofe ; and of which every 
 different kind of inftrument admits of a different collection. 
 
 117. The occafional infertion of other notes befides thofe 
 reprefented in the written mufic, when judicioufly managed, 
 adds greatly to the fpirit of a performance; but then, fuch 
 freedoms ought to be ufed with great difcretion, and feldom 
 with any other view than that of imitating the naturalinflexi- 
 ons of the voice; if this be difregarded, and a luxuriant fancy 
 let loofe, the true defign of the compofer is fure to be buried 
 under a heap of extravagant flourilhes and graces ; which are 
 neither confiftent with the original defign, nor with each o- 
 ther. 
 
 In written mufic, and efpecially in the vocal parts, we often 
 find notes of a fmaller fize inferted here and there, which are 
 not to be reckoned in the time, for the bars are complete with- 
 out them. Such a fmall additional note is called an appoggia- 
 tura, which word is derived from the Italian verb appoggiare, 
 to lean or reft upon, probably becaufe the voice ought to touch, 
 and make fome ftay upon that found, in its pafTage from the 
 preceding, to the following ■principal note. The appoggiatura 
 fhould always be tied to one of the principal notes; and, tho' 
 we are not ftrictly obliged to give it juft the time which its fi- 
 gure would require, yet whatever length of time is beftowed 
 upon it, muft be, as it were, borrowed from the principal note 
 with which it is tied ; fo that the appoggiatura and principal 
 note, both taken together, may be no longer than the principal 
 note would be of itfelf. See Exam. XXXII. 
 
 Thefe fmall notes may be looked upon as the finifhing 
 touches of the compofer, and the trueft fpecimens of fuch oc- 
 cafional infertions as are fuitable to the genuine fpirit of the 
 piece. 
 
4 o 
 
 1 1 8. Vocal mufic is diftinguifhed into 
 The word Recitative comes from recitare to recite or rehearfe; 
 and expreffes a particular' ilyle of mufic, nearly related to the 
 manner of fpeaking, which may be called a kind of mufical de- 
 clamation. 
 
 The chief difference between recitative, and declamation, lies in 
 the difference between a mufical tone, and natural fpeech, ob- 
 ferved in the former note to Art. 21. This fort of mufic is 
 wrote with as much regularity of time and meafure, as any o- 
 ther ; but yet the finger is more at liberty to tranfgrefs the 
 ftricT: rules of time, for the fake of expreffion. In general, the 
 fentiment ought 10 engage our attention, more than the mufic, 
 in Recitatives; and the mufic, more than the fentiment, in Airs. 
 
 1 19. Symphony is a term applied only to certain kinds of in- 
 ftrumental mufic; when this is interfperfed with vocal, the 
 parts belonging to each are diftinguiihed by the words fymphony 
 and fong, or Sy. and So. placed at the beginning of each. An 
 introductory or initial fymphony is called an Overture or Pre- 
 lude; and thofe which are interfperfed with the fong are called 
 interludes. 
 
 1 20. A Solo is an air for one fingle part, or one part and a 
 Bafs. A Duo is for two parts with, or without, a Bafs: the 
 diminutive Duetto is the fame as a fmall or fhort Duo. A Trio, 
 is for three parts. A Chorus is when feveral parts join toge- 
 ther in full harmony. 
 
 A Minuet is an air, in brifk triple time, generally confiding 
 of two parts, each being repeated. It ought to begin with an 
 accented note, i. e. with the beginning of a meafure, and to be 
 wrote in the mood of triple quavers: we frequently write them 
 in the triple of crotchets, which makes no material difference, 
 becaufe the name itfelf determines the quicknefs of the time. 
 
 A Saraband is much the fame air as a Minuet, but in a flow- 
 er movement. 
 
 A Gig is a very quick and fprightly air in triple time, and 
 
 JS/Iifcellaneous Explanations Part. I. 
 
 Recitative and Air. generally in the dodecuple, or nonuple, of quavers ; which of- 
 ten moves by large intervals or leaps, and is therefore fometimes 
 called Saltarella, from faltare, to leap. 
 
 A March, is a martial air in pretty brifk common time ; and 
 has its name from its ufe in military exercife. 
 
 A Gavot, is an air in common time, always equally divided, 
 and difiinclly accented, and for the molt part pretty quick. 
 
 An Allemand, is in flow common time. All thefe different 
 airs generally confift of two parts, each to be repeated, as was 
 obferved of the Minuet. 
 
 A Mufette, is an air in a particular flyle, fo called from the 
 French, une mufette, which is an inftrument of the fame fpecies 
 with our bag-pipe. Its movement is moderate, and molt fre- 
 quently in common time. 
 
 The names Hornpipe and Rigadoon, belong to certain fpecies 
 of dances, to which the tunes are adapted. 
 
 121. When a portion of mufic falls under no particular de- 
 nomination expreilive of the time and manner of performance, 
 the Italians ufe other words for that purpofe ; which being in 
 fome meafure adopted by all civilized nations, are become a 
 kind of univerfal language among muficians. We here fubjoin 
 a fpecimen of fuch as firlt occurred, ranked in alphabetical or- 
 der; acknowleging at the fame time that there are feveral, pur- 
 pofely omitted, which relate to particular inftruments only ; 
 and may, very probably, be others overlooked, of equal ufe 
 with fome of thefe : for, in truth, the Italians freely make ufe 
 of any word which feems moft fuitable for their purpofe ; and 
 almoft every new author introduces new terms of his own, fo 
 that the half of the Italian language would fcarce be too much, 
 for a mufician to underftand, in reference to his own art. • 
 
 dagio, q. d. at leifure, expreffes a flow, eafy movement. 
 When twice repeated, thus, Adagio Adagio, it expreffes a 
 very flow movement. 
 
Chap. V. Of Italian Words ufed 
 
 Ad libitum, [Latin,] at pleafure. Thefe words placed at the 
 beginning of certain ftrains, fignify, that the time may 
 be made quicker or flower, equal or unequal, at the per- 
 former's difcretion. 
 
 Affettuofo, affectionately. A moderate movement, with fre- 
 quent fwells and foftenings. 
 
 Allegro, gay. A pretty brifk and fprightly movement. 
 The diminutive Allegretto is lefs gay and fprightly, and 
 therefore not quite fo brifk a movement, a9 Allegro. 
 
 Amorofo, amorous. Nearly the fame as affettuofo. 
 
 Andante, walking. A regular, diftincT:, and moderate move- 
 ment. The diminutive andantino is fomewhat quicker 
 than andante, as if it were to be meafured by a little mm- 
 cingftep. 
 
 Andante allegro. A regular, diftincl, and pretty brifk move- 
 ment. A diftincT: allegro, 
 
 Animato, bold. A pretty brifk movement, with a firm and 
 bold manner : a bold allegro. 
 
 Ariofo, airy. A ftyle and manner refembling finging. 
 
 AJfai, enough. As AJJai allegro, fignifies a fufflcient degree 
 of brifknefs, but not too much. 
 
 Brillante, brilliant. A pretty brifk movement with a pleafant 
 and jocund manner: a mirthful Allegro. 
 
 Brio, vivacity. See Con. 
 
 Cantabile, that may be fung. Nearly the fame as Ariofo. 
 
 Con, with. As con brio, or con fpirito, with fpirit ; con dolce 
 maniere, in a fweet, agreeeable manner; con diligenza, 
 with care and exaftnefs ; con furia, with fury, or in a 
 violent and rapid manner. 
 
 Crefcendo, increafing. This word following piano, or pre- 
 ceding forte, fhews that the found muft increafeand grow 
 gradually ftronger. 
 
 Diligenza, care. See Con. 
 
 Dolce, fweet. See Con. 
 
 among Muficians. 41 
 
 E, or ed, or et, and. As dolce e piano, fweet and f6ft. An- 
 dante ed affettuofo, regular, diftincT: and affectionate. 
 
 Ecco, echo. The repetition of fome part of a ftrain, in a 
 foft manner, imitating a natural echo. 
 
 Furia, fury. See Con. 
 
 Fine, or ilfine, the end. This is fometimes ufed inftead of 
 the character called a Clofe. 
 
 Gratiofo, graceful. A moderate movement, nearly the fame, 
 but not fo pathetic, as affe'ttuofo. 
 
 Grave, grave A flow and folemn manner. Some authors 
 define it a movement, flower than Adagio; others, a de- 
 gree quicker than Adagio ■ but it ought, 'probably, to 
 be confidered rather as a particular manner, than a mood 
 of time. 
 
 Languente, or Languido, languishing. A flow, plaintive manner. 
 
 Largo, large, or ample A flow movement. There are 
 different explanations given of this word. Some will have 
 it, a large, and frequently, unreftricTed meafure, flower 
 than Adagio ; others, more conformably to its modern ac- 
 ceptation, define it, a flow Andante; but not fo flow as 
 Adagio: and in this fenfe Largo, compared With Andante, 
 is like an ample ftride, compared with art ordinary ftep. 
 The diminutive Larghetto, is fmaller, or lefs ample; and 
 therefore denotes a movement fomething quicker. 
 
 Lento, flow. A movement nearly the fame as Largo. 
 
 Ma, but. 
 
 Maeftofo, or Maeftuofo, majeftic. A flow movement, with 
 a bold and emphatical manner. 
 
 Men, or Meno, lefs. As men forte, lefs ftrong, or not fo 
 loud; men allegro, lefs gay. 
 
 Mezzo, half. As mezzo forte, a middling degree of flrength. 
 
 Moderate, moderate. This word is often put to exprefs a 
 moderate degree of any quality ; as Allegro moderato, mo- 
 derately gay. 
 
4 2 
 
 Mifc elk neons Explanations. 
 
 Non, not. 
 
 Phi, more. As piu piano, or P.P. more foft; piu allegro, 
 more gay. 
 
 Poco, or un Poco, a little. As un poco allegro, a little gay ; 
 meaning much the fame as Allegretto. Un poco piu, and un 
 poco men, exprefs a little more, and a little lefs, that is, 
 an increafe and diminution, of any quality, to which thefe 
 words are prefixed. 
 
 Prefto, quick. A quick movement. The fuperlative Pref- 
 tiffimo denotes a very quick movement. 
 
 Rondo, or Rondeau, [French,] a piece which concludes with 
 a repetition of the firft part. 
 
 Senza, without. As fenza Violini, without violins. 
 
 Siciliana, Sicilian; of, or belonging to Sicily. A particular 
 air in triple time, wrote in the fame mood as our Gig; but 
 much flower, and better adapted for finging. 
 
 Stentato, painful. A forced, (training manner; expreffive 
 of the crying of a perfon in pain. 
 
 Tar do, lazy. A flow, and feemingly idle, or negligent 
 manner. 
 
 Tempo, time. J tempo, or A tempo giujlo, placed after a Re- 
 citative, or other free and irregular piece, denotes that the 
 time muff be juftly kept in the part which follows. A 
 tempo ordinario, placed at the beginning of a piece, of a 
 well known ftile and movement, fhews that it is to be 
 performed in the ordinary and accuflomed time. Tempo 
 di Minuetto, di Cavotta, etc. denote that the following 
 part is to be performed in Minuet time, in Gavot time, etc. ; 
 though it be not properly a Minuet, Gavot, etc. : becaufe 
 not reftricted to the particular length of Arams, and the 
 repetitions which characterize thefe airs. 
 
 Timorofo, timorous. A faint and trembling manner ; expref- 
 five of fear. 
 
 Part I. 
 
 Troppo, too much. As Allegro ma non troppo, gay, but not 
 
 too much fo. 
 Tutti, all. This word is often placed after a Solo, or other 
 
 piece, for certain detached parts ; and fhews that all the 
 
 parts are to join, in what follows. 
 Vigorojo, vigorous. A bold and firm manner, expreffing re- 
 
 folution and fortitude. 
 Vivace, lively. A lively and fpirited manner; near the fame, 
 
 but not quite fobrifk a movement, as Allegro. 
 
 i 2 2. From this fpecimen it will eafily be obferved, that the 
 greateft part of thefe words rather denote the particular expref- 
 fion, or manner of performance, than the movement, or quick- 
 nefs of time : and indeed it would be to no purpofe, to attempt 
 to feparate thefe two particulars of expreffion and movement ; 
 becaufe the chief ufe of different movements is only to heighten 
 the different expreffions. 
 
 123. There maybe pieces of mufic, as there are pictures, 
 which are, limply, fine and pretty, without aiming directly at 
 any particular expreffion ; and among this clafs, it is to be 
 feared, too many of our inftrumental pieces may be juftly 
 ranked. 
 
 To give a better idea of the comparative degrees of move- 
 ment, denoted by thofe words which chiefly relate to the time, 
 the following lift may be of fervice ; where the floweft move- 
 ment is placed uppermoff, and the others are gradually quicker, 
 to the bottom. 
 
 Adagio, Adagio, very flow. 
 
 Adagio. , 
 
 Largo, or Lento. 
 
 Larghetto. 
 
 Andante. 
 
 Andantino. 
 
Chap. V. 
 
 Comparative Degrees of Movement. Of the 'Titles of entire Pieces. 
 
 Allegretto, or Poco Allegro, or Vivace. 
 
 Allegro. 
 
 Pre/h. 
 
 PreJliJJimo, very quick. * 
 
 i 24. The following titles of entire pieces deferve to be tak- 
 en notice of, before we finifh this Chapter. 
 
 A Sonata, or Suonata, is a piece of mufic for inftruments 
 only. The name is taken from fuonare, to found; intimating, 
 that it is a fucceffion of founds only, in contra diftinction to a 
 Cantata, which is a piece fet to words, and therefore to be 
 fung ; from cantare, to fing. 
 
 Both thefe titles are, by cuftom, appropriated to fuch pieces 
 as are of a confiderable length, and contain a variety of differ- 
 ent movements; the fhorter pieces are called tunes and fongs, 
 refpectively. 
 
 A Motetto is much the fame, in church mufic, as a cantata 
 in the moral kind. Our Anthems generally belong to this clafs. 
 
 An Opera, literally fignifies a work, or labor ; but is ap- 
 plied, by way of eminence, to mufical entertainments of the 
 dramatic kind. An opera is a dramatic poem, fet to mufic, 
 and exhibited on the ftage, by feveral performers, who perfonate 
 the characters of the drama. 
 
 A Burletta, is a comic opera. The word fignifies a burlefque, 
 or jocund performance. 
 
 An Oratorio is a kind of fpiritual opera. The mufic is in a 
 ftile fuitable for religious purpofes, and the fubjeft taken from 
 fome part of the Scripture hiftory. 
 
 A Concerto is a grand piece of harmony, employing feveral 
 inftruments, and generally of feveral different kinds. Among 
 thefe, there ought to be one principal part, to engage the 
 
 43 
 
 hearers attention more particularly, while the others ferve only 
 to fill up the harmony. The inlfrument, for which chis part 
 is adapted, is diftinguifhed by the epithet concertino; and the 
 reft are called concertanti. As Violino concertino, the firft or 
 principal violin: Flauto conceriante, the accompanying flute. 
 
 A Voluntary is a piece of mufic not refracted to any fixed 
 length of ftrains, or fimilarity of phrafes, like other regular 
 compofitions; but is, as it were, an extempore flight of the 
 performer's fancy ; and is therefore called, by the Italians, Phan- 
 tafia, a fancy ; Capricio, a caprice or whim ; Ricercata, a fearch- 
 ing, or feeling, for fomething ; Tajlatura, an eflay or trial, etc. 
 
 A Serenata, or ferenade, is a title given to nocturnal mufic, 
 performed in the open air, near to fome perfons door or win- 
 dow, whom the performers have a mind to entertain with an 
 unexpected concert. 
 
 A Fugue is a piece of mufic for feveral parts ; fo called from 
 fuga, a flight, or running away ; becaufe, after one leading 
 part has begun, and proceeded fome length with a ffrain, o- 
 ther parts fucceffively begin the fame ; and, as it were, fall in 
 behind, and purfue the leading part, which continues to flee 
 before them at a certain diftance. 
 
 The following part fometimes takes the very fame pitch 
 wherewith the leading part began ; fometimes an octave below, 
 or above it ; and fometimes a fifth, or fourth below, or above 
 it ; and hence arife the diftinftions of fugue in the unifon, in 
 the octave below, or above; or in the fifth, or fourth below, 
 or above. 
 
 A Canon is a felf-returning fugue; where each part fuccef- 
 fively re-commences at the beginning, as often as the perform- 
 ers pleafe; fo that the parts purfue each other, at ftated di- 
 ftances, in an endlefs round. 
 
 * The celebrated Mr. Roujfeau of Geneva, in the mufical articles of the French encyclopedic, fays, that the Italians have eftablifhed the four words, adagio, 
 andante, allegro, and frefto, for expreffing four principal degrees of llownefs and quicknefs in mufic. We have kept clofe to his explanation in this rc- 
 fpect, admitting the word largo, as a fpecies of andante. 
 
41 
 
 Of harmonica! Confonances. 
 
 Part I. 
 
 No more but the leading part of thefe piece9 is ordinarily marks are, properly, the caMni, or rules of performance ; and 
 wrote, affixing a certain mark, as a double bar, or a pointed hence the name canon came to be appropriated to this fort of 
 .S - , where each of the following parts is to fall in : thefe compofitions. 
 
 Chap. VI. OJHarmonical Consonances. 
 
 Article 125. HP' 
 
 H E nature and effects of harmonical confo- 
 nances muft be, in fome degree, underftood, 
 before we can properly proceed to explain the affinity of cer- 
 tain natural fcales, and the origin of the flat feries. 
 
 A Chord is a combination or various mufical founds, heard 
 together. When the feveral founds mix and unite, in a man- 
 ner agreeable to the hearer, it is called a confonance, or confonant 
 chord; when they do not unite, but feparately diffract the at- 
 tention of the ear, it is called a dijfonance, or ■ diffonant chord. 
 
 1 26. The moil perfect confonances are the octaves, as obferv- 
 ed, Art. 15. The fingle octave and its replicates, viz. double 
 octave, triple octave, etc. have each their peculiarities, which 
 ought not to be overlooked. 
 
 When the two terms of the fingle octave are founded, com- 
 pletely true, the upper term unites fo perfectly with the lower, 
 that the ear is fenfible of nothing more than the found of the 
 lower term, made louder and fuller, by the concurrence of the 
 upper term. 
 
 The cafe is different with the double octave, or the octave 
 of the octave. Here the upper term will be heard diflinctly, 
 as well as the lower, and the ear is abundantly fenfible of 
 two founds, but at the fame time, they are fo perfectly like to 
 each other, that we can attend to both without any difficulty 
 or diffraction. F 
 
 The terms of the triple octave are not fo eafy to he compared 
 together, without the interpofition of fome other found between 
 
 them. The triple octave is too large an interval to be conceiv- 
 ed undivided : and much more the quadruple, and the other 
 ftill more diftant replicates of the octave. 
 
 127. Along with the double o<ftave, the fingle octave may 
 be joined, and alfo the fifth above the fingle octave, and the 
 greater third above the double octave, without caufing any di- 
 ffraction ; and thefe five founds properly conltitute what is 
 called the perfeFt chord. The loweft term is called the junda* 
 mental, and the others are called harmonics. 
 
 In order to reprefent the founds of this chord, and others 
 which we fhall have to defcribe, we fhall refume the notation 
 eftabliined in Chap. III. where the founds were expreffed by 
 letters, and three feveral octaves diffinguifhed by Roman capi- 
 tals, Roman fmall, and Italic final], thus 
 
 r*A*BC*D*EF*G*a*bc*d*ef*g*«z *£<:*</**/*£ 
 
 Agreeable to this notation, the perfect chord of the fundamental C 
 
 is C — — — — c g — c — e 
 
 And that of the fundamental T is 
 
 r — — — — G d — g — b 
 
 The perfect chord of F F is 
 
 F_ F - - c — £ — a 
 
 and fo of others, 
 
Chap. VI. 
 
 Of the perfett Chord. 
 
 
 i 28. The intervals of mufical chords are often named by 
 counting them all upward from the loweft found, inclufive ; 
 and thus, the fifth, above the fingle octave, is often called a 
 twelfth ; the double octave, a fifteenth ; and the greater third 
 above the double octave, a greater feventeenth ; as will be found 
 by counting upward from the fundamental found, in the fame 
 manner as we formerly did from the key, in chap. I. and II. 
 
 It may be uleful to obferve, that, in this way of naming In- 
 tervals, the ordinal number, which exprefles any total interval, 
 is always lefs by one than the fum of the two numbers which ex- 
 prefs any two partial intervals which together make up the to 
 tal. Thus, in the above Chords, the octave, or eighth, as 
 from C to c, and the fifth from c to g, conftituteonly a twelfth 
 from C to g ; but the fum of 8 and 5 is 13. Again, the double 
 octave is only a fifteenth ; but 8 and 8 make 16 ; the fifteenth 
 and third is only a feventeenth ; but 1 5 and 3 make 1 8 ; and, to 
 defcend to fmaller numbers, a fifth and fourth, or a fixth and 
 third, or a feventh and fecond, make an octave ; two fourths, 
 or a fifth and third, or a fixth and fecond, make a feventh ; a 
 fourth and third, or a fifth and fecond, make a fixth ; two 
 thirds, or a fourth and fecond, make a fifth ; a third and fecond, 
 make a fourth ; and two feconds, make a third. In all thefe 
 infiances the fum of the names of the two partial intervals, is 
 one more than the name of the total interval, occafioned by our 
 including both extremes in the name of every interval, fo that 
 there is a middle term twice taken in, when we name two con- 
 tiguous intervals, and only once when we name the two toge- 
 ther. 
 
 If a total interval be divided into three parts, the number 
 which exprefles the total will be lefs by t-wo than the fum of the 
 numbers which exprefs the partial intervals ; and that becaufe 
 there are two middle terms, each twice taken in, in the names 
 of the partial intervals, and only once in that of the total. Thus 
 C to D is a fecond, D to F is a third, and F to b is a fourth, 
 
 Names of compounded Intervals. 45 
 
 and the total C'to b is only a feventh; whereas 2 and 3 and 
 4 make 9. The fame is to be obferved of all other intervals, 
 when named in this manner. 
 
 1 29. The number of degrees, or fteps of the fcale, contained 
 in any interval, is always lefs by one, than the name of the in- 
 terval : thus a fecond contains only one ftep ; a third contains 
 two fteps ; a fourth, three fteps ; a fifth, four Heps ; a fixth, 
 five fteps ; a feventh, fix fteps ; and an octave, feven fteps of 
 the fcale. In this way of conceiving intervals, the fum of the 
 numbers of fteps contained in all the partial intervals, is juff. 
 equal to the number of fteps contained in the total interval, and 
 as the name of the interval made lefs by one is always the num- 
 ber of fteps contained in it ; fo, on the contrary, the number 
 of fteps made greater by one is always the name. For infiance, 
 an octave contains feven fteps, and a fifth contains four fteps; 
 therefore the interval of an octave and fifth contains leven and 
 four, that is, eleven fteps, and is called a twelfth Two oc- 
 taves contain twice feven, that is, fourteen fteps ; and the 
 double octave is called a fifteenth. Two octaves and a ihird con- 
 tain fourteen and two, that is, fixteen fteps, and the interval 
 is called a feventeenth. Two octaves and a fifth contain four- 
 teen and four, that is, eighteen fteps ; and the interval is call- 
 ed a nineteenth The triple octave contains three times feven, 
 or twenty one fteps, and is called a twenty fecond, and fo of 
 all the reft. 
 
 130. Muficians alfo diftinguifh the founds of the firff, fecond, 
 and third octave above any fundamental, by the epithets fngle, 
 double, and triple refpeclively, annexed to the degree of the na- 
 tural fcale : thus, the G, which is next above C, is called its 
 fingle fifth ; the g, in the fecond octave above, is a double 
 fifth ; and the g, in the third octave, is a triple fifth to the 
 fundamental C. 
 
 This is the mod concife and intelligible way of expreffing 
 large intervals, and therefore we (hall, hereafter, chiefly make 
 M 
 
•bird, 
 
 46 Of Hannonical Confonances. 
 
 ufe of it. The reader is defired to obferve, that the word double and the fundamental E 
 or triple, when thus applied to any degree of the fcale, except 
 the octave, does not mean an interval of a double or triple fize, 
 but it means a replicate of fuch a degree in xhz-fecond or third 
 oclave, above. Thus, a double fifth is not an interval contain- 
 ing two fingle fifths, as from C to d ; for this is properly a 
 ninth, or a double fecond : and thus, a triple third is not the 
 interval of three contiguous thirds, as from C to b ; for this 
 is properly a feventh, but it is a replicate of the third, in the 
 third oclave above. 
 
 According to this way of naming the large intervals, the per- 
 fect chord to any fundamental confifts of its fingle octave, dou- 
 ble fifth, double oclave, and triple greater third. 
 
 131. In the formation of this chord to any fundamental, it 
 is to be obferved that the fifth and greater third mufl always be 
 ferfetl ; by which we mean, that they muff always be fuch as 
 would take place in the natural fcale of the fundamental, con- 
 fidered as a key note : and, in truth, this perfect chord never 
 takes place, in the harmony of a fharp ieries, but either upon 
 the principal key, or upon a note which can be confidered as an 
 occasional, or new key for the time, viz. the fifth or fourth of 
 the fcale, as was hinted in the firft note to Art. 31. In the har- 
 mony of a flat feries its ufe is only accidental, as will be fhewn 
 hereafter. 
 
 If the learner have well digefted the formation of the feveral 
 natural fcales, as explained Chap. III. he will be able readily to 
 determine the perfect chord of any nore in the fyftem, by con- 
 fidering that note as a key, and applying the fame-fifth and third, 
 which naturally belong to it as fuch. For infrance, in the chords 
 of C, T, and FF, reprefented above, Art. 1 27. the founds are 
 the fame which exift as thirds and fifths in the fcales of thefe fun- 
 damentals, according to the fcheme, Art. 48 : and, by the fame 
 rule, the fundamental D, will take a, for its double fifth, and 
 f% for its triple third ; according to the third fcale, Art. 48 : 
 
 Part I. 
 
 , will -take b, for its double fifth, and 
 according to the fifth fcale of the fame 
 
 g %. for its triple 
 fcheme. 
 
 132. This chord is very eafily [produced, in itsfull form, upon 
 an organ, or any other key'd inftrument, and if all the founds 
 be perfectly right tuned, and flruck juff at the fame initant, a 
 hearer can fcarcely imagine the whole chord to be any more than 
 one fingle found, of a remarkably full and bold kind. If the 
 feveral founds be introduced one after another, beginning with 
 theloweft, or fundamental, and proceeding upward, one may 
 plainly perceive every fucceffive found, as it were, entering and 
 uniting with the fundamental ; which feems.only to grow fuller 
 and bolder, by every addition. This expei iment is no lefs inflruc- 
 tive than entertaining. 
 
 133. The founds of this perfect chord may be arranged in 
 feveral other different ways, or fome of them may be left out, 
 and yet fbll the fame effect produced, but in a more imperfect 
 manner : for there is no other way in which the founds can be 
 arranged, to produce fo complete and fo perfect a confonance 
 as that already defcribed. 
 
 The leaving out of fome of the founds, belonging to a chord, 
 is called omiffion, or retrenchment ; and the placing one of the 
 harmonics in ithe bafs, inflead of the fundamental is called in- 
 verfion of a chord. 
 
 134. As the full perfect chord confifts of five founds, it is 
 plain that when fewer thar five parts are fet together, fome 
 omiffions become neceffary : and, in order to avoid diftant leaps, 
 and to give each part a more gradual and eafy fucceffion, as 
 well as to give a greater variety to the harmony, feveral diffe- 
 rent arrangements and inverfions of the fame chord are not only 
 allowable, but are alfo preferable to a confecution of chords all 
 equally full and perfect. 
 
 1? 5. In regard to the different arrangements, which we may 
 call dijlocations ; it is always efteemed a fpecies of the perfect 
 
Of the full Chord, and ofDiflo.cations, Omijjions, and Inverfions. 
 
 Chap. VI. 
 
 chord of any note, if its perfect third, fifth, and octave be join- 
 ed with it, in any pofition ; whether any or all of thefe three 
 be taken in tbefingle, or double, or triple octave above the fun- 
 damental, and which foever of them be uppermoft ,: but, in 
 general, the further we depart from the difpofition of the moff 
 perfect of this or any other kind of chord, the greater imper- 
 fection is thereby introduced. In every pofition where the fun- 
 damental note is in the bafs, the chord is denominated erect. 
 
 136. In legard to omijjions, we have to obferve, that not on- 
 ly one or two of the notes of this chord may be wanting, but 
 they may all be wanting except one, and yet ftill the fame idea 
 conceived by the hearer, as if they were all joined together ; e- 
 fpeciaily if the hearer be accuftomed to harmony, and have con- 
 tracted a kind of acquaintance with the proper fucceflions of 
 chords of which it confifts ; for, daily experience teftifies, that 
 by an acquired familiarity with the fucceflions of harmony, a 
 fkilful perfon becomes able to fupply the proper accompaniments 
 in imagination, when only one part is heard, if the pafTages be 
 not very uncommon or unnatural, 
 
 137. There is fomething in the very nature of mufical founds 
 which contributes fome aflifrance to the imagination in this mat- 
 ter. Every mufical found which we call fingle, becaufe produ- 
 ced by one firing, or one pipe, or one bell, <bc. is, really and 
 intrinfically, a -kind of chord, or combination of feveral founds, 
 uniting under the idea of one fundamental, by which we efti- 
 mate the tone or pitch of the whole. 
 
 Thefe combinations are very remarkable in the founds of large 
 bells, efpecially for a little time immediately after the ftroke is 
 given, and if the hearer be near the bell. The found is at firft 
 a perfect confufion of tones, and whatever particular tone we 
 have previoufly eftimated and fet our attention upon, we are al- 
 mofl fure to find fomething like it among the confonances of a 
 bell; and if it do not occur to us after the firft ftroke, it may 
 poflibly be found after a fecond or third ; for it may be obferv- 
 
 47 
 
 ed, that not only different bells have their founds differently com- 
 pounded, but alfo the fame bell, when ftruck feveral times, oft- 
 en yields feveral different combinations of tones, although the 
 fame fundamental ftill prevails. 
 
 If a founding body of any kind be fufpendedin a firing, and 
 the ftring applied to our ear, with one hand, and the body 
 ftruck upon with the other hand, we can perceive not only a re- 
 markably loud found, but alfo a very furprizing combination of 
 various tones in a fmall rod of iron, or a piece of thick wire, in a 
 flat piece ,of brafs, or other metal, and in numberlefs things, 
 which, if ftruck upon, without fome fuch device, would yield no 
 diftinguifhable tone at all. Experiments of this kind lie in eve- 
 ry one's power, without any troublefome apparatus. All we 
 would infer from fuch trials, at prefent, is, that although the 
 harmonics, belonging to the perfect chord of any found, be ge- 
 nerally, if not univerfally, predominant among its natural con- 
 fonances, when examined in this manner, yet there are not 
 wanting other founds to coincide with, and affift our imagina- 
 tion, in cafe we have occafion to regard the total found, not as a 
 fundamental, but as one of the harmonics of fome other funda- 
 mental. Thus when we attend tp the third of a natural fcale, 
 and confider it as an harmonic to the key as fundamental, we 
 either find fomething to induce us to join, or at lead we find 
 nothing to hinder us from joining, along with it, its lefs third 
 above, and its fixth, inftead of a greater third, and fifth : fo that 
 when E occurs in the key of C, we naturally join it, or imagine 
 it to be joined, with G and c, rather than G % and b, which laft 
 are its perfect chord. If we have begun with the key of C, the 
 rememberance of that key, and the fcale dependent upon it, will 
 not fuffer us toufe any found remarkably different from thofe 
 which exift in that fcale, fo that we cannot admit G % without 
 relinquishing, in fome meafure, the key of C : and rather than 
 do this, we chufe to give E a chord, borrowed from the perfect 
 chord of C, by inverfion. In the fame manner we fometimes, 
 
4 8 
 
 in the key of C, give G, which is the fifth 
 from the perfect chord of C by inverfion ; but G can take its 
 own perfect chord, without employing any note foreign to 
 the key of C, and the fame may be faid of F, as appears in the 
 fcheme, Art. i 27. 
 
 138. The chords borrowed from the perfect, 'by inverfion, 
 when the third or fifth is placed in the bafs, are called inverted 
 chords, and have their intervals alfo counted from the bafs note : 
 and hence arifes the chord of the fixth and third, called fimply 
 the fixth ; and the chord of the fixth and fourth : the former 
 of thefe exifting upon the third, and the latter upon the fifth 
 of the fundamental as bafs. 
 
 Of Harmonical Confonances. Part I. 
 
 a chord borrowed caufe it may be fo introduced, and focircumfianced, as to leave 
 nothing for the hearer to expect after it. 
 
 This is called the fiat pcrfefi chord, and gives rife to the flat 
 feries. We {hall, for brevity's fake, ufe the freedom to de- 
 note this chord by the characters [>P, and the other by the 
 character P. fimply, or rather %¥. to prevent miflakes. 
 
 Thus, from the, perfect chord of C, 
 
 which we count 
 arifes the Chord of the fixth on E, 
 
 which we count 
 and that of fixth and fourth on G, 
 
 which we count 
 
 viz. C- 
 
 -E— G c . 
 
 -3-5 8 
 
 E— G c 
 
 i-3 6. 
 
 G c— e 
 
 1 4—6 
 
 Thefe inverted chords are juftly efteemed imperfect reprefen- 
 tations of the fundamental chord, but ftill they are representa- 
 tions of it. The chord of the fixth is beft when the founds 
 are neareft to each other, in the fame order as they ffand above ; 
 but they may be farther diftant. That of the fixth and fourth 
 is better, when the fourth found, which reprefents the funda- 
 mental, is carried an octave higher, and fo made a double fourth 
 to the bafs note. 
 
 It is obfervable, that in all pofitions of this perfect chord, 
 whether creel: or inverted, the note which makes the third, at- 
 tracts our attention, more than that which makes the fifth to 
 the fundamental found. 
 
 139. There is another fpecies of chord which takes a lefs 
 third inftead of a greater, and is alfo denominated perfect, be- 
 
 The \) P chord takes place upon the fixth, or fecond, or 
 third of the natural fcale, without caufing any remarkable alte- 
 rations : for thefe three have a lefs third, and a perfect fifth a- 
 mong the founds of the fcale. 
 
 140. It may be obferved of the ^ P chord, that it is more 
 perfect when the third to the fundamental is covered by fome 
 higher found ; if the third be placed in the bafs, by inverfion, 
 it is ftill equally good : but yet, iri every pofition, the lefs 
 third, in fo far as we attend to it, caufes a kind of diffraction, 
 and is an abatement to the perfection of the chord';' fo that the 
 trueft perfection of this chord, is to omit the lefs third altoge- 
 ther, and take in only the fundamental, and its octaves and 
 fifth. 
 
 Thus, the [>P chord of A may be 
 
 or it may be C ■ 
 
 or beft of all, A : 
 
 c — e a 
 
 ■ — e a 
 
 - — e a 
 
 pro- 
 
 The order of this \ P chord may be varied feveral other ways, 
 and diflocations, omiffions, and inverfions practifed much the 
 fame as with the % P chord, and yet ftill the fame effect 
 duced, in fome degree. 
 
 141 . There is an eflential difference between the % P and the 
 [/P chords, in all pofitions. The >^P chord, and all its deri- 
 vatives are bold and commanding, the [j P, and its derivatives 
 are dejected and plaintive. The greater third of the $ P is a 
 found which attracts our attention, but the lefs third of the [, P 
 
Chap. VI. 
 
 Of the flat perfetl Chord. "Diftinclion of Chords Into final and medial. 
 
 is a kind of difregarded, or loft found F in all fituations; and our 
 attention is rather turned upon the fifth in its ftead. Further, 
 the third of the % P chord is a fupplicative tone ; but the third 
 of the \) P chord 1 is heavy and fpirit lefts. 
 
 142. In either kind of perfect chord, the founds unite bet- 
 ter, and the effect is more fatisfactory and conclufive, when 
 the fundamental is rather ft ronger, or more diftinguiftjablc than 
 the others : and the unity of the whole is deftroyed, when the 
 third or fifth attracts too much of our attention. 
 
 The joining of the octaves to any found renders it ft ronger, 
 and therefore the perfect chord is beft when the upper octaves 
 of the fundamental found are heard along with it, asreprefent 
 ed, Art. 127, and 140. But the joining octaves with the third 
 or fifth, which the muficians call doubling of thefe notes, is an 
 abatement to the perfection of a chord. 
 
 The reafon is this. It is one neceflary property of a fatisfac- 
 tory and conclufive chord, that the third and fifth be conceiv- 
 ed, only, as proceeding from, and ferving to fortify the natu- 
 ral confo?ia?ices of the fundamental ; and therefore they ought 
 not to be fo ftrong, or fo diftinguifhable as the fundamental. 
 The artifts fay, veryjuflly, that the doubling of the third or 
 fifth takes off* their dependence upon the fundamental. 
 
 Of the two, it is commonly better to double the fifth than 
 the third of a perfect chord ; becaufe the fifth has a more near 
 relation to, and is more prevalent among the natural confonan- 
 ces of the fundamental : the third of a [j P chord, efpecially, can- 
 not be doubled without diffracting the hearer's attention from 
 the fundamental, and fixing it too much on the fame third ; 
 which, as we have faid above, ought to be a difregarded, loft 
 found. 
 
 When we fpeak thus of doubling founds, it ought to be un- 
 derftood only in refpect to founds of equal ftrength, or nearly 
 fo ; for, as to founds much weaker, they pafs unregarded ; and 
 
 N 
 
 49 
 
 it is the fame thing in effect whether they be there or not. 
 Thus, a few womens voices may join with the tenor of a church 
 tune in the octave above, by which every note of the tenor will 
 be doubled in reality, and yet, if the mens voices be confiderably 
 ftronger, the harmony is not fenfibly hurt : and thus, in the 
 compound flops of an organ or harpficord, every note of a full 
 chord is accompanied by its upper octave, and frequently o- 
 thers of the harmonics of its own perfect chord, and yet, thefe 
 being in due fubordination as to flrength, do not at all confufe 
 or diffract the hearer. 
 
 143. Although the doubling of the harmonics of a chord, as 
 above, partly fubverts the natural order of the confonance, yet 
 that does not hinder fuch chords from taking place, and be- 
 ing of the greateft ufe in harmony, but then they are to be ad- 
 mitted as medial, and not as final or conclufive chords ; becaufe 
 there is always an expectation of fomething more fatisfactory to 
 follow ; and it is as neceflary to keep up the expectation of the 
 ear in the middle of a piece, as to put an end to it at the con- 
 clufion. 
 
 What is here faid of the founds of a chord, made ftronger' 
 by being doubled, is equally true in regard to founds, when 
 louder than the reft, though not doubled : fo that the ear is 
 not fatisfied if the third or fifth of a chord be louder than the 
 fundamental. 
 
 144. Some founds of a chord are more diftinguijbable than 0- 
 thers, by their fttuation. Thus our attention is carried either 
 to the loweft or the higheft found which exifts in a chord, ra- 
 ther than any of thofe which lie, as it were, concealed in the 
 middle. We expect to find the fundamental loweft, and the har- 
 monics, as hinted above, fortifying the natural confonances of 
 the fundamental i but when this is not the cafe, as in inverted 
 chords, it is beft to have the octave of the fundamental higheft, 
 that the hearer, being difappointed of it below, may yet find it 
 
5^ 
 
 Of Harmomcal Confonances. 
 
 I. 
 
 above ; and it is for this reafon, with the concurrent tefKmony 
 of experience, that we pronounce the inverted chords better, 
 as defcribed, Art. 138. 
 
 Hence alfo it is exprefsly forbidden to place the octave to the 
 bafs note of an inverted chord uppermoft ; except in paftages 
 where it would be no detriment to the harmony to confider that 
 bafs as being itfelf a fundamental. 
 
 All inverted chords may juftly be efleemed medial ,■ for how- 
 foever our attention may be carried to this or the other upper 
 found, through the courfe of a tune, it is fure to fall upon the 
 bafs at the end. See Art. 61 . If fome pieces of mufic end with 
 the chord of the fixth upon the third of the key as bafs, it is a 
 licence, allowed in adapting mufic to certain imperfect inftru- 
 ments, fuch as the Trumpet and French horn ; and is tolerated 
 only through cuftom, but is unfatisfactory to all who are 
 not accuftomed with it. 
 
 J4J. Some founds are alfo rendered more diftingiiifl.iable by 
 the fuccejjion in which they occur : but this muft be poftponed 
 till we come to confider chords in fuccefiion. We fpeak, at 
 prefent, only of chords, fingly confidered. 
 
 In Exam. XXXIII. Plate III. the chord of G is drawn out 
 upon the treble and bafs ftaves, in feveral different arrangements, 
 for the convenience of fuch as have practiced upon a key'd in- 
 ftrument, and of confequence can more readily ftrike the chords, 
 when reprefented in this manner, than by the letters as above. 
 This chord of G may ferve as a fpecimen of all the % P fort. 
 Exam, XXXIV. reprelents feveral arrangements of the chord of 
 A, in the fame manner, as a fpecimen of all the \) P fort. Thefe 
 chords arc diftinguifhed into final and medial, agreeable to the 
 preceding obfervations. Many more of each clafs might be 
 added, but thefe may ferve to evidence both the truth and the 
 importance of fuch obfervations ; which, till of late, have 
 been too much neglected, even by the moft fkilful writers on 
 this fubject. 
 
 146. Annotations to Exam. XXXIII. and XXXIV. 
 
 Thefe examples are divided into bars, marked with the let- 
 ters H. I. K. L. M. for the fake of references. 
 
 The firft chord of G, in the firft bar H, is the fame with 
 thofe defcribed Art. 127.; confifting of the fundamental, and 
 its fingle octave, double fifth, double octave, and triple greater 
 third. In the fecond chord, the fingle octave of the fundamen- 
 tal is omitted. In the third chord, the fifth is alfo omitted : 
 and the fourth and laft chord, in this bar, wants the fifth, and 
 both the octaves of the fundamental ; and the greater third on- 
 ly remains joined with the fundamental found. 
 
 If thefe four chords be ftruck fucceffively, in the order they 
 fland, a hearer will plainly perceive a deficiency in the fulnefs 
 of each fucceeding chord, compared with that which went be- 
 fore it : there is a kind of breach occafioned by the omiffion of 
 the intermediate notes of the chord, which renders the defective 
 chords unfatisfactory to the ear, while we retain the impreffion 
 of a more full and complete chord, which we have juft before 
 heard ; but yet fuch defective chords are abundantly fatisfacto- 
 ry and conclufive, when introduced after others which ate equal- 
 ly defective. A full chord is neceffary to conclude a full har- 
 mony ; but a defective chord ought to conclude a defective and 
 incomplete harmony. 
 
 The chords of A, in the bar H, differ from thofe above in 
 the difpofition of the founds ; for, in thefe, the fifth takes the 
 uppermoft place, whereas the third is higheft in thofe ; and in 
 the omiffions the third is here caft out, and the fifth retained to 
 the laft ; whereas there the fifth is caft out, and the third re- 
 tained : and in truth, if any refemblance be found between the 
 effects of the different founds of a ^ P, and thofe of a [3 P chord, 
 we muft compare the third of a ^ P chord with the fifth of a 
 \) P ; and the fifth of a $ P chord with the third cf a \j P ; as will 
 appear upon a careful examination and trial of thefe examples. 
 The bar I differs from the former, in both examples, only in 
 
I 
 
 PI. Ill Ex.XXVIII 
 
 vz*]P 
 
 fa<ty_r. 50 
 
 ^ r wm&& f ^gm ^mmg^ 
 
 R 
 
 „ Jx.iyj/a///l/'Y{. 
 
 1 K 
 
 L M „ N OP 
 
 ^^^^^Sfe^^se^^^ 
 
 E,. XXIX 
 
 3it; fct 
 
 J3Ei 
 
 .s* .s* 
 
 .s* .s' 
 
 3£=-^= 
 
 O 
 
 XE= 
 
 ^Bg H 
 
 S. T. 
 
 '../,,■ of //><■/!,> /.;. H H.I. 1. K. K. i. M. X. X. O.T. Q. P. Q. R. S. 
 
 Ki.JCXX. Aturnd-Shakc cxpl.aiifd. Ex.^XXl. A pi sin Sh»kr expltin'd . Ex. \Y^\\.At>Bogg/afu>-a. 
 
 i^ rff f f^fwffl llfr MfN 
 
 J 
 
 Bid. 
 
 ^-y^ V^^ gg 
 
 IT 
 
 Ei XXXIV. Hi 
 
 rrrrrrnr-rrrt 
 
 ~6 — 6—t> — o . ' . 4 — 4"" ~ ? — 4" ^ T* 
 
 Ve-fO.ordofj 
 
 
 •Si — r S 
 
 s 
 
 =i 
 
 ^^ffi^Wp 
 
 i^j 
 
 p pf^ pE 
 
 «-r-6— "6 -6 >' 4 — 4 — * 4" 
 
 m 
 
Chap. VI. Examples of 'final and medial Chords. Scheme of all the ferfetl Chords in the natural Scale, yi 
 
 the order of the founds, the harmonics being deprefled below 
 the double octave of the fundamental. 
 
 The chords in the bar K, in both examples, may be juftly 
 efteemed medial, on account of their imperfection, although 
 they are erebl : the fir ft of thefe is the whole chord brought down 
 into the firft octave above the fundamental, and in this fixati- 
 on the harmonics are too far deprefled below their natural places. 
 See Art. [35. The fecond chords of this bar K have only one, 
 aod that the more allowable one, of the harmonics, in this de- 
 prefled fituation : for it may be obferved, that the note which 
 mod naturally (lands uppermoft can leaft be allowed to be plac- 
 ed next the fundamental ; and therefore the fifth of a % P chord 
 is more tolerable, in this fituation, than the third would be ; 
 and the third of a jjP is more tolerable than the fifth. 
 
 The third and fourth chords of the bar K, have the fifth of 
 the $ P, and the third of the \, P uppermoft ; which circumftance 
 alone is enough to prevent them from being fatisfattory. We 
 fhall (hew the reafon hereafter. 
 
 The bar L confifts of chords of the fixth, and the bar M of 
 fixth and fourths by inverfion. In all the different pofitions 
 of thefe chords, it may be obferved, that the doubling of the 
 bafs note is avoided. See Art. 144. 
 
 At N is rep.refented a bad chord, being as much as pofiible 
 the reverfe of every thing which makes a chord good : and thus 
 much may ferve for the learners examination and amufement at 
 prefent. 
 
 We have begun the article of confonances with this fhort fur- 
 vey of the perfect chords, becaufe it is neceflary to conceive, 
 not only the compound founds of harmony, but alfo every 
 Angle found in melody, as belonging to fome perfect chord, in 
 order to account for their various effects, and the preference 
 due to fome fucceffions of founds rather than others : as will be 
 better feen hereafter. 
 
 147. It is neceflary to obferve, in the mean time, that the 
 
 natural fcale of any note affords fix different chords, three of 
 them being % P, viz. thofe of the key, the fifth, and the fourth ; 
 and the other three being fj P, viz. thofe of the fixth, and the 
 fecond, and the third : or, in other words, we have fix notes 
 in the natural fcale, ariy of which may be conftituted the funda- 
 mental of a perfect chord, without reiinquifhing the principal 
 key. 
 
 To avoid ambiguity in fpeaking of thefe chords, we fhall di- 
 ftinguifh the fundamental note of any chord by the letter/" an- 
 nexed to it. The neceflity of fome diftinition of this kind will 
 appear, becaufe, as we have already feen, Art. 138. there is a 
 chord called the chord of the fixth, arifing From the perfect by 
 inverfion, which is quite different from the perfect chord of the 
 fixth, as fundamental : and the fame ambiguity would happen 
 in the names of other chords, not yet fpoke of, unlefs the fun- 
 damental found be fome way diftiriguifhed. 
 
 See now the following fcheme of the perfect chords in the 
 natural fcale. 
 
 T Of the key/, contains the K — 3d — 5. 
 
 The $P 3 Of the 5th/. contains the 5th— 7th — 2d. 
 
 (_Of the 4th/. contains the 4th — 6th— K. 
 TOf the 6th/. contains the 6th— K— 3d. 
 
 The (j P } Of the 2d'/. contains the 2d — 4th — 6th. 
 
 (_ Of the 3d/. contains the 3d— 5th— 7th. 
 We fay thefe chords contain thefe notes, reflectively, by 
 which we would be underftood to mean, that thefe notes, not 
 only in the order they ftand here, but in any order, either eretl 
 or inverted, ftill constitute the fame chord, thongh in different 
 degrees of perfection, as fhewn above. 
 
 148. The feventh of the fcale is the only note which cannot 
 have a perfect chord applied to it, becaufe it has not a perfect 
 fifth in the fcale : for it may be remarked, that the perfect fifth 
 always contains four fteps of the fcale, three of which fteps are 
 tones, and one a femitone ; but above the feventh we have firft 
 
« 
 
 Of Harmonical Confoi 
 
 Part I. 
 
 a femitone, then two tones, and then another femitone, which 
 brings us to a fifth, which is only two tones, and two femitones 
 above the feventh ; and this is called afalfe fifth, being a femi- 
 tone lefs than the perfect fifth ; and is a difl'onance. 
 
 The famcconclufion follows from obferving, that the perfect 
 fifth always contains a greater third and a lefs third, and that 
 the greater third is fituated below in the $ P chord, and the 
 lefs third is below in the [j P chord, which makes the difference 
 between the two forts of perfect chords ; but that above the 
 fevertth we have two lefs thirds contiguous to each other, viz. 
 from the feventh to the fecond, and from the fecond to the 
 fourth ; fo that unleis either the feventh be flattened, or the 
 fourth fharpened, the interval between thefe two will be afalfe 
 fifth. Now, by flattening the feventh, we make it capable to be 
 the fundamental of a $P chord; and by (harpening the fourth, 
 we make the feventh fufceptible of a ^ P chord : but then either 
 of thefe alterations caufes a change of the key, as was obferved, 
 Art. 50.; fo that, without relinquifhing the key, we cannot give 
 the feventh a perfect chord, • that is, the, feventh cannot be a 
 fundamental note. 
 
 149. From the above fcheme it appears, that any note of 
 the fcale may occafionally belong to two or three different per- 
 fect chords. For inftance, the key may belong to three chords; 
 for, it may be the fundamental of its own perfect chord, or it 
 may be a third in the chord of the fixth /, or a fifth in that of 
 the fourth/. Again, the fecond may belong to two chords; 
 for, it may be the fundamental of its own chord, or a fifth in 
 the chord of the fifth/.- and thus every note may belong to 
 two perfect chords at lead, one of which is always % P, and 
 another & P. If the feventh had been fufceptible of a chord, 
 like the other notes of the fcale, then every note would have be* 
 longed to three perfect chords ; viz. each note would have been 
 the fundamental of one chord, the third of another, and the 
 fifth of another ; but becaufe the feventh cannot have a perfect 
 
 chord, therefore the feventh, and fecond, and fourth, belong 
 only to two perfect chords each ; while the key, the third, the 
 fifth, and the fixth belong to three. 
 
 1 50. It appears then, that when any note of the fcale is 
 founded, we have the choice of two or three chords to which 
 we may refer it ; and becaufe one of thefe chords is always $ P, 
 and another [j P, we have therefore two forts of harmony, call- 
 ed fiiarp and fiat harmony ; in the former of which, the $P 
 chords are more frequent, and in the latter, the \) P. If we have 
 begun with conceiving the key note, as bearing a $ P chord, 
 which is the cafe with the natural fcale, defcribed Ch. I. and II. 
 we chufe to refer all the following notes to fome $ P chord, as 
 often as we can doit, confiflent with other rules which we 
 lhall fpeak' of in due courfe » and if we have begun with a key 
 note, bearing a \) P chord, the fame preference is due to the ^ P 
 chords, fo long as we retain that key note. This may ferve at 
 prefent to give fome idea of the two different forts of harmony ; 
 which we lhall examine more minutely, after difpatching what 
 is farther neceffary to be previoufly obferved in regard to confo- 
 nances. 
 
 151. Within thefe laft fifty years, the greateft writers on mu- 
 flc have very affiduoufly purfued thefe enquiries concerning the 
 nature and propriety of fundamental fuecejjions ; which they oc- 
 cafionally exhibit in an additional ftaff of lines, placed below the 
 others, and called a fundamental bafs, This bafs is not intend- 
 ed to be performed along with the other parts, for the reafons 
 given, Art. 134.; but inftead thereof, another bafs is fubftitut- 
 ed, confifting of fundamentals and harmonics intermixed, which 
 is called the continued or thorough-bafs. The ufe of the funda- 
 mental bafs, when fubjoined, is only to fhew from whence eve- 
 ry confonance is derived, and to what note each of the founds 
 ought to be refered. 
 
 The fundamental bafs is never fubjoined to a piece of mufic, 
 except in examples defigned for beginners, but inftead of it there 
 
Chap. VI. Of the choice of Chords. Of the fundamental, and figured thorough-hafs . Concord and e Difcord defined. 52 
 
 The reader ought to be advertifed, that the method of iigur- 
 
 are figures, and other marks put above or below certain notes of 
 the thorough bafs, to denote the fpecies of every borrowed chord 
 which occurs, and this is called a figured thorough- bafs. 
 
 152. When the thorough-bafs takes the third of a perfect 
 chord* we have already feen, Art. 138. that the upper parts 
 then become fixth and third to this. bafs, and it is called a chord 
 of the fixth ; and when the thorough-bafs takes the fifth of a 
 perfect chord, the upper parts become fixth and fourth to this 
 bafs, and it is called a chord of the fixth and fourth. 
 
 The exiftence of a chord of the fixth is known by the figure 
 6.. and that of fixth and fourth, : by the figures 6 and 4. over 
 or under the bafs note. See the bars L and M of Ex. XXXIII. 
 and XXXIV. When the thorough-bafs coincides with the 
 fundamental, in a perfefr chords there is no occafion for any 
 figures. The accompanyment of the third, fifth, and octave, 
 belonging to a fundamental, is fuppofed to take place, either 
 with or without any of thefe figures, 3, 5, 8. annexed to the 
 note * 
 
 This is all that relates to the figuring of the perfect chords 
 in the fcale, and their derivatives ; and it is very eafy to re- 
 member, becaufe the inverted chords are figured juft as they are 
 named. The manner of figuring other chords will be explain- 
 ed, when thefe come under confideration. 
 
 thorough-bafs is not fo completely fettled among mufici- 
 ans, but that different mafters denote the fame chord in diffe- 
 rent manners. We fhall endeavour, in the courfe of this work, 
 to explain the method ufed by Signior Pafquali, in his Thorough- 
 bafs made eafy, being the fame which is molt generally received 
 in Britain ; and at the fame time add, in the fubjoined notes, 
 fuch remarks concerning the figuring of foreign mafters, as 
 feem pertinent and ufeful. 
 
 1 53. Two founds are faid to be concord betvieen themfelves, 
 when both of them can be refered to one and the fame funda- 
 mental perfect chord; and two founds are called difcord, when 
 they cannot both be refered to one perfect chord. 
 
 This is the molt fimple, and, at the fame time, the moft fa- 
 tisfadfory definition we can give of concord and difcord: for, al- 
 lowing that the mind naturally chufes to conceive every found 
 in mufic as belonging to fome perfect chord, it is plain, that 
 two founds willymtt to unite, when both of them are included 
 in the idea of one perfect chord ; and that they will not .unite, 
 but feparately diflracl our attention, when this cannot be done, 
 or when they muft neceflarily be refered to two different funda^ 
 mentals. . • 
 
 Neverthelefs, difcords, when judicioufly chofen, and proper- 
 
 * Though the third and fifth be implied, when no figures are annexed to a thorough-bafs note, yet there are cafes in which it is neceffary to figure 
 the perfect chord as well as any other. For inftance, when the fame note occurs feveral times over in the bafs, and having firft born fome other chord, 
 comes afterwards to bear a perfecSt chord : in this cafe it is necefiary to denote theenterance of the perfect chord by one or more of thefe figures, 8. j. 3. 
 becaufe otherwife the former chord would appear to be continued; for it is a rule, when any note is feveral times repeated, that, in playing thorough-bafs, 
 the chord of the firft note is. to be continued upon all the reft, unlefs contradicted by other figures. >j . 
 
 It is alfo proper to figure the perfect chord, in fome pafTagts, where it is introduced out of the ordinary courfe, and, of confequence, partly unexpected ; 
 becaufe the performer, feeing no figures to direct him, might follow the ordinary courfe, and fo run contrary to the defign of the compofer. 
 
 As it is fometimes necefiary to figure the perfect chord, it is alfo fometimes convenient to figure the chord of the fixth with 8 and 6, or with 6 and 3', ra- 
 ther than 6 fimply; and this often happens, when a former chord is to be contradicted upon the repetition of the fame note, as will be better underftood 
 hereafter. In the fame manner, and for the fame reafon, the figure 8 is fometimes added to the 6 and 4, which ufually denote the chord of the fixth and 
 fourth. 
 
 It is evident, that the occasional addition of thefe figures creates no difficulty, becaufe, whether they be there or not, the founds which they reprefent, 
 ire always implied in the ftrudture of the chord, as fhewn, Art. 138. 
 
 o 
 
ca Of Harmomcal Confonances. Part I. 
 
 ly inferted, are of great ufe in harmony, and produce the caufe no other chord contains both thefe notes, For the lame 
 moll agreeable effects f. We can frequently admit of, and re- reafon the 2d and 6th muft be refered to the if, and the 3d 
 ceive a particular pleafure from the intermixture of two funda- 
 mental perfect chords, if fuch dijjbnant or dov.bl' meaning chords 
 be properly preceded and followed by others ; or, in the mufi- 
 
 cian's language, if the difcords be properly prepared and refriv- 
 ed, of which more hereafter. 
 
 1 54. According to the above definition of concord, it will 
 eafily appear, that all the thirds which exift in the fcale-, whe- 
 
 and 7th to the 3/: and fo of all other perfect fifths and fourths. 
 When the K and the 3d ftand together, they conftitute a 
 greater third, when erect, andalefs fixth, when inverted ; and 
 thefe two may be conceived as belonging to the chord of the 
 Kf, or to that of the 6f; for either of thefe chords contains 
 both thofe notes : fo that here it is in our option to chufe either 
 the lower term of the third, viz. the K, or the note which is 
 
 ther they be greater or lefs ; and, by inverfion, all the fixths another third below this term, viz. the 6th-, for a fundamen- 
 
 in thefcaie are concords : as alfo all the fifths in the fcale are tal ; and the preference muft be given to owe or the other of 
 
 concords, except the falfe fifth-, which lies in afcending, be thefe, in compliance with other rules and reftlictions, to be treat- 
 
 tween the 7th and the 4'h ; and> by inverfion, all the fourths ed of hereafter. ' 
 
 are concord's, except the tritonc, fee Art. 1 1. which is the in- Again, when the 2d and 4th ftand together they conftitute 
 
 verfion of the falfe fifth, and lies in afcending between the a lefs third, when erect, and a greater fixth, inverted ; and 
 
 4th and the 7th. thefe two muft be refered to the if, and ho other, becaufe 
 
 155. It is alfo evident, that both the terms of every perfect were we to feek a fundamental, another third below this lower 
 
 fifth mult be refered to the lower term as fundamental, and of term, we fhould fall upon the 7th, which, as (hewn Art* 148. 
 
 confequence both the terms of every perfect fourth muft be re- cannot be admitted as fuch. 
 
 fered to the upper term as fundamental : but- in regard to thirds 
 and fixths, fometimes the lower term of a third, or the upper 
 term of a fixth, is to be efteemed the fundamental ; and 'fome- 
 times another found, which is a third below thefe terms, muft 
 be pitched upon, and oftentimes it is in our option to chufe ei- 
 
 On the contrary, when the 7th and 2d ftahd together as a 
 lefs third, we are obliged to defcend a third below the lower 
 rerm, which brings us to the £/". becaufe here the lower term 
 itfelf is the 7th. 
 
 From thefe inftances it will eafily appear, that the avoiding 
 
 ther of thefe for a fundamental ; but neither the upper term of of the 7th of the fcale in the fundamental bafs, is the fole caufe 
 
 fifth, Or a third, h6r the lower term of a fourth, or a fixth, 
 can be efteemed the fundamental, becaufe, in thefexrafes, theo- 
 ther term will not belong to the chord. 
 
 A few examples will make this fufficiently plain. When the 
 Kand the 5th ftand together they conftitute a perfect fifth when 
 erect, and a perfect fourth, when inverted, and in either pofi- 
 rion thefe two muft be refered to the Kf. arid no other, bc- 
 
 of all thefe varieties in the choice of fundamentals. 
 
 Thus much may fuffice, at prefent, in regard to confonances. 
 We muft crave the reader's farther patience and attention to a 
 few remarks conceE-ning diffonances, which are neceffary before 
 we can properly proceed to the application ; but fhall be as con- 
 cife as poffible. 
 
 f The definition commonly given of dlfcord in general, viz. that it is the efeft of two founds, ivhofc union difplcafes, or is difcgrceublc to the car, is not pro- 
 per; becaufe, though all difcords diftracf. our attention, yet they do not all difpleafe us, but, on the contrary, fome difcords are much more pleafing and 
 agreeable, in certain fituations, than concords would be. It feems neceflary to make a farther diftinelion between what only divides our attention, and what 
 difplcafes- The former of thefe is properly a difcord in mufic, and the latter a falfe or inharmonical relation. 
 
Chap. VII. Of Concords, and their Fundamentals. OfDifcords, and Chords dijjbnant by addition. 
 
 ss 
 
 C H A P. VII. Of D I S S O N A N C E S. 
 
 A 5 
 tervals ; fo alfo are all intervals of a feventh,. or 
 a fecond : the reafon is plain from the premifed principles, be- 
 caufe two notes, including either of thefe intervals, cannot both 
 belong to one perfect chord. 
 
 i 57. Seconds and fevenths are diftinguifhed, like thirds and 
 fixths, into greater and lefs. The whole tone is called a great- 
 er fecond, and the femitone a lefs fecond ; the feventh, which 
 is a femitone lefs than an octave, is called a greater feventh ; 
 and that which is a tone lefs than an octave, a lefs feventh. 
 
 The greater feventh, and lefs fecond, may either of them be 
 conceived as an inversion of the other ; in the fame manner the 
 lefs feventh, and greater fecond, are derived one from the other 
 by inverfion. So that we need only to confider the feconds in 
 particular, as whatever properties belong to them, may, for 
 the moft part, be equally attributed to the fevenths, only 
 changing the epithet greater for lefs, and vice verfa. 
 
 158. The lefs fecond, or femitone, is a much more harfh 
 and impracticable difcord than the greater fecond, or whole 
 tone. The former of thefe can only be admitted feldom, and 
 with great caution; but the latter is very often introduced. 
 Thus the 7th and K, or the 3d and 4th of the natural fcale, 
 are feldom joined in harmony ; thefe being intervals of a femi- 
 tone only: but the K and 2d, or the 2d and. 3d, or the 4th: 
 and 5th, or the 5th and 6th, or the 6th and 7th of the natu- 
 ral fcale, are often fet together ; thefe being all intervals of a 
 whoie tone. 
 
 1 59. A difTonant chord may be produced either by adding 
 another found to a perfect chord, or by omitting one of the 
 
 proper harmonics of a perfect chord, and fubftitutlng another 
 note in its ftead. The former of thefe is more commonly prac j 
 tiled, and therefore deferves to be firft considered. 
 
 Every erect perfect chord confifts of two contiguous thirds, 
 fituated next above the fundamental, and a perfect fourth above 
 thefe, which brings us to the octave of the fundamental. Now, 
 it is plain, that the addition of another found, to fuch a chord, 
 can be beft allowed within the limits of the fourth, in fuch 
 manner that this fourth may be divided by the added found in- 
 to a lefs third and a whole tone ; becaufe, by this means, the dif- 
 cords created among the mutual relations are the feweft and 
 moft tolerable ; and the degree of harfhnefs, in every difTonant 
 chord, may juftly be eftimated by the number and quality of the 
 difcords which exift together. 
 
 160. Hence it follows, that the added found, in a difTo- 
 nant chord, is generally either the lefs feventh, or the greater 
 fixth, above the governing fundamental ; that is, above the 
 found which would be the fole fundamental, if the difcord were 
 not heard ; for, it muft be obferved, that no one found can 
 properly be called the fole fundamental of a difTonant chord, be- 
 caufe of the divided attention which the difcord ereates ; but 
 notwithftanding we muft be fuppofed to have two fundamen- 
 tals partly in view, yet one of them may, for various reafons, 
 claim the greater fhare of our regard ; and therefore may be 
 properly called governing, or predominant. 
 
 161. All fundamentals are fufceptible either of a lefs feventh 
 or a greater fixth, according as the feventh above is a lefs fe- 
 venth, or the fixth above is a greater fixth. in the natural fcale 
 to which they belong. Thus the chords of the K/, and the 
 
5« 
 
 Of 1)i (finances. 
 
 Part L 
 
 4/, can have a fixth added, but not a feventh, becaufe they 
 have a greater fixth, but not a lefs feventh in the fcale. On 
 the contrary, the chords of the 3/, and the . 6f, can have a 
 feventh added, but not a fixth, becaufe they have a lefs feventh, 
 but not a greater fixth in the fcale. The chords of the if, 
 and the 5/, may have either a fixth or a feventh added, be- 
 caufe they have both a greater fixth and a lefs feventh in the 
 fcale. All this may be fuppofed obvious to the reader, being 
 drawn entirely from the fituation of the tones and femitones 
 in the natural fcale. It is plain, that if the added difcord mud 
 divide the interval of the fourth, which exifts between the fifth 
 of any fundamental and its octave, into a whole tone and a lefs 
 third, fuch added difcord muft either be a whole tone above 
 the fifth, or a whole tone below the octave of the fundamen- 
 tal : therefore, thofe fundamentals which have a whole tone 
 next above the fifth, are fulceptible of an added fixth; and thofe 
 which have a whole tone next below the fundamental, or its 
 octave, are fucceptible of an added feventh. 
 
 162. As we have formerly feen, Art. 147. that the natural 
 fcale affords fix perfect chords, viz. one for each degree of the 
 fcale, except the 7th;. fo we may here obferve, that the 
 fame fcale affords eight of this fort of chords diflbnant by addi^ 
 tion. This larger number of diflbnances by addition is occasi- 
 oned by the, two fundamentals, viz. if and 5/, which are 
 each of them fufceptible of both kinds of added difcords. 
 
 Of thefe eight diflbnant chords there are four which have 
 the added fixth, viz. the Kf, 4/, 5/, and if: and the other 
 four have the added feventh, viz. the 6f if 3/, and $f. 
 But it is farther to be obferved, that there are three with the 
 added fixth, and other three with the added feventh, which 
 contain the very fame degrees reflectively, and differ only 
 in the efiimation of the fundamental. Thus the Kf, with an 
 added fixth, contains the very fame degrees as the 6f with 
 an added feventh ; and the difference between thefe two lies 
 
 only in the manner of conceiving the founds : fo that the 6th 
 of the fcale, which is the added difcord in the former, is the 
 fundamental in the latter chord. In the fame manner the 4/, 
 with a fixth added, and the if, with a feventh, confift both 
 of the fame founds : thus alio the 5/, with an added fixth, 
 and the 3/, with an added feventh, differ only in the manner 
 of conceiving the founds ; fo that if we account the two chords, 
 which thus coincide together, only as one, we (hall reduce the 
 number of difTonances, by addition, in the fcale, to five ; of 
 which fee the following reprefentation, where the proper dif- 
 cord added, is denoted by a fmall figure put over the funda- 
 mental. 
 
 f 6 7 
 
 Kf or 6f contains the K — 3d — 5th,6th — 
 
 6 7 
 
 4/ or if contains the K 2d — 4th — 6th — 
 
 6 7 
 
 5th— 7th 
 
 1 — 7th 
 
 [ if contains the — 2d — 4th— 6th 7 th 
 
 5-f or 3/ contains the — 2d 3d 
 
 5/ contains the — 2d — 4th 5tl 
 
 In this reprefentation, we have taken the liberty of inverting 
 fome of the chords, fo as to comprehend all their founds with-' 
 in the compafs of an octave above the key, to render the fcheme 
 more compact, and alfo that the occurrence of the fame note, in- 
 different chords, may be more readily feen : and we fay, as be- 
 fore, Art. 147. that thefe chords contain thefe founds reipective- 
 ly, to intimate that different arrangements ffill constitute the 
 fame chord. 
 
 163. We may conclude, that there can be ordinarily no more 
 than five chords diflbnant, by addition, among the founds of 
 the natural fcale, from obferving, that there are but five de- 
 grees which are greater feconds, i. e. whole tones, and the dif- 
 
Chap. VII. Scheme of Chords diffonant by addition 
 
 fonant chord always contains a greater fecond, or, which is the 
 fame in effect, a lefs feventh, among the mutual relations : See 
 Art. 158.; and this lefs feventh is always divided by the other 
 accompanying founds, into three intervals of a third each ; one 
 of which three is a greater third, and the other two are lefs 
 thirds ; and the greater third is either below the two lefs thirds, 
 or between them, or above them, according as the fteps of the 
 natural fcale allow. 
 
 164. In regard to the fpecies of thirds, which refult from 
 the above-mentioned divifion of the fevenths, it will be ufeful 
 to obferve, that when the lefs feventh is added to a % P chord, 
 which can only happen in the chord of the 5/, the greater 
 third is fituated below two lefs thirds ; and when the greater 
 fixth is added to a \> P chord, which can only happen in the 
 chord of the if, the greater third is above two lefs thirds ; 
 but in all other chords diffonant by addition, the greater third 
 is between two lefs thirds. 
 
 165. The fpecies of the three contiguous thirds, would be 
 more eafily feen, if each chord were difpofed in an erect form, 
 fo as to bring the proper fundamental, which bears an added 
 feventh, into the loweft place, and the feventh itfelf uppermoft ; 
 for thus every chord would plainly appear to confift of three 
 fucceffive thirds, in afcending from the fundamental, and the 
 fpecies of each third could not fail to be obvious, becaufe the 
 very names of the founds exprefs their fituation in the natural 
 fcale, and of courfe, the places of the femitones, which always 
 conltitute one of the two fteps of a lefs third. 
 
 We (hall inftance the firft of the chords in the fcheme, Art. 
 162. which is that of the Kfi with a fixth, or the 6/, with a 
 feventh added ; therefore the 6th is to be brought into the low- 
 eft place, and then the chord ftands thus, 
 
 6th — K— 3d— 5th. 
 where we have three fucceffive thirds in afcending from the 6th 
 to the 5th. Now, the femitones being the next below the K, 
 
 . Of the mutual Relations of their founds. $j 
 
 and the next above the 3d, it is plain, that the intervals fith— K, 
 and 3d — 5th, are lefs thirds, becaufe they include one ftep 
 of a femitone : the interval K — 3d is a greater third, becaufe 
 both the included fteps are tones : fo that this chord is compof- 
 ed of a greater third between two lefs, as obferved above. 
 
 After the fame manner, the other chords, in Art. 162. might 
 be arranged and examined ; but this we fhall leave as a tafk for 
 the exercife of fuch readers as do not clearly apprehend the 
 ftructure of them, in the fevera! inverted forms there given. It 
 may reafonably be expected that few readers will have found 
 any difficulty in this matter. 
 
 1 66. By obferving the fpecies of the thirds, we come eafily 
 to the knowlege of the perfect and falfe fifths, which exift in 
 thefe chords : for if it be remembered, that two contiguous lefs 
 thirds con ftitute a falfe fifth, Art. 148. it will evidently appear, 
 that the chords which have the greater third, either below or 
 above both the lefs thirds, have inevitably a falfe fifth, or by 
 inverfion, a tritone, among their mutual relations: but the 
 others, which have the greater third between the two lefs, have 
 no fuch interval. Thus, the two loweft chords in the fcheme, 
 Art. 162.; viz. the 5/ with a feventh, and the 2/ with a 
 
 fixth added, have a tritone, viz. 4th 7th, which the 
 
 three uppermoft chords in that fcheme have not : and herein 
 confifts the moft material difference among this fort of chords, 
 as will be fhewn below. 
 
 167. The perfect chord of either kind, with an added fe- 
 venth, is called a chord of the feventh, and figured with a 7. 
 Sometimes one or both of the figures 5'. 3. are added below the 
 7, or even above it ; but this makes no difference, becaufe the 
 founds reprefented by them are always implied, whether they 
 be there or not. 
 
 168. The thorough-bafs may take any of the harmonics be- 
 longing to a chord of the feventh, viz, the third, the fifth, or 
 the feventh ; and hence arife three other chords by inverfion : 
 
 P 
 
i« 
 
 Of 'Dijfonances. 
 
 Part T. 
 
 thus, when the thorough bafs takes the third, the other accom- 
 panyments are fixth, fifth, and third, to this bafs, and it is cal- 
 led a chord of the fixth and fifth, and figured with 6 and 5, 
 one under the other. When the thorough-bafs takes the fifth 
 of the fundamental chord, the other accompanyments are fixth, 
 fourth, and third, to this bafs ; and it may be properly called 
 a chord of the fourth and third, and figured with 4 and 3, one 
 under the other ; the exiftence of the fixth being implied, tho' 
 not figured : but it is more ufual, among our Britifh muficians, 
 to add. the fixth alfo, both In the naming and figuring of this 
 chord. When the thorough-bafs takes the feventh of the fun- 
 damental chord, the other accompanyments are fixth, fourth, 
 and fecond to this bafs, and it is called a chord of the fecond, and 
 figured either with 2 fimply, or with 4 and 2, or 6, 4, and 2: 
 the exiftence of the fixth and fourth being implied along with 
 the fecond, although none but the fecond be expreffed either 
 in the naming, or the figuring of this chord. 
 
 169. For the better underftanding of the chord of the feventh, 
 and its derivatives, fee the following reprefentation of the chord 
 of A, which is formed in the fame manner as thofe already given. 
 
 The chord of the feventh to A, A— C— E— G 
 
 which we count 1 — 3 — 5 — 7 
 
 The fixth and fifth to C, C— E— G a 
 
 which we count 1—3 — 5 6 
 
 The fourth and third to E, E— G a— c 
 
 which we count 1 — 3 4 — 6 
 
 The fecond to G, G a — c — e 
 
 which we count 1 2 — 4 — 6 
 
 See alfo Plate IV- Exam. XXXV. the firft bar of the ex- 
 ample; where the fame chord of A, and its inverfions, are re- 
 prefented in the treble and bafs ffaves, with the fundamental 
 bafs fubjoined. It may be obferved, that when the bafs takes 
 
 the feventh of the fundamental chord, it is not proper to double 
 the bafs note in the accompanyment, as in the fourth chord of 
 this bar, becaufe fuch doubling would render the diflbnant fe- 
 venth too remarkable. 
 
 1 70. In this example, the fundamental bafs is put A with 7, 
 or C with 6 and 5 ; becaufe this, and its inverfions, pro- 
 duce, apparently, the very fame founds as the other, and 
 the difference lies only in the eftimafion of the fundamental. 
 Thus, if the note C, bearing fixth and fifth, were eftimated 
 the fundamental, the note A, bearing a feventh, would be one 
 of its inverfions, and the note E, bearing fourth and third, 
 and G, bearing a fecond, would be its other inverfions. 
 
 171. It is often more proper to omit fome of the founds of a 
 diffonant chord, in the accompanyment, and in feveral cafes 
 thefe omiffions may be, and ufually are, (hewn by a different 
 manner of figuring. Thus, for inftance, if it be proper to o- 
 mit the diffonant feventh in a chord of 7, or the fixth in a 
 chord of 6 and 5. the thorough-bafs is left without any figure, 
 or with fuch only as denote the perfect chord, Art. 152. 
 
 Thus alfo, if the fifth be omitted in a chord of 6 and 5, or 
 the fourth in a chord of 4 and 3, in fuch cafes either of thefe 
 chords is figured with 6 only, or with 6 and 3, to denote the 
 exiftence of the fixth and third. 
 
 Again, if the third be omitted in a chord of 4 and 3, or the 
 fecond in a chord of 2, the thorough bafs is figured with 6 and 
 4, becaufe in thefe cafes the fixth and fourth only remain. 
 
 172 We muft therefore take notice, that a chord in the 
 thorough-bafs may put on the appearance of the perfect chord, 
 or one of its inverfions, and yet in reality be derived from a 
 fundamental diffonant chord ; with the omiffion of one of the 
 two contiguous founds, which conftitute the difTonance : and 
 in order to comprehend this matter more fully, let the fecond 
 and third divifions of Exam. XXXV. be carefully examined. 
 The former of thefe divifions contains the derivatives of the 
 
Chap. VII. Of the Names and Ways of figuring the Chord of the Seventh, and its Liver fions. Of Sufpenfons. ^ 
 
 chord of A, with 7, which, by omiffion of the feventh, or of 
 the eighth above the fundamental, appear to be the produce of 
 the perfect chords of A, or C, refpectively. The latter divifion 
 reprefents the derivatives of the chord of G, with an added 7, in 
 the fame manner : and here it muft be obferved, that although 
 the original chord contains a falfe fifth, viz. from B to F. (fee 
 Art. 166.) yet when the octave of the fundamental is omitted, 
 as in the three laft chords of this example, the thorough-bafs is 
 figured in the fame manner as if the chords were derived 
 from the perfect of B ; although B is incapable of a perfect 
 chord, on account of the falfe fifth. In this, and fuch like in- 
 ftances, therefore, the falfe fifth, and the tritoneorgre.it fourth, 
 are denoted in the fame manner as if they were perfect ; and it 
 is left to the performer to apply fuch intervals as the fcale al- 
 lows * ; which ought to be well remembered. 
 
 173. In regard to the fecond fpecies of diffonances, Art. 159. 
 viz. thofe which have another found fubftituted inftead of one 
 of the proper harmonics, the fubftituted found for the moft part 
 only difappoints, or fufpends, the hearer's expectation, for fome 
 part of the time, and afterwards gives place to the proper found ; 
 and therefore thefe may be juftly called dhTonances by fufpenfwn. 
 
 The moft common diffonances of this clafs, derived from the 
 perfect chord, are, 
 
 ift, The chord of the fourth, which is figured with a 4, or 
 with 5 and 4 ; in which the fourth only fufpends the third, the 
 fifth and octave at the fame time taking place as ufual. 
 
 2d, The chord of the ninth, which is figured with a g, or 
 with 9 and 5, or 9 and 3 ; in which the ninth only fufpends 
 the octave, but the fifth and third take place as ufual. 
 
 3 J, The chord of the ninth and fourth, figured with 9 and 4, 
 or 9, 5, and 4 ; in which both thefe fufpenfions take place at 
 once, and the fifth only retains its proper place. 
 
 174. The fame kind of fufpenfion frequently takes place in 
 the chord of fixth, viz. 
 
 1/?, The feventh fufpending the fixth, in a defcending fuccef- 
 fion, which is called afequence of feventh s andfixths, and figur- 
 ed with 7, 6. one after the other. Care muft be taken to di- 
 ftinguifh this feventh, fufpending the fixth, in a defcending fe- 
 quence, from the fundamental chord of the added feventh ; 
 becaufe they are both figured in the fame manner with a 7. and 
 yet the former muft not be accompanied with the fifth, which 
 is an effential part in the latter chord. But the figure 6 fol- 
 
 * There is a material difference between our method of figuring, and that of the French, in this refpeet. The French, following their great leader, 
 Monf Ramcau, figure all their chords as if they were full, and leave the choice of omijfioiis entirely to the performer's difcretion ; hy which means the deri- 
 vation of each chord is more plainly fhewn, but the propriety and delicacy of the accompanyment is more obfeure and indetermined. 
 
 Thofe chords in particular, which imply a falfe fifth, or a tritone, fuch as the inverfions of the chord of G with a 7. in the laft divifion of Ex. XXXV. 
 are diftinguifhed both by peculiar names and figures. The chord of the 6 and j. which has E for its bafs, is called, accord de faujfe quintc, and figured with 
 5 h, or s with a dafli through the body ; and this is conftantly done, whether any omiilions take place in the accompanyment or not. In like manner, the 
 chord of 4 and 3 , which belongs to the note D, is called accord de fixte fcnfible, and figured with 6 $, or 6 with a dafli through the head : and the chord of 
 the fecond to F, is called accord de triton, and figured with 4 $, or 4 with a dafh through the horizontal ftroke. See the figures fubjoined to the thorough- 
 hafs, in the laft divifion of Exam. XXXV. 
 
 The productions of other chords of the feventh, which do npt include a falfe fifth, are named and figured much in the fame manner as ours ; and fome- 
 times the chord of the fixth and fifth is called accord de grande fixte : and that of the fixth, fourth, and third, accord de petite fixte. 
 
 Thefe are all the material differences between the Britifh and French methods of naming and figuring the chord of the feventh, and its derivatives, which 
 it feemed proper to explain in this place, more on account of the tbeorifis, than the compofcrs, which that nation has produced. 
 
 We flatter ourfelves, that, by following the Italians, we have adopted a better method of figuring, and of more immediate ufe to the performer ; although 
 the origin of fome chords may, in our method, be more uncertain. 
 
60 Of e DiJ/onances. 
 
 lowing upon the fame, note, fufficiently indicates the fe- 
 quence, 
 
 id, The fifth fufpending the fixth in an afcending fucceiTion, 
 which is called a fequence of fifths andfixths, and figured with 
 5, 6, one after the other. 
 
 2(1, The ninth may fufpend the octave and the feventh fuf- 
 pend the fixth, both at the fame time, in a chord of the fixth; 
 in which cafe the third only retains its proper place ; and this 
 is called a chord of the ninth and feventh, and figured with 9 
 and 7, or with 9, 7, and 3. 
 
 175. Some of thefe chords may be inverted, and from thence 
 other chords formed, which have alfo their proper ufe ; and 
 befides the above, there are feveral other fufpenfions practicable 
 in mufic, the fall detail of which would be too tedious in this 
 place. It may be obferved in general, that fuch paffages fome- 
 times affect the fundamental progreffion, and fometimes not : 
 and they are then only, properly fufpenfions, when the funda- 
 mental progreflions is not affected by them. 
 
 1 76. In order to judge of the fundamental to which a diffo- 
 nant chord ought to be refered, it is ufeful to obferve, that eve- 
 ry note which has its own harmonics, and efpecially its fifth ex- 
 iting along with it, will in fome degree attract our attention 
 as a fundamental, agreeable to what is remarked, Art. 155. ; 
 and if, in conjunction with this circumfhince, fuch a note be 
 rendered more dift inguifhable, by its fituation below or above the 
 reft, as explained, Art. 144. it will inevitably be regarded as 
 a governing fundamental, for the time : but if it be concealed 
 in the middle of the chord, another note, in a more confpi- 
 cuous fituation, may take the preference. 
 
 It will be neceffary alfo, in confidering the fucceffive chords, 
 as connected together in harmonic fhrafes, to take into account 
 the expectation, whether founded on nature or cujl.om, of hear- 
 ing fuch particular fundamentals in fucceffion, which often takes 
 place, and may be fo ffrong, in fome paffages, as to overcome all 
 oppofition in favour of a different fundamental note. 
 
 Partt . 
 
 177. Such enquiries as thefe are of the utmoft confequence 
 In mufic, and are yet perfectly new. The various effects of 
 different inverfions of the fame chord, which are circumftan- 
 ces well known among the practical muficians, but impene- 
 trable fecrets among the theorifts, are by this means accounted 
 for, with the greateft eafe and fimplicity. We fhall here add a 
 few inftances. 
 
 1 78. The chord of the feventh, when the fundamental bears 
 a lefs third, may be confidered as a mixture of two perfect 
 chords, whofe fundamentals are the two terms of the fame lefs 
 third. Thus the chord of A with a feventh is made up of the 
 \) P chord of A, and the % P chord of C ; becaufe each of thefe 
 notes has its perfect chord in the accompanyment. Now, if this 
 chord be arranged in its erect form, as at H, in Exam. XXXVI. 
 in this cafe the note C, being partly concealed in the middle, and 
 A at the bottom, the note A is undoubtedly the governing fun- 
 damental : but at K the fame chord is arranged with C upper- 
 moft, and A in the middle ; in which cafe C will command our 
 principal attention. At L, the note A is uppermofr, and C 
 loweft ; which caufes a divided attention, or double fundamental, 
 and the preference due to one or the other is to be determined 
 from other confiderations. The cafe is much the fame at I, 
 where both C and A are in the middle. 
 
 In thefe ambiguous cafes, if one of the notes which thus ffand 
 in competition, be doubled, or be made more remarkable by 
 falling in with the expectation of the hearer, the choice will be 
 determined by either of thefe circumftances. 
 
 In the fraff of lines fubjoined to Exam. XXXVI. the govern- 
 ing fundamentals are reprefented by notes, and the other by di- 
 rects : and the ambiguous or double fundamentals have two 
 directs. 
 
 1 79. Thefe ambiguities have no place in a chord of the fe- 
 venth, when the fundamental bears a greater third, becaufe the 
 fifth above that greater third isafalfe fifth. Thus, for infiance, 
 the chord of G with a feventh, and all its inverfions, muff in- 
 
Of the Fundamentals of dljfonant Chords. 
 
 Chap. VII. 
 
 conteffibly be refered to Gf, becaufe no other found has its 
 perfect fifth exifting along with it, in that chord. 
 
 The falfe fifth has alfo a very peculiar property of refering 
 the hearer to a fundamental note, at the diftance of a greater 
 third below its lower terra, even although that fundamental 
 note do not appear at all, in the chord. Thus the falfe fifth 
 B — F, will plainly point out G for its fundamental, though 
 G itfejf be altogether omitted, as at M, Exam. XXXVI.. The 
 fame is to be obferved of all other falfe fifths : for inftance, 
 
 F % C mufl be refered to Df 
 
 C* G to A/ 
 
 G« — D to E/ &c. 
 
 and this property mull be carefully attended to. 
 
 1 80. The tritone, which is the inverfion of the falfe fifth, 
 rather refers to a fundamental, at the diftance of a lefs third 
 below its lower term. Thus the tritone F B has a na- 
 tural tendency to point out D for its fundamental ; efpecially 
 if the note A be heard along with it, as reprefented at N : be- 
 caufe A is the perfect fifth to that fundamental ; and more- 
 over the fame A prevents G from taking place as a fundamental ; 
 whereas if A were not heard, and if the fucceffion of chords 
 could conveniently admit of it, G might be efteemed the 
 
 fundamental of the tritone F B, as well as of the falfe 
 
 fifth from which it is an inverfion. 
 
 There is fomething curious in thefe effects of the falfe fifth 
 and tritone, which we fhall endeavour to account for in the 
 theory. 
 
 181. The chord of the fourth, as reprefented at O, is a mix- 
 ture of the chord of the fufpending fourth, along with that of 
 the fundamental ; becaufe this chord confifis of the fundamen- 
 tal, with its perfect fifth, and the fourth with its perfect fifth ; 
 but then die fundamental, being both in the lowefl and higheft 
 place, will undoubtedly attract our principal attention : and 
 even if the fourth were to poffefs the higheft place, as at P, the 
 
 6l 
 
 fame fundamental would ftill prevail, becaufe it is more natural 
 to conceive the lowefl found as a fundamental, than the higheft. 
 But if the fame chord be inverted, as at Q, where the fourth 
 polTelTes the loweft place, the cafe will be altered, and in this 
 chord, fingly confidered, the loweft found will become the go- 
 verning fundamental ; and the chord be effectually a chord of 
 the ninth, to that fundamental ; but yet, in a fucceffion which 
 more properly leads to the fundamental of the original chord O, 
 even this arrangement at Q^may pais for an inverfion of that 
 chord, and in fuch cafes it ought to be figured with 5 and 2, 
 as it is there done, and care mult be taken not to double the 
 bafs note in the accompany men t, becaufe it is not one of the 
 proper harmonics of the chord. 
 
 The fame is to be obferved of the arrangement at R, which 
 is an inverfion of the fame chord of the fourth, having the 
 fifth loweft. This chord, fingly confidered, would feem to 
 belong to the higheft found as fundamental, but when it occurs 
 in a fucceffion wherein it may properly be taken for an inverfi- 
 on of the chord at O, it is then to be figured with 7 and 4. And 
 this, as will be fhewn, Art. 186. is the only proper introducti- 
 on of the feventh and fourth to any bafs note. 
 
 1 8 2. The chord of the uinth, at S, is a mixture of the chord 
 of the fifth note, along with that of the fundamental ; becaufe 
 the ninth ftands as a fifth to the fifth note : but here the fun- 
 damental will naturally take the preference, as being the loweft 
 found, and therefore this ninth i,s properly a fufpenfion. 
 
 The fufpending ninth may tje a falfe fifth above the fifth 
 note of the chord, in which cafe the fifth note is incapable of 
 being a fundamental ; and this happens in the chord of the 
 ■if of the natural fcale, as in phe chord of E, reprefented at T : 
 and here we have a right to conclude, from the property of the 
 falfe fifths, defcribed, Art. 179. that the attention of the hear- 
 er is divided between G and E, that is, between the fifth and 
 the third'of the natural fcale. 
 
61 Of T>iJJbnances. 
 
 There may be other chords formed by inverting the chord 
 of the ninth, but being very rarely made ufe of, we mail 
 pafs them over at prefent. 
 
 183. The chord of the ninth and fourth, as at U, tends dill 
 more directly than that of the fimple ninth towards introducing 
 the fifth note for a fundamental, in preference to the lowed note ; 
 becaufe here, the fufpending fourth, not only hinders the ente- 
 rance of the third, which is one of the proper harmonics of the 
 lowed note, but that fourth alfo ftands in the relation of a 
 feventh to the fifth, which circumffance alfo gives additional 
 influence to the fifth ; and therefore we conclude this chord, 
 even in its erect form, to have a dubious or double fundamen- 
 tal, as reprefented in the example. 
 
 184. The feventh fufpending the fixth in a defcending fe- 
 quence, when that feventh is a perfect fifth above the third, 
 tends to introduce the fame third for a fundamental, and the 
 fixth which follows and refolves the feventh, has a right by its 
 fituation to be regarded as the fundamental of the next chord, 
 as reprefented at W. ' 
 
 185. Upon the fame principles, the fifth fufpending the 
 fixth, in afcending, when it is perfect, will appear to introduce 
 along with it the lowed: note of the chord, for a fundamen- 
 tal, during its exidence; and when the fixth fucceeds to the 
 fifth, that fixth appears to have a right to be condituted a fun- 
 damental, as fhewn at X. But in both thefe fequences the ex- 
 pectation of the hearer often Counteracts thefe motives, and o- 
 bliges us to afiign other fundamentals to feveral of thefe chords, 
 as will be ihew when we come^tb treat of fundamental progreffi- 
 ons. ' r - 
 
 186. To finifli this difquifition at prefent, which probably 
 the reader, not acquainted with the application, will have thought 
 too long ; we fhall obferve in general, that the co-exidence of 
 the perfect fifth above, or fourth below any found, ferves to 
 
 fortify that found, and to draw the hearer's attention to it. 
 
 Part I. 
 
 much in the fame manner as if it were doubled ; and therefore 
 it is not proper to introduce a note, by fufpenfion, which has its 
 perfect fifth in the chord, unlefs that note could properly take 
 place as a fundamental ; and it is equally improper that the 
 fufpending note dand as a fifth to any of the other notes of the 
 chord, unlefs that other note could, on occafion, fucceed as 
 fundamental. Thus, for indance, the fixth mud not be made 
 to fufpend the fifth, in a perfect chord, at the fame time that the 
 third keeps its place, becaufe that third, exiding along with the 
 fixth, would in effect double the fufpending fixth, and render 
 it too remarkable ; but the fixth may fufpend the fifth, at the 
 fame time that the fourth fufpends the third, and this is very 
 frequently pra&ifed. Thus alfo, in a chord of 7, the fourth 
 mud not fufpend the third, becaufe that fourth would render 
 the didbnant feventh too remarkable by its affinity with it. The 
 fame obfervations may be extended to all the other chords. 
 
 187. The new principle we have chiefly to infid upon, which 
 accounts in the mod familiar and rational manner for the vari- 
 ous fucceffions of diffonant chords, practicable in mufic, is, that 
 almod all chords of this kind imply, in fome degree, a double 
 
 fundamental, and that it is neceffary that each of the implied fun- 
 damentals naturally follow what has preceeded, and alfo be na- 
 turally fucceeded by what follows: and this principle, joined with 
 a certain propenfity in fome founds to defcend, and in others 
 to afcend, which feems to depend partly upon their relations 
 to the key note, abdract from any regard to the fundamental 
 prefently exiding, and which we fhall endeavour to explain in 
 due courfe, will carry us with eafe and perfpicuity through a 
 fnbject which has hitherto perplexed and baffled the mod dili- 
 gent enquirers. 
 
 188. The following recapitulation of the chords explained 
 in this and the preceding chapter, will be of fervice to affid 
 the learner's memory. 
 
 j. Toe perfect, or common chord, confids of the 8 th, 5th, and 
 
Chap. VII. Of extraneous Fundamentals, introduced by Sujpenjion 
 
 3d, to the bafs note, and is denoted by all or any of thefe fi- 
 gures, or is underftood to take place when no figure is annexed 
 to the bafs — the 3d may be greater or lefs, and the fifth perfect 
 or falfe, according as the fcale allows, in which the mufic is fet. 
 
 1. The chord of thefixth, confifts of the 8th, 6th, and 3d,, 
 and is figured with 6, either with or without the other figures. 
 It is an inverfion of the perfect chord, the bafs taking the third- 
 note of that chord. 
 
 3. The chord of the ftxth and fourth, confifts of the 8th, 6th, 
 
 Recapitulation. 
 
 63 
 
 10. The chord of the feventh and fourth, figured with 7 and 4, 
 is an inverfion of the chord of the fourth, the bafs taking the 
 fifth note of that chord. 
 
 The chord of the ninth, is the perfect chord with the 9th 
 
 and 4th, and is figured with 6 and 4. It is an inverfion of the 
 perfect chord, the bafs taking the fifth note of that chord; 
 
 4. Tlie chord of the feventh, is the perfect chord with the 7th 
 added, and is figured with 7. either with or without any of the 
 figures 8. 5. 3. — It is the chief of all diffonances by addition, 
 and all the others may be deduced from it, by inverfion. 
 
 5.. The chord of thefixth and fifth, is the perfect chord with 
 a 6th added ; or it is an inverfion of the chord of the feventh, 
 the bafs taking the third note of that chord, and is figured with 
 6 and 5. 
 
 -6. The chord of the fourth arid third, commonly called fixth, 
 fourth, and third, and figured accordingly, is an inverfion of 
 the chord of the feventh, the bafs taking the fifth note of that 
 chord. 
 
 '.".'7. The chord of the fecond, confifis of the 6th, 4th, and 2d, 
 and is figured with 2, either with or withont the other figures. 
 Jt is an inverfion of the chord of the feventh, the bafs taking 
 the feventh, or diflbnant note, of that chord. 
 
 8. The chord of ', the fourth, is the perfect chord with the 4th 
 inftead of the 3d, and is figured with 4, or with 5 and 4. 
 Here the 4th only fufpends the 3d. for a time, and muft after- 
 wards defcend into it. 
 
 9. The chord of the fifth and fecond, figured with 5 and 2, is 
 an inverfion of the chord of the fourth, the bafs taking the 
 fourth or fufpending note of that chord. 
 
 inftead of the 8th, and is figured with 9, either with or with- 
 out the figures 3, 5. Here the 9th only fufpends the 8th for a 
 time, and muft afterwards give place to it. 
 
 1 2. The chord of the ninth and fourth, figured with 9 and 4, 
 or 9, 5, and 4, is the perfect chord with the 9th, fufpending 
 the 8th, and the 4th fufpending the third, both at the fame 
 time. Thefe notes mull; afterwards give place to the proper har- 
 monics. 
 
 13. The fequence of fevenths and fixths, figured with 7 and 
 6, one after the other, is a chord of the fixth, wherein the 
 7th fufpends the 6th, in a defcending fucceflion, for fome part 
 of the time, and afterwards gives place to it. 
 
 14. The fequence of fifths and fixths, figured with 5 and 6, 
 one after the other, is a chord of the fixth, in which the 5th 
 fufpends the 6th, in an afcending fucceflion, tor fome part of 
 the time, and afterwards gives place to it. 
 
 1 5. The chord of the ninth and feventh, figured with 9 and 7, 
 or with 9, 7, and 3. is a chord of the fixth in which the 9th 
 fufpends the 8th, and the feventh fufpends the 6th, both at the 
 fame time. Thefe notes muft afterwards defcend. 
 
 Signior Pafquali, in his Thorough- Bafs, enumerates and ex- 
 emplifies twelve chords, as being all which ordinarily occur. 
 The reader, who is acquainted with that valuable piece, wilt 
 obferve, that eleven of them are in this recapitulation, befides 
 two others at N° 9 and 10. which are not there. 
 
 The chord in Pafquali's, book, which we yet want, is that of the 
 2d, 4th, and 7th, and is only introduced by licence, when the 
 bafs holds upon the key-note, of which we fhall fpeak hereafter. 
 
H 
 
 Of Fundamental ProgreJJions. 
 
 Part I. 
 
 Chap. VIII. Of Fundamental Progress ions. 
 
 189. T N regard to fundamental progrefTions, it Is of the great- 
 JL eft importance to underftand well the pafTages called 
 cadences in mufic, and therefore we begin with thefe. 
 
 Cadences in mufic are like points in writing, or paufes in 
 fpeaking ; and ferve in the fame manner to diftinguifh the end- 
 ing of every fmaller portion, or phrafe, as well as of the 
 whole piece : and as there are different forts of paufes neceffary 
 to mark out the larger, and the lefs divifions of a fentence, fo 
 there are different forts of cadences, fome more and fome lefs 
 fatisfa&ory and conclufive. 
 
 190. The fpecies of a cadence depends on the mutual relation 
 of two fucceflive chords, the latter being always on the moft 
 accented or emphatical part of the meafure, as at the beginning 
 of a bar. The former of thefe is called the leading chord, and 
 the cadence iB faid to be made upon the latter ; which manner of 
 fpeaking is currently made ufe of among muficians, and muft 
 be well obferved in order to the underftanding of what fol- 
 lows. 
 
 191. When the fundamental bafs falls a fifth, or rifes a fourth, 
 upon an accented part of the meafure, that paffage is called a 
 regular cadence ; and when the leading chord of fuch a cadence 
 bears a greater third and a feventh, it is called a perfebl regular 
 cadence ; but when the leading chord bears a lefs third, the ca- 
 dence is imperfetl, and inconclufive. 
 
 The perfect reguhr cadence upon the principal key muft al- 
 ways conclude the piece, like the full flop in writing ; and there 
 
 is no other way in nature of coming into the chord of the key, 
 without leaving an expectation of fomething more to follow, 
 but by the chord of the 5/, bearing a greater third and feventh, 
 as a leading chord. * 
 
 192. In order that this cadence be fully fatisfaclory, it is ne- 
 ceffary alfo, that the thorough bafs coincide with the funda- 
 mental ; or, in other words, the chords which conftitute the 
 final cadence, muft not be inverted, but taken in their erect 
 form ; that is, the thorough-bafs muft pafs from the 5th upon 
 the K, either by a leap of a fourth afcending, or a fifth defcend- 
 ing. 
 
 We muft likewife obferve, that the harmonics, or accompa- 
 nyments of the perfect regular cadence, are alfo limited in their 
 progreffion. Thus the leading chord of the 5/ contains. 
 5th — 7th— 2d — 4th. 
 
 Now, as the bafs which takes the 5th muft pafs by a leap in- 
 to the K, fo the part which takes the 7th muft afcend by a ftep 
 of a femitone into the fame K, or its oftave, the 2d ought to de- 
 fcend by a tone upon the fame K, and the 4th muft defcenct 
 by a femitone upon the 3d. This is the final cadence in four 
 parts ; of which fee the following reprefentation. 
 5th — 7th — 2d — 4th 
 
 K K — 3d 
 
 where the meeting of the 7th and 2d, upon the fameK, in the 
 fubfequent chord, is denoted by the horizontal bracket. 
 
 * The French theorifts call the chord of the key la tonlqnc, and this leading j f, la domlnante tonique, as if th;"s chord governed the other, becaufe the other 
 i obliged to follow it ; but in truth the key governs all the reft, and therefore this method of naming the chords, though but lately adopted, feems to need 
 further reformation. The leading chord which bears a lefs third and feventh, is called by the French, dominant 'e fimple. 
 
Chap. VIII. Of the perfeft regular Cadence. 
 
 See alfo Exam. XXXVII. Plate IV. at the letter H, where 
 the fame cadence upon C is reprefented in the treble and bafs 
 ftaves, and the natural progreffion of the notes in the accom- 
 panyment is fhown by the direction of fmall ftrokes inferted be- 
 tween the two chords. 
 
 It may feem prepofterous to begin the article of fundamen- 
 tal progreffions with explaining the final paffage, but never- 
 thelefs it is the molt proper way, for this plain reafon, be- 
 canfe it is the molt fatisfactory of all paflages, and the only 
 one which puts an end to the expectation of the hearer; 
 and by examining what way this expectation may be fatisfied, 
 and the attention wholly fixed upon one chord, we come eafi- 
 ly to under/land how it may be difappointed or fufpended, and 
 the attention carried to different chords. Add to this, that the 
 iame kind of cadence, made fucceffively upon different notes, 
 proper for the purpofe, will be found to conftitute more than 
 one half of all fundamental progreffions. 
 
 193. The moft remarkable circumflance, and what inevita- 
 b!y attracts the chief attention of the hearer, in this regular ca- 
 dence is, the refolution of the falfe fifth, which exifts in the 
 •leading chord. It is an undoubted fact, that whenever a falfe 
 fifth is perceived, and fuch can fcarcely ever pafs unperceived, 
 it immediately becomes our principal concern to hear its terms 
 minced to a better agreement ; the jarring founds muft be re- 
 conciled before we leave them, and this is what we call the 
 reflation of the difcord. 
 
 The falfe fifth in the leading chord mutt be refolved by its 
 two terms approaching nearer together, and conftituting a third 
 In the foMowing chord ; thus in the leading chord defcribed a- 
 Ijotc, the falfe fifth exifts between the 7th and the 4th, and 
 thefe two founds approaching nearer together, each by a ftep 
 of afesnitone, conftitute a greater third, viz. K — 3d, in the 
 next chord ; and this fucceffion is fo conformable to the expec- 
 tation of the ear, that we cannot go contrary to it, with any 
 
 degree of fatisfaction. The 7th muft by no means defcend, 
 nor the 4th afcend, nor muft either of them keep their place 
 when another chord fucceeds. 
 
 194. The fecond is not fo ftrictly limited in its progreffion 
 to defcend upon theK, for in fome cafes it may with equal pro- 
 priety afcend into the 3d, as fhewn at I. Exam. XXXVII. In 
 the fituatiqn reprefented at H, it would have been improper to 
 have both the 2d and 4th meeting upon the 3d, becaufe that 
 3d being the higheft note, would by that occurrence be ren- 
 dered more remarkable than it ought to be in a final chord : 
 but this can be fafely allowed at I, becaufe here the 3d is 
 in a more obfeure place, being, as it were, concealed below the 
 octave of the K. 
 
 195. There may be two other parts added to the final ca- 
 dence as reprefented at I ; viz. one which takes the 5th and 
 continues upon it, and another which takes the 2d, and de- 
 fcends into the K ; and thus the cadence will confift of fix 
 parts, as fhewn at K : where the fmall ftrokes inferted between 
 the chords ftill fhew the progreffion of each part, and the two 
 which ifiue from the note D, in different directions, fhew that 
 two parts are fuppofed to have previoufly taken that note, and 
 muft go from it in different progreffions, one defcending upon 
 the K, and the other afcending upon the 3d. 
 
 196. We may obferve in general, that the natural defire of 
 hearing a difcordant interval properly refolved always limits, 
 in fome degree, the progreffion of the terms of fuch an in- 
 terval ; while the other founds, which are concordant, are at 
 liberty in that refpect. Thus in all the arrangements of the 
 perfect regular cadence, the two terms of the falfe fifth muft 
 take their limited courfe, viz. the fourth of the fcale, which 
 ftands as a feventh to the 5/", and is the upper term of the falfe 
 fifth, muft defcend ; and the 7th of the fcale, which is the 
 third to the 5/, and the lower term of the falfe fifth, muft a- 
 fcend : thefe motions may be called nece/fary in this cadence, 
 
Of Fundamental ProgreJJions. 
 
 66 
 
 while the progrefTion of the 2d, which ftands as a fifth to the 
 5/ and of the 5th, being the octave to the 5/", when thefe 
 notes take place in the leading chord, is unlimited, or acciden- 
 tal, and may be various according to the different defigns of 
 .the compoler. Thus if a. final cadence, or clofe, be intended, 
 the leading chord mufr.be fo difpofed as that the neceffary mo- 
 tions iury lead into one of the arrangements of the chord of the 
 key which ;is denominated final, in Art. 145, 146. and the 
 accidental motions muR be fo conducted as 'not to counteract this 
 defign, by fortifiying, or rendering too remarkable, the harmo- 
 nics of the K, which ought, ill this cafe, to make no other ap- 
 pearance than as dependents upon the principal. This will ap- 
 pear to be the rule obfervedin the arrangements defcribed above. 
 The fame confiderationsmuff be attended to in the forma- 
 tion of all other final cadences ; of which we cannot fail to meet 
 with fufficient examples, in the profecution of this fubject. 
 
 It the regular cadence be introduced, as it frequently is, in 
 the courfe of a piece, where the expectation needs not to be fully 
 fatisfied, and where a cadence may with equal propriety be made 
 upon one of the chords called medial, in thefe cafes the necef- 
 fary motions muff flili be conformed to, the falfe fifth mull ftill 
 be properly refolved, but the accidental motions may be taken 
 at pleafure, as beft fuits the air of the upper part in which they 
 occur: the chords may be inverted, and by this means the 
 thcrough-bafs may take the air of an upper part, and a vaff va- 
 riety of paffages be introduced, which, to an unfkilful perfon, 
 will appear widely different, and yet in reality all depend upon 
 the fame fundamental progreffion : for the underftanding of 
 which matter, the learner is defired to examine carefully the 
 latter part of the XXXVII example, with the following Anno- 
 tations, 
 
 197. Annotations to Exam. XXXVII. 
 In this, example, the reference letters H. I. K. <bc. are not 
 
 Part I. 
 
 affixed, as formerly, to a fingle chord, but are placed at the 
 bar between the two chords which form the cadence, to inti- 
 mate that it is not a fingle chord, but a fucceffion or pafTagc 
 from one to another, that is thereby refered to. 
 
 The paffages at H, I. K. reprefcnt the final cadence upon C 
 in different arrangements, asalready explained, the laft of which 
 at K contains fix different motions or parts; and thefe are after- 
 wards fmgled out, and extended into fix bars, reprefenting 
 as many fucceffive cadences in the treble, with a bafs put below, 
 wherein feveral inverfions are introduced in the four firff ca- 
 dences. 
 
 At L the treble afcends from the 7th to the K, which is one 
 of the neceffary motions in the regular cadence ; but the bafs, 
 infkad of falling upon the K, takes the 3d, bearing a chord of 
 the 6th, inverted from the perfect chord of the Kf, fo that the 
 fundamental progreffion is not affected by this change in the 
 bafs, altho' the cadence is thereby rendered inconclufive or me- 
 dial. — In this paffage the feventh is not added to the leading chord, 
 becaufe it is fuppofed that the ffrain begins here, and it would 
 be unnatural to begin with a diflbnant chord. The other notes 
 which would be neceffary to fill up the accompanyment in this 
 and the following paffages, are fhewn by directs, placed in the 
 treble ftaff. It may be obferved here, that the note D, or the 
 2d of the fcale, muff defcend into the K, as it is not allowed 
 to afcend upon the 3d, becaufe the bafs takes that fame note, 
 and being only confidered as a dependent upon the Kf, it rauft 
 not be doubled. 
 
 At M the fundamental chords are both inverted, and the 
 bafs, having taken the 4th of the fcale, viz. the feventh of the 
 leading chord, figured with a 2, is obliged to fall upon the 3d 
 in the following chord, this being one of the neceffary moti- 
 ons, as explained above. Thefe bafs notes are not doubled in 
 the accompanyment, for the reafons formerly given. 
 
 At N the bafs takes the 2d of the fcale, viz. the fifth of the 
 
PUte IV. 
 
 Full Chords 
 
 Om iff ions. 
 
 Ex. XXXV. 
 
 DifTotunces 
 byAdditic 
 
 o « ,3 
 
 
 
 5*« 
 
 5 «? 2 
 
 ai 
 
 ft" r 
 
 I 4 
 
 4-# r-w- — *-w- 
 
 V," - w w ' . J # Jf 
 
 6" • 
 
 6 6 
 
 -h='J 
 
 6 
 7 5l_ 
 
 S^ 7 ? * * 
 
 IS 
 
 Mpu££tyr\ 
 
 Ei.XXXVl. 
 
 Double 
 Fundamental? 
 
 ~fwu/JJ«/.> 
 
 
 Q t 7 
 
 4 
 
 7i 
 
 i- 
 
 U-4 
 
 il--g — J 
 
 Jrz^E 
 
 i^Eg^igii^Si 
 
 :E 
 
 ^^g^-ffe 
 
 ^^fel^sfe 
 
 M N 
 
Chap. VIII. 
 
 Of Paffages derived from the regular Cadence. 
 
 67 
 
 leading chord, figured with 4 and 3, and from thence defcends 
 by a ftep into the K, which is the only proper motion in this 
 place, becaufe the treble, taking the 4th of the fcale in the 
 leading chord, is neceffitated to defcend upon the 3d, and it 
 would be improper to have this note doubled, therefore the 
 bafs is not allowed to afcend. 
 
 At O the bafs4iavrr]g taken the 7th of the fcale, viz. the 
 third of the lea'd'hHf thord, is obliged to afcend into the K, and 
 the treble at the fame time afcends from the 2d into the 3d, up- 
 on which account it becomes improper to let the diffonant 4th be 
 heard at all, in the leading chord ; becaufe the refolution of 
 that fourth upon the fame third would render it too remarkable. 
 If the treble had defcended upon the K, which it might have 
 done with equal propriety, the 4th might then have been ad- 
 mitted in the accompanyment, as it is in the next cadence at P. 
 The' omiffion of the diffonance at O is (hewn, as obferved, 
 Art. 171, 172. by the figuring; for the full chord would have 
 been figured with 6 and 5. Befides this omiffion we muft ob- 
 ferve aifo, that the octave of the bafs note is left out in the ac- 
 companyment of the leading chord at O, becaufe it would be e- 
 qually improper to double that note, which is the third to the 
 fundamental, in the leading chord, as to double the third to 
 the K in the following chord ; fo that in this paffage the ac- 
 companyment of the leading note confifls only of two founds. 
 
 At P the bafs coincides with the fundamental progreffion, 
 while the treble defcends from the 2d upon the K, which cir- 
 cumftances agree with the charafteriftics of a final cadence, and 
 give liberty to add all the harmonics of both chords. The fame 
 is done at Q^ in the fulleft manner, which can properly be ad- 
 mitted. At P the cadence confifts of four upper parts and a 
 bafs; at Qj.ve have five upper parts and two baffes, beyond 
 which it is impoffible to go, without tranfgreffing the neceffary 
 rules of harmony. The cadence at Q^cannot be played full 
 upon the harpfichord, becaufe the fingers cannot be extended 
 
 to reach the note D, which is below the treble ftaff; but the 
 effect is fcarce fenfibly different if this note be omitted, becaufe 
 it is reprefented by the upper D, which is its octave 
 
 198. In the courfe of the preceding article, a queftion may 
 very probably have occurred to the attentive reader, which it 
 may feem neceffary to anfwer, before we proceed further. The 
 propriety of feveral omiffions there pointed out appears fuffici- 
 ently plain, when the different parts are allotted to different 
 voices or inftruments, in a choir or concert ; becaufe the coin- 
 cidence of two parts upon the fame found will evidently ren- 
 der that found ftronger than the reft ; but what occafion is there 
 for any fuch reftrictions, in filling up the chords of a thorough- 
 bafs upon a harpficord, where the accompanying founds, if 
 ftruck at all, can neither be ftronger nor weaker in whatever 
 fucceffion they occur ? In anfwer to this it may be obferved, 
 that the impoffibility of making one particular found in a chord 
 either ftronger or weaker, upon occafion, is one principal rea- 
 fon why thefe reftrictions muft be more carefully conformed to, 
 in playing thorough-bafs, than in compofing for feparate detach- 
 ments of a choir. An experienced ear will not fail to perceive 
 every accompanying found, and, as it were, analize the fuccef- 
 five chords ; and alfo will affign to each found its due progref- 
 fion, efpecially as far as relates to the motions called neceffary ; 
 and by this means the coincidence of two parts upon the fame 
 note comes to be plainly felt , even in the accompanyment of 
 a thorough-bafs, and the difappointment which we find in 
 hearing the found, upon which two parts are thus con- 
 ceived to meet, no ftronger in reality than we might expect ei- 
 ther part fingly to have produced, is more deftructive of the 
 good effect of the whole, than if fuch a coincidence were di- 
 ftinguifhed by the fuperior ftrength of the note. This is one 
 in fiance of a note rendered more diftinguif able by the fucceffion in 
 •which it occurs, as hinted, Art. 144. ; and to hear fuch a note on- 
 ly juft of equal ftrength with either of thofe which lead to it, is 
 
68 
 
 Of Fundamental 
 
 as unnatural, as to fee the trunk of a tree only juff. of equal 
 bulk with the branches that ifTLie from it. 
 
 199. This fhort furvey of the accompanyments, or upper 
 parts which properly belong to the regular cadence, in its dif- 
 ferent arrangements, will appear to be no digreffion from the 
 principal fubjeft of this chapter, if we conlider, that the melo- 
 dy or air of the upper parts of a compofition, is undoubtedly 
 the moll: apt to engage our attention ; and therefore if we would 
 avail ourfelves of the knowledge of the fundamental principles, 
 to the bell: purpofe, it is proper to conceive the fundamental 
 progreffions as fuggefted by, or at lead: fomeway connected 
 with, certain correfpondent motions or paffages in the upper 
 parts. A familiar acquaintance with the fundamental progref- 
 fions is undoubtedly the principal requifite in the fkill of mufic, 
 but the paffages of melody are more obvious and more engaging, 
 and therefore it is not improper to alfociate thefe two particulars, 
 fo that the moft ufefnl knowlege may, by this means, be drawn 
 from the moft familiar obfervations. 
 
 200. Befides the perfect regular cadence upon the principal 
 key, we can alfo admit the fame pafTage to be made occasional- 
 ly, upon the 4th, or the 5th, or the 6th of the natural fcale, as 
 dependents on the principal, without altogether adopting any of 
 thefe for a new key ; provided due care be taken not to dwell 
 too long, or recur too often upon them, without a proper in- 
 termixture of other paffages proper to keep the ear in fufpenfe, 
 or to renew the impreffions of the principal key. But we can- 
 not make a regular cadence upon the 2d or 3d of the natural 
 fcale, without previoufly adopting a new key, different from the 
 key of the natural fcale : and on the 7th we cannot make a re. 
 gular cadence at all. 
 
 In order to explain this matter in the moft intelligible way, 
 we {hall find it convenient to apply the terms adjunft 4th, 
 adjunft yk, and fubftituted 6th, to the notes upon which we 
 admit occafional cadences, as above. The reafon of thefe ap- 
 pellations will appear hereafter. 
 
 Progreffions. Part I. 
 
 201. Each of thefe cadences implies an occafionally altered 
 note in the leading chord which precedes it ; as will plainly ap- 
 pear from the characteriftics of a leading chord in a perfect re- 
 gular cadence, given above, viz. that it mult bear a greater 
 third, and a lefs feventh, and confequently comprehend a falfe 
 fifth between the fame third and feventh. Now, there is no 
 degree of the fcale, except the 5th, whieh.has this kind of 
 chord among the feven natural founds : fb/ajfJapugh the K and 
 4th have each of them a greater third, they have alfo each of 
 them a greater feventh in the fcale, and the other degrees, ex- 
 cept the 5th, which have a lefs feventh, have at the fame time 
 a lefs third. 
 
 This cadence on the 4th implies the flat 7th, in its leading 
 chord, viz. a flat 7th added to the $P of the K/. 
 
 The iame cadence on the 5th, implies the fharp 4th in its 
 leading chord, which fharp 4th Hands as a greater third to the 
 2/, with an added feventh. 
 
 The fame cadence upon the 6th implies a fharp 5th, which 
 flands as a greater third to the 3/, with an added feventh. 
 
 The two former of thefe cadences end with a ^ P chord on 
 the 4th and 5th refpeftively, but the laft ends more naturally 
 with a ft P chord on the 6th. 
 
 The reader will readily excufe our not entering here upon a 
 critical enquiry into the reafon why thefe perfect regular caden- 
 ces, and no others, can be admitted, without formally efta- 
 blilhing a new Key. This is rather to be expected in the the- 
 ory ; and it is fufheient at prefent to remember, that every al- 
 tered note by an occafional fharp or flat, will always be either a 
 fharp 4th, or a fharp 5th, or a flat 7th, in the natural fcale 
 which takes place at the time ; and that the (harp 4th belong* 
 to the chord of the 2f, as a leading chord in a cadence upon 
 the 5th : the fharp 5th belongs to the chord of the 3/, as a 
 leading chord in a cadence upon the 6th ; and the flat 7th be- 
 longs to the chord of the Kf, as a leading chord, in a cadence 
 
Of the perfect Cadences on the fourth, fifth, and fixth of the Scale. 
 
 Chap. VIII. 
 
 upon the 4th : And that although a regular cadence, made up- 
 on any of thefe degrees of the fcale, tends directly to the intro- 
 duction of that degree for a new key, yet fuch cadences may be, 
 and fome of them generally are, interfperfed through the whole 
 of a piece of mufic, even where the principal key is all along 
 retained in the ftricteft manner *. 
 
 202. The occafional flat or fharp note is fhewn, in the na- 
 tural fcale of C, by prefixing the proper character immediately 
 before the note ; and it is to be obferved, that as either of thefe 
 marks, placed at the clifF, affect: all the notes which follow on 
 the fame line or fpace, throughout the whole tune, Art. 57. ; 
 fo when either of them is occaflonally introduced in the middle 
 of a tune, it is generally underftood to affect all the notes 
 which follow, on the fame line or fpace, in that bar of the 
 mufic, but no farther ; fo that there is no need to repeat the 
 character in the fame bar, though the altered note fhould oc- 
 cur more than once ; but if another note fhould follow in the 
 fame bar, on the fame place with the altered note, and it be 
 proper that the latter be reftored to its natural place, it is ne- 
 ceflary to contradict, or take off the effect of the fharp, or the 
 flat, which hath formerly taken place ; and this we do by pre- 
 fixing another character, called a natural. See Plate V. Ex. 
 XXXVIII. with the following annotations. 
 
 203. Annotations to Exam. XXXVIII. 
 
 This example confifls of a treble part, with a fundamental 
 
 bafs fubjoined ; not with intent to be performed along with it, 
 
 but only to fhew the derivation of each paflage in the upper 
 
 part from the cadences which take place in this bafs. It is to 
 
 69 
 
 be obferved, that the mood of time is here fextuple, or double- 
 triple of crotchets in a bar, which is of the fame nature as the 
 fimple triple, the bars being only put at the end of every two 
 meafures, Art. 85. This renders the middle of every bar ac- 
 cented in the fame manner with the beginning, and confequent- 
 ly gives us the opportunity of making two cadences in each bar, 
 which are refered to by the letter H, I, K, <bc. as before. 
 
 At H, the treble afcending from the 7th to the K, implies a 
 regular cadence on the fame K : but the diflbnant feventh muft 
 not be added to the leading chord, becaufe it is the beginning 
 of the ftrain, and therefore the firft note, G, of the bafs bears 
 only its own perfect chord, and needs not to be figured. 
 
 At I, the fame cadence is again reprefented, though in a 
 more vague manner, by the treble afcending from the 2d to 
 the 3d ; this motion being only accidental in that cadence, 
 whereas the other, at H, is necejfary, Art. 196; and it is con- 
 fident both with reafon and experience, that the cadence be 
 moft plainly implied in fuch paflages as are mod ejfential to it. 
 In truth, this pafTage at I might have a different fundamental 
 bafs applied to it, without any impropriety; but as we have 
 yet fpoke of nothing but regular cadences, we take this pafTage, 
 at prefent, as belonging to a fecond regular cadence on the K. 
 
 At K, the treble afcends from the 3d upon the 4th, accent- 
 ed, which is one of the neceflary motions in a regular cadence 
 upon the fame 4th, as adjuntl to the principal key ; and there- 
 fore fuch a cadence muft naturally take place here, as fhewn in 
 the fubjoined bafs. For diftinguifhing the neceflary motions 
 in every cadence, it is to be obferved that they always bear the 
 fame relations to the note on which the cadence is made, as 
 
 * Notwithftanding the above affertion, that every occafwnally .altered note -will always be either the fiarp 4th, parp $th, or fiat jib of the naturalfcale, -which takes 
 place at the time, the reader need not to be alarmed at meeting fometimes with the fharp K, and even the lharp id of the original fcale, without any pre- 
 paratory paflages to denote the enterance of a new hey; for, in thefe cafes, he may fafely conclude, that the key is altered, be the tranfition ever fo abrupt 
 or unexpected. Some compofers have endeavoured, by fuch introductions, to give the hearers a happy furprize, but it ought to be done with great caution 
 and (kill. 
 
 s 
 
Of Fundamental Progrejfions 
 
 -jo 
 
 they do to the K, in the cadence upon it, which we have for- 
 merly defcribed ; it is the very fame circumfrance, viz. the refo- 
 lutionof afffe fifth, Art. 193. which renders thefe motions 
 neceffary in all fuch cadences ; and the falfe fifth being always 
 between the third and the feventh of the leading chord, that 
 third muft afctna by a ftep of a femitone, and become the fun 
 damental ; and that feventh muft defcend one degree, and be- 
 come the third, in the following chord: fo that the neceffary 
 motions always lead to the fundamental and its third, viz. to 
 the former by afcending a femitone, and to the latter by de- 
 fending either a femitone or a tone, as the fcale allows. Thus 
 in the cadence upon the adjunct 4th, fuch as that now before 
 us at K, the neceffary motions are from the 3d to the 4th 
 afendii'ig. as it is in the treble part here, and from the |p 7th 
 to the 6th defcending ; which latter motion is fecn in the treble 
 at P ; and though it does not appear here, yet it ought to be 
 fupplied by the accompany ment of the thorough-bafs. The 
 exiftence of the flat feventh in the leading chord, agreeable to 
 what was obferved, Art. 201. is mewn by the character pre- 
 fixed to the figure 7, over the bafs note; it being neceffary that 
 the exiftence of an altered note, when any fuch takes place, ei- 
 ther in the treble, or in any part of the accompanyment, mould 
 be denoted by the manner of figuring the bafs; as we fhall ex- 
 plain in courfe. 
 
 At L, the treble afcends from the % 4th to the 5th, which 
 reprefents a cadence upon the adjunct 5th, being a neceffary 
 motion in that cadence. The (harp character is here occafio- 
 nally prefixed to the 4th of the treble, which now ftands as a 
 greater third, in the leading chord of the if, as obferved, Art. 
 201 . and this occafional alteration of the third to the bafs note, 
 is (hewn, in the figuring, by the fimple character placed above the 
 note, it being undcrfiood that a fharp, or fiat, or natural, thus plac- 
 ed above a bafs-note, without any other figure following it, always 
 refers to the third of the chord. The principal key-note itfelf 
 
 Part I, 
 
 becomes a feventh in this leading chord, in the accompanyment, 
 and defcends upon the 7th of the fcale ; which is the other ne- 
 ceffary motion in this cadeifce. 
 
 At M, the treble defcends from the natural 4th upon the 
 3d, which motion fuggefts a cadence upon the K : the natural 
 character is prefixed to the 4th in the treble, to take off the ef- 
 fect of the fharp which appeared in the former part of the bar; 
 and the fame character is alfo prefixed to the 7, in the figuring 
 of the leading chord, becaufe the fharp, which took place atL, 
 proves to be the fame degree with the feventh of this bafs-note. 
 At N, the treble afcends from the % 5th upon the 6th, 
 thereby reprefenting a cadence upon the fubftituted 6th ; the 
 occafional fharp taking place in the treble, and alfo above 
 the bafs note, in the very fame manner as at L, only one de- 
 gree higher. This cadence differs from the others, in that the 
 2d of the fcale, which here ftands as a feventh to the ?>f m 
 the leading chord, defcends by a ftep of a full tone upon the 
 K, thereby making a lefs third to the 6f in the following 
 chord, whereas in all the other cadences the feventh of the 
 leading chord defcends only a femitone, and becomes a greater 
 third in the following chord. 
 
 At O, the 5th is again made natural, and the treble takes the 
 fundamental progrefhon of a cadence upon the K. Here, ofnecef- 
 fity, the fundamental bafs takes the fame notes with the treble, but 
 the natural needs not be put before the bafs note, becaufe the 
 fharp, which took place upon the fame degree, in the former 
 part of the bar, has not appeared in the bafs, but only in the fi- 
 guring; and as thefe figures relate only to the upper parts, which 
 are played with the right-hand, a change among the figures 
 is not fuppofed to influence the following real notes ot the bafs, 
 which belong to the left-hand, in playing a key'd inftrument. 
 At P, a cadence is made upon the 4th, as already obferved, 
 in fpeaking of the paffage at K. 
 
 At Q__the treble moves from the 6th to the 7th, which is 
 
Ex.xxxvBr 
 
 c'V 
 
 CC&dt07t4Uth 
 
 '/?/ a/terr?/' 1'r/r, 
 
 H T K _L M 1ST Q P _Gi_ R 
 
 ,' 7 i b.7 _ * ^7 I » 7 . V I 7 ■ .. 
 
 FundBafc 
 
 Exxxxix 
 
 jsP^P^S^ 
 
 6th K 4-tk 5 th K 
 
 ^ ^% H ^ ^^^ i 
 
 
 Ex .XL 
 
 YiS ,XLI 
 
 , 7_ Iff -.- _ ? \>>7 % 7 W _]L _ L__x£ .. 
 
 V* } --6 
 
 £ ^ 
 
 ?/-' 
 
 if 
 
 •vcW tn£e' , )u\'e 
 
 d 
 
 W 
 
 m 
 
 V, K 
 
 Hi 
 
 <o 
 
 fltie&cn-yr LJFizt&e y A 
 
 
 V ; H ■. \ 
 
 s.. ^ ^ 
 
 I It s 
 
 
 
 ! ^ I 
 
 t^ W s ^ # 4" 
 
Of occafionally altered Notes in the natural and tranfpofed Keys. 
 
 Chap. VIII. 
 
 not a neceffary motion in any cadence, but may accidentally re- 
 prefent a cadence upon the 5th, in the fame manner as that at 
 I, reprefented a cadence on the K, and for the fame reafon we 
 here confider it as part of fuch a cadence. Here, though the 
 7th in the treble be reftored to its natural place, which was af • 
 retted with a flat, the laft time it occurred, yet the interpofition of 
 the bar hinders that flat from affecting the prefent note, fo that 
 there is no occafion for the natural character here ; yet to prevent 
 the miftake which might other wife happen, thro' theinadvertency 
 of a performer, it is not amifs to infert the character in fuch a 
 place as this ; and we often meet with inftances, in the moft 
 correct pieces of mufic, of characters inferted, where, according 
 to the ftricteft rules, they might be left out. 
 
 The palTage at R, makes the final cadence on the key. 
 
 264. When mufic is fet in a tranfpofed key with fharps, as 
 in Exam. XXXIX. the occafional ^4th, and % 5th, are de-; 
 noted in the fame way as in the natural key ; but the \) 7th is 
 here made by rendering the 7th natural, which is originally 
 Iharp at the cliff". So that here the flat character is fupplied 
 by the natural contradicting a fharp at the cliff. 
 
 On the contrary, when mufic is fet in a tranfpofed key with 
 flats, as in Exam. XL. the [> 7th is denoted the fame way as 
 in the natural key ; but the ^ 4th is expreiTed by rendering the 
 4th natural, which is originally flat at. the cliff. The % 5th, if 
 there be no more than two flats at the cliff, will take its own 
 proper character, as in Exam. XL. but if there be three or 
 more flats at the cliff, the 5th of the fcale will be one of the 
 flat notes ; and then the natural, contradicting a flat, mufl be 
 ufed to exprefs the % 5th alfo. 
 
 Upon the whole, it is to be remembered, that the true mean- 
 ing of the natural character is ambiguous, and depends entire- 
 ly upon what precedes it. J natural, cent radioing a flat, is 
 to be looked upon as an occafional Jharp ; but when it contra' 
 dicls a Jharp, it is, in effect, an occafional 'flat. 
 
 71 
 
 205. The reader will take notice that our chief defign in 
 the preceding part of this chapter has been to point out what 
 motions in the trebie are necelTary in each regular cadence, and 
 what are only accidental ; and it is worth while to obferve, up- 
 on the whole, that the afcent or defcent of a femitone, upon an 
 accented part of the meafure, always is, or may be, efteemed 
 a necelTary motion in a regular cadence : the afcent or defcent 
 of a full tone is never more than accidental in fuch a cadence, 
 except in the cafe of the 2d defending upon the K, in a ca- 
 dence upon the 6th, as explained in fpeaking of the paffage 
 at N. 
 
 It may not be improper to obferve alfo that the example, a- 
 bove refered to, is very far from being a piece of good mufic, 
 the treble part is but barely tolerable along with the bafs, and 
 quite fhocking when alone. To make good mufic there muff, 
 be an agreeable variety of air and movement in the leading part, 
 and at the fame time a natural, eafy fucceffion of chords in the 
 thorough bafs ; the powers of melody and harmony mult con- 
 fpire together for this purpofe. Now, one of the principal re- 
 quifites in melody is variety of divifwn, which refults from the 
 breaking and fubdividing the fucceffive bars or meafures, in fe- 
 veral different manners ; of which the example before us is to- 
 tally deftitute ; and in regard to the harmony, the attention is 
 too much hurried from one cadence to another; we have no 
 fooner felt the cadence upon one key, than we are obliged to 
 quit it, in order to purfue another, and this next jnlt fervesus 
 the fame way. This makes the fucceffion of chords feemwild 
 and incoherent ; and, inffead of being pleafed, we are rather 
 puzzled to know what the founds mean, and how to reconcile 
 them one with another. 
 
 In order to obtain a more connected fucceffion of chords, 
 and yet not wanting variety, it is necelTary to admit of other 
 kinds of cadences, befides the perfect, already defcribed, fome 
 of which ferve to fufpend, and fome to difappoint the natural 
 
Of Fundamental ProgreJJio 
 
 Part I. 
 
 expedition of the ear, fo as not to fuffer our attention per- 
 fectly to fettle, either upon the principal key, or any other, 
 until fome certain phrafc or Jlrain of the tune, Art. 103. be 
 finifhed ; or until the fenfe of the words, fet to mufic, may 
 properly admit of a reft or paufe. 
 
 206. The attention is kept in fufpence, by the cadence ufu- 
 ally called irregular, which is juft the regular cadence revcrfed ; 
 viz. the chord of the K, preceding or leading into that of the 
 5th, inftead of the 5/, leading to the K/, as in the regular 
 cadence : and it is known by the fundamental bafs rifing a fifth, 
 or falling a fourth, upon an accented part of the meafure. 
 
 In this irregular cadence the diiTbnant feventh muft not be 
 added to the chord of the 5th, now ufed as a following chord, 
 becaufe the proper ufe of diifonances, in all cafes, is to excite 
 an expectation of another chord to follow, in which the diflb- 
 nance fhall be heard refolved, Art. 196. But here we are fup- 
 pofed to bring the harmony to a kind of paufe, though not to 
 a full concllifion ; fo that a diflbnance may here be more pro- 
 perly added to the chord of the K, now ufed as a leading chord : 
 and the proper diflbnance for this purpofe is the added fixth ; 
 of which we fhall fpeak below. 
 
 207. The regular cadence takes place occasionally upon the 
 fame degrees of the fcale, which are capable of leading in a re- 
 gular cadence, as (hewn above : fo that as we have four regular, 
 we have alfo four irregular cadences, reprefented, Exam. XLI. 
 where each irregular cadence is fo placed as to precede imme- 
 diately the regular from whence it is derived. In this example 
 the notes which form the irregular cadences bear only their per- 
 fect chords, in which cafe there is not any motion in the up- 
 per parts which can properly be called necejjary, becaufe there 
 is no difcord to be refolved ; and therefore we have put no tre- 
 ble part above the fundamental bafs. 
 
 208. The added fixth in the leading chord of an irregular 
 cadence is brought in by intermixing the irregular and a part 
 
 of the regular cadences together, in pafling upon the K, or the 
 ^th : as both of thcfe notes may be introduced by either kind 
 of cadence. Thus the paflage at I, in Exam. XXXVIII. which 
 we there take as part of a regular cadence on the K, Art. 203. 
 might be admitted in the treble, at the fame time that the fun- 
 damental bafs palfes, by the irregular cadence, from the 4th up- 
 on the K ; and the paffage at Q^ in the fame example, might 
 take place, while the fundamental bafs paffes, by the irregular 
 cadence, from the K upon the 5th. In both thefe cafes the 
 former notes of the treble would become added fixths in the 
 leading chords, and afcend by one degree upon the third of the 
 following chords ; and this motion is the proper refolution of 
 the difcord in the irregular cadence : we mall give examples 
 hereafter. 
 
 209. The falfe cadence is a regular cadence difappointed, by 
 the fundamental bafs afcending only one degree, inftead of a 
 fourth, and thereby introducing a fundamental note, a third 
 below that which would have taken place, if the falfe cadence 
 had not been admitted. 
 
 This paffage is properly a gradation or Jlepping from the lead- 
 ing chord of a regular cadence, upon the 'degree next above, 
 and is chiefly praclifed from the K upon the 2d, in place of the 
 regular cadence on the 4th ; and from the 5th upon the 6th, in 
 place of the regular cadence on the K. The leading chord in 
 this falfe cadence, or gradation, may bear an added feventh, 
 the fame as in the regular cadence ; becaufe the neceflary moti- 
 ons which refolve the difcord, as fhewn above, may in this cafe 
 be ufed, as they lead to the third and fifth of the following 
 chord. 
 
 210. The fame kind of gradation from the chord of the 4th/, 
 upon that of the 5th, is often ufed in bringing the harmony to 
 a kind o^imperf eel paufe, or reft, on the chord of the 5th : but 
 although this paffage is very frequently found in our mufic, yet 
 it has not been ranked among the clafsof cadences, by the wri» 
 
Chap. VIII. 
 
 Of irregular and falfe Cadences. Of the Gradation on the fifth. 73 
 
 upon the K, by the irregular cadence, our principal regard is 
 placed upon the 4th : but in making a gradation upon the 5th, 
 our attention is rather turned to the 2d ; which, as we have 
 formerly feen, ought to precede the 5th, in a regular cadence. 
 A direcT: is placed at I, in Exam. XLII. upon the 2d, in the 
 fundamental bafs, to exemplify this matter. 
 
 There are other ways of difappointing both the regular and 
 irregular cadences, but being very feldom admitted at the con- 
 clufion of a phrafe, where a formal cadence is expected, we 
 (hall pafs them over at prefenr. 
 
 213. To make an end of this difquifition concerning caden- 
 ces, let the following account of all the fundamental paflages, 
 which have already been explained, be well obferved. 
 
 From the K/, as a leading chord, we can make an irregu- 
 lar cadence on the 5th, a regular cadence on the 4th, or a 
 falfe cadence on the 2d, of the fcale. 
 
 From the 5th/*, as a leading chord, we can make a regular 
 cadence on the K, an irregular cadence on the 2d, or a falfe 
 cadence on the 6th. 
 
 From the 4th/, as a leading chord, we can make an irregu- 
 lar cadence on the K, or a gradation on the 5th. 
 
 From the 2&f we may make a regular cadence on the 5th. 
 
 From the 6th/, we may make an irregular cadence on the 
 
 From the 3d/, we may make a regular cadence on the 6th, 
 or a falfe cadence on the 4th, 
 
 In every regular and falfe cadence the proper difcord in the 
 leading chord is the added feventh ; and in the irregular ca- 
 dence and gradation, the added fixth : yet thefe added difcords 
 may be left out, and in many cafes it is neceffary to do fo ; of 
 which hereafter. 
 
 ters on this fubject, and therefore we (hall not prefume to make 
 innovations in this refpett, but (hall give it the title which firft 
 occurred, of a gradation on the 5//;. 
 
 211. In this pafTage, the leading chord of the 4th/ may take 
 an added fixth, after the fame manner as in the irregular ca- 
 dence on the K, mentioned, Art. 108;. but with this djffe 
 rence in the movement of the upper parts, in order to form the 
 following chord, viz. here the 2d of the fcale, which makes 
 the added fixth, muff keep its place, and become a fifth; and 
 the K muft defcend, and become a third, in the following 
 chord ; as reprefented at I, in Exam. XLII. Plate VI : where 
 as in the irregular cadence on the K, the 2d afcends upon the 
 3d, while the K keeps its place, and becomes the fundamental, 
 as (hewn at H, in the fame example. 
 
 The reader will obferve, that this example is in the fcale of 
 G, with a (harp upon F, at the cliff* and that the treble takes 
 one of the abovementioned movements, while the other is fhewn 
 by directs : and it is ufeful to remember, that in both thefe 
 pafTages, the difcordant interval, which exifts between the 2d 
 of the fcale and the K ({landing as fixth and fifth in the lead- 
 ing chord) is refolved by one of its terms moving a Jiep far- 
 ther from the other ; and, in both pafTages, the moving term 
 makes the third in the following chord: and that whenever 
 the fundamental bears an added fixth, one or the other of 
 jhefe motions becomes neceffary, for the refolution of the dif- 
 cord. 
 
 212. To account for this double ufage * of the chord of the 
 4/, with an added fixth, it may be obferved, that this chord 
 has an ambiguous, or double fundamental, as (hewn, Art. 1 78. 
 and Exam. XXXVI. Our attention is divided between the 
 4th of the fcale, and the 2d ; and in paffing from this chord 
 
 * This liberty of ufing the chord of the 4/, with an added fixth, either as leading; to the Kf, or the ? /", is called, by the French theories, U double 
 ph\ dehdijfonance. " J " •> 
 
 T 
 
74 Of Fundamental 
 
 If thefe paflages be well remembered, we fhall have little fur- 
 ther difficulty in affigning the proper fundamental progreflions 
 for the beginnings and middle parts of harmonic phrafes, where 
 no formal cadence is expected. 
 
 24. In an harmonic phrafe it is mod natural to difpofe the 
 fucceffive chords fo, that every fundamental paflage from one 
 chord to another, may be an mutation of fome of the above de 
 fcrfbe ' cadences; efpecially the paffiagesfrom an unaccented toan 
 accented note ; as from the end of one bar or meafure to the be- 
 ginning of the next : and fuch difpofitions become the more 
 nec'effary as the movement is flower. In the common pfalm 
 tunes, for inflance, there is ordinarily a cadence imitated at e- 
 very bar ; and thefe imitations are often introduced, without 
 altering any of the degrees of the fcale, although fome occafi- 
 onal alteration would be necefTary to make the cadence perfect 
 in its kind. Thus the paflage from the 3d/ to the 6th accent- 
 ed, or from the 6th to the 3d accented, is often admitted with 
 out altering the 5th ; though the former imitates a regular; and 
 the latter an irregular cadence, in both of which the % 5th is 
 an eflential note. Art. 20:, and 207. The fame liberty may 
 be ufed in other imitations, as we (hall fee below. 
 
 215. The movement of the fundamental bafs is lefs reftrict- 
 ed, in palling from an accented to an unaccented note ; and e- 
 fpecialiy afcer a cadence has been heard upon the former, becaufe 
 then the narmony is brought to a kind of paufe, and we aie 
 not much concerned about the connexion; yet even in thefe 
 cafes there are feveral cautions necefTary, in order to produce 
 an agreeable effect, which we fhall endeavour to explain in 
 courfe. It will not be difagreeable to the reader, in the mean 
 time, to fee a little of the application of what is already faid, 
 in fome familiar examples. We fhall begin with the fun 
 damenrai of the natural fcale afcending and dekending, as re- 
 prefented in the latter- part of Exam. XL1I. 
 
 ProgreJJlo 
 
 Part I, 
 
 216. The natural fcale, confiding of eight fucceffive founds, 
 when performed in a flow manner, as denoted by the minims, 
 divides into two phrafes of four founds each, both afcending 
 and defcending, as (hewn by the dotts in this example : and 
 the expectation of cadences falls particularly upon the conclufi- 
 on of each of thefe phrafes, which we here refer to, by the let- 
 ters K, L,'M, N. 
 
 Each of the phrafes in this example ends with a regular ca- 
 dence, VIZ: ■ 
 
 At K, a regular cadence is made on the adjunct 4th ; 
 
 At L, the fame is made on the principal key ; 
 
 At M, the fame is made on the adjunct 5th ; 
 
 At N, the final cadence is made on the key : 
 and it is plain, from what is formerly laid down, that the ca- 
 dences at K and L are implied in the motion of the treble, and 
 that at N is necefTary for making a conclufion : but it remains 
 to be (hewn, why the paffage in the treble, from the 6th upon 
 the 5th- of the fcale, at M, could not with equal propriety have 
 been refered to an irregular cadence upon the K ; as, by fo 
 doing, theenterance of the altered note in the aceompanyment, 
 denoted by the % in the figuring, would have been prevented, 
 and, of confequence, the original key more flrictly retained. 
 For this purpofe we may obferve, according to Art. 213 and 
 214. that every fundamental paffage in this example imitates 
 fome cadence ; as will eafily appear upon the flighted exami- 
 nation: and although this reflriction is not always indifpcnfible, 
 yet, in general, the lefs we deviate from it, the more eafy and 
 natural is the fucceflion. Now, having pafled, in imitation of 
 the irregular cadence, from the Kf, upon the 5th/", at the two 
 firft notes of the phrafe M, we cannot take the 4th for the 
 next fundamental note, in order to form the irregular cadence 
 on the K, without deltroying the connexion ; as the paflage 
 from the 5th/ to the 4th/, does not reprefent any cadence. 
 
Chap. VIII. 
 
 Of harmonic Phrafes. Examples 
 
 But this is not all ; for there are other reafons why the 5th muft 
 not be followed by the 4th in the fundamental bafs, of which 
 we fhall fpeak hereafter. 
 
 It is on the fame account, that We refer thedefcent from the 
 4th upon the 3d of the fcale, at the beginning of the next 
 phrafe N, to a regular cadence on the K, wherein the 4th be- 
 comes a fcventh in the leading chord of the 5th/~, rather than 
 to an irregular cadence on the K, wherein the 4th might have 
 been the leading fundamental. This latter would have intro 
 duced more variety, and therefore have been more eligible, had 
 not the preceding chord of the 5th/ at M, prevented its admif- 
 iion. The next example XLIII, will fufficiently illuftrate this 
 matter. 
 
 2t 7. This example confifts of phrafes of fix notes in each, 
 and the expectation of cadences falls upon the two laft notes 
 in each phrafe, as fhewn by the fubjoined letters O. P. Q^R. 
 
 The pafTage in the treble, from the 3d to the 4th, in the 
 phrafe O, which naturally reprefents a regular cadence on the 
 fame 4th/", as at K, is here difappointed, by introducing, in 
 the bafs, an imitation of the falfe cadence on the 2d ; and, by 
 this' means, the 2d is brought in as leading to the 5th/, at 
 the conclufion of this phrafe, which is the mod eligible bafs, 
 as it imitates a regular cadence on the 5th ; although the ca- 
 dence is imperfect in regard to the treble, becaufe the 4th is not 
 made fharp. In truth, this pafTage is a mixture of two funda- 
 mental progrefTions, the bafs moving by the regular cadence, 
 while the treble makes a gradation on the 5th, and the leading 
 chord has a double fundamental, as fhewn by the direct. 
 
 The diligent reader, will probably have obferved, that the 
 fame double fundamental is alfo reprefented in the phrafe L, 
 and a queftion may occur, which it will be very proper to an- 
 fwer, before we proceed, viz. Why is the 4th reprefented as a 
 governing fundamental at L, and the 2d at O.; for, in either 
 pafTage, any of the two is allowable, according to the rules 
 
 ofProgreJJions, imitating the Cadences. 73- 
 
 already laid down ? For anfwer, let it be obferved, that this 
 ought to be the cafe, whenever diflonant chords of an ambigu- 
 ous fundamental are introduced, as hinted, Art. 187. and to 
 account for the preference due to one of thefe fundamentals, 
 rather than the other, in the two cafes now before us, we muft 
 further obferve, that a confeculion of perfect concords, viz. 
 octaves or fifths, between the treble and the fundamental bafs 
 ought to be avoided, when it can be done, without tranfgref- 
 fing other rules. Now, had the 2d /taken place at L, there 
 would have been two confecutive fifths, at the pafTage from the 
 K to the 2d/; and if the 4th/ had taken place at O, there 
 would have been two confecutive octaves, at the pafTage from 
 it to the 5th/. 
 
 There is a fort of drynefs, and want of variety, in fuch con- 
 fecutions, which renders it neceflary to avoid them, as much 
 as pofhble. 
 
 The firft four notes of the phrafe P, are accompanied the 
 fame way as at L, but the laft two are here taken as part of a 
 regular cadence on the 5th, in which the K in the treble be- 
 comes a feventh in the leading chord, and defcends upon the 
 7th of the fcale, now taken as a third in the following chord. 
 The irregular cadence might have been ufed in this pafTage, 
 the fame way as at the beginning of the phrafe M; but the 
 regular is preferable, becaufe it is better to vary the harmony, 
 upon the repetition of the fame note in the treble, where it 
 can conveniently be done : and if the irregular cadence had 
 been ufed at the conclufion of this phrafe, the two fucceffive 
 K's, in the treble, muft have been both refered to the Kf. 
 
 The phrafe Q_ends with a cadence on the 4th, which obliges 
 us to take the preceding 5th, in the treble, as belonging to the 
 Kf: and in order to do this, we muft not refer the 6th, which 
 goes before it to the /d/, as at M, becaufe the defcent, from 
 the 2d to the K, could not be allowed in the fundamental 
 bafs, in this place, and therefore we here make the 6th itfelf.a 
 
7<s 
 
 fundamental, by means of the falfe cadence. And although 
 the pafiage, from this 6th to the K, does not imitate any ca- 
 dence, yet it is allocable here, becaufe of the divided attenti- 
 on which the falfe cadence creates ; as (hewn by the direct, ad- 
 ed in this place. In truth, the attention may at any time be 
 fuppofed to be divided between the fundamental, which would 
 naturally have taken place, and that which is introduced, part- 
 ly unexpected, by the falfe cadence, Art. 209, and it is of 
 confequence to obferve, that immediately after any fuch falfe 
 cadence, we may proceed, if it be convenient, in the very fame 
 manner as if the regular cadence had not been interrupted at 
 
 At the beginning of the next phrafe R, the fame 6th is re- 
 fered to the 4th/.. The 5th and 4th are taken as at Q^ and 
 the 3d, 2d, and K are ufed as at'N, which finimes the 1 train. 
 .It. is expected, that the reader will carefully obferve thefe dif- 
 ferent ufages of the fame note in different occurrences ; and 
 that, if he finds his ideas of thefe matters any way confufed, 
 he will not only take the trouble of reading over again, from 
 Art. 206, but alfo of copying carefully the fcale and the 
 Exam. XLIII. with his own hind;, which, if he neglects to 
 do, and attempts to proceed without a pretty clear knowlege 
 of what is pair, let him not blame the author for writing un- 
 intelligibly, if he (hould find himfelf bewildered hereafter. 
 
 218. Hitherto we have feen the treble moving only by fin- 
 gle degrees ; theExam XLIV. which is the fundamental of 
 York tune, (hews a little of the manner of proceeding when 
 the treble moves by leaps. This, and all other common mea- 
 fure pfaltn-tunes, being drams of 8 and of 6 fyllables alternate- 
 ly, the expectation of cadences falls upon the middle of the 
 firft and third lines, and at the end of every line, as (hewn by 
 the letters S. T. U. W. X. Y. 
 
 It is allowable, when the treble moves by leaps, to continue 
 the fame fundamental through a whole bar, if the notes of the 
 
 Of Fundamental Progrejjions. Part I, 
 
 treble will admit of it, as is frequently done in this example ; 
 but it is better that a new chord fucceed at the beginning of 
 the next bar, when it can properly be introduced. 
 
 At T and at X the fundamental bafs is made to afcend, by 
 taking the leading note an octave below that which went be- 
 fore it, in order to help the two confecutive fifths, between the 
 bafs and treble, becaufe the contrary motion partly compenfates 
 for the want of variety in fuch pafl'ages; but it would be frill 
 better if the courfe of the treble were altered, foas to avoid the 
 confecution. 
 
 At U there is a confecution of octaves, which is remedied 
 in the fame mannner by contrary motion ; but in this place wc 
 commonly fee the ^ 4th, in the place of the 2d, in the treble, 
 which is better. 
 
 219. Before we quit the examples now under confiderati- 
 on, it will not be improper to (hew why feveral omiffions take 
 place in the figuring of thefe baffes, which have not yet been 
 explained. 
 
 At K the leading chord of the regular cadence on the 4th is 
 left without an added feventh ; but at S and W the feventh is 
 added : and to account for this it mud be obferved, that the 
 [j 7th of the fcale ought not to take place in a chord, when the 
 natural 7th has been heard in the chord immediately preced- 
 ing ; becaufe the paflage from the natural to the l? 7th is difa- 
 greeable or mharmonical. The fame may be obferved of the 
 paiTages from the ^ 4th to the natural 4th, and from the $ 5th 
 to the natural 5th. Now, in the phrafe K, the natural 
 7th has been heard as a greater third to the 5th/, in the chord 
 preceding the cadence ; but at S and W it has not been heard. 
 
 The feventh is not added to the firft note of the phrafe L, 
 both becaufe a formal cadence is not here expected, and alfo 
 becaufe fuch a feventh muft have defcended, and met the treble 
 upon the 6th of the fcale, which 6th being employed as a third, 
 to the following fundamental, ought not to be doubled. 
 
Plate VI 
 ExXLTT 
 
 foh^JljJljJK 
 
 rJ/ite-- t>m^. 
 
 jsppglpf^ 
 
 JFactna Jay*?' ?{> 
 
 '^M 
 
 undMa/s^ | J" 
 
 fy mN j%^ sa =iE 
 
 fO ^Mpj 
 
 I 
 
 m 1 ' a° 
 
 K I 
 
 ExXLH 
 
 B 
 
 SEaeai 
 
 gBlil 
 
 ' i Ql u l l JlKU 
 
 nl Oll- G-K 
 
 I 
 
 tf r- cii e 
 
 33 
 
 ,FundJ3a/s' 
 
 i 
 
 §§| 
 
 XI 
 
 
 EiXLIT ^-W 5^«<s. 
 
 I 
 
 i 
 
 ^q-T^e 
 
 s 
 
 iFfifwy* 
 
 nzBpra^zizm 
 
 Fund _Bafi>> 
 
 H^ 
 
 1 
 
 €ha 
 
 iz 
 
 m 
 
 i 
 
 OMTQ 
 
 gBIM 
 
 in 
 
 €hQ 
 
 XT 
 
 W 
 
 I 
 

Chap. VIII. Various Reafons for Omiffions in 
 
 At Q__the fame feventh is omitted for no other reafon but to 
 prevent the cadence from being too conclufive ; for it requires 
 fome fkill and caution in the ufe of the regular cadence on the 
 4*h, left it Ihould deftroy the expectation of the principal 
 key ; and there is lefs hazard of this in the other regular ca- 
 dences. One reafon is, that the principal key, though it be 
 retained in memory, yet reconciles itfelf to our imagination, as 
 a fifth to the 4th/; if the cadence on the 4th be too formally 
 made, or too often repeated. 
 
 At T and X the fixth is not added to the chord of the K, 
 leading in the irregular cadence, becaufe that fixth, joined with 
 the 5th of the fcale, muft have afcended to the feventh, while 
 the 5th had kept its place; but here the 5th does not keep its 
 place, but defcends upon the 2d, in the treble part, and there- 
 fore the difcord is omitted, becaufe it could not be properly 
 refolved. 
 
 In the phrafe Qjhe feventh is not added to the 5th/, lead- 
 ing in a falfe cadence on the 6th, becaufe the treble defcends 
 from the 7th to the 6th of the fcale, whereas the addition of 
 the 4th of the fcale, as a feventh to the leading chord, would 
 require the 7th of the fcale to afcend into the K. 
 
 In the phrafe L, the 4th/, which precedes the cadence, is 
 %ured with 6 only, inftead of 6 and 5, as at I ; the fifth be- 
 ing left out in this chord ; becaufe, if ic were admitted, along 
 with the fixth, it muft neceflarily defcend upon the 7th of the 
 fcale, but the treble afcends upon the fame 7th, and therefore 
 this omiffion takes place here, to prevent the necefiity of dou- 
 bling the 7th. The fame is to be obferved of the 4th/, in 
 the phrafe P. 
 
 In the phrafes R. T. X. and Y. the 4th/, leading in the 
 irregular cadence, bears only its perfect chord, without the 
 added fixth, as at H : becaufe, in thefe paffages, the treble de- 
 fcends upon the 3d of the fcale, and if the added fixth were 
 ufed, it muff afcend upon the fame 3d. 
 
 U 
 
 the Chords of a Fundamental Bafs. 77 
 
 220. To fupport the afTertion, that not only fuch as are 
 complete judges of harmony, but even all who have any enjoy- 
 ment of mufic, although they be quite ignorant of its prin- 
 ciples, do naturally conceive the fucceffive founds as belonging 
 to fome fundamental progreffion, we may obferve, that while 
 a man of a good natural turn for mufic, without any far- 
 ther acquirements, fings a tune, Ihould another fupply the 
 proper fundamental of each found, as a bafs, the finger would 
 immediately find additional pleafure arifing from the perception 
 of the harmonious relations between his founds and this bafs ; 
 he would be able to attend to this bafs without any hazard of 
 lofing, or being put out ofhxs tune; on the contrary, fuch fub- 
 joined founds would rather fupport and guide him to a more 
 juft performance of his own part. Thefe are facts fufficiently 
 attefted by experience. 
 
 221. If an objector fhould fay, " I can fing feveral tunes as 
 " well as one of you who underftand mufic, and I know no- 
 " thing about it, neither am I fenfible of any regard paid to 
 " any other founds, except thofe of the tune, as they fucceed 
 " one another ; what influence then can thefe other founds you 
 " call fundamentals have upon my voice ?" Our anfwer is, that 
 their influence may take place, juft as effectually, when we are 
 not apprized of it, as when we are; there are numbers of cu- 
 
 Jlomary operations, both of mind and body, which we perform, 
 without ever ftudying the philofophical principles of the mat- 
 ter. Thus we ftand and walk, <bc. without any knowlege 
 of the laws of gravity and motion ; we converfe in our mother- 
 tongue, without ftudying grammar; argue, without under- 
 ftanding logic, <bc. ; and thus we fing, in conformity to cer- 
 tain fundamental progreffions, even although we are quite in- 
 fenfible of their influence. 
 
 222. By comparing the various effects of the founds of the 
 natural fcale, as mentioned, Art. 21 and 22, and the proper- 
 ties of founds conceived as parts of fome fundamental chord, 
 
Of Fundamental ProgreJfioHS. 
 
 Part I. 
 
 Art. 141, with the examples laft adduced, we (hall find abun- 
 daut reafon to conclude, that every finger who perceives thofe 
 effects, does really, in fome degree, adopt a fundamental 
 progreiiion. Thus, we fee, in Exam. XL.II. that the key- 
 note of the natural fcale has its own $P chord, and that the 
 5 th of the fcale ftands as a fifth in the chord of the Kf, in a* 
 {"pending, and has its own $P chord, in defcending; and that 
 the 3d and 7th of the fcale ftand as greater thirds in the $P 
 chords of the K and of the 5th refpeftively, both afcendingand 
 defcending: now, it was obferved, Art. 141. that the funda- 
 mental of a $ P chord, is bold and commanding, and its third 
 fupplicative ; and from hence we may fafely infer, that the bold 
 effect of the K, and of the 5th, and the fupplicative effect of 
 the 3d and 7th, as hinted, Art. 21, depend altogether upon 
 the fundamental progreffion. The plaintive effect of the 2d 
 and 6th, mentioned in the fame Art. 21. is mod perceivable 
 when either of thefe founds becomes a fundamental, bearing a 
 [j P chord, agreeable to Art. 141 . ; and the 4th is moft remark- 
 ably heavy and fpiritlefs when it ftands as an added feventh in 
 the % P chord of the 5th/", or as a lefs third in the fe P chord 
 of the 2d/~: but even in other paffages, if the principal key be 
 retained, thefe notes ftill keep fo mew hat of the fame properties. 
 Thus the 4th has a peculiar gravity ovfolemnity, even when 
 it becomes a fundamental, as in the afcending fcale ; and the 
 6th is, in fome meafure, plaintive, even when it ftands as a 
 greater third in the $ P chord of the 4th/, or as a fifth in the 
 chord of the 2df, 
 
 223. Another obfervation, which will frequently prefent it- 
 felf to all fuch as are engaged in teaching beginners to fing, 
 tends direclly to the confirmation of thefe principles. It will 
 often happen, that before a perfon begins to apply himfelf pro- 
 feffedly to learn to fing, he will have partly got feveral tunes 
 by the ear, in fome paffages of which it is ten to one that he 
 will fing falfe ; either copying the errors of thofe whom he has 
 
 been accuftomed to hear, or running into others of his own, 
 through the imperfection of his memory ; and if his erroneous 
 notes be fuch as require a fundamental fucceffion different from 
 the real tune, it is with the greateft difficulty he can be brought 
 to correct them ; but if the fundamental fucceffion be the fame, 
 he will fing the right notes at the firft trial, juft as eafily as 
 the wrong. 
 
 The difficulty of correcting the errors of fuch an untaught 
 finger is yet more increafed, if his falfe notes belong to a more 
 eafy fundamental fucceffion, than the true ones ; which often 
 happens. For inftance, a whole congregation will very fre- 
 quenly conclude a line of a pfalm-tune, by afcending from the 
 7th to the K, when the real tune defcends from the K to the 
 7th ; and in this cafe the wrong notes fuggeft a regular cadence 
 on the K, and the right notes belong to a cadence on the 5th ■ 
 and it is next to an impoffibility to correct this fault ; for the 
 regular cadence is fo natural in itfelf, and fo intimately con- 
 nected with that part of the tune, according to their way, that 
 nothing elfe will fatisfy them in its place. You may eafily bring 
 therri to ufe the 5th in place of the 7th, or to take any 
 of the other motions which reprefent the fame cadence ; but 
 nothing will be relifhed which cannot be reconciled to that 
 favourite paffagc. 
 
 224. Upon thefe principles we may give a rational account 
 of the various taftes of perfons who have a natural turn for 
 raufic, but unimproved by practice. Such perfons adopt, and 
 become familiarized with fome fundamental paffages, which 
 correfpond with the favourite tunes they are accuftomed to hear; 
 and when a new tune prefents itfelf, which can eafily be re- 
 conciled with their favourite fundamental paffages, they are im- 
 mediately pleafed ; but if this cannot be done, they are perplex- 
 ed and difgufted : though, at the fame time, perhaps the real 
 fundamental fucceffion of fuch a tune may be quite eafy and 
 natural in itfelf, and fufficiently obvious to another hearer. 
 
Chap. VIII. 
 
 Remarkable Circumjlances accounted for, on the foregoing Principles, 79 
 
 Hence we may alfo affign the reafon, why fo very few of the 
 more unfldlful fort are delighted with a new tune at the firft 
 hearing, and why it becomes more agreeable after feveral repe- 
 titions, when they begin to know the air of it : for, in truth, 
 the proper effect of every fucceffive found, which has a confi- 
 derable influence in the air of a tune, depends upon the fun- 
 damental progreflion ; and this is not always fo infallibly fug- 
 gefted at once, to an unexperienced hearer, but that his firft 
 conceptions are either vague and undetermined, or perhaps erro- 
 neous and inconfiftent with the true air of the tune ; and there- 
 fore need to be re-examined and corrected by frequent trials, 
 before he can fufficiently rely on them. 
 
 The more expert mufician is feldom at any lofs for a proper 
 fundamental progreflion, and therefore enters immediately into 
 thefpirit of any new piece which occurs. Monfieur Rameau 
 undoubtedly f peaks from expe.rience wben he fays, in )n\s Prin- 
 ciples qf 'compaction, that a man may feel the bafs which ought 
 to go along with the treble, while the latter is performed fin- 
 gly without any accompanyment : and this proficiency is the 
 very majler piece of the flail of mafic ; which, however rare, 
 is neverthelels, in fome degree attainable, by all who have a 
 natural tafle, if its principles be duly attended to, in the courfe 
 of practice. 
 
 225. The moft natural and eafy fucceffions of founds, for 
 the human voice, are thofe which afcend or defcend by the An- 
 gle degrees of the fcale, and not by leaps : and hence it follows, 
 that thefe fucceffions foon become familiar to every one, who 
 either practifes or hears vocal mufic; and, of courfe, the fun- 
 damental paflages belonging to fuch fucceffions, become equal- 
 ly familiar; though, perhaps, the finger is quite infenfible of 
 them. This remark may ferve to prove the ufefulnefs and 
 neceffity of a thorough knowlege of the proper fundamental 
 progreffions, while the treble moves by Angle degrees only ; 
 
 becaufe fuch progreffions cannot fail of being obvious, to every 
 hearer, for the abovementioned reafons. 
 
 226. We (hall now proceed to the explanation of fome other 
 paflages which are allowable in fundamental bafles. 
 
 Although thofe fundamental paflages which imitate the ca- 
 dences, as explained above, are generally the mod natural, yet 
 the fucceffion of chords is not fo ftriftly limited, but that, 
 where a formal cadence is not expected, and if neither diflb- 
 nances nor occafionally altered notes be introduced ; that is, if 
 every fucceffive fundamental bear only its own perfect chord, 
 fuch as the fcale allows, we may freely pafs from any one 
 chord to any other, provided there be one found common to 
 both; which, by continuing upon the fame degree, connecls 
 the two fucceffive chords ; and fuch a found is called, by the 
 artifts, a holding note. 
 
 Conformable to this principle of connexion by a holding note, 
 it is plain, that the fundamental bafs may either afcend or de- 
 fcend by a leap of a fixth, or fifth, or fourth, or third ; pro- 
 vided it does not touch the 7th of the fcale ; or, which is the 
 fame thing, the fundamental bafs may move by leaps of any 
 kind, but not by Angle degrees ; becaufe in all leaps, there 
 will, at leaft, be one holding note to connect the following 
 with the preceding chord. — If the fundamental bafs rife a 
 third, or fall a fixth, the third and fifth of the preceding chord 
 will be both continued, and become the fundamental and third 
 in that which follows: if the fundamental bafs rife a fourth, 
 or fall a fifth, the oftave of the former fundamental is held on, 
 and becomes a fifth to the latter : if the fundamental bafs rife 
 a fifth, or fall a fourth, the fifth of the former chord is held on, 
 and becomes a fundamental : and if the fundamental bafs rife 
 a fixth, or fall a third, the octave and third of the former fun- 
 damental are both held on, and become third and fifth to the 
 latter. 
 
So 
 
 227. This idea of the connexion of chords is well worthy 
 of the learner's notice, although it is not a fufficient foundati- 
 on for a fatisfactory account of all the fundamental progreffi- 
 ons which are prefered in practice ; becaufe we often pafs from 
 one chord to another when there is no holding note, as in the 
 falfe cadence • and even when there are holding notes, thofe 
 chords are found to be nearer related which have but one, as 
 when the fundamental bafs rifes or falls a fifth, than thofe 
 which have two, as when the fundamental bafs rifes or falls a 
 third ; contrary to what might be expected, if the holding note 
 were the fole caufe of the connexion. Befides, in many paf 
 fages, and particularly in the perfect regular cadence, it is a 
 matter of indifference whether the 5th of the fcale, which is 
 the holding note, be heard at all in the following chord, or 
 not. See Art. 192. And if the connexion of the two chords, 
 which form this cadence, were owing to this holding note, its 
 exiftence in the accompanyment might feem to be more ne- 
 ceffary For thefe and other reafons we have concluded it more 
 proper to reprefent the imitation of cadences as the moft natural 
 law of fundamental progreffions. in preference to the connexi- 
 on by holding notes ; as this latter may be fufficient in the 
 middle parts of a phrafe, but is a circumflance of little or 
 no importance in a cadence. 
 
 228. We account, then, every leap of a fourth or fifth, in 
 the bafs, to be an imitation of a regular or irregular cadence : 
 the leap of a third or fixth is allowed, by reafon of the hold- 
 ing notes: the rifing of a fecond imitates the falfe cadence 
 or gradation ; and the falling of a fecond, which occurs more 
 rarely than any of the former *, may be admitted only after a 
 chord of an ambiguous or double fundamental, or after an oc- 
 
 Of Fundamental Progrejjtons. Part I. 
 
 cafionally altered note has taken place, as we have now to ex- 
 plain. 
 
 220. It was formerly hinted, Art. 217. in fpeaking of the 
 phrafe Q, that immediately after the falfe cadence has been ufed 
 we may proceed, in the fame manner as if the regular cadence 
 had not been interrupted at all. Hence it follows, that after 
 the 6th has been introduced in the fundamental bafs, by means 
 of the falfe cadence, we may immediately defcend to the 5th/, 
 becaufe the palTage from the K, which in this cafe claims a 
 part of our attention, upon the fame 5th, is very natural. We 
 muff obferve farther, that fo long as we ffrictly retain the prin- 
 cipal key, the chords of that key, and of its two adjnntls, will 
 be fo ready to prefent themfelves to our imagination, on alloc- 
 cafions, that whenever the chord of the 6th/ is brought in, 
 either by means of the falfe cadence, or otherwife, the chord 
 of the Kf will in fome degree attract our attention as co-ex- 
 3 fling along with it. In the fame manner the chord of the 
 5th/ will claim a part of our regard along with that of the 
 2<if, when the 5th is not occalionally fharpened. And the 
 chord of the 4th/ will be felt along with that of the 2d/, when 
 the 4th is not occafionally fharpened. — So that in faff, we 
 have frequently chords of a double fundamental, when no dif- 
 fonances are heard, owing to a natural propenfity of reconcil- 
 ing every found we hear to one or another of thefe three prin- 
 cipal chords, viz. to the chord of the Kf, or to that of the 
 5th/, or that of the 4th/. 
 
 From hence it follows, that as the fundamental bafs may 
 defend from the 6th to the 5th, on account of the ambiguous 
 fundamental of the former chord; fo, by parity of reafon, it 
 may alfo occafionally defend from the 2d to the K, or from 
 
 * Monfieur D'-Alcii 
 leave to diiTent from 
 
 ; Element ic Mufique, concludes the falling of a fecond to be difallowed in a fundamental bafs on any occafion : but we beg 
 particular ; foi reafons which will probably appear fufficient. 
 
Of the Leap of a third or fixth, and the Step of a fecond. 
 
 Chap. VIII. 
 
 the 3d to the 2d, if the accidental fharp note do not take place 
 in the former chords of fuch paflages : and it is not difficult 
 to meet with examples of each of thefe progreflions among the 
 works of the very beft compofers. In like manner, the gradual 
 defcent from the 4th/ to the 3d/, is allowed in paflages 
 where the 6th may be fuppofed to attract part of our attention 
 as a fundamental, along with the 4th ; which never fails to 
 happen in the following chord of a falfe cadence on the 4th, 
 and in fome other cafes ; fo that, in fact, the fundamental bafs 
 may occafionally make a gradual defcent, in the middle of one con- 
 nehedftrainfrom any note except the K, and the $th. From the 
 K, fuch a defcent cannot be allowed, becaufe the 7th cannot be a 
 fundamental ; and from the 5th fuch a defcent caufes an inter- 
 ruption, or breach, in the harmony ; becaufe, according to the 
 above principles, the connection in fuch paflages is owing to 
 the third note in the former chord being taken as a co-exiftent 
 fundamental; which third being, in this cafe, the 7th of the 
 fcale, cannot be admitted as fuch. 
 
 InExam.XLV. Plate VII. are introduced inftances of each 
 of thefe defcents, marked by the ftars ; which example, as far 
 as it relates to this article, needs no other annotation, but that 
 the directs, placed in the bafs, (hew the co-exiftent fundamentals, 
 by means of which fuch defcents are allowed. 
 
 230. In regard to the liberties which may be allowed in a 
 fundamental bafs, when occafionally altered notes take place in 
 the accompanyment, we muft obferve, that fuch notes being 
 more diftantly related to the principal key, and therefore more 
 difficult both for the finger to execute, and for the hearer to 
 reliffi, do naturally require to be followed by the next neareft 
 degree of the fcale, as a kind of refolution of the difcord be- 
 tween them and the principal key; whatever be the fundamen- 
 tal of the chord for the time. Thus the $4th naturally tends 
 to the 5th, the % 5th to the 6th, and the ^ 7 th to the 6th of 
 
 81 
 
 the fcale : and the defire of hearing fuch notes properly follow- 
 ed is fometimes fo prevailing as to fuperfede all regard to the 
 common rules of fundamental progrefiions. For inftance, if 
 the % 5th take place in one chord, we are not much concerned 
 about the fundamental progrefilon, provided that particular 
 note may be heard refolved into the fixth, in the next chord : 
 fo that even in this cafe the fundamental bafs may fometimes 
 defcend from the 3d to the 2d without offence. Inftances of 
 this paflage, and others of the like kind, are to be found, au- 
 thorized by the moft eminent compofers. 
 
 In Exam. XL VI. the % 5th is introduced three times in the 
 accompanyment, and is always followed by a chord containing 
 the 6th, as obferved above. The two firft paflages reprefent 
 a falfe, and a regular cadence ; the laft, marked with a ftar, is 
 the defcent propofed to be particularly exemplified. 
 
 Tht: leap of a falfe fifth, or tritone, is difallowable in a fun- 
 damental bafs, and when fuch paflages are admitted in the 
 thorough bafs, they are to be efteemed licences. 
 
 231. There is fomething peculiar in the manner of accom- 
 panying thefe paflages which are in fome degree forced and un- 
 natural, in order to make amends for that defect, as much as 
 poflible, which deferves our notice the more, becaufe the beft 
 writers on thorough -bafs, in our language, feem not to have 
 been fully apprifed of its importance and ufe. 
 
 When the fundamental bafs falls a fecond, by means of the 
 third to the former chord being taken for a co-exiflent funda- 
 mental, as in Exam. XLV. at the ftars, it is better to omit the 
 octave to the former fundamental note, and alio to take the 
 third uppermoft in the chord, as often as is convenient, as is 
 Ihewn in that example ; becaufe, by that means, the third is 
 rendered more remarkable, and the pafiage into the next chord 
 more natural and eafy. 
 
 The movement of the uppermoft note of the chord, in a 
 
8 2 Of Fundamental 
 
 contrary direction to the bafs, is another well known method 
 of helping thefe paflages; for the management of which, in all 
 cafes where it may be neceffary, we muft refer to the authors 
 who have treated exprefly on that fubject. The Exam. XLV. 
 and XLV1. may ferve at prefent as fpecimens of the way of 
 proceeding. 
 
 232. It mud be well remembered, that thefe paflages in 
 which the fundamental bafs moves, either by the leap of a third 
 or fixth, or the defcent of a fecond, are not to be introduced 
 too frequently, neither muft they be taken indifcriminately, 
 and without defign. We moft ordinarily ufe liberties of this 
 kind for the fake of arriving more naturally at another chord 
 which is to follow ; and they generally indicate an attempt 
 either to pafs into a new key, or to return into a former key ; 
 and fnch an attempt, or motive, we can counteract in the ac- 
 companyment, by ufing only the founds belonging to the pre- 
 fent key, when the new key would require one of them to be 
 altered, or we can comply with it by admitting the altered 
 note, according as tafte directs, or as the air of the treble, or 
 leading part, requires ; when we are to adapt a fucceffion of 
 chords to a given melody. 
 
 For inftance, in Exam. XLV. the fir ft defcent of a fecond 
 aims directly at a cadence on the 5th, as at 11; with which 
 we io far comply in the accompanyment as to admit the %, third 
 of the leading chord ; but, forefeeing that the harmony is not 
 to reft any time in the new key of the 5th, we rather chufe to 
 omit the 7th in the leading chord, which renders the cadence 
 lefs decifive, and leaves the ear more at liberty to pafs into an- 
 other key. The fame cadence is counteracted at P, in Exam. 
 XLVI. to make way for the final cadence. 
 
 At I, the cadence on the 6th is introduced by the funda- 
 mental bafs falling a third, and at M the fame cadence is brought 
 in, by falling a fecond : the former of thefe is counteracted in 
 
 ProgreJJions. Part I. 
 
 the accompanyment by retaining the natural 5th as a lefs third 
 in the leading chord, but the latter is fully complied with. 
 
 At N, the cadence on the 4th is counteracted by giving the 
 following chord an added diffonance, to prepare for the final 
 clofe. 
 
 233. This judicious choice of proper founds, either for re- 
 taining or relinquifhing the key, is called, by the artifts, Modu- 
 lation ; which, although it may feem to be a fubject more im- 
 mediately concerning the compofer, yet it is no lefs neceffary 
 to be underftood by the finger in particular, becanfe it is only 
 by turning our attention upon the governing key notes, as they 
 fucceed one to another, that we can poffibly attain a mafterly, 
 or indeed a juft execution of vocal mufic. It is alfo very far 
 from being ufelefs to the performers upon instruments, and e- 
 fpecially to the players of thorough bafs, becaufe, if they do 
 not cb ferve what cadences are imitated, and what is the go- 
 verning key in every paffage, there is no chance of their exe- 
 cuting with proper tafte and elegance; although the mufic be 
 ever fo well compofed, and correctly copied. And for the 
 learners encouragement in this matter, he may be affined, that 
 if he begin with a little ftudy of the fundamental principles, 
 and keep the fame conftantly in view, in the courfe of his prac- 
 tice, he will find the neceffary knowlege becoming familiar 
 and obvious, long before the execution of thorough-bafs be 
 attained to any tolerable degree. 
 
 The truth is, as trie melody belonging to any one key, con- 
 fifts but of a few degrees of found, fo the harmony of one 
 key confifts but of a very few chords; being naturally no more 
 than three, viz. thofe of the key and its two adjuncts, viz. 
 4th and 5th ; the firft being always perfect, and the two laft 
 having their added diffonances, as fnewn above; and when. 
 ever another chord, different from any of thefe, is introduced, 
 it indicates an attempt to pafs into a new key ; with which we 
 
Chap. VIII. 
 
 may either wholly comply, both in the fucceeding movement of 
 the fundamental bafs, and the modulation of the accompany 
 ment ; or we may conform to it in the former, and counteract 
 it in the latter; or we may wholly fet it afide, and return to 
 the former key altogether. 
 
 234. In order to judge properly of the tendency of any 
 chord, which does not immediately belong to the harmony of 
 the prefent key, it is neceffary to take particular notice of the 
 natural relation of keys, on the one hand ; and of the fimilla- 
 rity or uniformity of proceeding in each key, which is called 
 imitation, and which is requifite for conflitu ting one connect- 
 ed whole, on the other hand : of both which the following 
 obfervations will give forme ufeful ideas. 
 
 235. In regard to the relation of keys, we have formerly ob- 
 ferved, that the principal key of a fharp feries is related to the 
 new keys of the 4th or 5th as its adjuncts, or of the 6th, as 
 its fubftitute : we have now to obferve farther, that the 2d of 
 the fcale may be brought in as a fubftituted 6th to the 4th, as 
 a new key ; and the 3d may be introduced as a fubftituted fixth 
 to the 5th: fo that, by conducting the harmony with judg- 
 ment, firft into thofe keys which are immediately related to the 
 principal, and afterwards into thofe which are only related me- 
 diately, or by the interpolation of another, we may occafional- 
 ly introduce all the founds of the fcale as keys, except the 7th; 
 if the piece be long enough to admit of them all. 
 
 236. To render this more familiar, it will be ufeful to re- 
 mark, that in the courfe of one connected piece of mufic, we 
 are not confined to the fcale of the principal key alone, but we 
 occafionally make a traafition into the fcale of either of its ad- 
 juncts. Thus from the fcale of the principal key C, we may 
 pafs either into that of G, with a $ upon F ; or into that of 
 F, with a \) upon B i but we cannot pafs into any other fcale 
 except one of thefe two, without deflxoying the rememberance 
 and expectation of the principal key ; and, of courfe, the con- 
 
 Of Modulation ; and of the Succeffion of occafional Keys. 
 
 83 
 
 nexion of the piece : and it is hardly to be doubted, but thai- 
 it was owing to fome obfervations of this fort, that Cv.ido Are- 
 tinus paid fuch a fuperior regard to thefe three notes, F. C. G. 
 in the fignature of his cliffs. See Art. 55. In antient times, 
 none but the fcale of C, and its two adjuncts, appear to have 
 been in ufe ; the artificial or tranfpofed keys being wholly the 
 device of the moderns. However, our principal fcale has frill 
 its two adjunct fcales, by whatever letter or mark it be denot- 
 ed ; and the nature of mufic is the very fame, though the con- 
 flmction of our inftruments, or the manner of our writing mu- 
 fic be ever fo widely different. 
 
 237. The arrangement of the feveral fcales in the general 
 fcheme, Art. 47. was made choice of on this very account, 
 that the two adjunct fcales belonging to any principal, might 
 always be found next to it, the one above, and the other be- 
 low, in that fcheme: and in this light it will be worth while 
 to perufe the fame fcheme, in order to attain a ready knowiege 
 of the relation between the feveral fcales, as they are there 
 placed. 
 
 Take any principal fcale at pleafure, fuppofe that of D, with 
 F and C fharp, N° 3. the fcale of its adjunct 4th will be 
 found next above, which in this cafe isG, with F fharp, N° 2. 
 and the fcale of its adjunct 5th will be found next below, 
 which, in this cafe, is A with F, C and G fharp, N° 4. 
 
 Again, taking the fcale of B (j for principal, N° 11. the 
 fcale of E\), N^ I o. will be found, for the adjunct 4th, and 
 that of F, N° 12. for the adjunct 5th. 
 
 Again, taking the fcale of C for principal, N° 1. and N° 13. 
 the fcale of F, N° 1 2, belongs to the adjunct 4th, and that 
 of G N° 2. belongs to the adjunct 5th. 
 
 238. To apply thefe remarks to our prefent purpofe, we 
 muff further obferve, that the natural fcale offords two notes, 
 upon either of which we can, with proper caution, make a 
 perfect regular cadence, or indeed a final conclufion : for we 
 
8 4 
 
 Of Fundamental ProgreffionS. 
 
 Part I. 
 
 can clpfe either with the K, as we have pretty largely explained 
 already, or with the 6th of the fcale, if this 6th be ufhered in 
 by the ^ 5th, as a leading note. The predominance of the K 
 conftitutes a tune of a (harp feries : but the predominance of 
 the 6th, and the conftant expectation of recurring upon it at 
 the ooneiu (ion, characterifes a tune of a flat feries ; of which 
 more hereafter. 
 
 In this view, the 6th becomes an occafional new key, with- 
 out introducing a new pofition of the natural fcale; being 
 brought in whenever the % 5th is ufed inffead of the natural 
 cth ; and this may be properly enough called a different modi- 
 fication of the fcale of the K. In like manner, the 2d takes place 
 as a key, in the fcale of the 4th, by fharpening the original K, 
 which is the fifth of that fcale ; and the 3d takes place as a key 
 in the fcale of the 5th, by fharpening the 2d of the original 
 fcale, which is the fifth of this fcale : fo that we have fix dif- 
 ferent keys, practicable in the courfe of one piece, though 
 only three different pofitions of the natural fcale, with the ad- 
 dition of an occafional % 5th in each fcale. 
 
 Thefe three fcales we (hall hereafter take the liberty of call- 
 ing collectively by the title of the fyjlem of modulation, of a giv. 
 en principal key. 
 
 239. Every key, as well occafional as principal, has its two 
 adjuncts; and the harmony chiefly dwells upon the chords of 
 the key, and of thefe adjuncts, fo long as that key is retained. 
 Thefe adjuncts are always the perfect fourth and fifth to the 
 key. Thus the principal K and the 2d are adjuncts to the 5th, 
 as a new key ; the \) ?th and K are adjuncts to the 4th as a 
 new key ; the 2d and 3d are adjuncts to the 6th ; the <;th and 
 6th are adjuncts to the 2d ; and the 6th and 7th are adjuncts 
 to the third as a new key. 
 
 It is to beobferved, that the 2d and 3d, [landing as adjuncts 
 to the 6th, may alfo become new keys, fubordinate to the 6th, 
 as principal, in tunes of a flat feries ; in the fame manner as the 
 
 4th and 5th become new keys, fubordinate to the K of the na- 
 tural fcale, in tunes of a fharp feries. 
 
 From hence it appears, that the fundamental bafs ought not, 
 upon any occafion, to touch any of the altered notes in the 
 fyftem of modulation, except the [7 7th, and this only as adjunct 
 fourth to the 4th of the fcale, as a new key. The fundamen- 
 tal bafs cannot touch the 7th of the fcale, but as adjunct fifth 
 to 3d as a new key : but the 3d and 4th may each take place 
 in two ways ; for, the 3d may come in either as a new key, or 
 as a fifth to the 6th ; and the 4th may be a new key, or a 
 fourth to the principal key : and the K, the 2d, the 5th, and 
 the 6th may each of them take place in three ways ; viz. ei- 
 ther as keys, or as fourths, or as fifths, to other keys. 
 
 240. For determining the fpecies of every fundamental chord, 
 we mufl take notice. 
 
 17?, That every key note takes its chord from the degrees of 
 the principal fcale, without allowing any alteration; fo that, 
 as the natural fcale affords three % P chords, viz. thofe of the 
 Kf, the 4th/, and the 5th/; and three \> P chords, viz. thofe 
 of the 6th/, the 2d/, and the 3d/: See Art. 147: fo thefe 
 degrees may all beaome keys ; and muff, in that cafe, bear the 
 fame fpecies of chords refpettively. 
 
 2d, That the adjuncts of every key naturally bear the fame 
 fpecies of chord as the key to which they belong ; except, in 
 particular occurrences, the adjunct fifth belonging to a key with 
 a \) P chord, bears its ^ P chord, for the fake of rendering the 
 cadence perfect. 
 
 Thus when the [j 7th and K, on the K and 2d, come in as 
 adjuncts to the 4th or 5th, refpectively ; each of thefe notes 
 bears its $P chord: but when the 2d and 3d, or the 5th and 
 6th. or the 6th and 7th, come in as adjunct fourth and fifth, 
 to the 6th, or 2d, or 3d, refpectively: the former, viz, the 
 adjunct fourth, bears its jj P chord ; but the latter, viz. the 
 
Chap. VIII. Of the Syftem of Modulation. 
 
 adjunct fifth, bears fometitnes a jj P, but more frequently a 
 % P chord, for the reafon hinted above. 
 
 241. The adjunct fourth of every key may take an added 
 fixth, and the adjunct fifth may take a feventh for their pro- 
 per diflbnances, after the fame manner as in the principal key. 
 But thefe diflbnances are to be admitted more fparingly in keys 
 which are more remote from the principal, for the reafon hint- 
 ed, Art. 232. in f peaking of the paflage at H; becaufe the 
 harmony muft not dwell long in fuch keys, but mull pafs from 
 them either into the principal key, or at leaft into one more 
 nearly related to it : otherwife the rememberance of the princi- 
 pal key would be loft. 
 
 242. Thefe confiderations will afford us clear and fatisfac- 
 tory ideas of the natural relation of keys one to another. We 
 have already faid, that the 4th, 5th, and 6th, are the degrees 
 which can moft naturally be made new keys, fubordinate to the 
 K of the natural fcale, as principal, in tunes of a (harp feries ; 
 and that the 2d and 3d may come in only asfubjiitutes, in the 
 fcales of the 4th and 5th respectively, or as adjuncts to the 6th. 
 We may now conclude, that the new key of the 5th is more 
 nearly related to the principal K, than that of the 4th, on account 
 that the latter brings with it an altered note, viz. the \j 7th, 
 as its adjunct fourth, whereas both the adjuncts of the former 
 are notes of the natural fcale. We may conclude alfo, that 
 the new key of the 2d is more eafy to be introduced than that 
 of the 3d, becaufe the latter brings with it the 7th of the 
 fcale as its adjunct fifth, and this 7th cannot naturally become 
 a fundamental ; whereas the former has both its adjuncts a- 
 mong the other founds of the principal fcale, which can natu- 
 rally become fundamentals. Thus, by taking into account 
 the adjuncts along with every key, we come at a folution of 
 this, otherwife, myflerious queftion, viz. Why is the new key 
 of the 3d more difficult to be introduced than that of the 2d ; 
 and yet the fcale of the 5th, in which that third takes place, 
 
 Practical Rules for the Succejjion of Keys. 85" 
 
 is eafier brought in than that of the 4th, upon which the fe- 
 cond depends ? Experience confirms this to be the cafe ; but 
 yet no writer, that I .know of, has attempted to account for 
 it. 
 
 243. It is neceflary to obferve farther, that the adjunct fifth 
 to any key is more eflentially neceflary to be ufed, than the ad- 
 junct fourth : for, without the chord of the fifth wc cannot 
 lead the attention to any key at all, but the harmony may pro- 
 ceed from the chord of the fifth upon that of the key, and 
 contrariwife for a confiderable time, without touching upon 
 the chord of the fourth ; fo that, to afcertain the natural fue- 
 ceflion of keys, with greater precifion., we are not to regard 
 the two adjuncts as equally connected with the key : the ad- 
 junct fifth mufi be introduced, in order to determine the key ; but 
 the adj unci fourth may take place or not, as the cmnpofer pleafes, 
 or as the given melody requires. 
 
 244. Thefe things being premifed, we may lay down the 
 following practical rules for retaining or relinquishing the prin- 
 cipal key. 
 
 (1.) For retaining the principal key, in the ftricteft manner, 
 we ought to ufe no chords except thofe of the K/, the 5th/i 
 and 4th/; (viz. the key and its two adjuncts ;) and the chord 
 of the 5th/onght to have an added feventh, and that of the 
 4th/an added fixth, as often as thefe can be admitted, and 
 properly refolved, as directed above. 
 
 (2.) We retain the principal key, in a larger fenfe, fo long 
 as we ufe none but the feven founds of the fcale belonging to 
 that key, in the accompanyment of the thorough-bafs. The 
 key is never effectually changed till an altered note appears ; 
 and thus any degree of the fcale may be a fundamental, except 
 the 7th, without introducing a change of the key, provided 
 its chord be fuch as the fcale of the key affords, as defcribed 
 Art. 147 ; and provided the paflages be chofen with difcretioa, 
 as hinted Art. 232. 
 
u 
 
 (3.) A change of the key is made by introducing one of the 
 adjuncts, (generally the 5th) of the intended key into the fun- 
 damental bafs, with the proper accompanyment which belongs 
 to it as fuch, Art. 246, 241. And if the 5th of the new key 
 be not firft introduced, it mud however be brought in as foon 
 as convenient, becaufe the key is never effectually determined 
 till the chord of its 5th be heard. 
 
 245. The order of the fucceffionof new keys depends much 
 on the fancy of the compofer : one principle however well de- 
 ferves our obfervation, viz. that the rememberanceof the prin- 
 cipal key may and ought to be kept up by means of a well 
 chofen fucceflion of new keys : and this is effected by making 
 fuch tranfitions from one key to another as could not be allow- 
 ed, if either of the two were the principal, and confequently 
 fuch tranfitions as plainly indicate the dependance of both the 
 new keys upon another principal. Thus, for inflance, it is 
 very common to pafs from the new key of the 4th to that of 
 the 5th, and contrariwife ; both of thefe keys bearing a $ P 
 chord : and fuch a tranfition plainly fhews that neither of thefe 
 is the principal key ; becaufe, if the lower were the principal 
 key, the upper would be its fecond, and in this cafe the latter 
 could not eafily follow as a key, and if it were brought in as fuch, 
 it ought to bear a [j P chord : and if the upper were the prin- 
 cipal key, the lower would be its flat feventh, and this, as fhewn 
 formerly, cannot be a key at all ; but yet thefe two, confider- 
 ed as adjuncts to another principal, have a fine effect in fuc- 
 cetfion. 
 
 In the fame manner the fubfUtuted 6th of the principal key 
 may eafily either follow or precede the adjunct 5th, as a new 
 key, or the 4th, 5th, and 6th may take place fucceffively. 
 Thus alfo the new key of the 3d may be introduced next to 
 that of the 2d, both of thefe keys bearing a \> P chord, though 
 this laft is a more difficult tranfition than any of thofe menti- 
 oned above ; and, in Short, as all the degrees of the fcale, ex- 
 
 Of Fundamental Progrejjlons. Part I. 
 
 cept the 7th, may become new keys, dependent on the princi- 
 pal, fo a tranfition may be made, on occafion, from any one 
 of them to any other, except from the 3d to the 4th, & vice 
 verfa ; between which the relation is fo diftant as to render the 
 quick tranfition from one to the other almoft, if not altoge- 
 ther, impracticable. 
 
 Upon the fame principle we can immediately return to the 
 piincipal key at any time, and from any of the new keys with- 
 out the leaf! difficulty. 
 
 246. But in order to fucceed in the enquiry concerning the 
 natural fuccelfion of keys, and the tendency of chords fo- 
 reign to the prefent key, we muff take particular notice of the 
 propenfity of the mind towards imitations; s or the defire of 
 hearing the harmony proceed in a new key, in a manner fome- 
 what fimilar to what has been heard in a former key. This 
 propenfity is fo prevailing, in many cafes, as to overcome all 
 other confiderations, and to tranfport usr, without any previ- 
 ous preparation, into keys which could not immediately fol- 
 low, without the utmoft difguft, were it not for the fake of 
 imitation. 
 
 Thefe imitations, being merely matters of tafte, are there- 
 fore not to be afcertained by rules, and will be beft learned by 
 carefully obferving the works of good compofers. One cir- 
 cumffance however is worth our notice, viz. that whether the 
 original phrafe or portion we imitate be longer or fhorter, 
 whether the imitation be an exact copy of the original, or whe 
 ther fome variations be allowed, yet ftill the imitating notes 
 muff be accented in the fame manner with the original ; that 
 is, the accent or emphafis muff fall upon fimilar parts of each 
 paffage, or otherwife the refemblance will be totally loft. Thus, 
 for inftance, a paffage from an unaccented to an accented note, 
 cannot be faid to be imitated by a paffage from an accented to 
 an unaccented note. 
 
 In Exam. XLV. the three notes at the firft ftar, are three 
 
Chap. VIII. 
 
 Of Imitations in the Fundamental Bafs. 
 
 ?<7 
 
 times imitated at the following ftars : but the paflage at the 
 ftar, in Exam. XLVI. though it confifts of the fame notes, 
 is not any imitation of that firft paflage, in Exam. XLV ; be- 
 caufe it is differently accented. 
 
 247. We fhall proceed to give further exam pies of the appli- 
 cation andufeof all that has hitherto been faid on this fubjecl:. 
 
 In Exam. XLVII. XLVIII. XL1X. and L. the imitations, 
 marked by the horizontal brackets below, confift but of one 
 cadence each : thofe in Exam. XLVII. and L. are regular; 
 thofe in Exam. XLVIII. and XLIX. irregular. 
 
 The defign of thefe examples is to fhew, what influence the 
 natural propenfity of the mind towards imitations has upon our 
 choice of fundamental progreffions. In Exam. XLVII. the 
 2d following a regular cadence on the 4th, is made leading 
 note in another fimilar cadence on the 5th ; but in Ex. XLVIII . 
 the fame 2d, following an irregular cadence on the 5th, is 
 made leading note in another fimilar cadence on the 6th. In 
 Exam. XLIX. the 6th following an irregular cadence on the 
 5th, is made leading note in another fimilar cadence on the 3d; 
 but in Exam. L. the fame 6th, following a regular cadence on 
 the 4th, is made to lead in another fimilar cadence on the 2d. 
 Thus the imitations are purfued throughout each example, ex- 
 cept in as far as the regular cadence which is always necefTary 
 at the conclufion, does not fall in with the preceding imitati- 
 ons, in Exam. XLVIII. and XLIX. 
 
 We need not to dwell any longer on thefe examples, as their 
 fcope will not fail to be obvious to every reader : only it may 
 be obferved, that the cadences are the fame with thofe fpecifi- 
 ed, Art. 207. and Exam. XLI. but neither the altered notes, 
 nor added dilTonances, are here admitted, in the middle ca- 
 dences ; by which means, the impreffion of the principal key 
 is more clofely retained, and the tranfition from one key to 
 another rendered more eafy, as remarked, Art. 232. There 
 was no abfolute neceffity for this, but we found them fo in 
 
 Mr. Lampe's Tborovgh-bafs, and in borrowing our examples 
 from theftudied compofitions of fo great a ma Iter, we give she 
 ftrongeft poilible proof both of the truth and importance of the 
 principles delivered above. 
 
 248. The Exam. LI. contains all the perfect regular ca- 
 dences which are practicable in the fyftem of modulation of G, 
 as principal of a {harp feries : and here the imitation is purfu- 
 ed, all along, from one cadence to another, by making the 
 bafs to afcend one degree from the following note of one ca- 
 dence to the leading note of the next; until we arrive at the 
 cadence upon B, in the feventh bar, where we are obliged to 
 flop ; becaufe, if we were to purfue the imiiation but one flep 
 further, the next cadence wouid fall uponF; which, as- we 
 have formerly feen, Art. 238 and 239. does not belong to the 
 fyftem of G: therefore, after forming the cadence upon B, 
 we return immediately to the principal key G, in order to make 
 the final clofe there. 
 
 We may alfo trace out another imitation in this example, 
 wherein each ftrain comprehends two cadences, as (hewn by 
 the brackets put below: and, in this view, the laft four notes 
 of the bafs may be confidered as an imitation of what is gone 
 before ; only thefe are to be differently accompanied in the up- 
 per parts, as directed by the figures, the note C taking an ad- 
 ded fixth, as adjunct 4th to the key G, in order to render the 
 final clofe more fatisfattory. 
 
 This example is alfo taken from Mr. Lampe's Thorough- 
 bafs. The reader will readily obferve, that the marks put im- 
 mediately below the notes of all the laft five examples, fhew 
 what degree of the principal fcale each note is : and further, 
 that in Ex. LI. the occafional new keys which take place, and 
 the names of the notes in relation to thefe new keys are added 
 in a line below, and that the changes, from one key to another, 
 are fhewn by the parenthefis J , occuring in this lower line. 
 There was no occafion for this additional line under the other 
 
Of Fundamental ProgreJJlo 
 
 22 
 
 examples, becaufe there is not, in reality, any change of keys 
 in them : for, although the movement of the bafs plainly aims 
 ,at a new key in every middle cadence, yet thefe motives are 
 counteracted in the accompanyments, as obferved, Art. 232. 
 
 249. There is another kind of fuccedion frequently admit- 
 ed in harmony called a fundamental fequence of ' fevenths ; which 
 we have purpofely omitted to fpeak of, till the propenfity of 
 the mind towards imitations had been confidered ; becaufe it 
 feems to be intirely founded thereon. 
 
 The fundamental fequence of fevenths is a fucceflion of re- 
 gular cadences frill moreclofely connected than thofein Ex. LI. 
 for in that fequence every paifage it? the bafs reprefents a regu- 
 lar cadence, but thefe cadences are continually dilappointed in 
 the accompanyment; by giving the leading/chord a lefs third, 
 when the fcale of the key requires it, and by continuing the 
 third of the leading chord, which becomes a feventh to that 
 which follows, and by that means turns the following chord of 
 one cadence, into a leading chord of another : and thus the bafs 
 proceeds by leaps of a fourth afcending, or fifth defending from 
 one chord of the feventh to another, till we come to the perfect 
 regular cadence on the key, which puts an end to the fequence. 
 
 250. This fequence may begin upon any degree of the fcale 
 as fundamental, provided the found which makes the feventh 
 in the firft chord of the fequence be a continuation of any of 
 the founds of the chord which precedes it ; and this is called 
 preparing the feventh, and muft be well obferved in this place, 
 though we have not fpoken of it before; becaufe though all 
 fevenths are beft introduced when they come in as a continua- 
 tion of a former found, yet that preparation is not abfolute- 
 ly BecefTu-y in the added feventh, which belongs to the adjunct 
 5th of any key ; and fuch were all the fevenths we have former- 
 ly had to do with. 
 
 251. The feventh to the firft chord of this fequence, and 
 every other feventh, will be found prepared, if the fund, bafs 
 
 Part I. 
 
 the note 
 
 ( rife a fecond, or a fourth, or a fixth, 7 f 
 1 fall a feventh, or a fifth, or a third, 3 Irom 
 which precedes it, and all the fucceeding fevenths are prepared 
 in the natural courfe of the fequence ; as may be feen in 
 Exam. LII. 
 
 252. Annotations to Example LII. 
 
 This example is in the natural fcale of C, and the fequence 
 of fevenths is four times introduced, and always ends upon 
 the principal key. At H the fequence begins upon the 2d of 
 the fcale, and the difcord is prepared by the bafs rifing from 
 the K. At I the fequence begins upon the 6th of the fcale, 
 and the difcord is prepared by the bafs falling from the K. 
 
 At L the fame fequence begins upon the 3d of the fcale, 
 and the difcord is prepared by the bafs falling from the 5th of 
 the fcale. At M the fequence begins upon the 4th of the fcale, the 
 difcord being prepared by the bafs rifing from the K ; and here 
 the bafs is obliged to run through all the notes of the natural 
 fcale of the key before the fequence come to a conclufion ; and 
 it is worth while to obferve that the length (or number of 
 ftepsj of the fequence always depends upon the place of its be- 
 ginning. 
 
 At K the feventh is prepared and introduced in the fame 
 manner as at H, but the occafional $ third being admitted in 
 the next chord, brings the harmony into the new key of G, 
 where a perfect regular cadence is made at the next ftep • and 
 it may be ufefully remarked that if the occafional $ third be 
 admitted in any chord of this fequence the next following note 
 of the bafs will become a new key, and there the fequence will 
 end with a perfect regular cadence. 
 
 The upper parts of this example are annexed in a ftafFof 
 chords, to give the reader an opportunity of feeing more di- 
 ftinctly how the fevenths are prepared and refolved, and after 
 what manner the courfes of the upper parts are reftricled ; as 
 fhewn by the direction of the ftrokes which occur among the 
 chords. 
 
Plate VH. 
 
 faevnytf g8. 
 
 Ex.XEVI. 
 
 Ba/J 
 
 ' : (S%fci:i; 
 
 ' V^H 
 
 iWj^JUMtJ^ S 
 
 E^XEVJL ExXLVIIL E^XLIK. 7 
 
 E3 ^£ ^ _g i_£ ft 4 , 2 g 4 k i k k: 1 6_^ 4 k: .5 ic 
 
 ^m^MiS 
 
 i I) 
 
 ■o 
 
 
 K. 4 & J2 ^ K KK 45 K. 2 ,5' 6 23 6*75 Kl ^5 K 
 ":)-) £ k)5 ls) /S g Ic)^ k K 
 
 \ „ 7 - ' -v- 
 
 ) ~1»~ . 1 7 . 7 t 7 . » z_ Z_a _Z j 7 V 7 21 7 z ;- ^ 
 
 X HI K. ' -' Ij ■ M ^ 
 
Of the fundamental Sequence offevenths, and the Preparation of 'Difcords. 
 
 Chap. VIII. 
 
 Thefe ftrokes take their rife fucceffively by two and two to 
 gether from the notes which make the feventh and fifth of one 
 chord, and tend downward to the third and octave of the next 
 chord, while the other two notes of the chord are held on, and 
 become feventh and fifth in the following chord : fo that in e 
 very ftep oi this fequence, there are two defcending and two 
 holding notes ; and the parts, which defcend at one ftep, hold 
 on at the next ftep, and foon, alternately defcending and hold- 
 ing, till the per feci cadence on theK takes place ; which is ac- 
 companied as directed formerly, Art. 196. and Ex. XXXVII. 
 and this courfe the upper parts are reftricted to take, when 
 the accompanyment is full (z e. when no omiffions take place) 
 except that on occafion the loweft note of the chord may be 
 allowed to afcend by a leap of an octave, inflead of holding on, 
 in order to avoid running too low. Thus the note E, at the 
 bottom of the firft chord of the fequence at L, is tranf 
 ferred, at the flair, to E at the top of the next chord, by a 
 leap of an octave. The fame note E keeps its place at the 
 ftar in the next fequence ; and hence arifes the difference in the 
 fucceeding chords of thefe two fequences. 
 
 253. The greater feventh to the chord of F at M is allow- 
 ed to defcend, like the other fevenths, though it is not its na- 
 tural courfe, for the fake of a regular fucceffion ; and, for the 
 fame reafon, the next note B has a chord of the feventh given 
 
 89 
 
 not to confider the noteB as the real fundamental of the chord 
 which is given to it, though it appears as fuch ; for the real 
 fundamental is G bearing the feventh, which belongs to the 
 adjunct 5th of the key, and a ninth by way of fufpenfion of 
 th° octave into which it afterwards defcends, Art. 173.; and 
 yet G is not admitted in the bafs, but B is placed inftead of it, 
 for prefeiving the uniformity of the fequence. A direct is put 
 upon G, and the figures 9 and 7, fubjoined in the example, to 
 fhew the true origin of the chord above it. 
 
 254. The introduction of all difcords, by preparation, is 
 founded on a natural principle, which we have formerly'had oc- 
 cafion to mention. Art. 137. viz. that when we have pre- 
 vioufly fet our attention upon any particular tone, we are rea- 
 dy to imagine we hear the fame tone exifting along with the 
 other accompanyments of the next following chord, if the 
 difcords thereby introduced be not too harfh : for inffance, an 
 added feventh along with a third and fifth ; or a ninth fufpend- 
 ing the octave, along with the fame third and fifth ; or a fourth, 
 fufpending the third, along with the octave and fifth : and, in 
 general, all fufpending notes, fuch as are enumerated in 
 Art. 188. ought to be prepared by exifting in the harmony of 
 the preceding chord. 
 
 255. The fundamental bafs is fometimes made to pafs by a 
 leap of a third defcending from one chord of the feventh to an- 
 
 to it, though it cannot naturally be a fundamental in the fcale other, and fometimes by a gradual afcent, fuch as the falfe ca- 
 of C, neither can the leap of a falfe fifth, which is between F dence or gradation, each chord bearing a ieventh ; but thefe 
 and B, properly take place in the fundamental bafs, Art. 230. paffages are to be efteemed licences, and to be ufed with cau- 
 To reconcile this pafitge to the rules formerly given, we are tion and fkill f . 
 
 f The modern French theorifts have been at fome pains to eftablifh another fundamental fequence, wherein the bafs proceeds by leaps of a fifth afcend- 
 jng, or fourth defcending, in imitation of the irregular cadence ; and each note is to be taken as an adjunct 4th, and to hive an added fixth : but if, ac- 
 cording to the 3d part of Art 144.. it be neceffary to introduce the adjunct jth, in order to determine the key, and if it be necefTary that every new key 
 which we admit be fufficiently determined, then we may prefume, that fuch a fequence as is defcribed above, where nothing but adjunct 4ths appear, 
 muft not enter into the lift of fundamental progreffions ; and this preemption is ftrongly fupported, by obferving, that among all the variety with which 
 the moft judicious modern compoier* ftudy to enrich their works, no one inftance of fuch a fequence can be found. 
 
 Z 
 
9° 
 
 Of the Flat Series. 
 
 Part I. 
 
 256. There is a Aill greater fundamental licence frequently introduced by the Italians, and adopted by other nations in ittri- 
 
 to be met with in modern mufic, where the feries is fuddenly ration of them: and will be better underflood when the flat fe-. 
 
 changed from fharp to fiat, and contrariwife, upon the fame ries has been more fully difcuffed in the next chapter. 
 key note, in the middle of a piece. This was probably firfr 
 
 C 11 a p. IX. Of the Flat Series. 
 
 Article 257.XTTAVING occafionally made mention of fe- 
 Xj|_ veral of the moff. important particulars re- 
 lative to the flat feries, in the preceding parts of this treatife, 
 we (hall not need to be fo prolix in treating exprefly of it in 
 this chapter ; but fhall rather briefly recapitulate and lay to- 
 gether the remarks formerly made, with the addition of fuch 
 further obfervations as feem neceffary or ufeful. 
 
 In the note to Art. 2gth, the flat feries is faid to be ' made 
 ' up by piece-meals out of two natural fcales, fo fit u a ted that 
 * the 5th of one fcale is the key of the other,' <bc. And having 
 fhewn at large, in the laft chapter, how nearly related the fcale of 
 the ad j unci: 5th is to that of the principal key, we may now fafely 
 fuppofe an intermixture of thefe two, in order to form a me- 
 lody which regards the 6th of the fcale as principal key, rather 
 than either of the other keys. 
 
 258. The intervals of the flat feries, when reckoned from 
 its key, differ from thofe of the fharp feries reckoned from its 
 key, in the 3d. 6th, and 7th degrees ; for the flat feries has 
 a lefs 3d, a lefs 6th, and a lefs 7th : and thefe intervals are 
 found in the natural fcale, by beginning from the 6th as a key; 
 for the K of the fcale becomes a lefs 3d, the fourth of the fcale 
 becomes a lefs 6th, and the fifth of the fcale becomes a lefs 7th : 
 
 and therefore, after obferving by the marks at the cliff what is 
 the key of the natural fcale, as directed, Art. 38. <b feq. if we 
 find a piece of a mufic, not ending upon this key, but two de- 
 grees lower, viz. upon the 6th of the fcale, we may conclude 
 fuch a piece to be in the fiat feries of that 6th as its key note : 
 and contrariwife if the natural fcale of any note be propofed, 
 and if it be required to find where the flat feries will go off 
 without altering any of the notes, we muft pitch upon the 
 6th of the fcale for that purpofe. For example, if a piece of 
 mufic have B(j,at the cliff, and yet do not end on F, but on D, 
 it is in the flax feries of D : and if it be required to fet the flat 
 feries with two fharps, \yiz, F and C, which make the natu- 
 ral fcale of D] we muff take B, which is the 6th above, or the 
 3d below D, for a key ; and the fame in all other cafes. 
 
 259. To fet a tune of a flat feries upon any degree of the 
 femitonic fyflem formerly defcribed, Art. 40. is feq. as its key; 
 we muff fo order 'the flats or fharps which are put at the cliff, 
 as to bring in the natural fiale of that degree which makes the 
 lefs third to our key note ; and, by this means, the general 
 fcheme, in Art. 48 will be made to ferve for the flat feries 
 alfo, if we firfr look for the degree intended to be made the 
 key in the general fyftem, then count three femitones forward, 
 
Chap. IX. 
 
 Of finding the Key Note, by the Marks at the Cliff*, and vice veria. $\ 
 
 which brings us to the lefs 3d, and then guiding the eye 
 downward till we come to the line in which the fame lefs 3d 
 ffands as a key, we (hall find on the right of this line the /harps 
 or fiats neceftary to be placed at the cliff. For example, to find 
 the fiat feries of A, we look, firft for A in the general fyftem, 
 then go forward three femi tones to C, and then looking down- 
 ward we find C, the firft of the natural fcale, (i. e. the key,) 
 in the very next line, and the words all natural on the right, 
 from which we conclude, that the flat feries of 'A muft alfo be 
 all natural rotes. ■ Thus alfo to find the flat feries of D, we 
 muft firft go forward to F, and then downward till we find 
 F, the firft or laft of the natural fcale, viz. in the loweft line 
 but one ; and the words B fat, on the right, are our directi- 
 on for fetting the flat feries of D. Again, to find the flat fe- 
 ries of G, we go forward to the ftar between A and B, and 
 then downward in the column under this ftar we find B jj, the 
 firft and laft of the natural fcale, in the third line from the 
 bottom, and the direction B and Ejlat, on the right. If it 
 were required to fet a flat feries upon thefemitone (or ftar) be- 
 tween A and B (which yet we know not whether to call A % or 
 Bj?,) as a key, we muft look three femitones forward to the ftar 
 between C and D, and then downward in the column under 
 this ftar we find, in the 3th fcale, C % with allfiarp on the 
 right, or D |j with BEAD and G flat on the right ; either of 
 which will anfwer our purpofe, only if we make ufe of the 
 (harps, our key note will be A % ; and if we make ufe of the 
 flats, which are preferable, our key note will be B |j. 
 
 260. After a little practice in mufic, the proper manner of 
 fetting either the (harp or flat feries of any of the more accu- 
 ftomed keys will become familiar and obvious, without con- 
 fulting any fcheme or table for that purpofe : and it may be of 
 ufe to obferve, by the way, that the taking away of three 
 fharps, or the adding of three flats ; or, which is the fame in 
 effeci, the taking away two fharps, and adding one flat ; or 
 
 taking away one (harp and adding two fiats, will reduce the 
 (harp feries of any key to the flat feries of the fame key ; for 
 this will always be found to bring down the 3d, 6th, and 7th 
 degrees of the fcale a femitone lower. For inflance, the (harp 
 feries of E takes four (harps [F, C, G, and D] ; and the flat 
 feries of F takes but one (harp [FJ. The (harp feries of G 
 takes one (harp [F] ; and the flat feries of G takes two flats 
 [B and E j . The (harp feries of F takes one flat [ B] ; and the 
 flat feries of F takes four flats [B, E, A, and DJ. And fo in 
 all other cafes. — And this remark is of more ufe fince the li- 
 cence mentioned in Art. 256. of palling from the (harp to the 
 flat feries, 6 1 vice verfa, upon the fame key note, in the mid- 
 dle of a piece of mufic, is become fo common ; which paflages, 
 though in themfelves unnatural, we are obliged to familiarize, 
 and learn to relifh, before we can enter into the fpirit of the 
 modern mufic. 
 
 261. Thus far the flat feries feems to be made up entirely 
 out of the natural fcale of that degree which makes the lefs 
 third of the feries, and to which, of courfe, its key bears the 
 relation of a fubftituted 6th; but as the flat feries has been 
 called an intermixture of the natural fcale of the key and ad- 
 junct. 5th, it may be proper, in this place, to take notice, that 
 this intermixture cannot be fully explained till we come to treat 
 of occafional temperament. Though the flat feries does not 
 ejjentially take in the $4th, which belongs to the fcale of the 
 adjunct 5th ; yet this $th accidentally takes place as a 6th in 
 the flat feries, much in the fame manner as the $ 5th occafio- 
 nally makes the leading note, or 7th, of that feries, Art. 138. 
 but the flat feries effentially takes in a part of the tempered 
 notes, called grave and acute, Art. 23. 24. of both fcales. 
 
 262. By admitting this intermixture of two fcales, and cal- 
 ling the key of the flat feries a found, partly fixth and partly 
 
 fecand,v!c reconcile the prefent method of fetting this feries with 
 that which was in ufe till within the laft 50 or 60 years j and 
 
5)2 Of the Flat Series. 
 
 which is ftill retained in the lateft editions of tbe works of Co- 
 relli, Rameau, he. Thefe authors kt the flat feries of Din 
 the natural fcale of C, and put an occafional flat to the note B, 
 which is the fixth of the fcale, in the courfe of the piece, 
 whenever a flat fixth is required. After the fame manner the 
 flat feries of G is fet with one flat only at the cliff, which makes 
 the natural fcale of F ; and fo of all others wherein we put a 
 \) at the cliff, upon the fixth of the feries, they omit this charac- 
 ter at the cliff, and place it occafionaliy among the notes; which 
 is a proof that they confidered the 6th of the feries as an ambi- 
 guous note, or, which is the fame thing, they confidered the 
 key of the feries. as the 2d of the natural fcale in thefe pieces. 
 But they fet the flat feries of A, of E, and of all other keys 
 which require lharps at the cliff, exactly as we do. 
 
 263. The flat feries ought to be pracYifed in the fame man- 
 ner, and with the fame view, as was directed in regard to the 
 natural fcale, in Chap. I. and II. taking along with us, a few 
 cautions, which follow. 
 
 (1.) The flat feries mufl be fung by gradual fteps defend- 
 ing, as reprefented, Plate VIII Exam. LIU. rather than a- 
 fcending: for it will be eafily perceived, that in afcending, 
 when we are come to the 6th note, which isalefs 6th, we can 
 not naturally afcend from it to the (harp 7th, becaule this is 
 an interval of a t rih emit one -, and if we take the natural (or lefs) 
 7th, and fo proceed to the key, the cadence is not fatisfactory 
 nor conclufive, for want of the leading % 7th : befides, the 
 lefs 6th is a heavy found (correfpondenc to the grave fourth of 
 the natural fcale) and on that account we defire to hear it de- 
 fcend into the fitth; therefore, in compliance with this defire, 
 and at the fame time to biing in the cadence on the key which 
 is expected at the eighth note, we naturally repeat the 5th in 
 place of the % 7th ; as (hewn, Exam. LIV. 
 
 (2.) The only practicable method of making a gradual a- 
 fteat in the fiat feries, fo as to clofe upon thekey note, is to take 
 
 Part I. 
 
 the % 6th and % 7th, as reprefented in' the treble part of Ex. LV; 
 and by this means the upper half of this feries becomes exact- 
 ly the fame with the tetrachords of the natural fcale, defcribed, 
 Art. 1 2.; and the only degree in which this afcending feries 
 differs from the natural fcale is the lefs 3d : fo that good care 
 muft be taken that this lefs 3d be properly founded, or other- 
 wife we fhall pafs altogether into the fharp feries. 
 
 264. From hence it appears, that although the lefs 6th and 
 lefs 7th do mod properly belong to the flat feries, yet the 
 greatefl 6th and % 7th are not to be excluded on many occafi- 
 ons ; and therefore the 6th and 7th may be juftly efieemed 
 variable founds, in this feries; the greater taking place in a- 
 fcending to the key, and the lefs in defcending from it. 
 
 265. For ascertaining the fundamental progreffion, in mu- 
 fie of a flat feries, it will be proper to look back to Art. 150. 
 for the definitions of fharp and fiat harmony : and to Art. 238 
 and 239, where the predominance of the 6th of the fcale as a 
 principal key, and the affinity between a key and its two ad- 
 juncts is explained. And, to apply thefe particulars to the 
 prefent purpofe, we may obferve further, that it is neceffary 
 not only that this 6th, which is the principal key of a flat fe- 
 ries be plainly indicated at the beginning, but alfo, that the 
 rememberanceof it be kept up by refering the following founds 
 to the chords of this key, and of the 2d/". and i&f. which are 
 its adjuncts, as often as convenient; and by frequently intro- 
 ducing the % 5th, which is the fharp feventh, or leading note 
 of this key, in the courfe of the piece : for otherwife the real 
 key of the natural fcale, and its adjuncts, will attract too much 
 of our attention ; and the expectation of recurring upon the 
 6th at the conclufion will ceafe, and the connexion of the 
 piece, Art. 31. will be defiroyed. 
 
 266. The mufic of the flat feries, having lefs of nature, re- 
 quires more of art, to render it fully fatisfactory. Yet we arc 
 not neceffuated to dwell conflantly upon the flat harmony of 
 
Chap. IX. 
 
 Of Fundamental Progrejjions, and of the Peculiarities in flat Harmony. 
 
 n 
 
 the key and its adjuncts, but may occafionally admit the (harp 
 harmony of the real key of the natural fcale and its adjuncts, 
 in the middle of a piece, provided care be taken to return to 
 the original key in due time. 
 
 267. Thefe particulars are briefly exemplified in the funda- 
 mental bafs, to Exam. LV. where the afcending and defcend 
 ing flat feries are refered to the very fame fundamentals in re- 
 lation to this key, as were affigned to the natural fcale in rela- 
 tion to its key, in Art. 216. and Exam. XLII. and thefe fun- 
 damentals bear the fame fpecies of chords refpectively , only 
 the accidental fharp 7th, and natural 6th of this feries are o- 
 bliged to be (hewn in the figuring, when they take place as 
 thirds in the chords of the adjunct 5th and 4th of this key. 
 And it may be obferved, that we give the adjunct 5th its fharp 
 third in all occurrences, except at I, where the treble takes the 
 lefs third of this fundamental, in order to defcend. The ad- 
 junct 4th has a greater third at H, becaufe the treble is oblig- 
 ed to take the fame greater third, in order to afcend; but o- 
 therwife the adjunct 4th ought to bear a lefs third as its pro- 
 per chord. The chord at K has a falfe fifth, which is allow- 
 ed, becaufe the treble moft naturally takes that fame note, in 
 defcending 
 
 268. The defcending feries at the end of this example is 
 refered to a different fundamental progreffion, by admitting 
 the fharp harmony at L, and returning to the original key, by 
 the falfe cadence atM : which latter progreffion is much more 
 natural than the former, but then it is not properly flat harmo- 
 ny, except the lirft chord and the final cadence. 
 
 260. It is neceffary to obferve further, that the fundamen- 
 tal paffages which appear fimilar, have yet a very different ef- 
 fect in fharp and flat harmony ; particularly the added feventh 
 in the leading chord of flat harmony is lefs natural, arifing in 
 a great meafure from the neceffity of its falling by a ftep of a 
 whole tone for refolution : which to avoid it was cuftomary, 
 
 in former times, to fall only a femitone, and by that means 
 introduce the % 3d to the final chord : but then this % 3d, fo 
 far as it is attended to, appears foreign to the mufic; and is 
 eafiefl: practifed, on the organ or harpfichord, where it may 
 pafs unregarded as one of the natural confonances of the final 
 key, Art. 1 37. We have not the fame opportunity when voic- 
 es or fmgle inflruments join together in concert, becaufe, tho' 
 it might pafs unregarded by a hearer, who attended only to 
 the united effect of the whole, yet it would be offenfive in 
 the higheft degree, to the particular performer whofe part it 
 might be to introduce it. It is almoft indifpenfibly neceffary to 
 omit the feventh in the leading chord of the final cadence in 
 flat harmony, in compofing vocal mufic, becaufe the refoluti- 
 on of that feventh, either upon the $ third, or upon the h third 
 of the key is an unfatisfaftory ending to the finger who per- 
 forms it. The third of a (? P chord ought to be a difregarded, 
 loft found, Art. 141. 
 
 270. The adjunct 4th/, when followed by the 5th/, which 
 is the gradation, in flat harmony, admits of a different accom* 
 panyment ; for that 4th/ may take a feventh for its added dif- 
 fonance, if the courfe of the upper parts require it, and this 
 feventh is refolved by falling a femitone, and becoming a fifth 
 in the following chord ; as fhewn at N in Exam. LV. where 
 the directs in the upper ftafF fhew the progrefs of that part 
 which makes and refolves the feventh as above. 
 
 271. The difallowance of a gradual defcent in the funda- 
 mental bafs, from the K or from the 5th, which takes place in 
 fharp harmony, as fhewn, Art. 229. does not extend to flat 
 harmony in reference to its key : for here the key, being the 
 6th of the natural fcale, we may make a defcent from the 6th 
 on the 5th, as obferved in the beginning of the fame article: 
 and the adjunct 5th of flat harmony, being the 3d of the na- 
 tural fcale, we may make a defcent from it upon the 2d of the 
 fame fcale, which is the adjunct 4th, as fhewn, Art. 230. and 
 
 a 
 
94 Of the Flat Series 
 
 at the flat, in Fx.-XLVI. The other peculiarities of flat har- 
 mony will be learned by practice. 
 
 27 2. The fucceffion of occasional new keys in mufic of the 
 flat feries is -alfo eafily accounted for, by obferving that the 
 very fame fyftem oi modulation, defcribed, Art. 238. takes 
 place in all mufic; and confidering, as we have all along done, 
 that the flat feries has the 6th of the natural fcale for its prin- 
 cipal key note ; from whence it will follow, that the adjuncts 
 of this key may become new keys fubordinate to the principal; 
 and that, of the two, the adjunct 4th, which is the 2d <jf the 
 natural fcale, iseafrer brought in, than the adjunct 5th, which 
 is the 3d of the fcale ; according to the remark at the end of 
 Art. 242. When the key of the natural fcale is allowed to 
 take place, the manner of proceeding has been already largely 
 explained. 
 
 273. Here it will be very proper for the reader to examine 
 other fundamental progreffions, by the rules delivered above; 
 and of this fort the firit five leflbns in Pafqitatis, Thorough bafs 
 are very good : thefe, being examples of different fucceffions 
 of common chords, are undoubtedly fundamental progreffions, 
 only the difcords are left out, becaufe the learner is fuppofed 
 able to flrike nothing but perfect or common chords. Thefe 
 leflbns, particularly the ift, and 5th, contain more variety of 
 modulation than is to be met with in any other pieces of the 
 fame length; becaufe the com pofer's chief aim was to intro- 
 duce as great a variety of paffages in as fmall a compafs as he 
 could, confident with the nature of harmony, and the rules 
 he propoied to exemplify. 
 
 274. The Exam. LVF. is Pafquali's firft leffon, or general 
 prelude, with marks fubjoiued, for a fpecimen of the way of 
 examining here prefcribed. Thefe marks fhew the keys which 
 take place, and the relations of the other notes to the keys to 
 which they belong. The principal key is diflinguifhed by a ca- 
 pital, and the fubordinate keys by fmall letters, and the changes, 
 
 Part I. 
 
 or paffages from one key to another, are (hewn by the paren- 
 thelis ) , in the the fame manner as at the foot of Exam. LI. 
 In this leffon all the different keys are introduced which are 
 practicable in the fyflem of modulation of F, (the key Of thfe 
 natural fcale) except C which is the 5th of the natural fcale, 
 and the 7th of the flat feries ; and it is plain, that this might 
 eafily have been brought in, but that it would have required 
 the ufe of that pofition which Pafquali calls the fecond chord of 
 G ; and this is not intended to be praclifed till the learner 
 come to the 3d and 4th leflbns of that well digefted and ufe- 
 ful fyftem of inflructions. 
 
 275. The frequent occurrence of the adjunct 5ths is re- 
 markable both in this and all other pieces of harmony, and it 
 may be obferved, that in every key the adjunct 5th appears 
 once at leaft, agreeable to the 3d part of Art. 244. and the 
 paffages, while the fame key is retained, are all of the eafieft 
 kind, viz. regular, or irregular, or falfe cadences, or gradati- 
 ons, agreeable to Art. 213 ; and the other paffages, in Which 
 the key changes are all agreeable to the rules in the preceding 
 chapter. 
 
 276. The fecond leffon in Pafquali's Thorough-bafs is in 
 the fharp harmony of G, as principal key; and the gradation 
 from the 4th/ to the 5th/, viz from C to D, is introduced 
 as often as poffible, becaufe this is the paffage propofed to be 
 exemplified — Note, that in the 8th bar of this leffon C is in- 
 troduced following D, which, according to Art. 229. is riot 
 to be allowed in the middle of one connetled jlrain : and, to re- 
 concile this paffage, it may be obferved, that the ftrain is 
 brought to a conclufion upon D, the former of thefe notes ; 
 and that altho' the introduction of the chord of C, at the next 
 note, makes a fenfible breach in the harmony, yet it is tolerat- 
 ed as being a return into the original key, agreeable to the laft 
 paragraph of Art 245, 
 
 The 3d, 4th, and 5th leflbns are perfectly conformable to 
 our rules, and therefore need no annotation. 
 
Plate vur 
 
 ExLE. 
 
 Fiat S'^tce^i OTLl 
 
 
 $ J&xLEr. s 
 
 /*"***%?/& jM-i 
 
 xl=3 
 
 ^^m 
 
 IE 
 
 Eocirv: 
 
 g^ s^^ ^j# M ^ yp 
 
 ExiiVJ m ^^^aA^yti^ -4&&&* 
 
 I K. 
 
 ^^^a 
 
 ^SB 
 
 
 mm 
 
 q_jl 
 
 S 
 
 
 Isf IN" 
 
 SP§I 
 
 F& 
 
 e 
 
 ■45 3c) k 5 k 4 &k 4 ,5 k) i B5T4 5K5 K 4 5 
 
 f 
 
 U 
 
 
 £ 
 
 *s 
 
 g ^Wriit g 
 
 I s 
 
 a 
 
 PP 
 
 * 
 
 k:) 5 Jc) 5 k) 5 k) 4" 3 K 5 E ) 5 k 5 i) 5 k I k) 4 5 K 
 
 -=F 
 
 4- 3 
 
 
 
[ 95 ] 
 
 C h a P. X. 0/ Chromatic. 
 
 Article 277.T T is a fa£t fufficiently proved and a'cknowledg- 
 X ed, thai; all melody is founded on harmony, and 
 that every finger conforms his founds to certain fundamental 
 progreffions, whether he be acquainted with the principles of 
 mufic or not, if he have a natural talent for it. It is as natu- 
 ral to conform our founds to certain fundamental progreffions, 
 in finging. as to conform our attitudes to certain laws of gra- 
 vity and motion , in walking. 
 
 If it be alked, why we do not moft naturally fing the fun- 
 damental bafs itfelf, our anfwer is, That the diftant leaps by 
 which this part ordinarily proceeds are not fo fuitable for the 
 voice, as the gradual fteps of the fcale which conftitute the main 
 part of our melody. The voice moves by fmall intervals, but 
 the fucceffive governing fundamentals are farther diitant one 
 from another. 
 
 278. This preference, which the very nature of the voice 
 obliges us to give to fmall intervals, accounts for a peculiar 
 fucceffion frequently introduced in modern melody, which a.- 
 fcends or defcends by feveral fteps of a femitone eachy and is 
 called Chromatic Mufic *. We ihall examine, in a few words, 
 the nature and origin of this fpecies of melody. 
 
 2^9. The fundamental progreffion of all melody, however 
 diverfified, muft ftill be agreeable to the rules formerly laid 
 down ; and, in general, the melody which is eafieft to be fung 
 ©r reliflied is founded on the moft natural fundamental pro- 
 
 greffion : and, conformable to this principle, we find fome paf- 
 fages of the chromatic kind fufficiently eafy and agreeable, while 
 others are extremely difficult. For example, a fucceffion of 
 three contiguous femitones afcending, from the 3d to the 5th, 
 may eafily be produced by two regular cadences in (harp har- 
 mony, as reprefented below ; where the fucceffive founds of the 
 bafs, and upper part, are denoted by the degrees of the fcale, 
 placed one after another, and the figuring of the chords of the 
 bafs is put over fuch degrees as require it. 
 
 r Upper part 3. 4. $4. 5. 
 
 £ Fundamental bafs K. 4. 2. 5. 
 See alfo Plate IX. Exam. LVil. where the fame paffage is 
 fet in notes, in the natural fcale of C. 
 
 280. A fucceffion of five contiguous femitones, afcending 
 from the 7th to the 3d, may alfo be produced by three regu- 
 lar cadences in ffiarp harmony, but with greater difficulty ; be- 
 caufe the cadences on which it it is founded, are more foreign 
 to the principal key. 
 
 r Upper part 7. K. $gk. 2. $2. 3. 
 
 ] *■.•:.,*.:.::.' 
 
 ( Fundamental bafs 5. K. 6. 2. 7. 3. 
 See Exam LVUI. 
 
 281. Thefe two fucceffions may be joined together by re- 
 peating the 3d of the fcale in the upper part, or by changing 
 
 * The Greeks, from whom we bortow this name, reckoned three kinds of Mufic, viz. Diatonic, Chromatic, and Enharmonic. The Diatonic proceed- 
 ed by the fteps of the natural fcale; the Chromatic, as above defcribed; and the Enharmonic, by ftill fmaller intervals, nearly fimilar to the occafional 
 temperament, of which we lhall treat hereafter. 
 
9 6 
 
 Of Chromatic 
 but not other 
 
 Part I. 
 
 the fundamental in the middle of that note 
 wife. 
 
 r Upper part 7. K. $k. 2. %2. 3. 3. 4. $4. 5 
 
 K. 4. 
 
 £ Fund, bafs 5. K. 6. 2. 7. 3. 
 
 Here the quick tranfition from the cadence on the 3d to that 
 on the 4th, which happens at the repetition of the 3d in the 
 upper part, is very difficult, (fee the lafl part of Art. 245;) 
 and therefore a fucceffion of chromatic, afcending all along 
 from the 7th to the 5th, is rarely to be met with in fharp har- 
 mony. 
 
 282. The fame paffages fucceed much better in flat harmo- 
 ny ; where the 2d and 3d of the natural fcale are more nearly 
 related as adjuncts to the principal key; and where the 4th 
 of the natural fcale, immediately following the 3d, has the 
 appearance of a falfe cadence : where alfo the fharp 5th fol- 
 lowed by the 6th naturally comes in at the clofe of the fuc- 
 ceffion, and fo brings the harmony to a conclufion on this 6th 
 as principal key. Thefe degrees, counted from the key of the 
 flat feries, ftand thus ; 
 r Upper part 2. 3. ^3. 4. $4. 5. 5. 
 
 c 
 
 6. ^6. 7. 
 
 Fund, bafs 5 
 See Exam. LIX. 
 
 K. k. 
 
 in the I 
 
 6. 
 
 4. 7. 
 
 K. 
 
 4- 2. 5. 3. 
 t feries of A. 
 
 283. A fucceffion of three femitones, defcending from the 
 key to the 6th, may be produced in the fharp harmony, by an 
 irregular cadence on the 5th, followed by a regular cadence on 
 the 4th. And three femitones, defcending from the 5th to the 
 3d, may be produced by an imitation of the fundamental fe- 
 quence of fevenths. 
 
 r Upper part K. 7. {? 7- 6. 5. ^4. 4. 3. 
 < \)7 7 % 7 
 
 £ Fund, bafs K. 5. k. 4. 6. 2. 5. K. 
 See Exam. LX. in the natural fcale of C. 
 
 284. It is not eafy to purfue either of thefe fucceffions any 
 further in fharp harmony, without admitting the fud- 
 den change from the fharp to the flat feries, mentioned, 
 Art. 256. ; but in flat harmony the defcending chroma- 
 tic may be carried a greater length, for the fame reafon which 
 was afflgned in regard to the afcending chromatic, in Art 28 1. 
 viz. the affinity between the principal key and the degrees on 
 which the cadences fall. 
 
 The flat feries admits of two defcending fucceffions of five 
 femitones each, as here reprefented ; counting the degrees from 
 the key of the feries. 
 
 Upper part K. K. $7. 7. %6. 6. 5. 5. 
 
 7 
 
 3- 2. K. 
 
 2. 5. 
 
 K 
 
 7 * 
 4-5- 
 
 K. 
 
 CFund.bafs K. 2. 5 k. 4. 7.3. ( 
 See Exam. LXI. in the flat feries of A. 
 
 285. Thefe fundamental fequences, which produce a de- 
 fcending chromatic melody, differ from the fequence of fevenths, 
 defcribed Art. 249. <&feq. in the fpecies of each alternate 
 chord, which here has the third accidentally fharpened, and 
 the feventh omitted. The fharp thirds are necefTary in con- 
 formity to the melody, and the fevenths muff, be omitted at 
 thofe occurrences, becaufe, if a feventh were added, the fharp 
 third would be neceffitated to rife, as obferved in the fecond 
 paragraph of Art. 252. whereas the fucceffion of the melody 
 requires that the fame note fall a femitone. 
 
 286. The difficulty of finging chromatic mulic is not owing 
 to the fmallnefs of the intervals, as is generally fuppofed ; for 
 on this account it ought rather to be eafier : but it is owing, 
 in a great meofure, to the quick, and frequently/a/W transi- 
 tions from one key to another, fo that we are apt to be at a 
 lofs for a proper fundamental progreffion, without which we can 
 neither reiifh this nor any other melody. Add to this, that 
 feveral of the altered notes are not allowed to take their natu- 
 
Chap. XL 
 
 ral courfe, according to Art. 23 
 
 culty, in defcending chromatic, the fevenths which occur are 
 
 not prepared as they ought to be, by Art. 250, and 254. 
 
 287. The fame chromatic pafTages may fometimes be refer- 
 ed to different fundamental progreffions, which the learner 
 will obferve m the courfe of practice. Thofe we have affign- 
 
 Of the Fundamentals, and the Peculiarities of Chromatic, 9 7 
 
 0. And to increafe the diffi ed above are the mod natural, and will give a fufficient idea of 
 
 the manner of proceeding in other cafes. 
 
 288. The gradual and often unexpected elevations of the 
 voice, in afcending chromatic, render it remarkably expreffive 
 of reviving hope ; and the contrary depreffions of defcending 
 chromatic are as aptly fuited to languishing and dejection. 
 
 Chap. XI. Of Plain Descant. 
 
 Article 289. 'TT* HE art of compofing, or fetting mvfic in fe- 
 jL veral parts, is. called Descant ; and this 
 is diftinguifhed into plain and figurative. Plain defcant is that 
 wherein the correfponding notes in all the parts are of equal 
 value, and is fometimes called counterpoint, becaufe this kind 
 of compofition was formerly denoted by points, difpofed in 
 columns, much like our manner of reprefenting the chords of 
 the thorough bafs. Figurative or florid defcant is that where- 
 in the notes of one or more of the parts are frequently fubdi- 
 vided and broken into feveral fhorter ones, according to the 
 fancy or tafte of the compofer, but under certain reftrictions. 
 It is neceffary firft to lay down the rules of plain defcant, the 
 other being juft the fame, with additional ornaments of me- 
 lody. 
 
 290. To handle this fubject in the moft intelligible and ufe- 
 ful way, we (hall, for the prefent, fuppofe, that a certain Am- 
 ple or plain melody is given, which may be called the princi- 
 pal or firft treble ; and we (hall lay down the moft material 
 rules for compofing a thorough-bafs, a fecond treble, and a 
 tenor or inner part to this given melody, in plain defcant. 
 
 291. The firft thing to be done, is toafcertain the proper 
 fundamental progreffion of the given melody ; which will not 
 be difficult, if the rules and obfervations in the three preced- 
 ing chapters be duly attended to ; as we have all along fup- 
 pofed the fundamental progreffions fuggejled by, or fome way 
 connected with, the paffages of the melody. See Art. 199. 
 
 292. A particular regard muft be had to the expectation of 
 cadences, as defcribed, Art. 216, 217, and 218. which de- 
 pends upon the natural divifion of all mufic into certain phraf- 
 esorftrains, Art. 103. and upon the refls or paufes which 
 the fenfe may require, when mufic is adapted to words: and. 
 here the expectation muft not be difappointed without a parti- 
 cular view to fome intended expieffion. For inflance, the ir- 
 regular cadence on the key is peculiarly emphatical in church, 
 mufic, in the Amen, at the conclufion of a petition ; for which 
 reafon fome chorifters call it the Amen cadence. The regular 
 cadence difappointed by a falfe cadence, as defcribed, Art. 209. 
 may be very properly ufed when a queftion isafked, the move- 
 ment of the bafs in a falfe cadence having fo near arefemblance 
 to the natural inflection of the voice in an interrogation. This 
 b 
 
Of Plain "Defiant. 
 
 Part I. 
 
 is well exemplified by Dr. Arne, in the beginning of the fong, 
 " Gentle Touth, ah! tell me why," &c. 
 
 293. In affigning the fundamentals of a given melody, tbofe 
 movements which we have formerly called neceffary, in the ca- 
 dences, muft be carefully obferved ; becaufe, if the given me- 
 lody take the former note of any of thefe reftricted paffages, 
 and not the latter, or if it take the latter note which fhould 
 refolve the difcord preceded by another note, and not the dif- 
 cord itfelf, the fundamental progreffion muft either be diffe- 
 rently affigned, or the difcord whofe refolution would occasi- 
 on the doubling of the refolving note, muft be left out of the 
 leading chord : and fuch occafional omiffions ought to be 
 fhewn in the figuring of the fundamental ; as explained, Art. 
 219. 
 
 The expreffion may fometimes be heightened in the mid- 
 dle of a phrafe, by a judicious choice of fundamental 
 chords. For inftance, thofe chords which include occafional 
 {harp notes, may be ufed to exprefs joy and elevation ; and, 
 on the contrary, thofe which include occafional fiat notes may 
 exprefs forrow and depreflion : a difibnant chord may exprefs 
 doubtfulnefs and perplexity; an unexpected chord, fur- 
 prize, etc. and it is very proper to have thefe particulars in 
 view, becaufe we fometimes have it in our option to chufe dif- 
 ferent fundamentals, though, for the moft part their progref- 
 fion is limited by the given melody ; and we often have liber- 
 ty either to admit, or not admit, an altered note, or an added 
 difcord. 
 
 294. Though the thorough bafs is the loweft part heard in 
 a concert, yet it is ftill to be conlldered as an upper part in 
 refpect to the fundamental ; and as fuch may take any of the 
 notes of the fucceffive chords, for the fake of an eafy~fucceffi- 
 011, only taking care that the following reftr'nftions be obferv- 
 ed, which are deduced from the principles already delivered. 
 (1.) The thorough-bafs muft coincide with the fundamen- 
 
 tal at the final cadence, and alfo at the cadences which con- 
 clude every principal fubdivifion or ftrain. 
 
 (2) In other places where the thorough-bafs differs from 
 the fundamental, the inverted chords thereby introduced muft 
 be fhewn by proper figuring, according to Art. 188 ; and the 
 paffages muft be ordered as follows. 
 
 (3.) A bafs note, bearing a chord of the fecond muft pafs in- 
 to the next lower degree, bearing either a chord of the fixth, 
 or of the fixth and fifth. 
 
 The former of thefe is the difcord 7th refolved in a re- 
 gular cadence ; and the latter is the fame ?th refolved 
 in a fequence of fevenths ; as at H and L, in Ex. LXII. 
 in the fharp feries of C. 
 (4.) A bafs note, bearing a chord of the fixth fourth and 
 third, muft pafs into the next lower degree, bearing either a 
 perfect chord, or a chord of the feventh. 
 
 The former of thefe is the limited courfe of the 5th in 
 the leading chord of a regular cadence ; the latter is 
 the fame in a fequence of fevenths. The fourth is ge- 
 nerally left out in this chord, and it is then figured 
 with 6 only, and paffes as a I or at N, in Ex. LXII. 
 (5.) A bafs note, bearing a chord of the fixth and fifth, 
 may pafs into the next higher degree, bearing either a perfect 
 chord, or a chord of the fixth and fourth, derived from the 
 fifth of the former bafs note ; or it may pafs into the fame fifth, 
 bearing a perfect chord ; or it may remain on the fame degree, 
 and take a chord of the fecond. 
 
 The firft of thefe is the courfe of the third of the leading 
 chord, in a regular cadence ; the next is the irregular- 
 cadence; and thejaft is the preparation of a difcord 
 7th, in the fequence of fevenths. See M, P, O, and 
 K, in Ex. LXII. 
 (6.) A bafs note, bearing a chord of the fixth and fourth, 
 may either remain on the fame degree till the fixth and fourth 
 
Plate JE, 
 
 
 JExLJZ. 
 
 S'ar^nfl*. 8« 
 
 ^IfTnffga i 
 
 
 Ji 
 
 ? d? 
 
 
 
 4 2 
 
 ■*4s 
 
 #7(?* 
 
 J*'tL9lti- 
 
 jb. y^-4 
 
 @ 
 
 cr 
 
 -P- 
 
 ^^^^^ ^B ^ ^^ ^^ 
 
Chap. XI. Rules to be obferved between the thorough-Bafs and the Fundamental. 
 
 have defcended, and become fifth and third ; or it may defcend 
 
 at the fame time into the next lower degree, bearing a chord 
 
 of the fecond. 
 
 This reftriflion is owing to the propenfity of the mind 
 to conceive the loweft found we hear to be the fun- 
 damental, if the progreffion will admit of it, and if 
 the accompanyments can be reconciled to it as fuch. 
 In the prefent cafe, the 6th and 4th are eafily taken as 
 fufpending the 5th and 3d ; and hence arifes the ne- 
 ceffity of their defcending as defcribed above. See S 
 and Q^in Exam. LXII. The directs in the funda- 
 mental bafs fhew the enterance of the fundamental 
 notes, by way of anticipation. 
 
 99 
 
 greater feventh fourth and fecond to the key, and this is a chord 
 
 not formerly fpoken of in this treailfe. See the end of Art. 
 
 188. Thefe pafTages are (hewn at H, I, and K. in Ex. LXIII. 
 
 in the flat feries of E. 
 
 In this, and all other pafTages, where two or more chords 
 are to be flruck to one -continued note, the feveral chords 
 are marked over the note ; and to know the precife time 
 when each chord is to be ftruck, it mult be obferved, that 
 when the note naturally divides into as many equal parts as 
 there are chords figured over it, they muff be flruck at each di- 
 vifion : but if only two chords be marked over a pointed note, 
 which naturally divides into three parts ; or if three chords be 
 marked over a plain note, which naturally divides into four 
 
 (7.) The movement of the thorough-bafs muff, not imitate parts, the firft chord muff, be continued on the twofirft parts ; 
 
 any of the more familiar fundamental pafTages, fuch as the 
 rifing or falling of a fourth or a fifth, 6c. unlefs the fame paf- 
 lages either be in the fundamental bafs, or at leaft might na- 
 turally be inferted there. 
 
 This reftridtion is alfo owing to the defire of hearing the 
 fundamental founds in the bafs ; fo that when the bafs 
 imitates any familiar fundamental paffage, we expect 
 accompanyment conformable to it. Thus the rifing 
 of a fifth at R, in Exam. LXII. obliges us to confti- 
 tute the next note a fundamental, as (hewn by the di- 
 rect. 
 295. A few licences are allowed between the plain thorough- 
 bafs and the fundamental ; which are reducible to three clafles, 
 called holding, fuppofition, and fitbflitution. 
 
 Holding is practifed only upon the key note, while the fun- 
 damental moves from the fame key upon either or both of the 
 adjuncts, and returns back to the key again : in which cafe 
 the adjunct 4th bears only a perfect chord, which makes fixth 
 and fourth to the key ; and the adjunct 5th bears a feventh, 
 but the oftave of the fundamental is omitted, which makes a 
 
 except this mark [— ] be found among them, which flands 
 inflead of a repetition of the preceding chord, and is called a 
 continuation. 
 
 296. Suppofition, (or placing below) is when the thorough- 
 bafs takes the third below the fundamental, and may°be 
 practiced in all forts of fundamental chords ; but there is 
 only one cafe wherein the chord thereby introduced differs 
 from thofe already fpoken of. This fuppofition below a per- 
 fect chord, or below a chord of the fixth and fifth, produces 
 a chord of the feventh. The fame fuppofition below a chord 
 of the feventh, when the third of that chord is a perfect fifth 
 to the fuppofed bafs, introduces a chord confifting of a ninth, 
 feventh, fifth, and third ; which chord is feldom ftruck full : 
 and if the ninth be omitted, it becomes a chord of the feventh ; 
 If the feventh be omitted, it becomes a chord of the ninth, 
 as in Art. 188. N°n.; and if the fifth be omitted it becomes 
 a chord of the ninth and feventh, as in Art. 188. N° 1 5. 
 
 If the third of the fundamental chord be a fuperfluous or 
 fharp fifth to the fuppofed bafs, which happens in fuppofition 
 below the leading chord of a flat key, it introduces a new fpe- 
 
loo Of Plain "Defiant. 
 
 cies of chord, confifting of a ninth, greater feventh, (harp 
 fifth, and lefs third, which is called a chord of the extreme 
 Jbarp fifth. The feventh is generally omitted to avoid the too 
 great complication of difcords, and when the chord of the key 
 fucceeds, the fharp fifth afcends, and the ninth defcends, 
 (which are their proper courfes) while the bafs continues upon 
 the fame degree, taking a chord of the fixth : as at H, in 
 Exam. LX1V. 
 
 297. Siibftitution is a term appropriated to the taking the 
 lefs 6th, inftead of the 5th of a flat feries, either in the bafs 
 or in any of the upper parts ; and this produces feveral chords 
 which we have not formerly had occafion to fpeak of. 
 
 (r.) By fubftituting the 6th of a flat feries in the bafs, in- 
 ftead of the leading 5th bearing a (harp third and feventh, 
 we have a chord confifting of a fixth, tritone, and extreme fbarp 
 fecond (or trihemitone,) as at H, in Exam. LXV. 
 
 (2.) The fame fubftitution in an upper part produces a flat 
 third inftead of a fecond, in the chord of the fecond, which 
 belongs to the defcending 4th of the feries, as at I : alfo a falfe 
 fifth inftead of a fourth, in the chord of fourth and third, 
 which belongs to the 2d, as at K : and an extreme flat feventh 
 inftead of a fixth, in the chord of the fixth and fifth, which 
 belongs to the leading ;^;th, as at L. 
 
 All thefe beiug borrowed from the leading chord of 
 a regular cadence, will, like the original chord, be followed 
 by the perfect chord of the key. 
 
 The like alterations may be produced in all other chords 
 which include the 5th of the flat (eries, by fubftitution ; but it 
 being feldom practiced, except in the chord of the adjunct 5th, 
 and its inverfions, as above, we (hall not here enumerate any 
 other of this fpecies. 
 
 298. There is another licence concerning which the theo- 
 rifts have been divided in regard to its origin, or fundamental : 
 it will be beft conceived as a fecond kind of fubftitution, which 
 
 Part I. 
 
 takes place in the chord of the adjunct 4th of a flat feries, by 
 taking the $4th inftead of the natural fundamental in a grada- 
 tion on the 5th. This fubftitution alfo introduces a new fett of 
 chords all of them containing extreme intervals ,• for 'mftance, 
 if the fubftituted % 4th be in the bafs, the chord will confift 
 of either an extreme flat feventh, or a lefs fixth, joined with a 
 falfe fifth, and extreme flat third, as at N, in Exam. LXV. 
 If the flat 6th be in the bafs, and the % 4th in an upper part, 
 the chord will confift of an extreme floarp fixth, with either a 
 fifth, or a tritone, and a greater third, as at M. This Lift chord 
 has been called the Italian fixth ; probably becaufe they firft 
 introduced it. Each of thefe chords will be followed by the 
 adjunct 5th. 
 
 299. The thorough-bafs, as well as any other upper part, 
 may take in the paflages of chromatic, as defcribed in the laft 
 chapter ; and in this cafe the proper figuring will be obvious, 
 from the confideration of the fundamental progreflion, and the 
 inverfions which take place. See Exam. LXVI. 
 
 300 After a little practice in compofition the real funda- 
 mental of every chord will become familiar, without formally 
 extending the fundamental bafs below ; which being of no o- 
 ther ufe, but to (hew the fucceffion of chords in the harmony, 
 may be omitted, as foon as the want of it can be fupplied by 
 the imagination : and in order thereto it will be proper for the 
 learner to practice the fubjoining of the fundamentals in letters 
 below any figured thorough-bafs, according to the fpecimens, 
 Exam. LXVI. LXVII. and LXVIII. 
 
 301. The movement of the thorough-bafs with refpefr to 
 the treble muft be regulated according to the following direc- 
 tions. 
 
 (1.) A confecution of perfecl: concords of the fame kind 
 muft always be avoided, when the parts move ; but they may 
 ftand ftill and repeat any concord, or even a difcord feveral 
 times over, while the fundamental remains the fame. The 
 
Plate X. E X.LXVL CAHnnaticMap. 
 
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 A. E .A. X) C G CA E A. 
 Ex-XXIX. ^ 
 
 A -** E A IE A D A r * A 
 
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Rules for the Thorough-Bajs in regard to the Fundamental, and to the Treble 
 
 Chap. XI. 
 
 leaping an oftave up or down is confidered the fame as Hand- 
 ing ftill. 
 
 (2.) While one part ftands ftill, the other may take any 
 motion which is confiftent with the fundamental harmony. 
 
 (3.) When both parts move, they ought to take contrary 
 motions as often as is convenient ; that is, the bafs ought to 
 afcend while the treble defcends, and vice verfa : and this ei- 
 ther by degrees or leaps; the variety thereby introduced hav- 
 ing always a good efFect. 
 
 (4.) Both parts may afcend or defcend together by fingle de- 
 grees, or by fmall leaps, in a fucceffion either of thirds or 
 fixths, as far as fuch a fucceffion does not interfere with the 
 other rules of harmony ; but feldom by any leaps larger than 
 a fourth. 
 
 (5.) Both parts moving the fame way by leaps of any kind, 
 except in the cafe of fucceffive thirds or fixths, as above, ought 
 to be avoided, when the fundamental changes ; unlefs the leaps 
 be in fome manner effential to the air of each part. Such paf- 
 fages always make a breach in the harmony, and are beft al- 
 lowed when the bafs takes a fundamental cadence, while the 
 treble leaps a third or fixth. 
 
 (6 ) When the fundamental ftands ftill, both parts may take 
 any leap which the chord allows, provided they do not meet 
 upon and double the third of the fundamental, or the difcord, 
 if it bear one. 
 
 (7.) The bafs muft not take a note to which the treble 
 ftands in the relation of a fourth, unlefs that fourth defcend 
 one degree when the chord changes : except in the chord of 
 feventh fourth and fecond, when the bafs holds upon the key, 
 for here the fourth may afcend on occafion. 
 
 Thefe (even rules are fucceffively applied, in Ex. LXIX. 
 
 (8.) A note which is a difcord to the fundamental, or which 
 refolves a preceding difcord, muft not be doubled. Neither 
 ought the third or fifth of the fundamental to be doubled, ex 
 
 Cc 
 
 cept it be to favour the contrary motion of the parts, or when 
 the fame third or fifth might eafily be admitted as a funda- 
 mental at that occurrence. 
 
 (0 ) Imitations, as defcribed, Art. 246. and fugues, de- 
 fcribed, Art. 1 24. are always agreeable, when different parts 
 can be made to fall fucceffively into them : there is alfo a beau- 
 ty called inverted imitation, where the defcent in one part 
 bears fome refemblance to the afcent in another, and vice ver- 
 fa ; alfo another called rccle <& retro, in which one part is 
 made to repeat backward, what another part has already done 
 forward. 
 
 All thefe varieties are eafieft introduced in figurative de- 
 fcant, where the compofercan avail himfelfof feveral liberties, 
 not yet fpoken of, to anfwer his defign ; yet as fomethiDg of 
 the fame kinds may be praclifed even in plain defcant, it is 
 proper to have them in view. 
 
 (10.) If no more parts thanafirft treble and bafs are intend- 
 ed, a good fhare of melody may be given to the bafs, and fu- 
 gues, imitations, and fequences of thirds or fixths, freely ad 
 mitted; efpecially in inftrumental muficj (for the baffies to 
 fongs ought to be fimple, that the attention may not be drawn 
 off from the finger;) but if a fecond treble is to be added, 
 it is beft to referve the greateft fhare of thefe beauties for that 
 part, and confine the bafs more clofely to the fundamental 
 progreffion : and for this purpofe, when they occur to a com- 
 pofer, it will often be proper to put them down immediately 
 in the fecond treble, and afterwards join a bafs to both tre- 
 bles. 
 
 302. The fecond treble and tenor ought generally to be fo 
 conducted as to fill up the fucceffive chords as much as is con- 
 venient. For as every perfecl chord contains three different 
 founds, and every diflbnant chord four, there will always 
 be one found in a perfefl chord, which neither the treble nor 
 bafs takes, and which therefore one of thefe parts ought to 
 
102 Of Plain Defiant. 
 
 take, while the other coincides with any part whofe found 
 can admit of being doubled ; and there will be two founds 
 wanting in adifibnant chord, which thefe two parts ought to 
 fupply. For inllance, if the bafs take G, bearing a perfect 
 chord, while the treble takes D (the fifth,) there is ftill B (the 
 third) left for one of thefe parts, while the other may take the 
 octave of the bafs : and if the bafs take F, bearing a chord of 
 fixth and fifth, while the treble takes C (the fifth,) there will 
 ftill be D (the fixth,) and A (the third,) left for thefe inner 
 parts to fupply, if it can be brought about without iranfgref- 
 fing any of the following rules. 
 
 303. Rules for the inner parts with refpect to the principal 
 parts, and to each other. 
 
 (1.) All the rules prefcribed, Art. 301. between the bafs 
 
 - and treble, ought to hold good between the bafs and any o- 
 
 ther upper part ; but a tranfgreffion of thefe rules is of lefs 
 
 confequence, according as the part in which it occurs is lefs 
 
 the object of a hearer's attention. 
 
 (2.) The neceflary motions in the tefolution of difcords, 
 mull: be ftriclly complied with, in whatever part the difcord 
 is introduced ; and fuch difcords as need to be prepared, mufl 
 be prepared in the fame part, and introduced by a continuati- 
 on of the preparing note. 
 
 (3.) The fecond treble ought in general to be kept below 
 the firft, and the tenor below the fecond, but above the bafs: 
 there are the natural places of the parts ; and if at any time 
 one of them go out of its natural place, the higheft part is 
 ftill to be efteemed the firft treble, and the loweft part the 
 bafs, for the time. 
 
 (4.) The upper parts ought not to lye too far afunder. In 
 general the larger and more perfect intervals ought to be below 
 in a chord, and the fmaller and more imperfect, above. See 
 Art. 127. and 133. 
 
 (5.) A confecution of perfect concords of the fame kind mufl 
 
 Part I. 
 
 be avoided between one upper part and another, as well as 
 between each of them and the bafs. When the firft and fe- 
 cond treble are made to go together, or the tenor to go on 
 with the octaves of the bafs, for fome time, fuch occurrences 
 are no exceptions to this rule ; for in thefe cafes the two parts 
 which coincide are confidered only as one. 
 
 (6.) The rules prefcribed between the bafs and treble, in 
 regard to doubling the harmonics of a chord, Art. 301. N° 8. 
 ought alfo to be obferved between one upper part and another; 
 though a fmall deviation from thefe rules, may be allowed 
 among the inner parts. 
 
 304. For examples of plain defcant the common fetts of 
 pfalm tunes in four parts may be taken ; obferving this pecu- 
 liarity, that the church part, ufually called the tenor, muft 
 here be confidered as the leading part, becaufe it contains the 
 principal melody, and is beft heard in the performance ; fo 
 that it is properly a firft treble carried an octave lower, in or- 
 der to fuit the generality of mens voices ; and the part ufually 
 called treble, which is adapted to womens and boys voices, is 
 properly a fecond treble in the defcant ; and the contra is the 
 inner part. 
 
 Exam. LXX. Plate XI. is a fpecimen of this kind of com- 
 pofition, with the fundamentals fubjoined in letters, and the 
 relation of each note to the thorough bafs marked over it, for 
 the learner's perufal. 
 
 305. The fequence of fevenths, as defcribed, Art. 249, 
 and Exam.LII. when the thorough bafs keeps the fundamen- 
 tal progrefhon, requires four upper parts to make it quite full ; 
 but all the fevenths are brought in by two parts, which take 
 alternately the 3d and 7th of the fundamental chords ; as will 
 appear by infpecting the example; while the other two parts 
 fill up the 5th and 8th of each chord by turns : and the effect 
 is much the fame though one or both of thefe parts be omitted. 
 
 The thorough bafs may take any of the parts of this fe- 
 
Plate xi. fcEAx 
 
 Cheirful 
 
 n b f> 3 6 5 
 
 5iU 
 
 S. MATTHEWS, or GLASGOW. j 
 
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 T2: 
 
 IS 
 
 Treble 
 
 ©-€> 
 
 <>Q 
 
 i 
 
 oct^ 
 
 € 5 8 g gpg. 
 
 0=^ 
 
 saftati te ffic^i^ iM^ 
 
 > 1 6 * fi~ 
 
 yji, 
 
 3 8 5 5 8 8 
 
 Continued 
 
 £ ff g 5 b 
 
 5 5, 2- 
 
 <^onr.muea , < s 
 
 S 5 ? 8 < 
 
 =3 
 
 ^2 
 
 E=T 
 
 ITf^F^ B g 
 
 ens* 
 
 §A J \ M 
 
 3 8 rjfllAj 55 8 7 g f8j7 
 
 ~T^ jta.7. 
 
 g . e g 
 
 #J 
 
 f ^jgjp 
 
 6 87 
 
 O: 
 
 * A "7 
 
 C C'd T T B i CJ 
 7 
 
 a- c a t a 
 
 6 .7 
 
Chap. XII. Rules for the inner Parts. Of Parages derived from the Sequence offevenths 
 
 quence, bearing fuch chords as are derived from the funda- 
 mental by inverfion : and if fome of the drfcords be omitted, 
 fuch omiffions give feveral liberties in the movement of the 
 parts ; from whence arife a vafl: variety of paffages, all origi • 
 nally founded on thefequence offevenths, which the ftudious 
 
 103 
 
 will obferve in the courfe of practice ; but it would be rather 
 tedious to take a full account of them here. In all cafes the 
 proper movement of the parts will be found out, as far as be- 
 longs to plain defcant, by a careful obfervation of the foregoing 
 rules. 
 
 Chap. XII. (^Figurative Melod 
 
 Article 3o6. r T"*HE liberties allowed in compofing zfigura- 
 JL five or florid melody, from a plain melody 
 and its bafs given or previoufly compofed, of which we mall 
 take a particular view in this chapter, are either fufpenfions, 
 fubflitutions, or breakings. 
 ' 307. The mod ufual fufpenfions are defcribed, Art. 113, 
 and 114.: and it is to be obferved in practice, that the fu- 
 fpending notes muft always be prepared ; that is, they muft 
 be brought in as continuations of former founds : which are 
 generally made to fyncopate, and by that means the difcords 
 are introduced on the accented part of the meafure. 
 
 308. The various kinds of fufpenfions muft alfo be ufed 
 with proper caution, that the fufpending notes may not im- 
 ply fundamentals which are foreign to the harmony ; and for 
 this purpofe it may be neceflary to read over again from the 
 181 to the 187th article; with the applications which here 
 follow. 
 
 309. The chord of the fourth is obferved, Art. 181. to be 
 «' a mixture of the chord of the fufpending fourth along with 
 ** that of the fundamental :" therefore this fufpenfion is moft 
 properly ufed on the adjunct: 5/, both in (harp and flat har- 
 
 mony, becaufe the fufpending note is the principal key. It 
 is alfo eafily allowed on the K/, becaufe the fufpending note 
 is the adjunct 4th. Thefe are the chords to which this li- 
 cence naturally belongs : but it may be ufed on all fundamen- 
 tals, except the 4th in (harp harmony, and the 2d and 6th 
 in flat harmony, for the fake of a regular fucceffion, or to fa- 
 vour an imitation of what is gone before; if the fufpending 
 notes be found prepared ; thefe fufpending notes in all cafes 
 being allowed to take place as co-exiftent fundamentals in the 
 harmony, 
 
 310. Neither this chord of the fourth, nor any other chord 
 diflonant by fufpenfion, may be inverted, except when they 
 are ufed in their molt natural place. The molt, practicable in- 
 verfion of any chord of this fpecies is that wherein the bafs 
 takes the difibnant note, viz. where the fufpenfion is practif- 
 ed in the bafs. But the chord of the fourth, being the moft 
 harmonious fufpenfion, admits of the greatefl liberties on that 
 account; and therefore both itsinverfions are practifed in mufic. 
 
 The 5th/ is the natural place of the chord of the fourth, 
 and in this fituation its inverfions produce the chord of fifth 
 and fecond, on the fyncopated key, followed by the 7th of 
 
104 
 
 the fcale ; and that of feventh and fourth 
 
 the fcale, followed by the fixth and third, or fixth fourth and 
 
 third on the fame note. 
 
 311. The chord of the ninth, Art. 182. tending to intro- 
 duce the fifth of the fundamental as a co exiftent fundamen- 
 tal, takes place properly on the Kf, and on the adjun<ft4th/, 
 in both forts of harmony ; becaufe their fifths may moll natu- 
 rally become fundamentals : but the fame fufpending ninth is 
 alfo ufed on any note whofe fifth can be admitted as a co- 
 exiftent fundamental at the time, for the fake of a regular fuc- 
 ceflion, if at the fame time the ninth be prepared by being ei- 
 ther a third or a fifth in the former chord ; that is, if the bafs 
 either afcend by a 
 
 ilngle degree, or £ defend* a°fifth j fr0m the former note# 
 
 The inverfion of this chord produces that of the feventh 
 fourth and fecond on the fyncopated 2d of the fcale, followed 
 by the key bearing its perfect chord. 
 
 When thefe fufpenfions are practiced in the bafs the move- 
 ment of the upper parts, in refpect of the bafs, is by fome au- 
 thors called anticipation : but this feems to be a needlefs in- 
 creafe of tech nica! terms. 
 
 Fxam. LXXI. Plate XII. is a fpecimen of the ufe of both 
 thefe kinds of fufpenfions, and their ufnal inverfions, in both 
 forts of harmony. 
 
 312 The chord of the ninth and fourth, Art. 183. which 
 introduces the fifth of the fundamental with an added feventh, 
 belongs properly to theK/"onIy. The inverfion of this chord 
 produces that of feventh fourth and third on the fyncopated 
 2d of the fcale, followed by the key with its perfecl: chord. 
 The chord of the ninth and feventh, Art. 174. N° 3. ought 
 to be fparingly ufed on fuch bafs notes as either really do 
 ftand, or at leaft may be fuppofed to ftand at the diftance of 
 a third below the fundamental, by way of fuppofition. See 
 
 Of Figurative Melody. Part I. 
 
 on the 2d of Art. 296. Therefore this chord belongs only to bafs notes 
 whofe thirds may be the governing fundamentals at the time. 
 This and the following fufpenfions, feldom or never admit of 
 inverfion. 
 
 N. B. All the ninths muff be prepared as defcribed, Art. 311. 
 
 313. The feventh fufpending the fixth in defcending, Art. 
 184. or the fifth fufpending the fixth in afcending, Art 185. 
 may be ufed on any bafs note which bears a chord of the fixth, 
 and has the fufpending note prepared : and when feveral bafs 
 notes, gradually defcending, bear e^ch a fixth, tho' their chords 
 be, in other refpe&s, differently formed, every fixth may be 
 held on and become a fufpending feventh in the next chord, 
 and every bafs note may be accompanied with its third, for 
 the fake of a regular defcending fequence. 
 
 The afcending fequence of fifths and fixths, be'mg lefs na- 
 tural, is more fparingly ufed, and feldom continued on more 
 than two or three bafs notes gradually afcending. 
 
 314. The fubftitutions defcribed, Art. 297. and 298. dif- 
 fer from fufpenfions in thefe feveral refpefts. 
 
 (1.) The fubftituted notes are generally introduced with- 
 out preparation, and on the unaccented part of the meafure, 
 provided that the courfe of the melody lead naturally to thefe 
 notes ; whereas fufpending notes are generally prepared, and 
 introduced on the accented part. 
 
 (2.) A chord, difionant by fubfiitution admits of inverfions 
 of all kinds, the bafs eafily taking either the fubftituted note, 
 or any other note of the chord ; whereas difionances by fu- 
 fpenfion are rarely inverted, except when the fufpending note 
 is introduced in the bafs. 
 
 (3.) The fubftituted notes are limited in their progreflion, 
 on account of the difcord which is to be refolved, much like 
 fufpending notes ; but in fubftitutions the chord changes 
 when the difcord is refolved, as it alfo does in the refolution 
 of fundamental difcords; whereas in fufpenfions the chord re- 
 
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 fr X J? 5# /5 
 
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Chap. XII. 
 
 Of Sufpenftons, Subflitutions, and 'Divifions, 
 per found has fucceeded ; that is. 
 
 mains the fame after the pr< 
 after the difcord is refolved. 
 
 Ex. LXXII. contains inftances of the ufe of all the licences 
 in this and the two preceding articles, in the flat feries of D. 
 
 315. In breaking, or fubdividing a note of an upper part 
 into feveral fmaller ones, it mult be obferved, that if the fun- 
 damental chord, which exifts at the time, be perfeft, we may 
 break any note into as many fmaller ones as we pleafe, ufing 
 none but the notes of the chord : but if the chord contain a 
 diflbnance, and confequently force of its founds be reftricled in 
 their motion, care mult be taken, that if the figurative melody 
 touch upon one of thefe reftri<fred notes, it may be heard re- 
 folved in the fame part, when the chord changes : and if the 
 figurative melody touch upon two reftricled notes in one 
 chord, the latter of thefe muff be heard refolved in the fame 
 part, when the chord changes. Such breakings are called divi- 
 fions by confonant intervals. 
 
 316. When the plain note which is to be broke is itfelf the 
 difcord to the fundamental, or is reftricled in its motion, on 
 account of the refolution of a difcord, care muft be taken that 
 the firft part of the difcording (or reftricled) note may be di- 
 ftinclly heard ; and after the remaining part of it has been 
 broke into fmaller notes, the divifion muft lead naturally into 
 the proper refolving note, that it may alfo be diftinftly heard. 
 
 316. In making the fame kind of divifions by confonant in- 
 tervals in the thorough bafs, the following cautions are alfo 
 neceflary. 
 
 (1.) It often happens, that in fubdividing the notes of a 
 plain bafs, we introduce a neceffity of ftriking the chords more 
 frequently with the right hand ; as for inftance, when a mi- 
 nim is broke into four quavers, it is natural to expect a chord 
 to be (truck once to the firft, and again to the third quaver ; 
 or, as the artifts fay, the third quaver -will bear a chord : and 
 therefore we muft take care that fuch parts of the divifion as 
 bear chords, may not counteract the original courfe of the 
 
 Dd 
 
 IOjT 
 
 harmony, by introducing inverted chords of a tendency dif- 
 ferent from that of the plain bafs. Thus, becaufe the chord 
 of the fixth and fourth, belonging to the 5th above a funda- 
 mental, requires to be followed by the perfect chord of the 
 fame bafs note ; therefore, in fubdividing a thorough. bafs 
 note, we ought not to place the 5th of the fundamental where 
 it will bear a chord, unlefs the fame note be to follow, with 
 its own proper chord. See Art. 294. N° 6. 
 
 (2.) We muft alfo take care, that all the fundamental ca- 
 dences, which take place in the plain bafs, be alfo diftinclly 
 perceived in the figurative bafs ; for thefe are the paflages 
 which characterize the bafs ; and in fuffering them to pafs 
 unperceived, we mould rather make a ne-w bafs than a varia^ 
 tion on the original. 
 
 A cadence will be diftinclly perceived, if the figurative bafs 
 either take the real notes of the cadence, or one of the moti- 
 ons called neceflary in the cadences ; as formerly defcribed, 
 Art. 196. and 211: But, in general, the figurative bafs co- 
 incides with the plain bafs in fuch occurrences ; and efpecially 
 in the cadences which conclude the principal phrafesor ftrains 
 of the piece. 
 
 318. Any note, either of the bafs, or an upper part, may 
 be broke into a fucceffion of fmaller notes, moving by fingle 
 degrees only, provided that fucceffion begin with the note it- 
 felf, or, at leaft, one of the notes of the chord which takes 
 place at the time, and lead naturally into the note which is to 
 follow ; or if the laft note of the fucceflion be a leap diftant 
 from that which follows, this laft uote muft alfo be one of the 
 notes of the chord ; or if a leap occurs in the middle of the fuc- 
 ceffion, both the notes of fuch a leap muft belong to the 
 chord. Thefe breakings are called divifions by pajfing notes. 
 
 319. Tafte obliges us fometimes to depart from the rule 
 prefcribed above, in regard to the firft note of a divifion by 
 pafling notes; and alfo in regard to the note which follows a 
 leap in the middle of fuch a divifion, this being, in effect, the 
 
io6 
 
 firfl note of a new fucceflion : but then, if thefe notes do not 
 belong to the chord, they pafs either as fufpending or fubfti- 
 tuted notes ; and therefore muff be immediately followed by 
 one which does belong to the chord, and which alfo ferves 
 for a proper rcfolution of the preceding difcord, according to 
 the rules formerly laid down. 
 
 320. When feveral breakings follow each other in regular 
 order, the delire of imitations often obliges us to tranfgrefs all 
 the rules which can be given ; but then it is to be obferved, 
 that the firffc of fuch breakings muft always be conformable to 
 the above rules ; and it is eafy to perceive when the attention 
 begins to be carried away by an imitation, that is, when the 
 rules may and ought to be fet afide, in compliance therewith. 
 
 321. It is ufeful to obferve, on this fubjett, that greater li- 
 berties may be taken in the breakings of fome chords than others: 
 Thus in the chords of the Kf, and of the 5th/, we can allow 
 of feveral paffages which would not be fuffered in chords more 
 diftantly related to the principal ; and this will eafily be ac- 
 counted for, by obferving, that moft of the fufpenfions and other 
 licences, formerly defcribed, naturally take place in the chords of 
 the Kf and of the 5th f. So that here almoft any fucceffion of 
 notes will pafs. A whole tune is often in reality no more than 
 a kind of divifion or breaking upon the key note as fundamen- 
 tal. The key might be held on in the bafs from firft to laft, as 
 in the mufette and bag-pipe mufic ; and although it fhould 
 not actually be fo held on, yet it is undoubtedly always kept 
 in mind in all connected pieces. 
 
 322. When feveral parts are fet together, if one of them 
 be divided into fmall notes, that part, fo ornamented, ought 
 generally to be conlidered as the leading part, or principal me- 
 lody, for the time, and it is proper that the other parts at the 
 fame time be plain, or that at moft only two of the parts go 
 on in divifions together, as firft and fecond tiebles ; to avoid 
 the diffraction which would inevitably follow, if too many ob- 
 jects were to demand our attention at once. 
 
 Of Figurative Melody, Part I. 
 
 323. The infertion of refts or paofes in one or more of the 
 parts, which are let together in mufic, may be properly ranked 
 among the liberties of figurative melody. Thefe, when judi- 
 cioufiy ufed, ferve greatly to diverfify the harmony ; and in 
 this matter the tafte of the compofer feems to be confined to 
 no other rules, except that when a reft is put in any part im- 
 mediately after a difcord, or any other note refrricled in its 
 motion, the proper refolving note, which was expefted in 
 place of the reft, muft inevitably follow : and when a reft is 
 put in the bafs, the note which follows will require a chord to- 
 be ftruck with it, or will bear a chord, though it fhould be in 
 an unaccented part of the meafure ; therefore the fame cauti- 
 on is necefTary in this cafe which was given in regard to the 
 divided bafs, Art. 317. M° 1. 
 
 324. There is no occafion to adduce any examples of thefe 
 divifions here, becaufe all our modern compofitions abound 
 with them; and he who examines mufic, fet in fcore, with 
 a due regard to the fundamental principles, cannot fail of meet- 
 ing with abundance of inftances which will aptly exemplify 
 each of the rules delivered above ; fo that, with a fmall degree 
 of attention, every piece of mufic which the ftudent either 
 pracYifes or hears, will tend to confirm the principles laid 
 down in this eflay, and to fix them more durably in his me- 
 mory. 
 
 3 25. In cafes where it may be doubtful whether a note be- 
 longs to the plain melody, or is a palling note ; which ambi- 
 guity will often occur where the bafs is not figured, the doubt 
 will be beft decided by trying whether fuch a note may be left 
 out, and yet the air of the tune remain the fame ; for if this 
 be the cafe, it may fafely be efteemed a pairing note : but if 
 the air be altered, or the connection interrupted, by leaving 
 out fuch a note, it is a fure proof that it ought to have place 
 ill the plain melody. The fame method ought to be taken 
 when a bafs is to be fet to a given figurative melody ; for, by 
 
Chap. XII. 
 
 Rules to be obferved In making "Divifions. 
 
 this means, the plain melody may be afcertained, and the bafs 
 fet to it, by the rules already given. 
 
 326. To finifh this chapter, it ought to be obferved, that the 
 liberties which we have clafled under the article of figurative 
 melody, when judicioufly ufed, contribute more than any 
 thing formerly fpoken of towards heightening the exprejjion of 
 mufic. Thus, the divifions by confonant intervals are gay and 
 fprightly when very quick, and are bold and majeftic when 
 a little flower. The divifions, by paffing notes, are pleafant and 
 airy when quick, and are foft and pathetic when (lower. The 
 divifion, beginning with a note which doss not belong to the 
 chord, has a particular affinity to the modulation of the voice 
 in fpeech, and efpecially in the language of the Italians, from 
 whom we have borrowed it ; and is remarkably tender and af- 
 fecting: the occafional infertion of refts, fometimes exprefles 
 diffidence or fufpence; but refts ferve to heighten every fpe- 
 ciesof pathetic expreffion. From thefe hints, joined with fuch 
 other remarks as the judicious will not fail to make in the courfe 
 of practice, it will eafily be obferved, that the fame funda- 
 mental progreffion may receive a vaft variety of melodies, which 
 may differ widely one from another ; fo that one (hall be 
 plaintive, another chearful ; one grave and majeflic, another 
 light and airy, &c. according to the different movement and 
 manner of dividing the melody. Thus we frequently find the 
 very fame progreffion in a folemn Pfalm tune, and in a joyous 
 catch. 
 
 327. It may alfo be remarked, that melody takes a con fi- 
 derable tincture from the cuftomary cadences of a language, 
 fo that there are national melodies in a manner peculiar to dif- 
 ferent countries, which are eafily diftinguifhed one from ano- 
 ther ; and that although the fundamental progreffion itfelf has 
 a (hare in this diverfity, the moft fimple and natural progref- 
 fions being moft clofely adhered to where mufic is leaft flu- 
 
 107 
 
 died and cultivated; yet the peculiar manner of fubdividing 
 the plain notes is the principal caufe of thefe differences. 
 
 3 28. The ftudious practifer will alfo perceive, with fome 
 regret, that the defile of exhibiting furprizing execution has 
 led feveral modern mafters, who otherwife have unqueftion- 
 able merit, to infert divifions which rather detract from than 
 add to the real beauties of their compofitions. 
 
 This fault is moft: confpicuous in the modern fongs. Thefe 
 being intended to make their firft appearance in fome of our 
 public places of diverfion, muft, of courfe, be adapted to the 
 abilities and manner of a particular finger, and fcope muft be 
 given to the finger to exhibit all her (or his) power of voice, 
 in trills, divifions, <&c. though the fentiment cannot proper- 
 ly admit of any fuch paflages at all. Thus true melody and 
 good fenfe are often facrificed to gratify the vanity of a per- 
 former ; and when that performer difappears, the favourite 
 fong alfo vanifhes of courfe. 
 
 329. The fancies of compofers are fo unbounded, and their 
 thirft for variety fo infatiable, that the reader ought not to be 
 furprized, if he fometimes meet with paflages which cannot 
 be perfectly reconciled to any of the rules delivered above : 
 much lefs ought he on fuch occafions to be hafty in cenfuring 
 either the irregularity of the compofer, or the imperfection of 
 thefe rules. Under the title of rudiments of practical mufic 
 he may juftly expect to find all fuch rules as beginners fhould 
 confine themfelves to, in order to correct the irregularities of 
 their fancies; and in this view it is hoped the preceding trea- 
 tife will be found to have fome merit : but then it muft be 
 confidered, that the expert mafter in mufic, as in the fifter 
 arts of poetry and painting, 
 
 may glorioiifly offend, 
 
 And rife to faults true critics dare not mend. 
 
ro 8 Of Figurative Melody. Part I. 
 
 We (hall therefore conclude this part with advifmg the rea nent compofers, particularly fuch as are printed in fcore, to 
 
 der, when he has advanced thus far, not to flop here, and ima- improve the hints and inflruttions here given, and to famili- 
 
 gine himfelf an adept in rriufic ; but to endeavour, by a fre- arize and perfect the ideas which, from a curfory reading of a. 
 
 quent perufal, and careful examination of the works of emi- treatife of this kind, cau at beft be but indiftin&ly conceived. 
 
 PART 
 
[ I op ] 
 
 The 
 
 part ir. 
 THEORY of 
 
 Music, 
 
 C h a p. I. Of Single Musical Sounds. 
 
 Article i. ¥ | "* HE defign of a theory ofmufic, is to explain 
 I the nature and properties of mufical founds ; 
 < and from thence to deduce fuch remarks and 
 directions as may be ufeful in the compofition of mufic, in 
 the ftructure or the tuning of mufical inftruments, or which 
 may tend, in any refpeft. to the improvement of the art. 
 
 We have no reafon to defpair of fuccefs, in the purfuit of a 
 fatisfactory theory of mufic, if we fet out in the right way ; 
 for, however ftrange in may feem, it is certainly true, that 
 the moft part of the difficulty and drynefs which is juftly com- 
 plained of in the ftudy of this fcience, is owing to the want 
 of right principles to proceed upon, and the laying too great 
 ftrefs upon circumftances of little importance, while others of 
 the greateft confequence are almoft, or altogether, difregard- 
 ed. 
 
 To account for this, it feems necefTary to fuppofe, that 
 feveral of thofe authors who have been belt verfed in the ma- 
 thematics, and moft intent on determining the effects of mu- 
 fic by its rules, have bellowed their labour on wrong mate- 
 
 E 
 
 rials, for want of fufficient practice in mufic, whereby to af- 
 certain what were the moft important particulars to be enquir- 
 ed into: and that others, though better qualified in this re- 
 fpect, have yet adopted erroneous principles, without fubmit- 
 ting them to accurate tryal ; and by proceeding upon them 
 have been difappointed and chagrined at the manifeft incon- 
 fiitencies between the theory and the practice, in feveral refpects, 
 and the insufficiency of the theory for eftablifhing, upon fatis- 
 factory principles, many rules which are found indifpenfibly 
 neceffary in practice. 
 
 2. We fliall in the following effay narrate, as briefly as poffible, 
 thofe properties of found which feem neceffary to be taken no- 
 tice of, and which have been fully difcuffed by other authors ; 
 but fhall more largely treat of other properties which have hi- 
 therto paffed almoft unobferved, and which feem to conflitute 
 a very effential part of the theory of mufic : fuppofing all along 
 that the reader is well acquainted with the common operations 
 of arithmetic, and with the particulars treated of in the firft 
 part of this book. 
 
j i o Of Single Mufical Sounds. 
 
 We fhall firil: lay down" the properties of mufical founds, 
 confidered both fingly and in fucceffion : fecondly, the pro- 
 perties of chords, or combinations of mufical founds: and, 
 thirdly, give fome fpecimens of the ufe and importance of the- 
 oretical knowledge. 
 
 3. Sound is produced by (or always accompanied with) a 
 vibratory, or undulatory motion of the air, firft impreffed on 
 the contiguous air by the vibrations of a founding body, and 
 fucceffively propagated through the air, towards all parts. 
 This undulatory motion agitates the tympanum, or drum of 
 the ear ; by which means a like motion is communicated to 
 the air contained in that organ, which, being carried to the 
 auditory nerve, excites in the mind the idea of found: this 
 we call hearing a found. 
 
 4. If the vibrations or pulfes of a found fucceed each other 
 at equal intervals of time, the found is mufical, but not other- 
 wife : and the hearing of a found whofe vibrations are equal 
 timed, conveys to our mind the idea of a determinate pitch or 
 tone. 
 
 5. The neuter or higher any found is, the quicker areits vibra- 
 tions ; and contrariwife, graver or lower founds have flower 
 vibrations: and becaufe the quicker any equal-timed vibrations 
 are, the greater .number of them will be made in a given time ; 
 therefore one mufical tone is higher than another, if it make 
 a greater number of vibrations in a given lime ; andcontrari- 
 wife. And it is by the numbers of vibrations made in the fame 
 time, that we propofe to compare founds one with another. 
 
 6: It concerns us very little to enquire whether the vibra- 
 tions here fpoken of, which occafion each particle of air to 
 move fucceffively forward, and back again, through very fmall 
 ipaces, be the immediate caufe of found; or whether that 
 caufe ought not rather to be attributed to fome more minute 
 motion or trembling of the particles themfelves. We have all 
 thereafon which the fubjecl is capable of, for concluding from 
 
 Part II. 
 
 the analogy with our other fenfes, that found is a motion of 
 fome kind, and that if it be not thofe very vibrations, yet it 
 is fome motion infeparably connected with them, and propor- 
 tional to them ; fo that we may fafely take the proportions 
 of the vibrations as the proportions of mufical founds ; efpe- 
 daily when the conclufions drawn from thence are found a- 
 greeable to experience. 
 
 Every mufical found has one principal or total vibration, 
 
 by which we eftimate its tone, and alfo feveral other fubordi- 
 nate or partial vibrations exifting along with it, which produce 
 a combination of founds, asobferved, Parti. Art. 137. which 
 are there called the natural con finances of a mufical found. Thefe 
 natural confonances confift wholly of founds, which are acut- 
 er than that of the principal vibration ; and among them 
 thofe founds, whofe vibrations coincide moft frequently with 
 the principal vibration, are generally found to be predominant ; 
 but this depends on accidental circumftances which it will be 
 worth our while to examine a little. 
 
 8. That the air can receive a great number of different vi- 
 brations at once, and properly comply with each of them, we 
 are afTured from daily experience, by the multiplicity of 
 founds we frequently hear at the fame inftant, and each of 
 them as perfect in its kind, as if no other interfered with it : 
 and the moft familiar way of accounting for this is by compa- 
 rifon with the waves on the furface of a pond of water ; for, 
 if feveral Waves be raifed at different places, and of different 
 fizes, we may obferve them to meet, and pafs over, and inter- 
 mix with each other, without fuffering the leaf! alteration as 
 to their fize, direction, or velocity ; fo long as their motion 
 continues to be perceivable. Here the furface of the water 
 properly complies with all the different waves as they pafs, and 
 the motion of each fmall particle is the refult of all the various 
 motions which the feveral waves would fingly produce, com- 
 pounded together. Now, the analogy between the fpreading 
 
Chap. I. 
 
 Of the Pulfes of Mufical Sounds. 
 
 of waves, faifed from a centre in water, fuch as proceed from 
 dropping a pebble, etc. and the expanfion of founds in the 
 air, is obvious ; only obferving, that as the motion of a wave 
 on the furface of water is that of a circle, regularly extending 
 itfelf round the point where it was firft produced ; fo the mo- 
 tion of found in the air is that of a globe expanding itfelf equal- 
 ly in all directions round the founding body, which is its cen- 
 tre : and that what the water does by means of its gravity and 
 fluidity, the air does with much greater freedom by its fpring 
 or elafticity. 
 
 9. But though the air can receive and tranfmit founds of 
 all kinds at the fame time, yet neither our mufical inftruments 
 are adapted to produce an indifcriminate variety at once, nor 
 our faculties capable of attending to, and reliftiing them, were 
 they fo produced. If a firing or pipe, when founded, pn> 
 duce a combination of tones offenfive to the ear, we fay it 
 founds falfe, or is miftuned, and on that account reject it; 
 therefofe the real choice of the ear might be afcertained, pret- 
 ty fatisfactorily, by examining carefully what are the properties 
 of thofe founds which we receive into our mufic : which en- 
 quiry Monf. Rameau purfued with great ardour in his Genera- 
 tion Harmonique, printed at Paris, 1737, and other following 
 works. But ftill it may be objected, that this is only prov 
 ing that the ear chufes fuch founds becaufe it chufes them, and 
 no philofopher ought to reft fatisfied with this method of rea- 
 foning. We Want, if poffible, to know why the ear chufes 
 and refufes, as we find it doe6 ; and, for this purpofe, fome 
 new principle muft be adopted. The following one, which, 
 to propofe and fupport, was one principal defign of this part, 
 feems to promife good fuccefs in this enquiry. 
 
 10. The new principle we have here to propofe, as being 
 that whereby the various choices of a mufical ear are beft ac- 
 counted for. is that ot our diftributing the vibrations of mu- 
 fical founds by ifochronous, or equal-timed parcels, fomething 
 
 \A netv Principle propofed. 1 1 1 
 
 very fimilar to the diftributions we find natural to be made a- 
 mong quavers, or other fhort notes, in the timing of mufic. 
 
 11. To fupport this principle we may obferve, in various 
 inftances, where equal and equidiftant Objects affect our fenfes, 
 that there is a certain propensity in our mind to be Subdividing 
 the larger numbers into fmaller equal parcels ; or., as it may be 
 juftly called, compounding the larger numbers of feveral fmall 
 factors, and conceiving the whole by means of its parts. 
 
 When we caft our eyes on nine equidiftant windows 
 in a row, they are no fooner feen than fubdivided into 
 three times three : eight appears at firft fight to be two fours, 
 and each of thefe fours, two twos : feven we conceive as two 
 threes disjoined, and one in the middle ; fix moft naturally di- 
 vides itfelf into two threes; but if feen along with nine, or 
 immediately after it, we then trifect it, in conformity with 
 nine, and it appears three twos : five becomes two twos dif - 
 joined, and one in the middle ; four is two twos, and fingle 
 three or two need no fubdivifion. 
 
 1 2. In the above cafe, the whole number prefent themfelves 
 to our fenfe at once ; but in the timing of mufic, the notes 
 occur in fucceffion, which occafions fome difference : for here 
 we have to do not only with the choice of the mind, but alfo 
 with the memory. The mind infifts that all our notes be made 
 up, as it were, into ifochronous parcels, which we call bars, 
 or meafures, and that the number of equal fhort notes which 
 conftitutes each meafure, be a number fomeway compounded 
 of the fmall factors two and three multiplied together, andrate- 
 ly admits any larger factor than thefe. 
 
 13. Thus the firft, or primary divifion of the reeafu re/in 
 common time, is into two parts, and the primary divifion of 
 the triple meafure is into three parts ; thefe may be again fub- 
 divided, either into two or three each, and thefe fecondary 
 divifions again fubdivided into two or three, and fo on. The 
 divifion into four parts refolves into two and two ; and an e- 
 
Ill 
 
 qual divifion into five parts is not eafily admitted * in timing 
 of mufic, and much lefs feven. Thefe particulars are largely 
 difcuffed in Part I. Chap. iv. of Time. 
 
 14. Befides this diftribution of mufic into meafures, the 
 mind extends its views, and as far as the memory can be fup- 
 pofed diftinftly to retain, goes on to conftitutefome number of 
 meafures into ifochronous phrafes, or ftrains of a tune; and thefe 
 ftrains may contain a greater number of meafures in quick 
 time than in flow time, becaufe of the inability of the memo- 
 ry ; but here, as before, the number of meafures in a ftrain 
 muft always be either two or three, or fome produft of thefe 
 numbers : for here five bars in one ftrain is not ufed, and 
 feven proves much more intolerable. 
 
 For an ioftance of this it may be obfeived, that in the flow 
 tedious way of finging pfalms, which is in ufe among us, the 
 memory has enough to do to retain the length of one fingle 
 bar ; as the number of pfalm fingers, who do not keep true 
 time, plainly evidences : here we cannot conftitute any larger 
 portions or ftrains, and one bar is all we regard at once ; 
 fo that if the firft line employ four bars, and the next line 
 only three, we are not fenfible of any impropriety : but let us 
 try to fing a pfalm tune, adapted to the common meafure, 
 where the lines are alternately 8 and 6 fyllables, much quicker 
 than ufnal, and we (hall find ourfelves under a neceffity either 
 of holding on the laft notes of the fecond and fourth lines, or 
 of reding at the ends of thefe lines, till the time of the firft 
 and third lines be elapfed : becaufe in this way of finging the 
 memory retains the whole length of the line, and therefore we 
 muft make thefe lengths all equal, they muft be ifochronous 
 ftrains. 
 
 Of Single Mufical Sounds. 
 
 Part II. 
 
 Common minuets and marches generally confift of ftrains 
 containing 4 bars each ; and here we are fenfible of no impro- 
 priety, although the firft part of the tune fhould contain only 
 12 or 16 bars ; that is, 3 or 4 ftrains, and the next fucceed- 
 ing part perhaps 20 or 24 bars ; that is, 5 or 6 ftrains ; for 
 we do not commonly advert to any greater portion than 4 bars 
 at once. 
 
 Where the movement is much quicker, as in jigs, <bc, we 
 ftill include a greater number of bars into one ftrain; or, to 
 fpeak more properly, we make up two, three, or four ftrains 
 into one ftill larger parcel, which we call one part of the tune : 
 and here thefe parts, although containing 8, 9, 12, 16, or 
 more bars each, are yet as clofely reftrifted to be each of an e- 
 qual length, or number of bars, as any fingle bar is reft rifted 
 to contain juft the time of fo many crotchets, becaufe the me- 
 mory includes a whole part at one view. 
 
 15. Thefe, and numbei lefs other familiar and obvious in- 
 ftances, which the reader will not fail to fuggeft to himfelf, 
 on many occafions, tend direftly to the confirmation of our 
 new principle, viz. that among the ifochronous fingle vibrati- 
 ons of mufical founds, the mind naturally feeks to conftitute 
 ifochronous compound parcels . And if it fhould be objefted, 
 that although experience plainly proves this to be the cafe with 
 the fhorteft notes ufed in mufic, yet perhaps the vibrations of 
 a found are much too quick for the mind to lay hold on ; and 
 that where the fucceffion of pulfes is much quicker than can 
 be counted, or fenfibly diftinguifhed one from another, as they 
 pafs, which is the cafe with the pulfes of a found, we cannot 
 conceive any preference due to one number above another, be- 
 caufe we fee no poffibility of afcertaining the precife number. 
 
 * The divifion of a crotchet into five equal parts, though not ufed formerly, is introduced by fome of the belt compofers of the prefent age. The 
 works of the light honourable the Earl of Kelly contain feveral inftances of it, denoted by live femiquavers, figured over with a 5 — : neither does it 
 feem to be any more difficult either to execute or to relifh this kind of fubdivifion into 5, than what may jufily be attributed to the want of that habi- 
 tual familiarity which we have contracted with the other fubdivifions. 
 
Chap. I. The Principle of ifochronous compound 
 
 To remove this objection, which feems to be the mod weighty 
 one that can be cart in our way, let it be obferved, that it is one 
 thing for the fenfe to be pleafed with a number, and another 
 thing to count it ; and that we may perceive the equality, or ifo- 
 chronifm, not only of certain compound parcels, but alfo of 
 each fingle pulfe, when a drummer beats much quicker than can 
 be counted; and an inequality among the fingle pulfes.or falfe 
 and irregular time among the larger compounded parcels would 
 difguft us as much in this cafe, nay more than if the fucceffi- 
 on of pulfes were flower. 
 
 The cafe is much the fame in objects which pleafe the fight. 
 There is a certain fymmetry which ftrikes us immediately with 
 delight, in the profpect of a regular and well-defigned piece 
 of architecture ; neither have we any occafion to count and 
 meafure the proportions of the feveral parts, before we can en- 
 joy the profpedl. The pleafure arifes inftantaneoufly from the 
 perception of certain due proportions, though it is a work 
 of time and ftudy to determine exactly what thofe proportions 
 are. This may ferve to apologize for the ufe of numbers in 
 the theory of mufic; becaufe, in all probability, the due propor- 
 tions which exift among the vibrations of mufical founds, are 
 the very things from whence our pleafure is derived ; and thofe 
 proportions are bed examined by means of their conftituent 
 numbers. 
 
 1 6. The riming and parcelling of the beats of a drum is an 
 inftance which approaches neareft to that of the vibrations of 
 a found ; and, I think, we need not defire any thing nearer : 
 for here the quick fucceffion renders it impoffible to count the 
 fingle pulfes, all we can here do is to judge of, and I may fay 
 feel the ifochronifm of each ; and that this is what we can alfo 
 do among the fingle vibrations of founds is plain from Art. 3. 
 becaufe we find a neceffity of rejecting all fuch founds as want 
 this uniformity or ifochronifm. 
 
 If a drummer were to give a number of quick, but equal- 
 timed beats, each with juft an equal force, a perfon attending 
 
 Ff 
 
 Parcels further illuflrated and defended. \ \ ? 
 
 to them would moft naturally parcel them by 4 and 4 toge- 
 ther, by giving a greater regard to every 4th beat; by which 
 regard theie beats would to him acquire a kind of emphafis or 
 accent, as if the drummer did really flrike them harder than 
 the others, which indeed he could fcarce avoid doing, becaufe 
 his own imagination, as well as that of a hearer, would moft 
 readily lead him to this kind of parcelling. But the cafe would 
 be the very fame with a hearer, if an inanimate machine were 
 made to beat the drum. The fame circumftance is alfo well 
 exemplified in the timing of mufic. See Part I. Art. 94, 95. 
 Let it be obferved, that we only fay it is moft eafy and natural 
 for us to parcel fuch equal timed pulfes by 4 and 4 together, 
 when there is no attendant circumftance to influence our choice 
 either one way or another : for it is very obvious, that we can 
 with great eafe imagine thofe pulfes to proceed by 3 and 3 to- 
 gether ; and this we fhould certainly do, if we were to turn 
 our attention from one fett of pulfes, which we had firft heard 
 and mentally parcelled by 4 and 4, upon another fett of pulfes, 
 juft fo much flower as that three of the latter fliould exactly 
 take up the fame time as four of the former ; for here the de- 
 fire of continuing the fame equal meafure of time, would ine- 
 vitably lead us to parcel the latter by 3 and 3. We can alfo, 
 upon occafion, go on further, and conftitute ifochronous com- 
 pound parcels, each confifting either of 4 or of 3 primitive 
 parcels ; but here again it is more natural for us to quadru- 
 ple than to triple our primitive parcel ; and if the fucceffion of 
 pulfes be quick enough, we can proceed yet further, either by 
 quadrupling or tripling the compound parcel. 
 
 17. The well known rules of time in mufic afford us ftill 
 more familiar inftances, for confirmation of the fame principle. 
 Suppofe a piece of mufic to begin in pretty quick common 
 time, fuch as gives us liberty to parcel the bars by 4 and 4 
 together, into ifochronous (trains, (Art. 14.). In this cafe, the 
 time of one (train will be mentally fubdivided into 4 equal 
 parts, or bars, and each of thefe into 2 or 4 equal parts, viz. 
 
H4 
 
 2 minima or 4 crotchets-, and each crotchet into 2, or 4, or 8 
 equal parts, viz. 2 quavers, or 4 femiquavers, or 8 demi- 
 femiquavers : fo that here we have an eflablifhed meafure of 
 time divided into 4 femibreves, or 8 minims, or 16 crotchets, 
 or 32 quavers, or 64 femiquavers, or 128 demifemiquavers; 
 and the fame mental fubdivifion will ftiil take place, if the mu- 
 fic proceed by any of thefe notes, any way combined and inter- 
 mixed. Now, ifthemufic, for the fake of variety, begin af- 
 terwards to move by quavers, figured over with 3, viz. each 
 being the exact third part of the crotchet, (fee Part I. Art. 67 ) 
 which it very frequently does, this makes no alteration in the 
 length of the crotchet, nor in the length of the whole ftrain 
 of 4 bars, but only in the fubdivifion; for now the whole 
 ftrain or meafure of time muft be mentally divided into 4 
 femibreves, or 8 minims, or 1 6 crotchets, as before ; and 
 thefe laft fubdivided into 48 quavers, or 96 femiquavers. 
 We fee then, that in the timing of mufic, we often pafs 
 
 Of Single Mufical Sounds. Part TI. 
 
 from the divifion of a flrain into 64 or 128 equal parts, to 
 another kind of divifion into 48 or g6 parts, and refume again 
 the original divifion at pleafure, and from thence derive a ve- 
 ry agreeable variety. 
 
 18. Before we purfue thefe fpeculations any farther, the 
 reader will not be difpleafed to fee a few ufeful inferences, re- 
 lative to the properties of mufical founds in fucceflion, drawn 
 from what is already pafl. If there be really an analogy be- 
 tween the diftributions which we fenfibly makeamong quavers, 
 and other fliort notes, in the timing of mufic. and thofe which 
 we infenfibly make among the vibrations of mufical founds ; 
 and if the pleafure which refults from the hearing well-tuned 
 founds owe its rife to the fame principle which influences us to 
 be delighted with well-timed notes : then we may, by means of 
 the known practice of dividing time in mufic, bring to light 
 the hitherto unknown principle which guides us in the choice 
 and arrangement of mufical founds. 
 
 Chap. II. O/Musical Sounds in SucceJ/ion. 
 
 Article 19. TI 7 HEN two founds differ from each other by 
 V v the interval of an octave, it is well known 
 that the acuter of them makes two vibrations for one of the 
 graver; and when the interval is a doubie octave, their vibra- 
 tions are 4 to 1; and when a triple octave, 8 to i; and fo on, 
 continually doubling the vibrations of the acuter term, for eve 
 ry octave which is added to the interval : and that the effects 
 of fuch founds are to usfo perfectly alike, that muficians have 
 mriverfaily agreed to call them all by the fame name, and de- 
 note them by the fame letter, and in all refpects to confider 
 
 them as fimilar founds. Now, in the timing of mufic, when 
 a bar, or flrain, contains any certain number of crotchets, we 
 can at pleafure imagine each crotchet to contain two quavers, 
 or 4 femiquavers, or 8 demifemiquavers, without fuppofing 
 any manner of change in the nature of the time ; fo that the 
 very fame bar which contains three crotchets, may, at plea- 
 fure, be imagined 6 quavers, taken by 2 and 2 ; or 12 femi- 
 quavers, taken by 4 and 4 ; etc. without any change in the 
 triple mood of time : and the very fame bar which contains 
 o crotchets, may be 18 quavers, taken by 2 and 2; or 36 fc- 
 
Of the Si 'miliar ity ofOffaves, and the Remembrance of the Key Note. 
 
 Chap. II. 
 
 miquavers, taken by 4 and 4 ; without any change in the no- 
 nuple mood of time. Analogous to this, in the distribution 
 of mufical vibrations into ifochronous parcels, whatever num- 
 ber of vibrations of one found be parcelled together, the vibra- 
 tions of its octave above, taken by 2 and 2, and thofe of its 
 double octave above, taken by 4 and 4, or of its triple octave 
 above, taken by 8 and 8, will each constitute a parcel the ve- 
 ry fame in all refpefts with the firft. So that as the divifion 
 of a crotchet, or other fhort note, into 2, or 4, or 8, makes 
 no alteration in the nature of the time in mafic, fo the dou* 
 bling and redoubling the number of vibrations in a parcel, 
 makes no alteration in the effect of fuch parcel on our fenfe : 
 and this accounts in the mod Satisfactory manner for the well 
 known fimilarity of all octaves, each to other. 
 
 20. Purfuing the fame analogy, the faculty of remembering 
 the key note, and the conftant expectation of returning to it 
 at the conclufion, which' is fo remarkably perceived by the 
 mufician (fee Part I. Article 29.) refolves immediately into 
 that of retaining the idea of a fmall portion of time, divided 
 and fubdivided in fome eligible manner, by the vibrations of 
 the fame key, or its octaves ; and agrees exactly with the re- 
 membrance of the length of one bar or (train, and of the mood 
 of time, in the timing of mufic. 
 
 21. To difcufs this matter more distinctly, we (hall, in 
 this treatife, give the name module to that fmall portion of time, 
 by which we fuppofe the vibrations not only of the key, but 
 alfoof every other found which we admit into our mufic, while 
 we retain that key, to be meafured and distributed into ifo- 
 chronous parcels. 
 
 22. It will eafily be obferved, that whatever fmall portion 
 of time be taken for a module, there may always be a certain 
 pitch or tone of found, whole vibrations (hall divide the fame 
 module in any affignable manner whatever. Thus, for an in- 
 stance in fmall numbers, as one found may divide the module 
 
 115 
 
 into 1 6 equal parts ; fo another found, a little graver, (hall 
 divide the fame module into 1 5 ; another (till graver, into 14; 
 another into 1 3, and another into 1 2 parts ; and fo on down- 
 wards. Thus alfo, in afcending above that found which di- 
 vides the module into 16, there will be found anoiher, a little 
 acuter, which (hall divide it into 17; another, Still acuter, 
 which (hall divide it into 1 8 ; another into 1 9 ; and another 
 into 20 ; and fo on upwards. But then it mutt be remarked, 
 that thofe founds only are of ufe in our mufic which divide 
 the module in certain fimple and intelligible manners, and fuch 
 we fay have eligible parcels. 
 
 23. Thofe founds are alfo moft eafily and naturally admit- 
 ted, which aSTume the moft fimple parcels, viz. fuch parcels 
 as are compounded of the powers or products of the fmalleft 
 numbers, 2 and 3. Other founds are admitted, but with 
 more difficulty, and generally only as dependants, or harmo- 
 nics, whofe parcels involve the higher numbers 5 and 7 ; but 
 thofe whofe parcels involve any of the higher prime, or un- 
 compounded numbers, 11, 13, 17, etc. are totally rejected. 
 
 24. Now, fuppofing our ear to be entirely unbialTed, and 
 not retaining the leaft impreffion of any former heard found, 
 when a mufical found is firft propofed : in this cafe, we (hall 
 moft naturally regard this firft heard found as a principal key 
 note of a (harp feries, and (hall mentally parcel its vibrations 
 by continual reduplication, or by the powers of 2, rather than 
 in any other practicable manner ; and thus we (hall constitute 
 a module divided and fubdivided by continual bifecYion, like 
 the meafure in common time. 
 
 Every ones own experience will fuffkiently teftify, that 
 when the firft found of a tune is not the principal of a Sharp 
 feries, but perhaps a fifth, or a third, or a fecond, or a fixth ; 
 in all fuch cafes, there is a neceSfity of Something of a previous 
 preparation of our mind, by thinking over a few of the firft notes 
 of our intended tune, in order that we may readily fall upon 
 
Of Muftcal Sounds in Succejjion. 
 
 116 
 
 the proper manner of parcelling fuch firft found, as foon as ever 
 it is propofed ; but when the firft found is to be itfelf the prin- 
 cipal of a (harp feries, there is no need of any fuch preparation. 
 
 The church-clerk, or precentor, who begins the pfalm-tune 
 immediately after the founding of a pitch-pipe, will be abun- 
 dantly fenfible of this : for if the intended tune be one of a 
 flat feries, which has the fixth of the natural fcale for its key ; 
 or if it be one of a (harp feries, and yet the firft note be not 
 the principal key ; he will, in either of thefe cafes, find that 
 it requires fome care and attention to begin his tune right: 
 whereas, if the firft note be the key note of a fharp feries, he 
 cannot eafily go wrong. When the tune is ufhered in by an 
 organ prelude, or any other fymphony, thefe difficulties are 
 entirely evaded by the impreffions of the preceding notes. 
 
 Granting that it be natural and neceftary for the purpofes 
 of mufic, to diftribute the vibrations of founds into certain par- 
 cels, there is no doubt but we fhall moft naturally parcel the 
 vibrations of the firft found we hear in the moft eafy and fi tu- 
 ple manner ; for it muft certainly be the prejudice of an al- 
 ready conceived module, which puts us upon the more difficult 
 and complex ways of parcelling, in order to make certain fol- 
 lowing founds commenfurate with thofe which have already 
 preceded them. 
 
 25. It is a curious and interefting fpeculation, to attempt to 
 determine the real number of vibrations which we fhall include 
 in one module, when a found of a determinate pitch is con- 
 ftituted a key ; and this may be done pretty well, by obferv- 
 ing, that the whole compafs of audible found is limited, tho' 
 its limits are fomething different, according to the different per- 
 fection of the organs of hearing. One perfon's ear can efti- 
 mate an acuter found than another's, much like as one per 
 fon's eye can difcover a minuter object than another's. The 
 whole compafs of mufical founds which the ear can eftimate 
 without any help of art, is generally found to be more than 
 
 Part II. 
 
 fix, but lefs than feven fuccefiive octaves. Suppofe it to be 
 feven octaves complete ; as from F, a double octave below the 
 bafs cliff line, to F, a triple octave above the uppermoft line of 
 the treble ftaff at concert pitch. The higheft of thefe F's 
 will then make 1 28 vibrations, (viz. the 7th power of 2) 
 while the loweft makes 1 vibration. Now it is reafonable to 
 imagine, that the vibrations of this higheft F are as quick as 
 the fmalleft mental fubdivifions which we can make in a given 
 module, which may be the caufe why no found ftill acuter can 
 be admitted into our mufic. And on the other hand, it is e- 
 qually reafonable to imagine that no found can be eftimated 
 till three, at leaft, of its pulfes have been felt, becaufe three 
 pulfes only determine two of the fmall intervening particles of 
 time ; and till we have perceived two or more of thefe particles, 
 we can form no judgment concerning their equality ; that is, 
 we cannot eftimate the tone of the found. Therefore, as in 
 attempting to eftimate a very grave tone, we undoubtedly take 
 our module as long as poffible ; fo it is probable that the long- 
 eft module we can take will not contain more than 3 vibrati- 
 ons of this loweft F. 
 
 The conclufions deduced from thefe two premifes are, 
 1. That the longeft module we can take may poffibly contain 
 3x1 28, or 384 fmall fubdivifions, either real or imaginary, 
 but cannot contain more. 2. That the longeft module 
 we can take may poffibly contain 6 vibrations of the double F, 
 which ftands next below the bafs ftaff; or 1 2 vibrations of 
 the F which ftands on the bafs clilf line ; or 24 vibrations 
 of the F which ftands next below the treblecliff line ; or 48 
 vibrations of the F which ftands on the uppermoft line of the 
 treble ftaff, etc. continually doubling the number for every 
 upper octave. 
 
 26. If the above particulars be granted, it will follow, that 
 when the F below the treble cliff line, is heard, and confti- 
 tuted a key, our module may include juft 16 of its vibrations 
 
Chap. II. 
 
 Of the Manner of parcelling the Key, and the fifth of the Scale. j i y 
 
 (for it cannot be fo long as to include 32. ;) and thefe 1 6 we 
 mentally divide and fubdivide like the crotchet in brifk com- 
 mon time, Art. 17. : but it muft be obferved, that the expert 
 mufician is able both to conftitute a longer module, and to de- 
 fcend to more minute fubdivifions than the beginner ; which 
 we find is exactly the cafe in the timing of mulic : fo that 
 where the expert mufician conftitutes a module including 16 
 ■vibrations, and mentally fubdivides each fingle vibration into 8 
 or 1 6 parts, the beginner will perhaps include only 4 vibrations, 
 or 8 at mod, in his module, and will only mentally fubdivide 
 each fingle vibration into 2, or 4 at mod. But if thefe par- 
 ticulars mould be found to want fomething of the undeniable 
 evidence which a phibfopher defires, it is of little importance, 
 as to what follows ; becaufe all the infeience we (hall here 
 draw from them is, that whatever number of real vibrations 
 be contained in a module, an expert mufician can proceed by 
 mental fubdivifions, as far as the 256th part of the whole 
 module; and this will readily follow from the confederation of 
 the fubdivifions of the flrain of mafic, Art. 17th, by obferv- 
 ing, that as the vibrations of mufical founds are vafHy quicker 
 than the fhorteft notes of mufic, fo we have it in our power to 
 parcel together a ftill greater number of them, or their parts, 
 in one module. 
 
 2.7. Aftei the key note, the firft and moft natural change of 
 fouud is into the fifth of the fcale, for the rifing or falling an 
 oclave makes no change. Now, the fifth makes 3 vibrations 
 for every 2 of the key note below it, and confe^uently 3 for 
 every 4 of the key note above it ; and this correfponds exactly 
 with the change from 2 quavers or 4 femiquavers in a crotch- 
 et, to 3 equal quavers, figured over with a 3, taken notice of, 
 Art. 17 th, with this difference only, that there the change 
 from the half to the third part takes place only in the fmaller 
 fubdivifions, viz. the 16th part of the whole flrain, being 
 beft admitted in fhort notes ; whereas, in turning pur atten 
 
 Gg 
 
 tion from the key to the fifth, it is moft probable that the 
 change takes place in the primary divifion of the module ; fo 
 that as the key prefents a module divided into 2, and thefe 
 fubdivided by the powers of 2, fo the fifth prefents the fame 
 module divided into 3, and thefe fubdivided by the powers 
 of 2. This circumflance alone might be fufficient topreferve 
 our new principle from being rejected without further tryal; 
 and, joined with the other particulars already fpoken of, will fuf- 
 ficiently apologize for our endeavouring to purfue the analogy 
 a little further, before we truft its fate to the critical world 
 
 28. In order to make out the whole fcale of mufic, it is ne- 
 ceflary to fuppofe that the fubdivifion into 5 equal parts is 
 alfo eafily admitted, though not quite fo natural as the fub- 
 divifion into 2 or 3; and this will readily be granted fince 
 this kind of fubdivifion is now taking place in our mufic; fee 
 the note to Art. 13th. We muft alio go yet further, and al- 
 low the fubdivifion into feven equalparts, on fome occafions, 
 to be practicable; of which I believe we have as yet no piece- 
 dent in the timing of mufic ; and this gives occafion for fome 
 farther enquiry, whether this kind of fubdivifion might not 
 alfo be introduced there without offence. 
 
 29. It is evident, that when we fee 5 or 7 windows in one 
 row, (Art. 11.) there is nothing too complicated in the idea, 
 but that we readily conceive 5 by its affinity to 4, and 7 by 
 its affinity to 6. This remark will be of farther ufe. There 
 is a perceivably greater difficulty in reconciling the number 1 1, 
 which is the next prime, or undividable number, greater than 
 7 ; fo that, at the firfr fight, 1 1 appears as a kind of irregu- 
 lar confufed multitude, too large to be conceived without fome 
 fubdivifion, and yet not capable of any regular diflribution into 
 equal fmaller parts. This may poffibly have partly been the 
 occafion of the firft arithmeticians flopping at the number ten, 
 and reprefenting the next number eleven, by conjoining tea 
 and one together, becaufe they found it altnoft impolfible clear- 
 
n8 
 
 ly to conceive the number 1 1, all at once, while the other fmal- 
 ler numbers were abundantly fimple and eafy to be underftood. 
 30. This leads us to take notice of a very remarkable dif- 
 ference between the arithmeticians and the muficians methods 
 of conceiving the larger numbers. In the ufual method of 
 expreffing numbers by units, tens, and hundreds, etc. their 
 very names import a mixed idea of feveral diftinct parts con- 
 curring together in one total fum ; and as we conceive an hun- 
 dred by its being ten times ten ; fo when a number is expref- 
 fed in hundreds, tens, and units, we conceive it in three parts, 
 the firft containing fo many hundreds, the next fo many tens, 
 and the laft fo many units; and thefe three we join together, 
 in order to get the idea of the total number, without any re- 
 gard to the factors by which fuch number may be compofed, 
 except it be juft one of the powers of ten, viz. an hundred or 
 a thoufand, etc. In the notation of numbers we regard no 
 other products but fuch as have the number ten, or fome of 
 its powers, for one factor, and one of the fingle digits for the 
 other ; and all numbers which confift of more than one figni- 
 ficant digit, are conceived by piece meals. But in mufic thefe 
 piece meal conceptions have nothing to do : here all large numbers 
 mud be received as compounded of fmall and intelligible fac- 
 tors, or elfe rejected. Thus the number 64 takes place as it 
 is compounded of the fmall factor 2; and 27 is received as 
 compounded of 3, without any regard to the tens and units 
 which they contain. Thus alfo 1 5 is received as being 3 times 
 5 ; and 21, as being 3 times 7 ; and fo of the reft. 
 
 31. Now, though it is impoffiblc to foretel whether ever 
 the fubdivifion into feven equal parts will be introduced in 
 the time of mufic ; yet it is eafy to perceive, that fuch a fub- 
 divifion would be beft admitted in very (hort notes : and that 
 as c equal fhort notes are introduced, where we naturally ex 
 peeled 4 ; fo 7 might probably be introduced where we natu- 
 rally expected 6 : fo that after hearing one crotchet broke into 
 
 Of Mufic al Sounds in Succejfion. 
 
 Part II. 
 
 fix equal femiquavers, figured over with 6, the next crotchet 
 might be broke into 7, upon occafion ; and though this fubdi- 
 vifion (hould be for ever rejected in the timing of mufic, yet 
 it may fafely be fuppofed to take place in the parcelling of mu- 
 fical vibrations, on account of the very quick fucceflion of the 
 latter, which gives us a greater latitude in that refpecl. 
 
 32. We (hall now take a view of the ifochronous parcels of 
 each degree of the natural fcaleof mufic, which, in their frn<il- 
 left numbers are, key 32 
 
 feven th, 30 
 acute fixth, 27 
 fifth, 24 
 grave fourth, 21 
 fundamental great third, 20 
 fecond, 1 8 
 key, 1 6 
 Thefe numbers are fuited to the defcending fcale, in which 
 the grave fourth takes place, as obferved, Part I. Art 33, etc. 
 33. We have already feen, Art. 27th, that after the key 
 note, the moil natural change of found is into the fifth; 
 but when we propofe to defcend by the fmalleft practicable de- 
 grees, in order to make a complete fcale, we take the feventh 
 inflead of the fifth; that is, we take 30 vibrations inftead of 24 : 
 and this prefents a module divided, like the fifth, into 3 ; but 
 each of thefe fubdivided into 5, inftead of 4; or 10, inftead 
 of 8. And thus, as we conceive of 5, by its affinity to 4, 
 Art. 31. fo, while we found the feventh of the fcale, the filth 
 is eftentially implied, and is our fundamental. 
 
 34. The interval between two founds, whereof the acuter 
 makes 5 vibrations for 4 of the graver, is ufually called a great- 
 er third. We (hail here give it an additional epithet, and 
 call it a fundamental great third : becaufe, for the reafons giv- 
 en above, the lower term of fuch an interval will always be e- 
 fteemed the fundamental; and becaufe there are other great 
 
Chap. II. 
 
 thirds, not in the fame proportion 
 not the fundamental?. 
 
 3 5. The next note which we take in defcending, after the 
 feventh, is the acute fixth, which has 27 vibrations, or parts, 
 in the module ; and this we are obliged to refer to the fecond 
 as a fundamental, which has 18 vibrations in the fame time, 
 and to which it becomes a fifth. To account for this, it muft 
 be obferved, that after the module has been heard divided Am- 
 ply into 3 parts, we may, at the next ftep, proceed to trifect 
 each of its third parts ; and, by this means, introduce adivi- 
 fion into 0, which correfponds with the fecond of the fcale : 
 but we cannot eafily reconcile the making of two fuch trifecYi- 
 ons at one flep ; that is, we can fubftitute 3 vibrations inftead 
 of 4; and contrariwife, 4 vibrations inftead of 3: but we 
 cannot immediately fubftitute 9 vibrations inftead of 8, nor 8 
 inftead of 9 ; which is exactly the cafe in the timing of mu- 
 fic alfo. Now, the acute fixth has 9 vibrations for 8 of the 
 fifth, which was the laft object of our attention : therefore, 
 though we take this note, we are obliged to regard it only 
 as one of the harmonics of the fecond, which has 3 vibrations 
 for 4 of ihe fifth ; and which, on that account, becomes our 
 fundamental : fo that here we have the module divided into 9 ; 
 and though each of thefe is really again fubdivided into 3, yet 
 by refering to another fundamental, which has the fame ninth 
 parts fubdivided only into 2, we reconcile this divifion with 
 the fimple trifection which preceded, and which muft alfo fol- 
 low in the next note. 
 
 36. The next note is the fifth itfelf, which we naturally 
 take as a fundamental, and thereby reftore the fimple trifection 
 of the module. The fame fifth, in other fucceflions, is often 
 refered to the key as a fundamental ; but this we cannot here 
 do, becaufe it would require the palling from a module divid- 
 ed into 9, to one ftmply bifeited, which we have already feen 
 is unnatural. 
 
 The natural defcending Scale afcertained and examined. 
 whofe lower terms are 
 
 II 9 
 
 37. Next follows the grave fourth, which divides the mo- 
 dule into 21 parts, viz. into 3, and each of thefe into 7. 
 Now, according to the analogy formerly laid down, thefe 7 
 parts (hould be admitted inftead of 6; and confequently the 
 grave fourth, which is 21, necefTarily implies the fecond, 
 which is 18, to which it ftands as a lefs third ; and this would 
 have been our fundamental, if the fucceffion would have ad- 
 mitted it; but perceiving that the third of the fca.'e is next to 
 follow, which having 5 vibrations for 4 of the key, will ine- 
 vitably require the key for its fundamental, Art. 34. and that 
 if we here take the fecond as fundamental, we cannot take the 
 Kf immediately after it : we therefore refer the grave fourth, 
 and its implied fecond, to the fifth fundamental which exift- 
 ed in the preceding note, and to which the fourth becomes an 
 added 7th, as the fecond is its 5th. 
 
 38. The third, which next follows, takes the K/, and the 
 fecond neceffarily takes the $f, as is obvious from the preced- 
 ing conliderations : and this arrangement of fundamentals to 
 the defcending fcale is exactly agreeable to the practice. See 
 Part I. Art. 216. and Exam. XLII. 
 
 39. Inftead of taking up our time in examining any othec 
 particular paflages of melody, we mayobferve from the above, 
 that it is not in the fucceflions of notes themfelves, but in the 
 fundamental progreflions, that the analogy exifts between the 
 diviCons of the time in mufic, and the divifions of the mo- 
 dule : and fo it ought to be, as we have largely fhevn in the 
 fir ft part of this treatife that the fundamental progreffion is 
 the bafis both of harmony and melody, and that every finger 
 conforms his founds to certain fundamental progreflions, only 
 diverfifying the firain, and rendering it more melodious, by 
 the occafional ufe of the harmonics, inftead of the fundamen- 
 tals themfelves. We may alfo obferve, in the defcending 
 fcale, that the founds which include the number 5 in the 
 compofition of their parcels naturally become great thirds, and 
 
Of Muftcal Sounds in SucceJJJon. 
 
 the found, which includes the number 7, will be either a lefs 
 third, or an added feventh ; while the real fundamentals 
 themfelves include only the numbers 2 and 3 ia the compofition 
 of their parcels : (o that the introduction of the more complex 
 fubdivifions of the module in mufic may be attributed to the 
 liberty of taking certain more difficult founds, as harmonics to 
 a more eafy fundamental *. 
 
 40. It will therefore be fufficient, in this place, to obferve, 
 that the preference due to fundamental cadences is a neceffary 
 confequence of this new principle ; for the regular cadence, in 
 the fundamental bafs, always implies a change from trifection 
 tobifeclion, either of the whole module, or of fome determi- 
 nate part of it ; and that, on the contrary, the irregular ca 
 dence implies a change from bifccVion to trifecVion : and that 
 to determine upon u hat degrees of the fcale thefe cadences 
 may fall, it is neceffary to examine what degrees can be form- 
 ed by trifections only, and how oft this may be repeated ; for 
 all thofe founds, whofe parcels can admit of another infecti- 
 on, have their perfect fifths exifting in the fcale, and therefore 
 may be made fundamentals : but that note, where we can tri- 
 fect no farther, cannot have its perfect fifth, and therefore 
 cannot be a fundamental. 
 
 4i.Now,from the principles alreadylaid down, it will follow, 
 that the higheff power of 3, which we can admit in the par- 
 celling of mufical vibrations, is its fifth power 243, fee Art. 24, 
 25, and 26. ; for the fixth power, which is 729, is unquefti- 
 onably beyond our limits. The founds reprefenting each of comma. 
 
 * That fome founds are move difficult to be taken than others, will not be denied by any who has tried to fing at all : and this difference is entirely 
 owing to the different degrees of iinplicity in the fubdivifion of the module; for if a finger do not peiceive and relifh the proper lubdivifion of the mo- 
 dule, he cannot found the note with any fatisfadtion. Thus although the defcending fcale, as defcribed above, is perfectly eafy and natural to a profi- 
 cient, there are many beginners who cannot give the grave fourth its proper found ; and fome who cannot relifh either the feventh, or the third of the 
 icale, for want of being habituated with the fubdivifions into j or 7, which thefe founds imply. And the belt way to help a beginner over thefe diffi- 
 culties is to let them frequently hear the proper fundamentals fubjoined, by way of a bafs, 
 
 Part II. 
 
 the powers of 3 are eafily found, by proceeding from the key 
 by fucceffive fifths afcending, or fourths defcending; thus, the 
 key itfelf reprefents a continual bifeclion, or a divifion by the 
 powers of 2. The fifth of the fcale reprefents a tingle infection. 
 The fecond reprefents a compound trifeclion or divifion into 9 : 
 and the fixth reprefents a divifion into 27. Thus far we agree 
 with the numbers in the fcale, Art. 32; but we muft go 
 yet further, and fuppofe the third of the fcale to reprefent a 
 divifion into b't ; and the feventh of the fcale a divifion into 
 243. This third and feventh differ a little from thofe in the 
 fcale, as will be difcovered by comparing their numbers toge- 
 ther t and in order to do this, we may freely double and re- 
 double the fmaller numbers, to bring them nearer to the larg- 
 er ; this being only the taking of the fame found fome octaves 
 higher, or rather only enumerating the fmaller fubdivifions of 
 the module, which exift in the fame found. See Art. 26. Thus 
 the fundamental great third of the fcale, which, in Art. 32. 
 is expreffed by 20, may alfo be exprefled by 40 or 80 ; and 
 the third which arifes from continual trifeclion, as above, is 
 exprefled by 81 : fo that this latter is acuter than the former, 
 in the proportion of 81 to 80, which fmall difference is called 
 a comma. After the fame manner the feventh of the fcale, 
 which, in Art. 32. is exprefled by 30, may be redoubled to 
 240; and the feventh which is produced by continual infecti- 
 on, as above, is 243 ; being alfo acuter than the former in 
 the fame proportion of 243 to 240, or 81 to So, which is a 
 
Of fundamental Cadences } and of the Variation of a Comma in form Sounds. 
 
 Chap. II. 
 
 42. The inferences we (hall, in this place, draw from the 
 preceding obfervations, are, 
 
 (1.) The key, the 5th, the 2d, and the 6th of the fcale 
 are the eafieft and mod natural founds; and in the tone of 
 thefe, as they occur in the defending fcale, all who have any 
 tafle for mufic perfectly agree. But there are fome, not de- 
 llitute of a natural turn for mufic, but unimproved by ac- 
 quaintance with the modern ftyle of harmony, who, not con- 
 ceiving any thing of the agreement of a great third ; that is, 
 not being accuftomed to conceive 5 by its affinity to 4, admit 
 no fuch proportion among the founds of their fcale ; but make 
 their third a comma too acute, and alfo their feventh ; which 
 lair they admit but rarely, on account of its parcel being ra- 
 ther too complicated for them ; and fuch fingers have no fourth 
 at all. This is the true origin of the genuine old Scots tunes, 
 fuch as have undergone no changes by the interpolations of 
 foreigners : and it is not eafy to join a moving bafs to one of 
 this fort, becaufe every note muff either be a fundamental, or 
 a fifth, and this affords not fufficient variety to entertain a 
 modern ear : and the joining of a fecond treble to a tune of 
 this fort is'ftill more impracticable, becaufe the fundamental 
 great third cannot take place at all. 
 
 (2 ) Though the modern finger can, on occafion, take his 
 third in the relation of 5 to 4, viz. a fundamental great third 
 to the key ; and his feventh in the fame relation to the fifth ; 
 and though he generally does fo, in order to confine his fun- 
 damental progreffion more clofely to the original key, and the 
 founds which are neareft related to it ; yet he can alfo, with 
 the greatefl eafe, take his third as a perfect: fifth above the 
 6th f; and, in this cafe, it will be a comma more acute than 
 ordinary : and he can take his feventh a perfect fifth above 
 this acute third, as a fundamental; which will alfo render it 
 a comma more acute than ordinary. For in thefe he does no 
 more than what nature fuggefts to the mod uncultivated ling 
 
 H 
 
 121 
 
 er. — By this means we fee how the fixth and third of the 
 fcale may alfo take place in a fundament.il bafs, and that the 
 feventh of the fcale cannot be admitted there. 
 
 (3.) Thefe different ways of conceiving the third and 
 feventh of the fcale may tfleclually take place, although no 
 alteration be made in the real pitch of the founds. When the 
 difference of two founds is only a comma, we readily take one 
 inflead of the other, in a fucceflion of fingle founds, or melo- 
 dy •. which may be accounted for by observing, that in cafe 
 our imagination Ihould lead us toconflitute a parcel of Hi, by 
 repeated triplication, though, at the lame time, 80 vibrations 
 of the iound which flrikes our ear, would fill up the whole 
 time of our module, yet we may vary as much in the length 
 of our module, as to include 81. without perceiving any ine- 
 quality ; juft in the fame manner as in the timing of mufic, 
 one bar may be, and always will be iome very fmall matter 
 longer than another, perhaps much more than one eightieth part 
 of the whole bar, and yet we are fenfible of no impropriety, 
 if each bar be properly divided.— The mind allows not the 
 leaft deviation from the proper method of dividing and fubdi- 
 viding each parcel ; but the equality of the whole fucceflive 
 parcels, being determined by the fenfe, is not fo perfectly efli- 
 mated. Hence it is that iounds rot exactly but nearly in the 
 proportions of the fcale ate. freely admitted into our mufic, 
 and produce the fame effects as if they were perfectly propor- 
 tioned, becaufe we adapt our module to them without any 
 
 fenfible inequality : and hence the expert mufician becomes 
 more critical in regard to the tune of founds than the beginner, 
 becaufe he is more critical in regard to the equality of the 
 time of fucceffive modules. Hence alfo the finger who, in the 
 courfe of his performance, lofes or gains fomething in the real 
 pitch of his key note, is like a mufician who plays fomething 
 flower or quicker time tovards the conclufion, than at the 
 beginning of a movement. 
 
 h 
 
Of Mufical Sounds- in Succejfwn. 
 
 Part II. 
 
 43, It remains yet to be faewn, how the perfect fourth of 
 the Scale is introduced into our melody, of which we. have 
 hitherto feea no instance, and which neveithelefs takes place 
 in our modern mufic almoft as eafily, though not fo frequent- 
 ly as the fifth. In order to this it is moft natural to SuppoSe* 
 that after a very little acquaintance with the modern ftile of 
 mufic, we can occafionally distribute the vibrations of the key 
 into 3 parcels, each containing 8, or 16, or 32 Smaller parts;, 
 which three parcels together constitute a module, Simply tri- 
 Sected, like the bar in triple time, or like as we Suppoled the 
 fifth of the fcale to be parcelled, in the foregoing articles. 
 And it will readily be granted, that as triple time, after a 
 very little practice, becomes as eligible and as eafy as common 
 time, fo may alfo this method of parcelling the vibrations of 
 the key become as eligible as that formerly defcribed : and in 
 this way the module will contain 24, or 48,, or 96 vibrations 
 or parts of the key note. 
 
 44. See now the ifochronous parcels of each degree of the 
 afcending and defcending fcale of mufic, according to this way 
 of conceiving the key. 
 
 key, 
 
 Seventh, 
 
 acute Sixth, 
 
 fifth, 
 
 perfect fourth, 
 
 fund, great third, 
 
 Second, 
 
 key, 
 
 key, 
 
 96 
 
 Seventh, 
 
 9° 
 
 acute Sixth, 
 
 81 
 
 fifth, 
 
 72 
 
 grave fourth, 
 
 63 
 
 fund, great third, 
 
 60 
 
 Second, 
 
 54 
 
 key, 
 
 4 3 
 
 The intervals of this deScending Scale agree exactly with 
 thofe of Art. 32. for the numbers here are each juft triple of 
 the other ; and the only difference between this afcending and 
 defcending Scale is in the fourth, which is 64 in aScending, and 
 
 only 63 in defcending : that is, the perfect fourth is acuter 
 than the grave fourth in the proportion of 64 to 63 : which 
 Small interval, not having formerly been fuppofed to exift a- 
 mong mufical founds, has no eStabliShed name ; we Shall there- 
 fore call it a bearing ; not chufing to borrow from any other 
 language the, name of an interval which a Briton firfr. intro- 
 duces. 
 
 45.: In every paSfage of mufic our attention is partly divid- 
 ed, either between the key and its adjunct fifth, or between 
 the key and its adjunct foUfth, as fundamentals. Now, when 
 the 5th takes place along with the key, we take the method 
 firft defcribed of parcelling the key by continual reduplication, 
 like the meafure in common time ; but when the fourth at- 
 tracts a part of our regard, we afSume a module of a different 
 length, comprehending either three halves or three fourths of 
 the other module, and which contains 24, or 48, or 96 parts, 
 as in the la.ft article: and by conceiving the principal key oc- 
 casionally ip one or the other of thefe ways, the whole fyftem 
 of modulation may be eaSily and Satisfactorily aScertained ; of 
 which we Shall Shortly take a minute view. 
 
 46. The moft material objections which may here be made 
 to this theory, Seem to be Such as theSe. 
 
 Objeft. 1. Though it muft be acknowleged, that if the men- 
 tal diftributions of the pulfes of mufical Sounds into iSochro- 
 nous parcels be granted, as above deScribed, the Similarity of 
 octaves, the remembrance and expectation of the key, the 
 origin of the whole natural Scale, the dependence of the more 
 difficult founds on other more eafy fundamentals, and the pre- 
 ference due to the paSSages called Sundamental cadences are 
 well accounted Sor ; yet what need is there of entering into 
 abftrufe. Speculations about the diviSions and Subdivisions of an 
 imperceptibly Small portion of time called a module? which, 
 though they may take place, yet we never can be aSSuredof it; 
 nor can we Support the hypothcSis otherwiSe than by its ana- 
 
Chap. II. 
 
 Of the Introduction of the per feB fourth. Objections anfwered. \ 2 3 
 
 ing 1 a perfect fifth above, and the other a perfect fifth 
 
 logy with the choice of the mind in other cafes, which feem to 
 bear a refemblance to it. When the mod part of thofe con- 
 clufions may be drawn from circumftances which are obvious 
 to every one, and of which the ear is a fufficient judge. 
 
 Now to enter into an impartial examination of the fufficien- 
 cy of the principles formerly adopted, would require us to ex- 
 plain thofe principles more fully than is confiftent with our 
 prefent defign ; we fhall therefore, in anfwer to the above, 
 content ourfelves with making a few remarks, addrefied to fuch 
 readers as are acquainted with, and prejudiced in favour of 
 the theories formerly given. 
 
 "(i.) The perfect fimilarity of all odlaves has never before 
 been fatisfactorily accounted for. If the perfection of the con- 
 cord between two founds be taken as the reafon of their fimi- 
 larity, then the fingle oclave above or below any found, will 
 have the greateft degree of refemblance, as it really has ; but 
 then the twelfth above or below, being a more perfect concord 
 than the double octave, ought to have the next degree ; and 
 the greater feventeenth ought to be more fimilar than the triple 
 octave. Now the greater feventeenth, v. g. the greater third, 
 below the key, is a lbund which does not enter into our fcale at 
 all, and the greater feventeenth, or greater third, above the key, 
 has no fort of refemblance to the key : befides, from this way 
 of reafoning, it would follow, that no two founds are perfect- 
 ly fimilar, except they be unifons. No other property of founds 
 has been examined into, in order to account for this moft obvi- 
 ous circumftance of the fimiliarty of all octaves ; but the fact 
 has been taken for granted, as ellablifhed by experience, and 
 the univerfal confent of muficians. 
 
 (2.) In regard to the origin of the feven founds, which con- 
 ftitute our natural fcale of mufic, thefe have been deduced 
 from the perfect chords of the key, and of its adjunct 5th, and 
 4th, as fundamentals; and the preference due to thefe chords 
 has been attributed to their near relation to the key ; one be- 
 
 beldw the key. Thus the K, 3d, and 5th belong to the 
 chord of the K/; the 5th, 7th, and 2d belong to the 
 chord of the 5th/; and the 4th, 6th, and K belong to the 
 4th f. So that all the feven founds are fuppofed to exifr in 
 thefe three perfect chords : but then, what becomes of this 
 fuppofition,when it appears that the fourth itfelf is frequently 
 a deficient, or grave fourth, and therefore more than a perfect 
 fifth below the key; and that the fixth is frill more frequent- 
 ly an acute fixth, and therefore more than a great third above 
 the perfect fourth ? 
 
 If, in order to affign a more fatisfactory origin to the real 
 founds of the afcending and defending fcale, we purfue the 
 enquiry into the proper fundamental progreffions, the added 
 difTonances, etc. as in Chap. VI. VII. and VIII. of Part I. 
 we may at laft obtain the true fundamentals of thefe founds, as 
 in Art. 216. but then the queftions why fuch particular notes 
 enter into each chord, and why fuch and fuch cadences are 
 admitted, remain ftill to be decided by the Theorift. See 
 Art. 201. 
 
 (3.) If we endeavour to account for the preference due to 
 cadences above all other pafTages in the fundamental bafs, by 
 the near relation of any note to its fifth above or below, we 
 are (o far in the right; but then what is. the caufe of this 
 near relation ? and how fhall we determine the affinity of 
 founds which may properly fucceed each other in a funda- 
 mental bafs ? If the degree of perfection of the concord be- 
 tween two founds be affigned as the caufe, then it will follow, 
 that the leap of a great third, which is the belt concord, except 
 the octave and 5th, ought to be the moft eligible fundamental 
 pafTage, except the cadences, which is quite contrary to ex- 
 perience ; for the leap of a great third is feldom admitted, 
 while leaps of lefs thirds, and fteps of feconds are very common. 
 
 (.4.) Not to dwell longer on the infufHciency of principles 
 
12 4 
 
 formerly received, than what feems abfolutely neceffary in de 
 fence of our new principle againft one fort of readers, it ftiall 
 fufHce at prefent to afk, whether it be more incredible to fup- 
 pofe, that the mind meafures the proportions of rounds by di- 
 flributing them into ifochronous parcels, and perceiving the 
 formation of fuch parcels, as d^fcribed above; or to fuppofe 
 that thofe proportions are perceived without any fuch mental 
 diftribution, by fome other unknown way ? It being a well 
 known fact, that in the cafe of imperfect unifons, the 
 ear perceives a better agreement when the real coincidences 
 happen feldomer ; and it is therefore more than probable, 
 that the real coincidence of the vibrations is not a neceffaiy 
 circumftance even in fuch concords as the ear approves; 
 much lefs in fuch founds as are heard in fucceffion only, 
 and compared with other founds which have no exiflence but 
 in the imagination, as fundamentals. Since then it is not by 
 perceiving the coincidence of vibrations that we get the idea 
 of thefe intervals, can any other way be thought of more fimple 
 than ours, and more analogous to the known choice of the 
 mind in other like inftances : and if the principle which we 
 have adopted be as fimple as any other, the Conclufions refult- 
 ing from it are much more fo. Nothing can be imagined more 
 familiar than to reprefent the fcale of mufic by a like fcale of 
 natural numbers; and to a fee r tain the properties of each found 
 by thofe of its correfpondent number. 
 
 47. Objetl. 2. If this method of conceiving the vibrations 
 of founds as meafured by an eftablifhed module do really take 
 place, one would imagine, that *-hs proportion between the 
 reai time of that module, and the time of one of the fhorteft 
 notes of the mufic lhould alfo rv perceived; and that this 
 would in fome manner regulate the real time of the mufic : for, 
 as the pitch of the key note determines the length of the mo- 
 dule, fo the length of the module fhould alfo be exactly fome 
 proper part of the length oi the fhort note. Thus, for in- 
 
 Of Mufecal Sounds In Succe felon. 
 
 Part II. 
 
 fiance, one would imagine that the femiquaver containing 2, 
 or 3, or 4, or 6, or ti, or 9 modules, ought to be moft na- 
 tural ; the fame note, containing 5 or 7 modules, ought to be 
 lefs natural, and containing any other number, not admitted 
 in the parcelling of mufical vibrations, ought to be quite in- 
 tolerable : and if this were the cafe we fhould have a deter- 
 minate number of moods, or degrees of quicknefs of time, be- 
 longing to any given key-note, in which all muficians would 
 agree as perfectly as they do in the degrees of found, which 
 conftitute the natural fcale of a given key. But we find, by 
 experience, that we are under no fuch reftrictions in the tim- 
 ing of mufic, and therefore fuTpect the authenticity of a prin- 
 ciple, which muft be fuppofed to be ftrictly regarded in the 
 parcelling of the fingle vibrations in a module, and difregard- 
 ed in the comparifon of the module with one of the fhort 
 notes, and again ftrictly regarded in the parcelling of the 
 fbort notes into bars and ftrains. 
 
 In anfwer to this objection, we obferve the following par- 
 ticulars. 
 
 (1.) That it is fuppofed to be enough for the mind to per- 
 ceive the divifions and fubdivifions of any fingle module, with- 
 out going on to compound feveral modules together, in order 
 to make up the time of a fhort note in mufic ; and that we 
 are as little concerned about the number of modules in one 
 note, as we are about the number of 4 bar ftrains in one part 
 of a march or minuet, Art. 14. Neither does it make any 
 material difference though a fmall portion of unmeafured time 
 fhould intervene between the lafl: module in one note, and the 
 firft in the following note, this being like an unmeafured 
 paufe at a double bar, fee Part 1. Art. 109. and this accounts 
 fufficiently for the liberty we take in making the time of a 
 note of mufic any length we pleafe, provided it be proved 
 that the fhorteft note which enters into our melody will con- 
 tain more than one module. 
 
Chap. II. 
 
 Objections anfwered. 
 
 \1$ 
 
 (2.) In regard to the real length of one module we may ob- 
 ferve, that if the longeft module we can take include 1 2 vi- 
 brations or pulfes of the note F on the bafs cliff line, at con- 
 cert pitch, fee Art. 25. the fame note F is found to make 
 about 320 pulfes in a fecond [Mr. Emerfon, in his Fluxions, 
 Seel. III. Prop. II. determines the noteE, a femitone below, 
 to make 300,] it will follow, that the quotient of 320 divid- 
 ed by 1 2, viz. 26 and two thirds, will be the number of mo 
 dules in a fecond, when they are the longeft poffible. And 
 if, as we have afterwards fuppofed, Art. 26. the fame note 
 F be conftituted a key, and 8 of its pulfes included in the 
 module, we fhall then have 40 modules in a fecond. Now, 
 the fhorteft note which enters into our melody, even in a 
 fhake, can fcarce be lefs than the fixteenth part of a fecond, 
 and will therefore contain more than two of thefe modules. 
 
 (3.) Mr. Emerfon in the inveftigation and experiment here 
 refered to, reckons the complete vibration to be the time in 
 which a founding firing moves from one fide to the other, 
 like as we alfo reckon the vibration of a pendulum : but 
 Dr. Smith, in his Harmonics, reckons the complete vibration 
 to be the double of this, viz. the time in which a firing 
 moves forward and back again ; and in Prop. XXIV. deter- 
 mines the note D, in the middle of the open diapafon of the 
 organ at Trinity-College, which is the note below the treble 
 ftaff, to make 262 fuch vibrations in a fecond : and upon this 
 he builds his directions for tuning a tempered fyftem of 
 founds. Now, this D, being about the Roman concert pitch, 
 and more than a femitone below the prcfent opera pitch (Har- 
 
 monics, Prop XX. Schol. 1. and Appendix to the fecond 
 edition, p. 271.) will be more than a great third below the 
 lower F of the treble. fbff. at the opera pitch : and the bafs 
 D an oftave below, which has 131 vibrations in a fecond, 
 will be more than a great third below the F, which is on the 
 bafs cliff line. Suppofe it exactly a great third, then fay, as 
 4:5:: 131 : 163.75, the complete vibrations of Fin 
 this cafe : which is a little more than the half of the above 
 number, refulting from Mr. Emerfon's experiment; owing, 
 very probably, to this caufe, that the virginal with which 
 Mr. Emerfon made the experiment has not been tuned quite 
 fo high as the prefent opera pitch. 
 
 (4.) Granting, according to Dr. Smith, that the ear per- 
 ceives only one vibration during both the courfe and recourfe 
 of a founding firing; it will ftill follow, that the fhorteft 
 note which enteisinto our melody will contain more than one 
 module, which is all we want to prove. 
 
 (5.) Though we can take the liberty of making the time of 
 a note any length we pleafc, yet ftill there is a preference due 
 to certain degrees of quicknefs, when the key note is afcer- 
 tained, and to certain keys, when the degree of quicknefs is a~ 
 fcertained, of which the moft experienced muficians are a- 
 bundantly fenfible ; and which peihaps will never be account- 
 ed for, otherwife than by a critical enquiry into the real length 
 of the module of every key, and its proportion to the time of 
 a fhort note. But this we leave to fuch as have more leifure, 
 and more experience in the practice. 
 
[ »<$ ] 
 
 C h a p. III. O/Harmonical Arithmetic 
 
 Article 48.T) E FO R E we proceed to examine the nature and 
 jLJ effect of combined founds or chords, it is ne- 
 ceffary to lay down a few rules for the management of num- 
 bers, in computations relative to mufical intervals ; or afmall 
 fketch of Harmonica! Arithmetic. This will neither be long 
 nor intricate with us; becaufe, in the way which our new 
 principle leads us to conceive and exprefs the founds of mufic, 
 we have little to do with the confideration of arithmetical, or 
 geometrical, or even hannonical proportion ; and ftill lefs 
 with the properties of geometrical magnitudes, which have 
 been needlefsly introduced in alrooft all the theories of mufic 
 .extant, to the great perplexity, but fmall improvement of the 
 readers. 
 
 40, The fize or magnitude of a mufical interval is eflimat- 
 ed by the inequality of the ifochronous parcels of vibrations of 
 its two terms. Thus when there is.no inequality in the ifo- 
 chronous parcels of two or more founds, thofe founds in- 
 clude no interval, and are called vnifons : and in proportion 
 as their ifochronous parcels differ more from equality, the 
 included interval becomes greater. 
 
 This proportion of the inequality of two terms i? called 
 their ratio. An expreffion much more frequently ufed than 
 clearly under ftood. We {hall not need to trouble ourfelves 
 with any other than the plaineft and moft fimple idea of it, 
 which the practice of the rule of three conveys. 
 
 When we fay in the rule of three ; as the firft term is to 
 the fecond, lb is the third term to the fourth, we mean that in 
 
 what manner foever the whole, or any affignable part of the 
 firft term is contained in the fecond ; in the fame manner muft 
 the whole or the like part of the third term be contained in 
 the fourth : or, in other words, we mean that the quotient 
 of the firft term divided by, or dividing, the fecond, muft be 
 the fame with the quotient of the third term divided by, or 
 dividing, the fourth : and when this is the cafe, we fay the 
 ratio of the firft to the fecond is the fame as the ratio of the 
 third to the fourth term. 
 
 This is obvious, that the equality of the quotients refulting 
 from a divifion of the terms, denotes an equality of the ratios 
 of fuch terms : and that the greater or lefs quotient denotes 
 a greater or lefs ratio, and confeqnently a greater or lefs 
 mufical interval, which will ferve all our purpofe at prefent. 
 
 50. Thofe who have a mind to look farther into the ana- 
 logy between the magnitude of mufical intervals, and that of 
 the ratios of the ifochronous parcels of their terminating 
 founds, will find, that'although the ratio increafes when the 
 quotient increafes, and vice verfa ; yet the augmentation of 
 the ratio does not keep due pace with that of the quotient, fo 
 as to remain always in one certain proportion with it. Sup- 
 pofe the terms equal, and therefore the quotient unity, the 
 ratio is nothing, and" the mufical interval is alfo nothing. — if 
 the terms be. as 1 to 2, and the ratio, or, which is the fame 
 thing, the interval be called one (fay one octave), then the 
 terms 1 to 4 will include a double interval (two octaves) ; the 
 terms x to 8 will include a triple interval (three octaves) ; and 
 
Chap. III. Mufical Intervals and Ratios compared. 
 
 i to 1 6 a quadruple interval; and fo on: where the lefs 
 term being always unity, .the greater termisalfo the quotient; 
 and it is obfervable, that for every one ottave which the in- 
 terval increafes, the quotient of the terms becomes the double 
 of what it was. The quotients increafe by a redoubled geome- 
 trical progreffion, i. 2. 4. 8. 16. fac. while the intervals in- 
 creafe only in the natural order of the numbers, 1 . 2.3. 4. <bc. 
 Thus it will be, whatever other interval we pitch upon, in- 
 ftead of the odtave. The double interval will always exift 
 when the quotient of the terms becomes duplicate, that is, the 
 fquare of that which arifes from the terms of the fingle inter- 
 val ; and the triple interval will require a quotient triplicate, 
 or the third power of the firft, and fo on. 
 
 Thus the progreffion of intervals, and their correfpondent 
 quotients, exactly anfwers to that of Logarithms, and their 
 correfpondent natural numbers; as thofe who underftand the 
 nature of logarithms will immediately perceive from the above. 
 
 The line of numbers, called Gunter's line, is flill more ea- 
 fily applied to the ufe of attaining proper ideas of the magni- 
 tudes and proportions of intervals : for here the real magni- 
 tude of every interval is reprefented by the diftance between 
 the two numbers which exprefs its terms ; fo that nothing 
 could be devifed more fuitable for this purpofe. 
 
 51. The mufical interval, or ratio, between two numbers, 
 is not altered by multiplying or dividing both its terms by a- 
 ny one number whatever ; the effe£t of which is only making 
 the terms larger or lefs, but flill retaining the fame propor- 
 tion : therefore a mufical interval may be reduced into fmaller 
 numbers by divifion, if both its terms have any common di- 
 vifor. Farther, as all harmonic numbers mult be compofed 
 of the fmall primes 2, 3, 5, and 7, therefore a mufical inter- 
 val will be reduced into its fmalleft numbers by firft dividing 
 both its terms by 2, or fome power of 2, continually, as often 
 as this can be done without any remainder ; afterwards by 3, 
 
 Reduction of their 'Terms, Addition. \ 2 7 
 
 or fome power of 3, as often as this can be done ; and after- 
 wards by 5, or 7, if either of thefe will divide both terms. 
 
 For example, the interval between the key and feventh, 
 which, in Art. 44th, is 96 to 90, by dividing both terms by 2, 
 becomes 48 to 45 : thefe cannot both be divided by 2 any more, 
 but divided by 3, give 1 6 to 15; which numbers we can re- 
 duce no lower, becaufe they have no common divifor ; there- 
 fore the interval, or femitone, between the key and the fe- 
 venth is exprelTed, in its fmalleft numbers, by the ratio of 
 16 to 15. 
 
 This procefs of reducing ratios to their fmalleft numbers, 
 we fhall not hereafter infert at length, becaufe our readers 
 may be fuppofed well enough acquainted with it, from its 
 correfpondence with the method of reducing vulgar fractions 
 to their leaft terms : and fuch as are not, need only to follow 
 the above directions, which are fufficient for all cafes that can 
 occur among harmonic numbers. 
 
 52. Addition of intervals teaches how to find an interval, 
 which (hall be equal to the fum of two or more given ones, and 
 anfwers exasftly to addition, or rather compofition, of ratios. This, 
 when the greater term of one interval is the fame with the lefs 
 term of another to be added ; or, which comes to the fame thing, 
 when the intervals to be added lie contiguous one to another in 
 the fcale, is no more than comparing the ifochronous parcels of 
 the extreme terms one with the other. Thus, the interval be- 
 tween the key and third, 48 to 60, added to that between 
 the third and fifth, 60 to 72, evidently makes the interval 
 between the key and fifth, viz, 48 to 72 ; which reduces to 
 that of 2 to 3. 
 
 53. Any two intervals maybe added, though .the greater 
 term of one be not the fame with the Ie(s term of.jthe other, 
 in the given expreffions ; but then they muff, firfl kp reduced 
 to this form, by feeking a common multiple of thefe two 
 terms, and multiplying both terms of one ratio by fuchanum- 
 
128 
 
 Of Ha 
 
 'leal Arithmetic. 
 
 Part II. 
 
 ber as will raife its greater term to that multiple, and both 
 terms of the other ratio by fuch a number as will raife its lefs 
 term to the fame multiple ; after which their fum will be had 
 by comparing the extremes, as before. This procels agrees 
 exactly with multiplication of vulgar fractions, confidenng 
 the lefs term of each ratio as a numerator, and the greater 
 term as its denominator; but as the addition of intervals 
 which are not to be found contiguous to each other in fome 
 part of the fcale, ferves only to find expreffions of unnatural 
 or inharmonical relations, wefhall not need to infill upon this 
 particular. 
 
 54. The addition of one or more octaves to any given in- 
 terval is performed by halving the lefs, or doubling the great- 
 er term, for every octave to be added. Thus the interval of 
 a perfect fifth being 2 to 3, that of an octave and fifth, or a 
 double fifth, fee Part. I. Art. 130. is i,to 3 ; and that of a 
 triple fifth is 1 to 6. Again, the interval of a fundamental 
 great third being 4 to 5, that of a double great third is 2 to 
 5 ; and of a triple great third is 1 to 5. 
 
 55. Subtraction of intervals teaches how to find an 
 interval which fhall be equal to the difference of two given 
 ones; and anfwers to fubtraction, or rejbluiion, of ratios. 
 Here, if either the greater or the lefs terms be the fame in both 
 expreffions, or if one interval makeeither an upper or lower part 
 of the other in the fcale, the difference will be had byinfpecti- 
 on : thus, to find the difference between a fundamental great 
 third and a perfect fifth ; as the former of thefe is 48 to 60, 
 and the latter is 48 to 7 2, in the fcale ; fo their difference is 
 60 to 72, which reduces to 5 to 6. 
 
 56. The difference of any two intervals may be found, 
 though neither the greater nor the lefs terms be the fame in 
 the given "-ixpre/fions, by a previous reduction, like that in 
 addition, Art. 53. Thus, feek firft a common multiple of 
 the lefs terms of each interval, then multiply both terms of 
 
 one ratio by fuch a number as will raife its lefs term to that 
 multiple, and both terms of the other ratio by fuch a number 
 as will raife its lefs term to the fame multiple; after which 
 their difference will be had by comparing the greater terms one 
 with the other : and this procefs agrees with divifion of vulgar 
 fractions, as that in Art. 53. did with multiplication. This 
 method of fubtraction being of as little ufe, in our way of 
 representing founds, as the addition to which it correfponds, 
 we fhall only obferve, by the vvav, that fince mufical inter- 
 vals are proportional to the logarithms of the value of their 
 terms placed fraction- wife, Art, 50; it plainly follows, that 
 addition of intervals muft anfwer to multiplication of thefe 
 fractions, and fubtraction of intervals to divifion. 
 
 57. To fubtract one or more octaves from any given inter- 
 val, which is properly only bringing the terms nearer to each 
 other, by fleps of an octave each, we may either double the 
 lefs, or halve the larger term, for every octave to be fubtract- 
 ed. This, being extremely obvious, has been already exem- 
 plified, in Art. 41. 
 
 58. To find an interval which fhall be double, or triple, 
 or quadruple, etc. of a given one, which may be called Mul- 
 tiplication of an interval by 2, or 3, or 4, etc. we muft 
 raife both the terms of the given interval to the fecond, or third, 
 or fourth power, etc. refpectively. This rule is a direct con- 
 fequence from the foregoing, but is of no ufe with us, except 
 in cafes where the required powers of both numbers, which 
 exprefs thefimple interval, are alfo fuch as confiitute eligible 
 parcels, and on that account are harmonic numbers. Thus, 
 the interval of a perfect fifth being exprtffed by 2 to 3, that 
 of two contiguous fifths, or a ninth, will be 4 to 9 ; three 
 contiguous fifths, or a thirteenth Mill be 8 to 27 ; and fo 
 on. 
 
 59. To find an interval which fhall be the juft half, or 
 third part, etc. of a given one, may be called Division of 
 
Subtraction, Multiplication, 'Divifion, and Comparifon of Intervals. 
 
 Chap. III. 
 
 an interval by 2, or 3, etc. ; and this is done by extracting 
 the fecond or third root, etc. of both the terms, refpectjvejy. 
 This rule proves alto ufelefs with us, except the terms which, 
 exprefsthe total interval be j u ft powers of the fame denomina- 
 tion with the part required. T bus the interval £4 to 8 : may 
 be halved, and its half will be 8 to 9 ; becauie 64 is the fe- 
 cond power of 8, and dl is the fecond power of 9. Thus 
 alfo the third part of the interval 8 to 27 will be 2 ,to 3, or a 
 perfect fifth ; becaufe- thofe terms arejulf the third roots of 
 the former, refpecliveiy: but the ha f of an interval,, whofe 
 terms are not juft Iquare numbers,. or the third part of an in- 
 terval, whofe terms are not juft cube numbers cannot be ex- 
 preiTed in harmonic numbers, nor indeed in any others ; and , 
 is therefore a relation which, in our way of conceiving founds, 
 has no place in mufic. 
 
 60. It is farther to be obferved, in regard to the compara- 
 tive magnitudes of different intervals, that when the terms of 
 one interval are not fome juft power of thufe ol another; or, . 
 when one interval does not exa6tly divide another, as above, 
 it is impoffible to exprefs the comparative magnitudes of fuch 
 intervals exactly, by the proportions ot one -number to ano 
 ther But the magnitudes of mufica! intervals being exactly ' 
 proportional to the logarithms of the- quotients of their terms, 
 Art. 50. that is, to the differences of the logarithms of their 
 terms ; we can, by this means, eafily obtain numeral expref- 
 iions corref ponding to each interval, to a fufficient degree of 
 exaclnefs. For example, fuppofe the logarithm of 2 (which 
 is the quotient of the terms of an octave) to teprefent an oc- 
 tave; the difference of the logarithms of 5 and 4 will repre- 
 fent a fundamental great third. Thus, 
 
 The log. of 2, viz. .3010300 reprefents an octave. 
 
 The log. of 5 .61,89700 
 
 The log. of 4 ,6o.o6co fubt. ' 
 
 The differ. viz. .0969100 reprefents a fund, gr. third. 
 
 Kk 
 
 129 
 
 Therefore the interval of an octave, is to a great third, as 
 30103 to 96gi. .After the fame manner all other intervals 
 may be reprefented and compared ; and the arithmetician will 
 eafily fee that by this means all queftions relating to their com- 
 p.irative magnitudes may be expeditioufly folved, to great ex- 
 act nefs. 
 
 61. Computations of this kind may be {fill (hortened, but 
 with lefs exaclnefs, by ufing fuch fmall numbers as fhall ap- 
 proach neater than any other, not exceeding them, to the 
 ratio of the large -numbers obtained, as above, from the lo- 
 garithms. 
 
 The method of finding fuch fmall numbers, being of ex- 
 cellent ufe in many other pbdofophical calculations, and tho' 
 very eafy, yet not commonly taught, we (hall here infert the 
 procefs, neaily the fame as directed by Mr. Cotes in his Har- 
 mania Menfurarum, Prop. I. Schol. 3 
 
 Divide the grearer term by the lefs ; then the divifor by the 
 remainder; then the laft divifor by the lafr. remainder ; and (o 
 on, after the fame manner as in feeking the greateft common 
 divifor of the two terms : the quotients of thefe fucceffive di- 
 vifions will be the numbers from which to form two columns, 
 one containing ratios greater, and the other ratios lefs than 
 true ; but both approximating nearer to the tr.ue, as the num- 
 bers grow larger ; and each expreffion being nearer than any 
 other which could be formed ot fmaller numbers. Thus for 
 the firft ratio greater than true, put 1:0; and for the firft 
 lefs than true put the firft quotient : 1 . Then for the next 
 ratio greater than true, multiply the terms of this lafr ratio 
 by the fecond quotient, and, carrying the products over to 
 the other column add them to the terms of the ratio already 
 put in that column: and, for the next ratio lefs than true, 
 multiply the terms of this laft found ratio by the third quotient, 
 and carrying the products over again to the other column, 
 add them to the terms already put in that column : and fo 
 proceed always .multiplying the terms of the laft found ratio 
 by the next quotient, carrying the prod u (its over to the other 
 
130 Of Harmonic al Arithmetic. Part II. 
 
 column, and adding them to the terms of thelaft ratio in that From this procefs we find that the ratio of 3 to 1 is lefs; 
 
 column, for a new ratio. that of 28 to 9 is greater ; that of 59 to 19 is lefs; that of 
 
 For example : by proceeding in this manner with the ratio 146 to 47 is greater; and that of 643 to 207 is lefs, than the 
 
 of an octave to a great third, found above, viz. 30103 to true ratio of an octave to a great third. Or, we find that an 
 
 969 r , we get firft this f ucceffion of quotients, 3, 9, 2, 2, 4, etc. octave contains 3 great thirds, and lefs than one ninth part of 
 
 a great third, but more than two nineteenth parts, lefs than 
 five 47th parts, but more than twenty-two 207th parts of a 
 
 whence we derive the following exprefllons. 
 
 Ratios greater than true. Ratios lefs than true. 
 
 I : 
 
 27 . 9 
 
 28 : 9x2 
 118 . 38 
 
 3 : 1x9 
 56 . 18 
 
 59 : 19 X 2 
 584 . 188 
 
 146 : 47 x 4 
 &C 
 
 643 : 207 
 &C. 
 
 great third. 
 
 62. For a particular application of the rules delivered a* 
 bove, we have here annexed a fcheme of the fyftem of modu- 
 lation of C, as a principal key ; which may equally ferve for 
 that of any other key, by only changing the letters, 6c. ac- 
 cording to the rules of tranfpofition, formerly given. 
 
 9 I 
 
 SCHEME of the System of Modula tion of G. 
 
 'IV. t.g. V. t.l. Vl./.d.YU^. t.r. K.t.g. 11././. 111. /.p. IV. 6c. Scale of the adjunct fourth. 
 
 32. 
 F. 
 
 Scale of the prin. Key 
 48. 50. 
 
 40. 42. 
 
 A. ty. 
 
 Medial 
 
 Final 
 A. 
 54- 
 
 45. 48. 50. 54, 
 
 B. C. c& D. 
 
 48. 54. 
 
 60. 
 E. 
 60. 
 
 64. 72. 80.84. 9°- 9 "* IOO « IQ 8. 
 
 F. G. A. ty. B. C. c%. D. 
 
 90. 96. 
 
 63. 72. 81. 
 
 \ i 
 
 K. t.g. II. /. /. HI./ d. IV. t.r. V. t.g. VI. t.I.Vll.f.p.K. 6c. 
 
 B. 
 
 64. 
 C. 
 
 60. 63. 
 
 72. 80. 
 
 D. d%. E. 
 72. 75- 8l 
 
 84. 
 F. 
 
 96. 108. 
 G. g%. A. 
 
 1 20. 1 28. 
 B. C. 
 
 90. 96.100.108. 120. 126. 
 
 D. 
 
 144- 
 
 40. 5U. 54. uo. 03, j &. 75. oil yu. yu. iuu, iuo. » *<->. 1 in. lJ \'\' 
 
 V. t.g. VI. t.l. VLIJJ.K. t.r. II. t.g. III. t.l. %lV.f.p.V. 6c. Scale of the adjunct fifth. 
 
Chap. III. The Syflem of Modulation examined by the preceding Rules. 
 
 63. For underftanding this fcheme it muft be carefully ob- 
 ferved, 
 
 (j.) That the degrees of the fcale are here denoted by nu- 
 meral letters, in (lead of figures, to avoid the confufion of too 
 many figures. 
 
 ( 2.) That fuch of thefe degrees as occafionally become grave 
 or acute, in fome of the fcales, are marked with the proper ac- 
 cents over them. 
 
 (3.) That the names of the intervals, or fteps, from each 
 degree to the next are denoted by fmall Italic letters put be- 
 tween them : thns t.g. ftands for a greater tone ; 1. 1, for a 
 lefs tone; and t.r. for a redundant tone: f.fi ftands for a 
 proper femitone ; and/, d. for a deficient femitone. We (hall 
 fpeak below of the magnitude of each of thefe. 
 
 (4.) That the numbers which ftand in every alternate line 
 (hew the ifochronous parcels of vibrations of the degrees of 
 the fcale which ftands over or under them : and becaufe the 
 fcale of the principal key is one way formed with regard to 
 the adjunct fourth, and another way with regard to the ad- 
 junct fifth, Art. 45 ; the numbers belonging to the fcale of 
 the key, in the former cafe, are put above, with the title of 
 fnedial; and thofe belonging to the fame fcale, in the latter 
 cafe, are put below, with the title of final ; this being the 
 fcale with which we are always obliged to conclude. 
 
 (5 ) The meaning of the lines of letters is obvious, only it 
 may be obferved, that the (harp fifth of each fcale is added a- 
 mon<* the letters which (land below it, and its ifochronous 
 parcel among the numbers ; though no notice is taken of them 
 in the fcales themfelves. Thus, among the upper row of let- 
 ters we have put c^;, being the (harp fifth in the fcale of 
 the adjunct fourth ; and among the lower row, g$, and al- 
 fo d$ ; the former of thefe being the (harp fifth in the fcale 
 of the key, and the latter the (harp fifth in the fcale of the 
 adjunct fifth. 
 
 '1* 
 
 64. Thefe things being premifed, it will be proper for the 
 ftudent firft to feek out the component factors of each num- 
 ber in the fcheme, in order to be aftured that they are all har- 
 monic numbers, according to the characteri flics given in the 
 firft and fecond chapters. Thus, 32, 64, and 128 are juft 
 
 powers of 2 48, and 96 are powers of 2, multiplied 
 
 by 3. 36, 72, and 144 are powers of 2, multiplied by 
 
 the fquare of 3 (9) 54, and 108 are powers of 2, multi- 
 plied by the cube of 3 (27) 81 is the fourth power of 3. 
 
 All thefe numbers are therefore compofed of the fmalleft 
 primes 2 and 3. 
 
 Farther, the numbers 40 and 80 are powers of 2, mul- 
 tiplied by 5. 60, and 1 20 are powers of 2, multiplied by 
 
 3x5 (15). 45 is 3 x 3 x 5, and yo is the fame doubled. 
 
 5 50, and 100 are powers of 2, multiplied by 5x5(25). 
 
 75 is 3, multiplied by 5 x 5. Some of thefe numbers 
 
 include 5, and others the fquare of 5, along with 2 or 3. 
 
 Again, 42 and 84 are powers of 2, multiplied by 3X 7 (21). 
 
 63 is the fquare of 3, multiplied by 7, and 1 26 is the 
 
 fame doubled. Here we have 7 included along with 2 or 3. 
 
 Thefe are all the numbers in the fcheme, and thus they 
 are all found to be harmonic numbers. 
 
 65. It is farther to be obferved, that if we exclude the third 
 power of 5, the fecond power of 7, and the product of 5 x ->, 
 which it is reafonable to fuppofe we ought to do, becaufe the 
 larger factors cannot be fo often compounded as the fmaller 
 ones, without becoming too complex to be received ; we (hall 
 then find that the fcheme contains all the poffible harmonic 
 numbers between 32 and 144, viz. between the lowed and 
 the higheft number in it; except fuch as might be formed of 
 the powers of 2, multiplied fimply by 7 ; and thefe numbers 
 we (hall hereafter (hew, have no place in our mufic. 
 
 66. In regard to the intervals, or fteps of the fcale, it muft 
 be remembered, that a great tone may always be reprefented 
 
132 
 
 by 8 to 9 ; a lefs tone by 9 to lo ; and a proper femitone, by 
 15 to 16: a redundant tone, by 7 to 8 ; and a deficient- femi- 
 tone, by 20 to 2f. And if the ifochronous parcels of any 
 two contiguous degrees in the fcheme, be reduced into their 
 fmallett numbers, as directed in this chapter, the refult will 
 a'ways be iome one of the above proportions, viz. that which 
 correiponds with the fpecies of tone or femitone, which is 
 marked between the lame degrees, in the fcheme. Thus, the 
 two firft numbers in the fecond line, viz. 32 and 36, b\ re- 
 duction, become 8 and 9, correfponding to a great tone,: as 
 maiked between the IV and V in the firft line. Proceeding 
 forward in the fame lines, 36 and 40, by. reduction, become 
 9 and 10; anfwering to the lefs tone between V and VI : 
 Again 40 and 42, by reduction, become 20 and 21 ; anfwer- 
 ing to the deficient femitone between VI and VIi[j : and 42 
 and 43 are reducible into 7 and 8, anfwering to the redun- 
 dant tone between VI ty and K ; (the number 45, which ftands 
 between the other two, not being taken any notice of, in the 
 fcale above it). After the fame manner 48 and 54 (neglect- 
 ing the 50 which ftands between them) are reducible into 
 8 and o, anfwering to the great tone between the K and-II. 
 
 In this manner the ftudent may, at his leifure, examine all 
 the other contiguous degrees in each of the three fcales, by 
 their numbers ; and fuch an examination will be the more ufe- 
 ful as it will tend to familiarize the compofition of harmonic 
 numbers, and to fix the numbers themfelves in the learner's 
 memory. 
 
 67. The fteps of each of the three fcales are exactly the 
 fame, and in the fame order, viz. t, g; t, /,• f, d ,• t, r ; t,g; 
 t, I -, f. p. This order agrees exactly with that which has 
 formerly been eftablifhed by the writers on this fubject, ex- 
 cept in the fourth of the fcale, which we make a grave, in- 
 fread of a perfeB fourth ; and hence arifes the deficiency of 
 the femitone next below, and the redundancy of the tone next 
 
 Of Harmomcal Arithmetic. Part II. 
 
 above it : for if the fourth were made perfect, the femitone 
 
 below it would be a proper femitone, 1 5 to 16, and the tone 
 above it would be an exact geat tone, 8 to 9 : as will eafily 
 appear by fubftituting 64 inftead of 63, in the medial fcale 
 oi" the principal key, viz. the 4th line of the fcheme. Our 
 reafons for this innovation will be feen in the following chap- 
 ter. 
 
 68. From this arrangement of the fteps of the fcale there 
 arife two different forts of great thirds, and three different 
 forts of lefs thirds ; all of which have their ufe in mufic. The 
 proportion of 4 to 5 we call a fundamental great third, and 
 this exifts between the K and III, and between the V and VII 
 of the fcale The proportion of 7 to 9 we call a redundant 
 great third, and this exifts only between the grave IV and 
 the acute Vl of the fcale. Here it is neceffary to obferve, that 
 the flat VII of the fcale is naturally and • immutably a grave 
 fourth; and the II of the fcale is naturally an acute fixth. in, 
 refpett to the adjunct IV as a new key ; and that every grave 
 fourth includes the number 7. The proportion of 5 to 6 we 
 call a fuperftructed lefs third, becaufe it always lies between 
 the fundamental great third and the fifth ; and this exifts be- 
 tween the III and V, and between the Vll and II o! the fcale. 
 The proportion of 6 to 7 we call a deficient lefs third, and 
 this is found only between the II and the grave IV of the 
 fcale, The proportion of 2" to 32, which exifts between the 
 acute VI and the K, reprefents the fundamental lefs third of 
 the flat feries, of which we have to (peak hereafter. 
 
 69. The fame arrangement produces one redundant and one 
 deficient fifth, befides that which is well known by thetitle 
 of the falfe fifth. The redundant fifth exifts between the 
 grave IV and the K; and has its terms in the proportion of 
 21 to 32. The deficient fifth is between the acute VI and 
 the III, and is as 27 to 40. The falfe fifth, between the Vil 
 and the grave IV, is as 5 to 7. . , 
 
Chap. IV. e tbe Intervals of the Scale afcertained and compared. 
 
 All the other fifths in the fcale are perfect, in the propor- 
 tion of 2 to 3 : and, in practice, the redundant and deficient 
 
 »33 
 
 fifths, mentioned above, are alfo rendered perfect , by the oc- 
 cafional temperament of one of their founds. 
 
 70. The inverfions of the different forts of thirds produce 
 as many different forts of fixths ; bat as all inverted chords 
 are of the fame nature and produce much the fame effects as 
 the erect chords from which they are derived, we fhall not 
 trouble the learner with a particular name to each of thefe 
 fixths; but rather denominate them by the thirds from which 
 they are derived. 
 
 Now, to exprefs the inverfion of any chord lefs than octave, 
 we have only to compare half the upper term, if it be an even 
 number, with the fame lower term ; or to compare double 
 the lower term, with the fame upper term, if it be an odd 
 number: this being in effect either carrying the upper term 
 an octave lower, or the lower term an octave higher. Thus 
 the fixth inverted from the fundamental great third 4 to 5, 
 is 5 to 8 ; and this exifts between the III and K, and between 
 the VII and V. The fixth inverted from the redundant great 
 third 7 to 9, is 9 to 14 ; and this exifts between the acute 
 VI and the grave IV. The fixth inverted from the fuperftrudt- 
 ed lefs third 5 to 0, is 3 to 5 ; and this exifts between the V 
 and III, and between the II and "V II. After the fame manner 
 7 to 1 2 is the inverfion of the deficient lefs third, and 16 
 to 27, that of the fundamental lefs third, of Art. 68. 
 
 71. We fhall next take a furvey of the fmall differences of 
 the tones, femitones, and other intervals, mentioned above, 
 
 (r.) The acute note exceeds the natural note of the fame 
 name by the difference of a cvmma, 80 to 8 1 . This appears 
 at the fecond A in the third line of the fcheme ; which, as a 
 natural VI, takes the parcel 80, above it, and as an acute VI 
 the parcel 81, below it. The fame difference of natural and 
 acute appears again at E, in ti^e feventh line; and is denoted 
 by the fame numbers above and below it. See alfo Art. 41. 
 
 LI 
 
 (-.) The grave note falls fhcrt of the natural note of the 
 fame name by the difference of a bearing, 63 to 64. This 
 appears at the fecond F, in the third line; and again at the C, 
 in the feventh line of the fcheme. See alfo Art. 44. 
 
 (3.) The greater tone exceeds the lefs tone by a comma. 
 This may be found by (ubtracling 10:9 from 9:8, by the 
 ruie, Art. 56 ; and appears in the fcheme between the fecond 
 occurrence of G and A in the third line ; and between D and 
 E in the feventh line ; for in thefe occurrences the natural up*- 
 per note makes a lefs tone, and the acute upper note a greater 
 tone, while the lower note remains the fame. 
 
 (4.) The redundant tone exceeds the greater tone by a 
 bearing. This is feen at C, D, in the feventh line of the 
 fcheme; where the natural C is a greater tone, and the grave 
 C a redundant tone, below D. 
 
 (5.) The proper femitone exceeds the deficient femitone by 
 a bearing. This is feen at E, F, in the third line ; where the 
 natural F is a proper femitone, and the grave F a deficient fe- 
 mitone, above E. 
 
 72. The differences of all larger intervals of the fame name 
 will ffill be either a comma or a bearing, or both thefe, fome 
 way compounded ; becaufe all larger intervals are made up 
 of fingle degrees, and muft therefore be affefted with the fame 
 differences as the degrees which compofe them. We fhall 
 juft mention a few of thefe differences, which are of the molt 
 importance to be known and remembered ; and leave it to the 
 ftudents own indufiry to find them out in the fcheme, and 
 to extend the fame enquiry to other intervals, as he thinks 
 proper. 
 
 (1.) The redundant great third exceeds the fundamental 
 great third, by the difference of 35 to 36; which is the fum 
 of a bearing and a comma. 
 
 ( 2.) The deficient lefs third falls fhort of the fuperfiruct- 
 ed lefs third, by the fame quantity. 
 
 (3.) The fundamental lefs third falls fhort of the fuper- 
 
134 
 
 ftructed, by a comma 
 
 Op Jiarmomcal Arithmetic, 
 and exceeds the deficient,, by a bear 
 
 Part IT. 
 
 (4.3 The redundant fifth is a bearing) more, and :the defi- 
 
 cient fifth is a comma lefs than the 
 
 per 
 
 fth. 
 
 . Thefe are forne of the moft material./;^//, and hitherto al- 
 mo(i unob'fcrved differences of intervals ; and he who has the 
 method of ascertaining thefe can be at. no lofs for the larger 
 differences between intervals of different, names; So that we 
 fhall not give any particular detail of them at pre.fent. 
 
 73. To finifh this chapter with a praxis to the rules relat- 
 ing to the comparison of mufical intervals, Art. 60 and 6t, 
 fuppofe it be required to find how many commas and deci- 
 mals of a comma,, are in every ftep of the natural fcale. 
 
 This may be done by fir ft feeking how many. commas make 
 a bearing, a deficient femitone,. and a lefs tone; and after- 
 wards all the other ileps are eafily made up, from thefe, by 
 
 addition, 
 
 (1.) For the Comma. 
 Log. 81 1.9084850 
 Log. 80 1. 9030900 
 
 M . ■ - 5395° 
 
 (3.) For the def. femit. 
 Log. 21 1. 3222193 
 Log. 20 1. 3010300 
 
 Diff. 211893 
 
 (2,.).. For the Bearing. 
 Log 64 1.806(800 
 Log. -63 f. 799340 5 
 
 Diff. 
 
 6S39S 
 
 (4.) For. the lefs tone. 
 Log. 10 i. 0000000 
 Log. 9 0.954-2425 
 
 Diff. 457575 
 
 Now, the quotient of 68395, divided by 53950, viz. 1.2677, 
 
 (hews the commas and decimal in a bearing. 
 
 The quotient of 21 1893, divided by 53950, viz. 3.9276, 
 
 fhews the commas and decimals in a def. femitone. 
 
 The -quotient of 457575, divided by 53950, viz. 8.4815, 
 
 fhews the commas and decimals in a lefs tone. 
 To the def. femit. .3,9276 
 add the bearing 1.2677 
 
 The fum 
 
 To the lefs tone 8 
 add one comma ' 1 
 
 5.1953 the commas in a proper femitone. 
 4815 
 
 The fum is '9.481 5 the commas in a great tone, 
 
 add the bearing 1.2677 
 
 The fum is 
 
 10.7492 the commas in a redundant tone. 
 
 For proof of the above work, range thefe quantities one be- 
 low another, as their correfpondent intervals occur in the 
 fcale, and then add them together ; thus, 
 
 Commas. 
 Proper femitone 5.1953 
 Lefs tone ■ 8.4815 
 
 Greater tone 9-4815 
 
 Redundant tone 10.7492 
 Deficient femitone 3.9276 
 Lefs tone - 8.4815 
 
 Greater tone 94815 
 
 C to B. 
 
 B to A acute. 
 
 A to G. 
 
 G to F grave. 
 
 F to E. 
 
 E to D. 
 
 D to C. 
 
 The fum 557981 is the commas in an octave. 
 
 Divide now the ofrave 3010300, (fee Art. 60.) by the 
 comma 53950, and it will yield the fame quotient, within 
 one five thoufandth part of a comma j which proves the truth 
 of the whole. 
 
I -ns .■]■ 
 
 G H A P. IV. Of GO M B I N E D SOU N D S. 
 
 Article 74.VT /H EN two different founds are heard toge- 
 W ther, their combination always either really 
 produces, or. efleotiaSJy implies, a third found, whofe vibra.- 
 tions are equal to the difference of the vibrations of the two 
 founds, in the fame time. ,;Thus C, 64, heard together with 
 G„ 96, (fee the 6th and 7th lines of the fcheme, Art; 62.) 
 produces 32 ; which as C an octave below : and G, 64, heard 
 along with E, .80, produces 16, which is C, a double octavg 
 below. We take thefe examples in the large numbers of the 
 fcheme ; bat the refelt will be. the fame if the intervals be ex- 
 prefled in their fraalleft numbers. Thus, the perfect fifth, 
 whofe terms are as 2 to. 3, produces 1, which is an octave. 
 below the lower term ; and the fundamental great third, 
 whofe terms are as 4 to 5, produces, 1, which is a double 
 octave below the lower term. . After the fame manner, the 
 produced or implied found of every other internal may be af- ; 
 
 certained. • . -.- 1 
 
 75. We are the Jefs follidtons to account for this effect up- 
 on philofophical principles, .becaufe the fact -is confirmed, by, 
 experience, and is introduced as an eff-ntial principle of har- 
 mony by M. Serre of Geneva, in a fyltem printed at Paris. 
 1753, intitkd, Efals fur ks principes de I' barmonie ; and by 
 M. Tartini, to whom the Erft difcovery of it is attributed, 
 and whofe experiments are related in the Encychpedie, under 
 the article Fundamental, and his whole fyflem briefly explained 
 in Mr. Rouffeaa's Diiffimaire de Mujiqtte, under the article Sy- 
 
 Jleme. The fenfe of his affertion relating to this matter is as 
 follows. 
 
 ."Whenever two bud, uniform, and continued founds are 
 *' heard together, there refults from their claming [cbocj a 
 " third found, more or lefs perceivable, in proportion to the 
 *' fimplicity of the relation of the two former founds, and to 
 '■' the delicacy [JineJJe ] of the ear of the hearer. 
 
 " To render this experiment the mod perceivable, fet two 
 M hap tbois,. well tuned, fome paces afunder, and place your- 
 " felf between the two, at equal diftance from each Inftead 
 " of Hautbois one may take two violins, which, although 
 % .their founds are weaker, will, by playing firong and uni- 
 '.'. f,orm,.[>2 touchant avec force et jujle£e\ be fuffkient to make 
 " the third, found diftinguifhable." 
 
 . M. RoufTeau then proceeds to fpecify the third found pro- 
 duced by each of the concords, according to M. Tartini; and 
 in every one of them, it is affigned juft an octave higher than 
 that which arife's from the difference of vibrations, as above, 
 Art. 74. We (hall leave it to the curious to determine whe- 
 ther this can be accounted for by confidering the courfe and 
 recourfe of each particle of air, which together conflitute on- 
 ly one pulfe, as each producing a fimilar and equal effect on 
 our organs of hearing, fo that we perceive two coincidences 
 for every one pulfe which the acuter term gains ; and that 
 thefe coincidences become, as it were, the pulfes of a fee- 
 ble grave found, generated by the combination of the other 
 
1 3 6 Of Combined 
 
 two; which may be actually heard in the fituation and cir- 
 cumfiances above defcribed. M. Tartini attempts no phyfi- 
 cal explication of this matter, but M. Serre exprefly derives 
 the generated third found from the coincident vibrations of 
 the two generating founds : and to remove an objection which 
 naturally occurs, and which M. Rouffeau adds at the foot of 
 the article Battemens, againft this hypothefis, viz. that the 
 real coincidences of vibrations occur rnoft rarely when two 
 founds approach neareft to the true ratio of a concord, but yet 
 differ fome very little from it; and therefore one would think 
 that the generated third found fhould vanifh away, as the two 
 founds drew near to a perfect concord, and prefent itfelf again 
 at the inffant the concord became juft : let it be obferved, that 
 although the raz/ coincidences happen feldom, yet the appa- 
 rent or Jenfib le coincidences fall out nearly in the fame order, 
 whether the ratio of the vibrations be the exact ratio of a con- 
 cord, or only near to it : with this difference only, that when 
 the ratio is perfect, thefe coincidences follow each other in a 
 regular and uninterrupted fucceflion ; whereas, if the ratio be 
 imperfect, the fucceflion of apparent coincidences, after con- 
 tinuing for a little time, is interrupted for a little time, and 
 then takes place again, and fo on : and thefe periodical con 
 tinuances and interruptions occur more frequently as the dif- 
 ference from the true ratio is greater ; and are the very caufes 
 of the fenfation which Dr. Smith calls the beating of imperfect 
 confonances. 
 
 Now, according to the hypothefis of M, Serre, the third 
 found ought to prefent itfelf during the continuance of the 
 fenfible coincidences, and to difappear during their interrup- 
 tion ; fo that the difference between a perfect and aa imper- 
 fect concord is no more than this, the perfect concord pro- 
 duces ah uniform and uninterrupted third found, and the im- 
 perfect pioduces the fame third found periodically interrupt- 
 
 Sounds. 
 
 Part II. 
 
 ed ; and this being precifcly the cafe in nature, it is a ftrong 
 proof that the hypothefis is right. 
 
 If we fuppofe any analogy to exift between the perception 
 of different combined founds, and the viewing of different 
 motions in one and the fame diiedtion, it may be infered, that 
 as the velocity with which one moving body approaches to- 
 wards or recedes from another (which may be called its rela- 
 tive motion, and is equal to the difference of the two abfolnte 
 motions,) is a circumftance which always attracts part of our 
 regard ; fo the difference of the vibrations, or the relative ve- 
 locity of the pulfes of one found in comparifon to thofe of the 
 other, may fome way be perceived, abfiract from all confide- 
 rations either of real ox fenfible coincident pulfes : and the fci- 
 ence of mufic may probably receive confiderable improve- 
 ments from a critical enquiry into thefe matters, by fuch as 
 are qualified for it. 
 
 76. We fhall proceed to fhew the confequences of thefe ge- 
 nerated or implied founds in the chord called the perfetl chord, 
 Part. I. Art. 127. which is the final or concluding chord of 
 all fharp harmony. The perfect chord in its moft complete 
 and natural arrangement, has the fundamental found, or key 
 note, for its bafs ; above which ftand the tingle octave, double 
 fifth, double octave, and triple greater third; fee Part I. Art. 
 130. to 134. The proportions of thefe founds, expreffed iu 
 their fmalleft numbers, are, 1, 2, 3, 4, and 5, respectively; 
 viz. fuppofe the fundamental to be reprelented by 1 ; its tin- 
 gle octave will be 2 ; its double fifth, 3 ; its double octave, 4 ; 
 and its triple greater third, 5. 
 
 Thefimplicityof the mutual relations of he founds expreffed 
 by this progredion of fmall numbers is, in part the reafon of their 
 being heard together with fatisfaction, or conftitutingan agree- 
 able chord ; but befides this, it is alfo neceffary to take into con - 
 fideration the implied founds, arifing from the mutual differ- 
 
Chap. IV. Of generated or implied Sounds. 
 
 ences of the terms, in order to prove that this chord brings 
 with it no idea foreign to the natural conffitution of the key, 
 and therefore is perfectly conclufive when applied to the key 
 as fundamental 
 
 Now, it is plain that the difference of any two terms in 
 this progreffion, i. 2. 3. 4. 5, any way taken, will always be 
 equal to fome one of the terms themfelves ; viz. either 1, or 
 2, or 3, or 4, Thus the difference of any two adjacent 
 terms is always 1, and confequently any two adjacent founds 
 will always generate the fundamental. The difference of any 
 two terms, feparated by an interjacent term is always 2, or 
 the octave above the fundamental. The difference of any 
 two terms, feparated by two interjacent terms, is always 3, 
 or the doable fifth to the fundamen;al : and the difference of 
 two terms, feparated by three interjacent terms, is 4, or the 
 double octave to the fundamental, 
 
 From hence it appears that the implied founds, in this per- 
 fect chord, produce no other effect than that of fortifying, 
 or doubling, all the real founds of the chord, except the high- 
 en;. It is further to be obferved, that the difference between 
 two adjacent terms will always be more remarkable than that 
 between two terms which are feparated by one or more inter- 
 jacent ; and alfo that the generated found is more perceivable 
 when it falls below both the generating founds, than when 
 it falls between them; confequently as every two adjacent 
 terms, in the above chord, generate the fundamental, or 
 loweft found, 1 ; this fundamental will be much more remark- 
 ably fortified than any of the other founds. 
 
 77. All thefe circumftances are exactly agreeable to the na- 
 tural conffitution of the key. For, 
 
 (1.) The octaves in the accompany ment are perfectly na- 
 tural to the key ; becaufe the manner of parcelling the vibra- 
 tions of the key, by continual reduplication, as formerly de- 
 fcribed, effentially implies the co-exiftence of all its octaves, 
 
 M 
 
 Of the full perfeSt Chord. j 3 ? 
 
 both above and below, not exceeding the limits of audible 
 found. 
 
 (2.) The key muff be the principal object of our attenti- 
 on ; which is inevitably the cafe when the perfect chord is 
 applied to it; not only on account of the octaves in the ac- 
 companyment, but alfo on account of the generated founds, 
 as above. 
 
 (3.) The perfect fifth and fundamental great third are ad- 
 mitted along with the key, in modern harmony, but only as 
 dependent founds, or harmonics ; becaufe it is eafy to admit 
 three inftead of two, or five inffead of four-, as already fhevvn : 
 and the double fifth, and triple great third, are preferable to 
 the fingle intervals of the fame names, becaufe of the regular 
 progreffion which is thereby formed. 
 
 78. It maybe afked why this progreffion is not farther con- 
 tinued ? and why the founds represented by 6 and 7 are not 
 admitted in the perfect chord, along with thofe reprefented 
 by the fmaller numbers ? To this it is fufficient to anfwer, 
 that 6, being the octave of 3, viz. the fifth, cannot be ad- 
 mitted without rendering the fame fifth too remarkable, and 
 thereby taking off its dependance on the key. See Part I. 
 Art. 142. And that 7, being a difficult factor in the par- 
 celling of vibrations, produces a found difficult to be fung, 
 and unfatisfactory, except when it can be properly followed 
 by another more eafy found, in the way of refolution : on 
 which account it cannot have place in a final chord. 
 
 79. The predominancy of the founds of this perfect chord 
 among the natural confonances of every mufical found, obferv- 
 ed, Part I. Art 137. is a neceffary confequence of the fmall- 
 nefs of the numbers which exprefs their lelations : for in the 
 production of every mufical found the air is agitated by a va- 
 riety of other impulfes, befides thofe of the principal and uni- 
 form vibrations by which we eftimate the tone; as will plain- 
 ly appear by considering what muff inevitably happen to the 
 m 
 
I g 8 Of Combined 
 
 air within the found box of any ftringed inffrument : for as 
 the vibrations of founds are again and again reflected by ftrik- 
 ing againft the back, or the fides, or the belly of the found 
 box, this will occafion the air within the box to partake of an 
 infinity of different vibratory motions, croffing and intermix- 
 ing with each other in the mod complicated manner imagin- 
 able ; aud the fame complicated motions will be communicat- 
 ed to the contiguous air, and propagated along with the uni- 
 form vibrations of the firing, from whence will arife a multipli- 
 city of founds, acuter than the principal, and in due fubor- 
 dination one to another : thofe whofe relations to the princi- 
 pal found aie exprefied in the fmaUefl: numbers, and whofe 
 vibrations therefore do moft frequently coincide with the 
 principal, being thereby kept up and preferved, while thofe 
 whofe vibrations more rarely coincide with the principal are 
 defiroyed. 
 
 One may venture to afftrt, that except the air be thus agi- 
 tated, and put into a kind of ferment, no mufical found can 
 be produced ; elfe why does not a ftretched firing yield its 
 found when apart from a founding box or board ? as when 
 a weight is hung by it from a hook in the middle of the roof 
 of a room. The firuclure of different infiruments occafions 
 their founds to be more or lefs compounded ; thus the founds 
 of a German flute are lefs compounded than thofe of a com- 
 mon flute, and thefe lafi much lefs than thofe of a haut- 
 bois, becaufe the trembling of the edge, which cuts the blaft 
 in wind-infiruments, has a confiderable fhare in introducing 
 thefe conlonant vibrations; and this, joined with the additi- 
 onal agitations which the air afterwards fuffers within the 
 pipe, ferves them infiead of any other found box : apd the 
 edge which cuts the blafi in a German flute is lefs pliable 
 than that in a common flute, and this laft is much lefs pli- 
 able than the reed of a hautbois. 
 it may be further obfeived, that the confonanccs of any 
 
 Sounds. 
 
 Part II. 
 
 mufical found, whether inflrumental or vocal, are much lefs 
 diftinguiihable in the open air than in a room ; becaufe the 
 reflection of the found, from all parts of the room, contributes 
 towards the breaking and fubdividing the total vibrations. 
 The confonances of every fpeciesof mufical founds alfo become 
 more perceivable as the principal fouud itfelf is made graver 
 in its tone. Thus we can fcarcely efiimate the pitch of the 
 lowefl firing of a fpinnet, or even of a violoncello, but by 
 means of their confonant upper offaves. 
 
 Thefe remarks feemed pertinent in this place, not only 
 that the reader may have the pleafure of obferving the perfect 
 agreement between the nature of found, and the choice of 
 the human ear -, but alfo becaufe feveral modern theorifis have 
 not hit upon the true caufe of the multiplicity of found. 
 
 80. The famenefs of effect of the different arrangements 
 and inverfions of this perfect chord, taken notice of in Part I. 
 Art. 13} to 136, is to be accounted for by obferving that we 
 eafily fubftitute the octaves above or below any found infiead 
 of the found itfelf, or we mentally carry any found fome 
 octaves higher or lower, upon occafion ; and thus we form 
 the natnral arrangement of the perfect chord in imagination, 
 though the real founds of the chord which we hear be diffe- 
 rently fituated. 
 
 8r. The implied founds, arifing from the differences of vi- 
 brations, do alfo contribute, in the moft part of thefe cafes, 
 towards forming the idea of the perfect chord. Thus, when 
 the fingle great third and fifth are joined with the fundamen- 
 tal, and confequently the founds of the chord are reprefent- 
 ed, in fmall numbers, by 4, 5, and 6; the difference Between 
 4 and 5, and alfo that between 5 and 6, generate 1, or the 
 double octave below the fundamental 4 ; and the difference 
 between 4 and 6 generates 2, or the fingle octave below the 
 fame fundamental. 
 
 Again, in the inverted chord, called the chord of the Jixth 
 
Of 'the different Arrangements, Liver fions, and Parts of the per feci Chord. 
 
 Chap. IV. 
 
 upon the third of the fundamental asbafs, reprefented by 5, 6, 
 and 8; where the fundamental is carried an octave higher, and 
 made the uppermoft fonnd, the difference between 5 and 6 
 generates 1, or the triple octave below the fundamental 8; 
 that between 6 and 8 generates 2, or the double octave be- 
 low the fame fundamental ; and that between 5 and 8 gene- 
 rates 3, or the octave below 6, which is the fifth to the fun- 
 damental. 
 
 If the loweft found of the fame inverted chord be deprefTed 
 another octave lower, the chord will then be reprefented by 
 5, i 2, and 16 : in which cafe it becomes much lefs harmoni- 
 ous, both becaufe the fundamental great third, reprefented 
 by 5, is farther removed from its natural place, and alfo be- 
 caufe the difference between 5 and 1 2 generates 7, which is 
 an unfatisfactory found, as obferve J, Art. 78; and the dif- 
 ference between 5 and 16 generates 1 1, which is an unnatu- 
 ral found : and although thefe generated founds are lefs per- 
 ceivable becaufe the relation of the founds themfelves is lefs 
 fimplethan before, yet fiill they contribute fomething towards 
 the imperfection of the chord. 
 
 Thus alfo in the inverted chord called the chord of the fixtb 
 and fourth, upon the fifth of the fundamental as bafs, repre- 
 fented by 3, 4, and 5; the difference between 3 and 4, and 
 alfo that between 4 and 5 generate 1, or the double octave 
 below the fundamental 4 ; and the difference between 3 and. 
 5 generates 2, or the fmgle octave below the fame fundamen- 
 tal. If the middle found or fundamental of this inverted 
 chord be carried an octave higher, the chord will then be re 
 prefented by 3, 5, and 8 ; in which cafe it becomes more har- 
 monious, becaufe the difference between 3 and 8 generates. 5, 
 and the difference between 5 and 8 generates 3, fo that thefe 
 generated or implied founds are the very fame which really 
 exift in the chord ; while the difference between 3 and 5 ge- 
 nerates 2, or the double octave below the fundamental 8. 
 
 139 
 
 Thefe confiderations may ferve to account for the effects ta- 
 ken notice of in Part I. Art. 138. 
 
 82. To afcertain the nature and effect of any two combin- 
 ed founds, it feems necelTary, in all cafes, to confider thefe 
 three particulars. (1.) The formation of the parcel of each 
 found. (2.) The fimplicity of the ratio, or the fmallnefs of 
 the numbers by which the int^^al between the two given 
 founds may be exprefled. (j.f°the implied third found, 
 arifing from the difference of vibrations of the two given 
 founds, which difference may be doubled and redoubled, when 
 neceffiry, in order to bring the implied found as near as pof- 
 fible to the other two, and to obferve what kind of chord re- 
 fults, at lalt, from all the three. 
 
 For example, the interval of a perfect fifth is exprefled by 
 the ratio of 2 to 3 ; aud on account of the fmallnefs of thefe 
 numbers we may fafely conclude that the terms of a fifth are 
 concord to each other ; but we cannot determine which of 
 the two ought to be efieemed the fundamental without con- 
 fidering the formation of each parcel ; from whence we im- 
 mediately obferve, that as 2 is fimpler than 3, fo the lower 
 term of the fifth has always a fimpler parcel than the upper 
 term, becaufe this- latter involves 3 in place of 2 of the form- 
 er, whatever be the other factors which enter equally into 
 both parcels, therefore the lower term naturally takes place 
 as the fundamental : and this conclufion is farther confirmed 
 byobferving, that the implied found, reprefented by 1, is al- 
 ways the octave below the fame iower term. 
 
 Thus alfo the interval of a fundamental great third, 4 to 5, 
 is a concord, becaufe of the fmallnefs of the numbers; and 
 the lower term is the fundamental, becaufe it is fimpler than 
 the upper, and becaufe the implied found is the double octave 
 below the fame lower term : but the interval of a perfect 
 foutth, 3 to 4, refpects the upper term as a fundamental, be- 
 caufe here the upper term is fimpler than the lower, and be- 
 
||vjj|| 140 Of Combined 
 
 MB '■ - caufe the implied found is the double octave below the fame 
 ™3 upper term. 
 
 Again, the interval of a fuperftrudted lefs third, 5 to 6, is 
 alfoefteemed a concord, though in a more imperfect degree; 
 but here the implied found, reprefented by 1, is not the octave 
 of either of the terms, but being brought near by doubling 
 and redoubling, becomes £~ zK viz. the fifth below the upper 
 term, and the fundamental gfreat third below the lower, and 
 this implied found is itfelf the fundamental. 
 
 In all cafes it may be obferved, that when the difference of 
 the vibrations of two founds, exprefled in their fmalleft num.- 
 
 Sounds. Part II. 
 
 bers, is I, or 2, or 4, the implied found, reprefented by that 
 difference will always be the fundamental, en account of the 
 fimplicity of its parcel, whether it be or be not the oftave of 
 either of the given founds, 
 
 83. Hitherto we have examined only the $ perfect chord 
 and its parts, whofe places in the fyftem of modulation may 
 be fuppofed well enough known to the reader. It will not be 
 improper here to fubjoin the fcheme, in order to exhibit more 
 clearly the fituation of the other chords which remain to be 
 examined. 
 
 SCHEME of the System of Modulation ofC. 
 
 "IV. t.g. V. t. I. Vl.fJ.VU]). t.r. K. t.g. II. t.l. 111./. p. IV. 6c. Scale of the adj unci: fourth. 
 
 32. 
 F. 
 
 36. 
 
 40. 42. 
 
 45- 
 
 48. 
 
 50- 54- 
 
 
 60. 64 
 
 72. 
 
 80.84. 90. 96. 100. 108. 
 
 G. 
 
 A. ty. 
 
 B. 
 
 C. 
 
 c% D. 
 
 
 E. F. 
 
 G. 
 
 A. ty. B. C. c%. D. 
 
 
 C Medial 
 
 
 48. 
 
 54- 
 
 
 60. 63. 
 
 72. 
 
 81. 90. 96. 
 
 prin.Key 
 
 
 
 K. 
 
 t.g. II. 
 
 t.l. 
 
 lll./i.IV. 
 
 t.r. V. 
 
 t.g. VI. t.l.Vll./.p.K.&c. 
 
 
 C Final 
 
 
 64. 
 
 72. 
 
 
 80. 84. 
 
 9 6. 
 
 108. 120. 128. 
 
 G. g& 
 
 A. 
 
 B. 
 
 C. 
 
 D. 
 
 d& 
 
 E. F. 
 
 %. G. 
 
 g*. A. B. C. D. 
 
 48. £0. 
 
 54- 
 
 60. 
 
 63. 
 
 72. 
 
 75- 
 
 81. 
 
 90. 96. 
 
 1 00. 108. 120. 126. 144. 
 
 V. t.g. 
 
 VI. t.l. 
 
 VII./^K. 
 
 t. r. II. 
 
 t>S 
 
 III. t.l. 
 
 *lV./f.\ 
 
 . 6c. Scale of the adjunct fifth. 
 
Chap. IV. 
 
 the Syjlem of Modulation further explained. 
 
 141 
 
 84. Befides the explanations of this fcheme, given, Art. 
 63 and 67, it may be ufeful here to add the following, viz. 
 (1.) Each of the three fcales in the fyftem has a grave 
 fourth and an acute fixth in reference to its own key ; tho' 
 this circumftance does not fully appear in the adjunct fcales 
 of the fcheme, becaufe the degrees of thefe fcales are all deno- 
 minated from their relations to the principal key. Thus 
 the VII [j is naturally and immutably a grave fourth in refe- 
 rence to the IV, as a key, and the II is naturally an acute 
 fixth, in the fcale of the IV, N° 1,2, 3. Again, the grave K 
 is a grave fourth, and the acute III is an acute fixth in refe- 
 rence to the V, as a key, N° 7, 8, 9. And it will be very 
 convenient for the reader to familiarize the com pari fons of 
 each degree of the principal fcale with both the adjunct keys, 
 in order to enter fully into the meaning of this fcheme, and 
 of what we have farther to add in regard to the nature and 
 conftitution of chords. He ought alfo to fix in his memory 
 the altered, or occajionally tempered degrees which belong to 
 each fcale, thefe not being any way diftinguifhed in written 
 mufic, nor known but by a nice attention to the different ef- 
 fects of the fame note in different occurrences, on many of 
 our modern inftruments. Thus the fcale of the adjunct IV 
 brings in a perfect IV, and a natural VI, befides the VII \, ; 
 though this laft is the only altered note which appears in our 
 mufic; while the fcale of the principal K takes a grave IV, and 
 an acute VI ; and the fcale of the adjunct V introduces a grave 
 K, and an acute III, befides the % IV which takes place in 
 our written mufic. 
 
 (2.) It muff alfo be well remembered, that the epithets 
 grave and acute, which we here annex to certain degrees of 
 the fcale, do not at all import, that fuch degrees are to be 
 depreffed or elevated in an unnatural ox forced manner : they 
 only ferve to diftinguifh the different founds of notes, which 
 are apparently the fame, on different occafions, according as 
 
 one or another of the three fcales takes place at the time. 
 Thus the grave IV and acute VI, in the fcale of the princi- 
 pal key, are as natural as the perfect IV and natural VI, ia 
 the fcale of the adjunct IV ; and fo of others : and if it fhould 
 feem improper to denominate one ftate of thefe variable founds 
 natural, when the other ftate is aflerted to be equally na- 
 tural in its proper place, let it be obferved, that we have cho- 
 fen thefe epithets with a view to conform, as near as poffible, 
 to the language of muficians, and to the fcale formerly efta- 
 blifhed by theorifts. We alfo gain another advantage by this 
 application of the terms grave, natural, and acute, viz. that 
 the real quantities of the occafional temperament are thereby 
 eafily remembered, as obferved in the firft and fecond para- 
 graphs of Art. 71. 
 
 We now proceed to examine the proportions of other 
 chords. 
 
 85. The chord of the 7th, with a greater third, which is 
 the leading chord of every perfect regular cadence, is pro- 
 duced by admitting the found which divides the module into 
 7, along with the other harmonics of the perfect chord, as al- 
 ready explained. This chord is alfo the production of the 
 natural confonances of a mufical found, as obferved, Art. 79; 
 for the coincidence at every feventh pulfe is fufficient to ren- 
 der this added feventh perceivable ; and Mr. Rameau, in his 
 Generation Harmonique, acknowledges, that he diftinguifhed 
 it among the reft, but calls it a loft found [fon perdu]. But 
 however loft it may have been to the theorifts, the practitio- 
 ners have not failed to make great ufe of it, and to relifh its 
 effect very highly. This added feventh, in a leading chord, 
 naturally refolves by falling into the third of the following 
 chord. Thus the leading chord of the regular cadence on 
 the principal key confifts of the V, VII, II, and grave IV, 
 which, taken in the numbers of the principal fcale, intitled^SW, 
 viz, the 6th line of the fcheme, and reduced, become 4, 5, 6, 
 
 Nn 
 
1^2 Of Combined Sounds 
 
 and 7, refpettively. The fame may be obferved of the K, 
 III, V, and VII |j, which form the leading chord to the ad 
 junct IV, taken in the numbers of the 2d line ; and alfo of 
 the II, ^IV, acute VI, and grave K, which form the lead- 
 ing chord to the adjunct V, taken in the numbers of the 8th 
 line. 
 
 86. We have alfo the fame kind of chord upon the III of 
 the fcale leading in a regular cadence on the fubfiituted VI as 
 a key; fee Part I. Art. 200: by joining together the III, 
 ■% V, VII, and II ; with this difference only, that here the II, 
 {landing as an added feventh to the III/", ought to be re- 
 prefented by 70, inflead of 72, as in the 6th line of the 
 fcheme ; or elfe the proportions of 4, 5, 6, and 7, which be- 
 long to every perfect leading chord, will not take place; the 
 confequence of which is, that the II, in this occurrence, muft 
 either be depreffed below its natural place, in the proportion 
 of 35 to 36, by way of accidental temperament, or, at leaf!, 
 its parcel muft be conceived as formed of the factors 7x5, 
 in order to reconcile this chord to the imagination : and it is 
 very probable, that the difficulty of conceiving fuch a parcel, 
 as obferved, Art. 65, is the only reafon why beginners in fmg- 
 ing find it much harder to join a feventh to the leading chord 
 of a flat than to that of a fharp feries. 
 
 87. The fame chord alfo occurs again in each of the ad- 
 junct fcales, taking jud the fame degrees, in reference to the 
 keys of thefe fcales, as that above mentioned does in reference 
 to the principal key. Thus the VI, ^K. Ill and V, form 
 a leading chord to the II, as fubfiituted fixth in the fcale of 
 the adjunct IV, viz. the 2d line of the fcheme ; and the VII, 
 %l\, $IV, and VI, form a leading chord to the III, as fub- 
 fiituted fixth in the fcale of the adjunct V, viz. the 8th line 
 of the fcheme : but as each of thefe requires the note which 
 makes the added feventh to be accidentally tempered, like the 
 II in the preceding article ; and as fuch temperament is found 
 
 Part II. 
 
 very difficult, even in the fcale of the principal key, where 
 the greateft liberties can be taken, it would be extremely dif- 
 ficult, in an adjunct fcale, unlefs the key of that fcale 
 were really taken as a new key, for the time ; we therefore 
 conclude, as in Part I. Art. 200. that the key muft effectu- 
 ally be changed, and the IV, or the V, muft be received as 
 a new key, before either of their fubfiituted fixths can be in- 
 troduced as fuch. 
 
 88. In regard to the VI note, which we have all along cal- 
 led the fubfiituted VI, as being made the principal key of a 
 flat feries without introducing any change in the fcale, ex- 
 cept the $ V and tempered II, which are properly only acci- 
 dental alterations ; fee Part I. Art. 200, &feq. It muft be 
 obferved, that this is always the acute VI of the fcale; and 
 that the principal chord of the flat feries, which we call the 
 \) P chord, Parti. Art. 139. confifts of the acute VI, theK, 
 and the acute III of the fcale : concerning which chord the 
 following particulars muft be obferved. 
 
 (1.) This chord is not to be found in any one of the fcales 
 in the fyftem, but takes its fifth note (viz. the acute HI) 
 from the fcale of the adjunct V, while the third note, (viz. 
 the K) belongs to the fcale of the principal key, and the fun- 
 damental note, (viz the acute VI,) belongs indifferently to 
 both thefe fcales This is one inftance of a chord formed by 
 the intermixture of two fcales, which may be fuppofed to di- 
 vide our attention at that time, as obferved, Art. 45. and 
 hence it appears, that the flat feries is really a mixture of two 
 fcales, and its key note is a found partly fixth and partly fe« 
 cond. See Part I. Art. 34, 257, 261, and 262. 
 
 ( 2.) The proportions of this chord, taken in the numbers 
 of the fcheme, in the 8th and 6th lines, are 54, 64, and 81 ; 
 the two firft of which reduce to 27 and 32, but no lower; 
 and the firft and laft reduce 10 2 and 3, viz. a perfect 
 fifth. The largenefs of the numbers 27 and 3 2 has induced 
 
Of leading Chords, and of the flat perfecl Chord. 
 
 Chap. IV. 
 
 all former theorifts to conclude the interval expreffed by them 
 to be afalfe, or difcordant lefs thi. J ; and fuch a conclufion 
 would be inconteftible, if the perfection of a concord were al- 
 ways to be truly eftimated by the fmallnefs of the numbers, 
 or the frequency of coincidence of its terms : but this is not 
 the cafe, for the implied found, as we have already feen, 
 muff be taken into account ; and it is more than probable, in 
 regard to the fimplicity of the ratio, which is another necef- 
 fary circumftance, fee Art. 82. that a ratio may be fimple 
 or intelligible to the mufician, if its two terms be fufficient- 
 ly fimple and eafy to be conceived, though not reducible in- 
 to very fmall numbers. However this be, we find by experi- 
 ence, that the principal key of the flat feries is really the a- 
 cute VI of the natural fcale ; and that the K of the fcale, when 
 it is taken as a fundamental lefs third above this VI, is not 
 perceivably altered in its pitch, like the other occafionally tem- 
 pered notes, but only becomes a difregarded found, for the 
 time, and therefore we conclude the fundamental lefs third 
 of the flat feries to be reprefented by 27 to 32, and not by 
 5 to 6, as has been fuppofed by all writers on this fubject ; 
 in defence of which material innovation we obferve, 
 
 (3.) That the proportion of 27 to 32 generates 5, which, 
 by doubling, becomes 20; that is, the acute VI and K, when 
 heard together, generate the III, which we take as a perfect fifth 
 to the fame VI, and on that account the VI becomes a funda- 
 mental. I fay, " we take it as a perfect fifth," becaufe it is 
 nearly fo, being only deficient by a comma. Were the K to 
 make 32 vibrations and one fixteenth part, for 27 of the VI, 
 the implied found would become 20 and one fourth, or 81 ; 
 viz. the acute III, which is the perfect fifth of the key of the 
 flat feries : and in this cafe, the proportion of the key to its 
 lefs third would be 432 to 513, which reduces to that of 16 
 to 19. Now 16, 19, and 24, are probably the very proporti- 
 ons which flrike our fenfe on hearing the \) P chord, though at 
 
 m 
 
 the fame time we do not ceafe to regard the key and its third 
 as confifting of 27 and 32 vibrations or parts refpectively, 
 agreeable to their natural conftitution in the fcale : fo that the 
 third of the flat feries becomes a difregarded found, becaufe 
 the parcel 19, which it partly aflumes at that time, is not 
 eligible, neither can be received into mufic otherwife than 
 with refpect to the implied founds; that between it and the 
 key of the feries, reprefented by 3, being the perfect: fifth of 
 the feries ; and that between it and the fifth, reprefented by 
 5, being a fundamental great third to the key : but if our at- 
 tention be in any degree fixed on the fame lefs third, it imme- 
 diately becomes the principal key of the fyftem, agreeable to 
 its natural conftitution. 
 
 (4.) This fuppofition is further confirmed by obferving, 
 that as the VI of the fcale is never conftituted a key without 
 the affiftance of the $V, either in the chord immediately pre- 
 ceding, or in that following ; fo this ■$. V being a fundamen- 
 tal great third to the III, and the III being taken as an acute 
 III, in order to its being the perfect fifth to this key, the $V 
 of confequence feems to bear to this key the exact proportion 
 of 1 5 to 16, like the VII and K of the fharp feries. Now, 
 this proportion being both more fimple and more familiar to 
 us than that of 25 to 27, is eafily perceived, tho' the latter 
 is the true proportion ; fo that the paflage from the $V to the 
 VI, and vice verfa, naturally inclines us to give fome atten- 
 tion to 1 6 vibrations of the VI ; and this imperfect and diftant 
 refemblance between the keys of the two different feriefes pro- 
 bably firft gave rife to thofe fudden tranfitions from the fharp 
 to the flat feries, and contrariwife, upon the fame key, taken 
 notice of in Part I. Art. 256 : it feems alfo to have fuggeiled 
 the joining of a fharp third to the final chord of flat harmony, 
 called by the French, la tierce dePicardie. At the fame time 
 we are not to fuppofe that a new module is really adopted con- 
 taining 16 vibrations or parts of the acute VI, when the flat 
 
1 44 Of Combined 
 
 feries is introduced, but only that the abovementioned pro- 
 portions are perceived as accidentally taking place, while the 
 module continues the fame as in the fharp feries. In fhort, 
 the flat feries is juftly reputed a kind of impofition or trick 
 put upon the hearer, to make him acquiefce in a conclufion 
 which is not in itfelf the moft natural. 
 
 (5.) That the key and the lefs third of the flat feries are 
 not in the proportion of 5 to 6, or not conceived as fuch, may 
 be infered from thefe confiderations. If they were conceived 
 by the proportion of 5 to 6, the implied found, reprefented 
 by 4, would inevitably prefent itfelf to the imagination along 
 with 5 and 6; that is, the lefs fixth of the feries would always 
 take place along with the key and its lefs third ; and further, 
 . this lefs fixth would be efteemed the fundamental, according 
 to Art. 8 2 ; and the lefs third, having a Ampler parcel than the 
 key, would therefore attract the greater part of our attention : 
 how widely different thefe circumftances are from the truth no 
 one need be told who has read this treatife, or indeed who 
 has any fenfibility of the effect of mufic. Several other argu- 
 ments might be brought in defence of this ne-iv proportion af- 
 figned to the fundamental lefs third, which we here omit ; 
 prefuming, that if truth be on its fide, we have faid enough ; 
 and if not, too much. 
 
 89. Thus far we have treated only of fuch chords as are 
 rezMy fundamental chords, in the ftricteft fenfe of the word ; 
 but there are feveral other chords which enter into our har- 
 mony, which we have alfo called by the fame name, as con- 
 fiding of a fundamental, and its third and fifth, though the 
 attention is not fo entirely placed on the fundamental found. 
 Of this fort is the chord of the IV of the fcale, leading in a 
 gradation on the V, Part I. Art. 210. and the chord of the 
 II, with its lefs third, leading in an imperfect cadence on the 
 fame V. The IV in this gradation is always a grave IV, and 
 accompanied with the acute VI, (being its redundant great 
 
 Sounds. 
 
 Part II. 
 
 third,) and the grave K, (being its perfect fifth.) Now, tho' 
 the perfect fifth always generates the octave below its lower 
 term, yet the redundant great third, 7 to 9, generates 2, or 
 the double octave below the intermediate tone 8 ; and the de- 
 ficient lefs third 6 to 7, which lies between the upper term 
 of this third and the fith, generates 1, or rhe triple oclave be- 
 low the next fuperior tone 8 : that is, though the grave K 
 in effect doubles the grave fourth, yet the acute VI, joined 
 with the fame grave IV, generates the V ; and the fame VI, 
 with the grave K, generates the II of the fcale ; fo that the 
 V and II of the fcale being effentiafly implied in this combi- 
 nation, it follows, that this chord has a dependence on, and 
 connexion with the chord of the V. and this, in all probabi- 
 lity, is the reafon why the muficians have generally reprefent- 
 ed this chord as derived from that of the V, and arifing from 
 the feventh, ninth, and eleventh of the fame V. 
 
 In like manner, the chord of the 11/, confifting of the II, 
 grave IV, and acute VI, has been found, by the artifts, to 
 bear fome relation to the fame V ; and has therefore been e- 
 fteemed as its fifth, feventh, and ninth : the reafon now ap- 
 pears to be, becaufe the V is the implied found in both of 
 the thirds which conftitute this chord. 
 
 90. The only chord which has the lefs third 5 to 6 be- 
 low the great third 4 to 5, or which, according to our me- 
 thod of defigning chords, has a fuperftructed lefs third below 
 a fundamental great third, is the chord of the III/, confifting 
 of the III, V, and VII of the fcale ; and here the attention, 
 inftead of being wholly fixed on the III as fundamental, 
 as it ought to be, if this were really the \>? chord of the 
 key of the flat feries, is partly carried away to the V of 
 the fcale, and partly to the key, as coexiftent fundamentals. 
 Such a multiplicity of fundamentals claiming our attention at 
 once, is a fufficient reafon why this chord ought to be feldom 
 ufed j and accordingly it is but rarely found in the compofi- 
 
Chap. IV. 
 
 tions of the beft matters, except in paffages where it is taken 
 as an attempt to pafs into the new key of the fubftituted VI, 
 counteracted, by retaining the natural V in the accompany- 
 ment. See Part I. Art. 2:4. 
 
 91. It appears then that we have no lefs than three diffe- 
 rent forts of chords, each confiding of a fundamental with 
 its lefs third and fifth, and two forts confifting of a funda- 
 mental with its great third and fifth ; which differences are 
 entirely owing to the differences of the thirds, as defcribed 
 above : for every fundamental fifth is to be fuppofed perfect ; 
 and this affords us a quite new and extenfive profpect of va 
 rieties in harmony, which, if found upon impartial trial to be 
 real, will deferve a more minute furvey. 
 
 The public is defired to confider what we have here at- 
 tempted as a kind of rough draught of a defign, which we 
 are in fufpenfe whether to finifh or cafl away, till their judg- 
 ment fhall afcertain its defert. 
 
 This being the cafe, we fhall not here pretend to fay whe- 
 ther there may not be yet other forts of chords fometimes ad- 
 mitted, different from any of the above, or their inverfions : 
 for, though it feems pretty clear, that the generality of chords 
 belong to one or another of thefe forts, yet it is not impof- 
 fible but that other proportions may be perceived, and efpe- 
 cially in the fubftitu tions ufed in fiat harmony, by way of 
 accident, fomething fimilar to what is obferved of the princi- 
 pal chord of this feries, in Art. 88, N° 4. 
 
 It is for the fame reafon, and to avoid the cenfure, often 
 toojuftly due to the propofers of new fyftems, of forcibly de- 
 ducing every thing from a favourite principle, that we have 
 omitted to afcertain the fmalleft interval which can be a con- 
 cord, from the principle of implied founds ; though it is very 
 probable that as an interval of three octaves is known by ex- 
 perience to be too large to be conceived without fome inter- 
 mediate found, fee Part I. Art. 1 26. fo no combination 
 
 Of other Chords, improperly called per feci. \^ 
 
 ought to be efieemed a concord, however /imple and eligible 
 its terms may be in every other refpect, if the implied found 
 be three octaves, or more, below the lower term ; and this ex- 
 cludes the interval of a greater tone 8 to 9, and every other 
 lefs than this, from the clafs of concords; and alfo fhews us 
 why thedifcord necefiarily becomes greater, as the founds ap- 
 proach nearer, becaufe the implied found becomes more and 
 more remote, till their difference be fo inconfiderable that 
 they pafs foi unifons. 
 
 It is alfo very probable, that the fame principle of implied 
 founds may be the reafon why the more perfect concords ought 
 to be below, and the lefs perfect above, in a chord ; and whyaa 
 interval, which is concord in the upper parts, is often no con- 
 cord when taken in the bafs : for if we lay itdoWn as a rule, 
 that the implied found of a concord ought always to be with- 
 in the limits of audible found, and that the loweft audible 
 found is an octave below the FF of the bafs, fee Art. 25 ; it 
 will follow, that if the lower term of a fifth be below FF ; 
 or the lower term of a fundamental great third below F, the 
 bafs cliff line, thefe intervals will ceafe to be concords. Af- 
 ter the fame manner might the fituationsof all other concords 
 be afligned, but we fhall not infift on thefe particulars at 
 prefent. 
 
 92. If this theory be received, the imperfections of the or- 
 gan, and all inftruments conftructed on the principles of a 
 femitonic fyfiem of fixed founds, will become much more con- 
 fpicuous than they have hitherto been. It mult immediate- 
 ly appear furprizing, that fuch great thirds as are tuned ex- 
 actly in the proportion of 5 to 4, do not prove quite intoler- 
 able, in fituations where we conceive them by the proporti- 
 on of 9 to 7. The fame may be faid of the lefs third 6 to 5 
 in fituations where we conceive its terms by the proportion of 
 7 to 6 : as in thefe occurrences the error of the interval is 
 36 to 35: Art. 72. N° 1 and 2, which is a complete quar- 
 
 Oo 
 
146 
 
 Of Combined 
 
 ter of a tone. And although by tuning the fifths fomething 
 too fmall, which the nature of the fyftem renders inevitable, 
 and alfo the great thirds a fmall matter too large* which many 
 muficians have thought eligible, thefe errors sire every where 
 dimini(hed ; yet as ail the diflocation or temperament which 
 the ear can allow in a fundamental great third, or even in a 
 fifth, does not exceed a quarter of a comma ; fo no reme- 
 dy for the abovementioned errors can be obtained by any 
 fcheme of tuning. And yet for all this imperfection, joined 
 with all others, which arife from thefe principles, or have 
 been obferved by former theories, the oigan exhibits a delight- 
 ful variety of fine harmony ; though it muft be acknowlegeu, 
 that the variety is much greater, and more fti iking in a con- 
 cert of violins, &c. whofe founds are variable at the difcie- 
 tion of the performers. 
 
 93. Shall we then venture to fay, that on attending to the 
 harmony of a well-tuned organ, where thefe redundant and 
 deficient thirds cannot really exift ; as for inftance, in the key 
 of C natural, we do not admit any idea of fuch proportions ; 
 but conceive every great third by the proportion of 5 to 4, 
 and every lefs third by that of 6 to 5, to which they nearly 
 correfpond ? The fallacy of fuch an afiertion would be too 
 glaring; for if the idea of the proportion of 6 to 7, have 
 place no where elfe, yet it muft certainly exift between the 
 II and IV of the fcale, when this latter comes in as an added 
 feventh in the leading chord of a final cadence on the key ; 
 and the falfe fifth between the VII and the fame IV is un- 
 doubtedly taken as 5 to 7 : and accordingly this proportion 
 of the falfe fifth has been afiigned by feveral late authors ; 
 though the confequences necefiarily refulting therefrom, and 
 which we have endeavoured to trace out in this ElTay, have 
 not been attended to. In the fame leading chord, when the VI 
 is alio joined, as an added ninth, the proportion of 7 to 9 
 muft unqueftionably exift between the IV and VI. Now, if 
 
 Sounds. Part II. 
 
 thefe proportions could not be received into organ mufic, the 
 confequence would be, that added fevenths and ninths Ihould 
 be totally excluded in the leading chords of regular cadences 
 in compofitions for that inftrument ; but this is far from be- 
 ing the cafe, and the eftecl: of fuch added diflonances is much 
 the fame on the organ as with violins. 
 
 94. The moft fatisfaclory folution of this difficulty is got 
 by obferving that the lefs fimple any ratio '19, the lefs follici- 
 tousweare concerning the juftnefs of the terms between 
 which that ratio is conceived to exift ; and confequently al- 
 though octaves, fifths, and fundamental great thirds, are 
 obliged to be nearly juft before we acquieice in them ; yet 
 lefs thirds of all kinds, falfe fifths, redundant great thirds, he. 
 may differ widely from the true proportions, and yet pafs 
 without offence ; though at the fame time the nearer they ap- 
 proach to the true proportions, the more agreeable is the ef- 
 fedT: ; and this is the only fuppofition which will rightly ac- 
 count for our tolerating the. falfe relations which often inevit- 
 ably occur in mufic ; though perhaps no one ever fufpe&ed 
 that errors fo large as a quarter-tone were thus paffed over. 
 
 Now, to prove, by a familiar inftance, that errors even larg- 
 er than this are tolerated in fome chords of mufic, we need 
 only to obferve, that from the very nature of the French- 
 horn and Trumpet, the IV of the fcale of thefe inftruments 
 really bears to the key the proportion of 1 1 to 8. (See Phil. 
 Tianf. N° 195. andabrigedby Mr. Low thorp, vol. I. p. 607.) 
 while the II bears to the fame key the juft proportion of 9 to 
 8. therefore the interval between the II and IV is exactly 9 
 to 1 1 : and yet this interval, in compofitions for a firft and 
 fecond horn, is very frequently ufed where the ear muft ine- 
 vitably conceive the founds by the proportion of 6 to 7. Here 
 the difference of the interval really exifting, from that which 
 is conceived to exift, is no lefs than 22 to 21, viz. about three 
 eighth parts of a tone ; and though it is true, that we perceive 
 
Chap. IV. 
 
 Of ImperfeSlions in the Semitonic Syftem. Experiments propofed. 
 
 this IV to be too fharp, yet it is alfo true that we are not 
 Shocked with it, in this occurrence, as we fhouid be with the 
 III, or V, it either of thefe were diflocated but the third part 
 of the above quantity. In the cafe here inftanced, it is fur- 
 ther to be obferved, that the whoie error of the interval 1 arifes 
 from the error of the IV, which is fo much too fliarp, while 
 the II is perfectly juft ; whereas, when an error of a quarter- 
 tone occurs in the organ, it always takes its rife from two 
 fmaller errors added together, one of the founds being a little 
 toofharp, and the other a little too flat, at that occurrence; 
 and on this account the error of either of the founds will be 
 much more inconfiderable. 
 
 95. Though the imperfections of our femitonic fyftem are 
 by this means fet in a new light, yet it will appear that the 
 received method of tuning the organ and harpficord is ftill the 
 beft of which the fyftem is capable ; for we ought to make 
 all our fifths and fundamental great thirds good, if it could 
 be done; but as this is abundantly proved to be impoffible, 
 and appears very plainly in the Ample inftance of the hnpof- 
 fibility of dividing a perfect. octave into three tolerable funda- 
 mental great thirds ; it is beft to make eight of the moft ac- 
 cuftomed great thirds good, and let the remaining four bad 
 ones be between F $ and B [>, between G% and C, between B 
 and E\), and between C$ and F, which are lefs ufed : and this 
 may be effected by making all the fifths in the fcale one fifth 
 part of a comma lefs than perfect, except that between G $ 
 and E[j, which is leaft ufed, and which will, by this method, 
 be very bad, being more than a comma and a quarter too 
 large. The good great thirds will be one fifth of a comma 
 greater than perfect, and the bad ones will be a comma and a 
 half too large. And although by tuning the fifths a quarter of 
 a comma lefs than perfect, the good great thirds might be ren- 
 dered quite perfect, yet the four bad great thirds, and the one 
 bad fifth, would then be larger and worfe. Indeed, if care were 
 
 M7 
 
 taken to compofe and fet mufic always in fuch keys that thefc 
 large great thirds might come in, where the proportion of 7 
 to 9, or that of 64 to 81 were wanted, one might venture to 
 pronounce them perfections rather than otherwife, and to 
 afcertain the beft method of tuning from our new principles; 
 but as this is not often the cafe, nor likely foon to become fo, 
 we can only obferve, at prefent, what effect thefe too large 
 great thirds have, when they happen to come in places where 
 the redundant thirds ought to exift. Now, the moft proper 
 keys for fuch trials are thofe with three flats at the cliff; viz. 
 the fharp feries of E [?, and the flat feries of C : for here the 
 great third between the IV and VI of the fcale, that is from 
 G $, (now taken for A [/,) to C, is a large great third, on 
 account of the IV being too grave ; for the interval from E»h 
 to A \j is lefs than a perfect fourth, by as much as the inter- 
 val from A \) to E [j exceeds a perfect fifth, which, in the moft 
 approved method of tuning, comes very near to the quantity 
 which we have all along called a bearing. Let it then be tried, 
 on fome fimple flop of a well tuned organ, whether the note 
 A[j, when it occurs as an added feventh to B[j as fundamen- 
 tal, leading in a regular cadence on E \/ as a key, have not a 
 finer effecl: than any other feventh in the organ ; and if it be 
 found preferable, this fingle inftance will ferve to prove the 
 eligiblenefs of the proportions which we have affigned, in this 
 Treatife, to the leading chord of every cadence. It may be 
 further tried, whether the large great third between A \> and C 
 be any way offenfive, when thefe two are joined as feventh 
 and ninth to B [> ; or when they ftand as third and fifth to F, 
 leading in an imperfect cadence on B[>, as adjunct fifth, in 
 the fame fcale of E fy ; or when the fame chord F, A |>, and C 
 occurs as chord of the adjunct fourth, in the flat feries of C. 
 In all thefe occurrences, and many others of the fame kind, 
 the large great thirds were found not only tolerable, but con- 
 feifedly preferable to the perfect fundamental thirds which 
 
1 48 Of Combined 
 
 occur in fimilar fituations, in other keys, according to the 
 judgment of all the expert muficians whom the author has had 
 opportunity of confulting : and he earneftly defires the moft 
 fkilful, into whofe hands this Effay may come, to endeavour 
 to fatisfy themfelves and the world in regard to the truth of 
 the principles above delivered, by thefe or any other trials, 
 which their knovvlege may fuggeft as proper and fufficient. 
 
 96. It is not fo eafy to prefcribe proper trials whereby the 
 juft proportion of the fundamental lefs third maybe afcertaia- 
 ed, becaufe, as hinted above, the ear is not very critical in 
 regard to this cihord, the upper note is a difregarded found, 
 and, as fuch, may vary greatly from the juft proportion, be- 
 fore we difcover any error: but if the implied founds, as 
 formerly defcribed, do really take place, and, if the manner 
 of perceiving the pulfes, by certain eligible parcels, prove to 
 be really the director of the human mind in the choice of mu- 
 fical founds, it feems impoffible to affign any other proporti- 
 on to the fundamental lefs third, than that which is given a- 
 bove, confiftent with thefe principles. 
 
 97. The fcale of the organ, tuned as ufual, has nine lefs 
 thirds, each falling fhort of the proportion of 5 to 6, by a- 
 bout two fifth parts of a comma, and the three other lefs 
 thirds are very near the proportion of 6 to 7 ; viz. thofe from 
 Efr to ¥% from F to G% and from Bfc, to C$ : fo that 
 the organ does not afford any lefs third in, or near, the pro- 
 portion of 27 to 32, (or 16 to 19) whereby to make trial 
 whether fuch a third be more fatisfactory in the final chord 
 of flat harmony ; but yet the trial may eafily be made by flat- 
 tening, for the time, that note which makes the lefs third of 
 the feries. 
 
 Sounds. 
 
 Part II. 
 
 98. Before the author came to a refolution of publifhing 
 this (ketch of a theory, he made feveral accurate trials of the 
 different forts of thirds, by the well-known method of put- 
 ting two firings of the fame fize on a violoncello, tuned ex- 
 adtly unifons, and flopping off one of them after the manner 
 of a monochord, by proper divifions on the finger board ; 
 and he found that altho' among the lefs thirds, 5 to 6, is the beft 
 concord, yet the mind is naturally led to regard the upper 
 found rather than the lower. In the lefs third, 16 to 19, 
 the lower found inevitably attracts our attention, and the up- 
 per is difregarded. In the deficient lefs third, 6 to ?, the 
 lower found alfo prevails, but the upper feems dejected and 
 unconclufive. The fame dejected effect attends the lower 
 found of the redundant great third, 7 to 9 : and the great 
 third, 19 to 24, taken without any other accompanyment, 
 is intolerable difcord ; probably becaufe the lower found, re- 
 prefented by 19, cannot, in this fituation, be difregarded, 
 as it would be, if another found reprefented by 16 were alfo 
 added. As a farther trial of thefe principles, he alfo publifh- 
 ed, fome time ago, a fmall collection of Pfalm-Tunes, in parts, 
 with the grave and acute notes marked by accents, according 
 to the view he then had of the matter, of which copies may 
 be had, by applying to thofe who fell this Treatife. 
 
 As the Author has not been fparing, either of time or flu- 
 dy, in order to difcover, and publifh, what appeared to him 
 to be truths of the greateft importance in the fcience of mu- 
 fic ; fo he expects his Work will not be haftily cenfured, nor 
 condemned, without as fair and impartial a trial as the cafes 
 in queftion will admit of. 
 
 FINIS.