AUGENER'S EDITION, No. 0182. 
 
 HARMONY: 
 
 ITS THEORY AND PRACTICE. 
 
 BY 
 
 EBENEZER PROUT, B.A. Lond. 
 
 Professor of Harmony and Composition at the Royal Academy of Music, 
 
 LONDON : 
 
 AUGENER & CO. 
 
 [ENTERED AT STATIONERS' HALL.] 
 COPYRIGHT FOR ALL COUNTRIES.] [RIGHTS OF TRANSLATION RESERVED.
 
 UNTWISTING ALT. THE CHAINS THAT TIE 
 THE HIDDEN SOUL OF HARMONY." 
 
 MILTON, L' Allegro.
 
 Stack 
 Annex 
 
 K 
 
 PRE FACE. 
 
 So large a number of works on Harmony already exists that the 
 publication of a new treatise on the subject seems to call for 
 explanation, if not for apology. The present volume is the out- 
 come of many years' experience in teaching the theory of music, 
 and the author hopes that it contains sufficient novelty both in 
 plan and in matter to plead a justification for its appearance. 
 
 Most intelligent students of harmony have at times been per- 
 plexed by their inability to reconcile passages they have found in 
 the works of the great masters with the rules given in the text- 
 books. If they ask the help of their teacher in their difficulty, 
 they are probably told, " Bach is wrong," or " Beethoven is 
 wrong," or, at best, "This is a licence." No 'doubt examples of 
 very free part-writing may be found in the works of Bach and 
 Beethoven, or even of Haydn and Mozart ; several such are 
 noted and explained in the present work. But the principle must 
 surely be wrong which places the rules of an early stage of musical 
 development above the inspirations of genius ! Haydn, when 
 asked according to what rule he had introduced a certain harmony, 
 replied that "the rules were all his very obedient humble servants"; 
 and when we find that in our own time Wagner, or Brahms, or 
 Dvorak breaks some rule given in old text-books there is, to say 
 the least, a very strong presumption, not that the composer is 
 wrong, but that the rule needs modifying. In other words, 
 practice must precede theory. The inspired composer goes 
 first, and invents new effects ; it is the business of the theorist 
 not to cavil at every novelty, but to follow modestly behind, and 
 make his rules conform to the practice of the master. It is a 
 significant fact that, even in the most recent developments of the 
 art, nothing has yet been written by any composer of eminence- 
 which a sound theoretical system cannot satisfactorily account for; 
 and the objections made by musicians of the old school to the 
 novel harmonic progressions of Wagner are little more than 
 repetitions of the severe criticisms which in the early years of the 
 present century were launched at the works of Beethoven.
 
 iv PREFACE. 
 
 It is from this point of view that the present volume has been 
 written. The rules herein given, though in no degree inconsistent 
 with the theoretical system expounded, are founded, not upon 
 that, nor on any other abstract system, but upon the actual 
 practice of the great masters ; so that even those musicians who 
 may differ most widely from the author's theoretical views may 
 still be disposed to admit the force of practical rules supported by 
 the authority of Bach, Beethoven, or Schumann. 
 
 The system of theory propounded in the present volume is 
 founded upon the dictum of Helmholtz, quoted in Chapter II. 
 of this work ( 42), that "the system of Scales, Modes, and 
 Harmonic Tissues does not rest solely upon unalterable natural 
 laws, but is at least partly also the result of sesthetical principles, 
 which have already changed, and will still further change with the 
 progressive development of humanity." While, therefore, the 
 author follows Day and Ouseley in taking the harmonic series as 
 the basis of his calculations, he claims the right to make his own 
 selection, on aesthetic grounds, from these harmonics, and to use 
 only such of them as appear needful to explain the practice of the 
 great masters. Day's derivation of the chords in a key from the 
 tonic, dominant, and supertonic is adhered to, but in other respects 
 his system is extensively modified, its purely physical basis being 
 entirely abandoned. It will be seen in Chapter II. ( 44) that by- 
 rejecting altogether the eleventh and thirteenth notes of the har- 
 monic series, and taking in their place other notes produced among 
 the secondary harmonics, the chief objection made by the oppo- 
 nents of all scientific derivation of harmony that two of the most 
 important notes of the scale, the fourth and the sixth, are much 
 out of tune has been fully met. In the vexed question of the 
 minor tonic chord, Helmholtz is followed to a considerable extent ; 
 but Ouseley's explanation of the harmonic origin of the minor 
 third is adopted. 
 
 Truth is many sided ; and no writer on harmony is justified in 
 saying that his views are the only correct ones, and that all others 
 are wrong. No such claim is made for the system herein set forth ; 
 but it is hoped that it will at least be found to be intelligible, 
 perfectly consistent with itself, and sufficiently comprehensive to 
 explain the progressions of the advanced modern school of com- 
 posers. 
 
 It has been thought desirable to separate as far as possible the 
 practical from the theoretical portions of this work. The latter 
 are therefore printed in smaller type; and it will be found
 
 PREFACE. v 
 
 advisable for beginners, who may take up this work without any 
 previous knowledge of the subject, to omit at least Chapters II. and 
 III., dealing with the Harmonic Series and Key or Tonality, until 
 some considerable progress has been made in the practical part of 
 the volume. The exact point at which the student will do well to 
 return to the omitted portions will depend upon his progress and 
 his general intelligence, and must be left to the discretion of the 
 teacher. 
 
 In the practical part of the work an attempt has been made to 
 simplify and to codify the laws. With a view of effecting these 
 objects, many rules now obsolete, and contravened by the daily 
 practice of modern writers, have been altogether omitted, and 
 others have been greatly modified ; while the laws affecting the 
 chords, especially the higher discords the ninths, elevenths, and 
 thirteenths have been classified, and, it is hoped, materially sim- 
 plified. It is of the utmost importance that students who wish to 
 master the subject should proceed steadily and deliberately. For 
 example, a proper understanding of the chords of the eleventh will 
 be impossible until the student is quite familiar with the chords of 
 the ninth, which in their turn must be preceded by the chords of 
 the seventh. The learner's motto must be, " One thing at a time, 
 and that done thoroughly." 
 
 In preparing the exercises a special endeavour has been made 
 to render them interesting, as far as possible, from a musical point 
 of view. With this object they are, with a few exceptions, written 
 in the form of short musical sentences, mostly in four-bar rhythm, 
 illustrating the various forms of cadence. To stimulate the pupil's 
 imagination, and to encourage attempts at composition, many 
 exercises are in the form of double chants or hymn tunes. Each 
 bass can, of course, be harmonised in several different positions ; 
 and the student's ingenuity will be usefully exercised in trying to 
 write as melodious an upper part as possible for these little pieces. 
 
 Not the least interesting and valuable feature of the volume 
 will, it is believed, be found in the illustrative examples, consider- 
 ably more than 300 in number. These have been selected chiefly, 
 though not exclusively, from the works of the greatest masters, 
 from Bach and Handel down to the present day. Earlier examples 
 are not given, because modern harmony may be said to begin with 
 Bach and Handel. While it has been impossible without exceed- 
 ing reasonable limits to illustrate . all the points mentioned, it is 
 hoped that at least no rule of importance has been given without 
 quoting some recognised author in its support. It may at all
 
 vi PREFACE. 
 
 events be positively said that, had want of space not prevented 
 their quotation, examples might have been found to illustrate 
 every rule laid down in the volume. 
 
 It was originally intended to have included in the present work 
 chapters on Cadences, and on Harmonising Melodies. The 
 volume has, however, extended to so much larger dimensions than 
 was at first contemplated, that these chapters, which belong rather 
 to practical composition than to harmony in its strict' sense, have 
 been reluctantly omitted. It is intended to follow the present 
 work by a treatise on Composition, in which these and similar 
 subjects will be more appropriately dealt with. 
 
 The author desires to acknowledge the valuable assistance he 
 has received in the preparation of his work, first and foremost from 
 his son, Louis B. Prout, to whom he is indebted for a very large 
 number of the illustrative examples, and who has also written 
 many of the exercises. Valuable aid has also been received from 
 the late Rev. Sir Frederick Ouseley, with whom, down to the time of 
 his lamented death, the author was in frequent correspondence on 
 the subject of this work. To his friend, Dr. Charles W. Pearce, 
 also, the author must express his thanks for much generous interest 
 and many most useful suggestions, as well as for his kind assistance 
 in revising the proof-sheets of the volume. 
 
 It would be unreasonable to expect that the present work 
 will meet with universal approval ; but it may at least claim to 
 appeal to teachers and students as an honest attempt to simplify 
 the study of harmony, and to bring it down to date. 
 
 LONDON, June, 1889.
 
 TABLE OF CONTENTS. 
 
 [.V./?. The numbers refer in every instance to the sections, not to the pages. ] 
 
 CHAPTER I. INTRODUCTION- page Li 
 
 Amount of knowledge presupposed, I A musical sound, 2 Melody and 
 harmony denned, 3 Interval defined, 4 Semitone, 5 Diatonic and 
 chromatic semitones, 6- A Tone, 7 Scales, 8 The diatonic scale, 
 9 Nature of the Octave, 10 The chromatic scale, n -Names of the 
 degrees of the diatonic scale, 12-14 Consonance and Dissonance defined, 
 15-17 Discords, 18 How intervals are reckoned, 19 Compound in- 
 tervals, 20 Different kinds of intervals, 21 Perfect, major, minor, 
 augmented, and diminished intervals, 22-25 Inversion of intervals, 26, 
 27 Consonant and dissonant intervals. 28 Perfect and imperfect con- 
 sonances, 29 The scientific distinction between consonance and dis- 
 sonance, 30 Table of intervals, 31. 
 
 CHAPTER II. THE HARMONIC SERIES page ^4 
 
 The nature of Harmonics, 33, 34 Pitch defined, 35 The Harmonic 
 Series from C, 36, 37 Ratios of intervals, 38 Compound tones, 40 
 Secondary harmonics, 41 A selection made from these, 43 What 
 decides our choice, 44 Which primary harmonics are used for chords, 
 45 Calculation of difference of ratio of two intervals, 47-49 Calculation 
 of secondary harmonics, 50, 51 The " enharmonic diesis," 52. 
 
 CHAPTER III. KEY, OR TONALITY page 32 
 
 Definition of Key, 54-56 Development of key from its tonic. 57 A Chord, 
 58 The major common chord, 59, 60 The fundamental chord of the 
 seventh, 61 Chords of the ninth, 62, 63 Chords of the eleventh and 
 thirteenth, 64 The complete fundamental tonic chord, 66 The dominant 
 fundamental chord, 67, 68 The supertonic fundamental chord, 69 Why 
 the subdominant is not used, 70 The materials of the key of C, 71 
 Diatonic and chromatic notes in the key, 72 The chromatic scale, 
 73 The diatonic scale of C major, 74 All the triads derived from the 
 three fundamental chords, 75, 76 The whole key developed from the 
 tonic, 77 Other keys than C, 78, 79 Tetrachords, 80 Keys wjth sharps, 
 81-83 Keys with flats, 84, 85 The relationship of keys to each other, 
 86, 87.
 
 viii CONTENTS. 
 
 CHAPTER IV. THE GENERAL LAWS OF PART-WRITING ... page 42 
 
 Definition of a Part Progression of parts, 88 Rules of melodic progression, 
 89-92 Harmonic progression ; similar, oblique, and contrary motion, 
 93 Four-part harmony; names of the voices, 94 Rules of part- 
 writing, 95 Consecutive unisons and octaves, 96-98 Consecutive fifths, 
 99-101 Hidden octaves and fifths, 102-105 Consecutive fourths, 106 
 Consecutive seconds, sevenths, and ninths, 107-108 Progression from 
 a second to a unison, 109 Five recommendations as to part-writing, 
 110-116. 
 
 CHAPTER V. THE DIATONIC TRIADS OF THE MAJOR KEY : 
 
 SEQUENCES ... .. page oi 
 
 Diatonic and chromatic notes in a key, 117 The diatonic scale, 118 A 
 chord: its root, 119 Triads and common chords, 120 Major and 
 minor keys, 122- Major and minor chords : the diminished triad, 123 
 Doubling of notes, 124 Compass of the voices, 125 Close and 
 extended position, 126 The best position of chords, 127, 128 The 
 "root position" of a chord, 129 What notes may be doubled, 130 
 Progression of the leading note, 132, 133 Omission of a note of a 
 chord, 134 Effect of chords in their root position, 135 A doubled 
 part, 136 Sequence, 137 Tonal sequence, 138 Real sequence, 139 
 Irregular progressions justified by sequence, 140 Harmonising a se- 
 quence, 141, 142 Short score and open score, 143 An exercise 
 worked, 144-147. 
 
 CHAPTER VI. THE INVERSIONS OF THE TRIADS OF A MAJOR Km page 65 
 
 Inversion of a chord defined, 150 The first inversion of a triad, 151 
 Figured bass, 152 Figuring of a first inversion, 153 A series of first 
 inversions, how treated, 156 Which note to double, 157 The second 
 inversion of a triad, 158 The fourth with the bass a dissonance, 159 
 Examples of second inversions, 160-163 Rules for approaching a 
 second inversion, 164 Rules for leaving a second inversion, 165 Which 
 note to double in a second inversion, 166. 
 
 CHAPTER VII. THE MINOR KEY page 74 
 
 Difficulty of scientific explanation of the minor key, 168 Forms of the 
 minor scale, 169-171 Signature of the minor key, 172 Relative keys, 
 173 Enharmonics, 174 Tonic major and minor keys, 175 The 
 harmonic basis of the minor key, 176 The derivation of its diatonic 
 triads, 177
 
 CONTENTS. ix 
 
 CHAPTER VIII. THE DIATONIC TRIADS OF A MINOR KEY AND 
 
 THEIR INVERSIONS ... page 79 
 
 The triads of a minor key, 180, 181 The different character of major and 
 minor keys, 182 The diminished triad on the supertonic of the minor 
 key, 183 The augmented triad on the mediant, 184, 185 Employment 
 of major chords on ihe subdominant and supertonic, 1 86 Progression 
 between the dominant and submediant, 187 First inversions in the 
 minor key, 188, 189 A first inversion on the minor seventh, when 
 used, 190 Second inversions, 191, 192 The "Tierce de Picardie," 
 193, '94- 
 
 CHAPTER IX. THE CHORD OF THE DOMINANT SEVENTH ... page 87 
 
 Mental effect of tonic and dominant harmony, 195 Cause of the difference, 
 196 The chord of the dominant seventh, a "fundamental discord," 
 197 Its resolution, 198, 199 Resolution on the tonic chord, 201 
 Ornamental resolution, 202 Resolution on the submediant chord, 203 
 Resolution on the subdominant chord, 204 First inversion of the 
 chord of the seventh, 205-208 Second inversion, 209-212 Third inver- 
 sion, 213-215 Changes in the position of chord of the seventh, 216 
 Exceptional resolutions, 217 Reason of the importance of the chord of 
 the dominant seventh, 219. 
 
 CHAPTER X. KEY RELATIONSHIP MODULATION TO NEARLY 
 
 RELATED KEYS FALSE RELATION ... page 101 
 
 Modulation, 220 What constitutes Key Relationship, 222-224 Unrelated 
 keys, 225 The most nearly related keys, 226-228 How to effect a 
 modulation, 229 Immediate and gradual modulation, 231 Modulation 
 from a major key to the supertonic minor, 232 Modulation from minor 
 keys, 233 Examples of modulation, 235-242 Which modulations are 
 generally preferable, 243 False Relation, 244. 
 
 CHAPTER XI. AUXILIARY NOTES, PASSING NOTES, AND AN- 
 TICIPATIONS page 112 
 
 Auxiliary notes defined, 246 How taken and left, 247 When a tone and 
 when a semitone from the harmony note, 248 Chromatic auxiliary 
 notes, 249 Auxiliary notes in more than one part, 250 Examples 
 referred to, 252 Auxiliary notes quitted by leap of a third Changing 
 notes, 253, 254 Passing notes, 255 Ditto in the minor key, 256 Two 
 successive passing notes, 257 Passing notes quitted by leap, 258 
 Chromatic passing notes, 259 Examples referred to, 260 Auxiliary 
 notes cannot make false relation, 261 Anticipation, 263, 264 Ex- 
 ceptional treatment of auxiliary notes, 265.
 
 x CONTENTS. 
 
 CHAPTER XII. THE CHROMATIC SCALE CHROMATIC TRIADS 
 
 IN A KEY page 121 
 
 The Chromatic Scale, 266 How obtained from the major scale, 267 The 
 true form of the chromatic scale, 268 Convenient notation, 269 De- 
 finition of chromatic chords, 272-274 Chromatic triads in the minor 
 key. The chord on the minor second, 275 Its first inversion, the 
 " Neapolitan sixth," 276 Its second inversion, 277 Its harmonic de- 
 rivation, 278 The chromatic chord on the supertonic, 279 Treatment 
 of the third of this chord, 280, 281 Examples, 282 The inversions of 
 this chord, 284 Examples of the chord, 285 Its harmonic derivation, 
 286 The tonic major chord in a minor key, 287-289 Chromatic triads 
 in a major key, 290, 291 Table of chromatic triads, 292 Modulation by 
 means of chromatic chords, 293. 
 
 CHAPTER XIII. THE FUNDAMENTAL CHORDS OF THE SEVENTH 
 
 ON THE SUPERTONIC AND TONIC page 134 
 
 The supertonic and tonic sevenths both chromatic chords, 294 The super- 
 tonic seventh, 295 How distinguished from a dominant seventh, 296 
 Its inversions, 298 Doubling of the seventh, 299 Progression of the 
 chord, 300 Change of position, 301 The same in major and minor 
 keys, 302 Examples, 303-30677^ tonic seventh, 307 Why used with 
 major third in minor key, 308 Progression of the third and seventh, 310 
 It resolves on another discord, 311 Examples, 313-317 Modulation 
 by chords of the supertonic and tonic seventh, 318 General laws for 
 the treatment of all chords of the seventh, 319 Resolution, 320 Pro- 
 gression of the third, 321 Progression of the seventh, 322. 
 
 CHAPTER XIV. CHORDS OF THE NINTH ENHARMONIC MODU- 
 LATION page 147 
 
 The chords of the ninth, 324 Which note omitted, 325 Which chords not 
 used in minor keys, 326 Figuring of the chord, 327 Inversions, 328 
 How to distinguish inversions of chords of the ninth from chords of the 
 seventh, 330 How to find the generator of a chord, 331 Resolution 
 of a chord of the ninth on its own generator, 332-338 Resolutions on 
 other generators, 339, 340 False notation of the minor ninth, 341 
 Examples, 342-352 Chord of the diminished seventh, 354 How used 
 in modulation, 355 Enharmonic change of notation, 356 Enharmonic mo- 
 dulation, 358-360 Analysis of passage from Bach's "Chromatic Fantasia," 
 361, 362 The "Leading Seventh," 365 Chords of the ninth without 
 generator or third, 366-368 Summary of rules for treatment of chord of 
 the ninth, 369-371. 
 
 CHAPTER XV. CHORDS OF THE ELEVENTH page 168 
 
 The chord of the eleventh, 372 Which notes mostly omitted, 375 Figuring, 
 376 Resolution, 377 First inversion, 379 Second inversion, 380 
 Third inversion, "Chord of the Added Sixth," 381 Fourth inversion, 
 385 Fifth inversion, 386 Summary of rules for treatment of dominant 
 eleventh, 387, 388 The tonic eleventh, 390-392 The supertonic 
 eleventh, 393-395.
 
 CONTENTS. xi 
 
 CHAPTER XVI. CHORDS OF THE THIRTEENTH page 178 
 
 The major and minor thirteenths, 396 Which unavailable in a minor key, 
 398 Numerous forms of the chord, 399 The thirteenth a consonance 
 with the generator, 400 Resolution of the chord, 402 Treatment of 
 the thirteenth, 404, 405 The inversions, 407 How to recognise chords 
 of the thirteenth, 408 Chords with the generator fresent : (i) Generator, 
 third, and thirteenth, 410 False notation of minor thirteenth, when used, 
 413 (2) Generator, third, fifth, and thirteenth, 417 (3) Generator, 
 third, seventh, and thirteenth, 418 (4) Generator, ninth, eleventh, and 
 thirteenth, 423 Chords of the thirteenth without the generator, 424 
 (5) Third, minor ninth, and thirteenth, 425 (6) Third, fifth, ninth, and 
 thirteenth, 428 (7) Fifth, seventh, ninth, and thirteenth, 429 (8) Third, 
 fifth, seventh, ninth, and thirteenth, 430 (9) Fifth, ninth, eleventh, and 
 thirteenth, 431 (10) Seventh, ninth, eleventh, and thirteenth, 432 (n) 
 Ninth, eleventh, and thirteenth, 435 Unused form of the chord, 436 
 The chord in its complete form, 437 Use of the chord in modulation, 
 438 Examples, 439-441 Summary of treatment of the chord, 442-445. 
 
 CHAPTER XVII. THE CHORD OF THE AUGMENTED SIXTH page 197 
 
 The chord of the Augmented Sixth made from the dominant and supertonic 
 sevenths, 448 Harmonic derivation of the chord, 450-452 The chord 
 is always chromatic, 453 Its resolution, 454 On which degrees of 
 the scale used, 455 The Italian Sixth, 457 Its resolutions, 458 Its 
 inversions, 459 Examples, 460 The French Sixth, 461 Its resolutions, 
 462 Its inversions, 463 Examples, 464-467 The German Sixth, 468 
 Its inversions. 469 Examples, 470 475 Rarer forms of the chord of 
 the augmented sixth, 476 Modulation by means of this chord, 477-484. 
 
 CHAPTER XVIII. THE SO-CAT.LED "DIATONIC DISCORDS" page 214 
 
 The early use of discords, 486 All essential discords can be used without 
 preparation, 487 Distinction between the root and the generator of a 
 chord, 488 Definition of diatonic discords, 489 Diatonic sevenths; their 
 derivation, 490 The old rule for their treatment, 492 Examples, 493, 
 494 Disregard of the rule by old masters. 495, 496 Modern practice, 
 497 General principles, 498 Diatonic ninths, 499. 
 
 CHAPTER XIX. SUSPENSIONS page 221 
 
 Definition of a suspension, 501 Difference between suspensions and diatonic 
 discords, 502 Which notes can be suspended, 504 Rules for prepara- 
 tion of a suspension, 505 Position in the bar of a suspension, 506 
 Length of its preparation', 507 When it may be sounded with the note 
 of its resolution, 508 Incorrect progressions, 509 Always resolved on 
 the same chord, 510 Practical limitations, 5 1 1 Figuring, 513-516 
 Ornamental resolutions, 519 Examples, 521-524 Suspensions resolving 
 upwards, 525-532 Double suspensions, 533, 534 Suspensions of com- 
 plete chords, 535.
 
 xii CONTENTS. 
 
 CHAPTER XX. PEDALS page 237 
 
 A Pedal defined, 536 Which notes are used as Pedals, 538 Treatment of 
 the harmony when the Pedal is not a note of the chord, 540 Examples 
 of dominant pedals, 541 Tonic pedal, 542 Inverted pedals, 543 Pedal 
 above and below, 544- Pedal in a middle voice, 545 Generally ends 
 with a chord of which the pedal is a part, 546 Modulation on a pedal, 
 547-550 Ornamentation of a pedal note, 551 The double pedal, 
 552 A curious pedal, 553. 
 
 CHAPTER XXI. HARMONY IN FEWER AND MORE THAN FOUR 
 
 PARTS page 245 
 
 Harmony wholly in four parts rare, 554 Three-part harmony, 555 
 Characteristic notes to be retained, 556 Broken chords, 557 Position 
 of chords in three-part harmony, 558 The cadence, 559 The motion of 
 the separate parts, 560 Examples, 561 Two-part harmony, 562 
 Examples referred to, 563 Examples given, 564-567 Harmony in 
 more than four parts, 568 Greater freedom of part-writing allowed, 
 569 Five-part harmony, 570 Six-part harmony, 572 -Seven-part har- 
 mony, 574 Eight-part harmony, 575 Conclusion, 578.
 
 HARMONY: 
 
 ITS THEORY AND PRACTICE. 
 
 CHAPTER I. 
 
 INTRODUCTION. 
 
 1. A certain amount of elementary knowledge of music will 
 be necessary to the student before beginning the study of the 
 present work. It will be assumed that he is acquainted with the 
 names of the notes, the mranings of the various musical signs 
 (accidentals, &c.), the relative time values of notes of different 
 lengths, and such other matters as are treated of in ordinary text- 
 books on the Elements of Music. 
 
 2. A musical sound is produced by the periodic vibration of 
 the air, that is to say, its motion at a uniform rate. When the 
 air moving at a uniform rate comes in contact with the nerves of 
 hearing, there is produced, provided the motion is sufficiently 
 rapid, what is called a musical sound, or note. The pitch of 
 a sound (that is, its being what is called a high or a Imv note), 
 depends upon the rapidity of the vibration. This will be seen 
 later (see Chapter II., 35) ; all that is needed now is that the 
 student shall understand what is meant when we speak of a 
 musical sound. 
 
 3. If sounds of different pitch are heard one after another, we 
 get what is called MELODY ; if sounds of different pitch are heard 
 together, we get HARMONY. It is the laws of harmony that we 
 shall explain in this book ; but it will be seen as we proceed that 
 the question of melody is often so closely connected with that of 
 harmony, that it is impossible to treat of one without also paying 
 some attention to the other. 
 
 4. If two different notes are sounded, either in succession or 
 together, it is clear that one of the two must be the higher, and 
 the other the lower. The difference in pitch between two sounds 
 is called the Interval between them. This difference may be so 
 small as to be hardly recognisable by the ear ; or it may be as 
 great as between the lowest and highest notes in a large organ, 
 or anything between the two. An infinite number of intervals is 
 possible ; but in music we make a selection, the nature of which 
 will be explained later. For the present we are merely defining 
 the meaning of the word " Interval."
 
 14 HARMONY : [Chap. i. 
 
 5. The smallest interval used in music is called a SEMITONE.* 
 We may define a Semitone, as the distance between any one note, 
 and the nearest note to it, above or below, on any instrument which 
 has only twelve sounds in the octave. For example, on the piano, 
 the nearest note to C is B on the one side (below), and C$ on 
 the other side (above). From B to C, and from C to C? are 
 therefore both semitones. Similarly from F| to Ftf, and from F$ 
 to G will be semitones ; but from G to A will not be a semitone ; 
 for A is not the nearest note to G; G# (or Ab), comes between 
 them. 
 
 6. If the semitones on each side of C be compared, 
 
 it will be seen that there is a difference between them. C and B 
 are on two different places of the staff; one is on a line and the 
 other on a space ; but C and C$ are both on the same place in 
 the staff; but the latter note has an accidental before it. A 
 semitone of which the two notes are on different degrees of the 
 staff is called a diatonic semitone ; the word " diatonic " means 
 "through the tones, or degrees of the scale." A second meaning 
 which is attached to the word will be explained later ( 72). 
 When the two notes of the semitone are on the same degree of the 
 staff, and one of the two is altered by an accidental (e.g., C to C:$) 
 the semitone is called chromatic, a word literally meaning 
 'coloured." This use of the word will be further explained later. 
 7. As the word semitone means " half tone," it is evident that two 
 semitones together will make a tone. Thus in the example given 
 in 6, as we rind a semitone from B to C, and another semitone 
 from C to C, the whole interval from B to Cj is a tone. But as 
 a tone is always composed of two notes on adjacent degrees of the 
 staff, one being always on a line, and the other on the next space, 
 above or below, it is necessary that of the two semitones one must 
 be diatonic and one chromatic^ For if we take two diatonic semi- 
 tones one above another, _ ^_ ^-* the resulting interval 
 will be from B to D? ; gp^=j^= which is not a tone 
 as the two notes are not 5^ on the next degrees of 
 
 the scale to one another. And if we take two chromatic semitones, 
 
 it is equally clear that they will not make a tone ; 
 ?_E for now the resulting notes C? and C$ are both 
 
 on the same degree of the stave. 
 
 * In one sense this statement is not strictly accurate, as the "enharmonic 
 diesis " (} 52) is sometimes used in modulation. For ordinary purposes, however, 
 the statement in the text is correct. 
 
 t The two semitones composing a tone are not of exactly the same size. It 
 will be seen later ( 50) that a diatonic semitone is larger than achromatic ; neither 
 semitone is therefore exactly half the tone ; but as the difference is of no practical 
 importance in harmony, the student need not regard it. It is only mentioned here 
 for the sake of accuracy.
 
 chap, i.] ITS THEORY AXD PRACTICE. 15 
 
 8. A SCALE is a succession notes of arranged according to 
 some regular plan. Many different kinds of scales have been 
 used at various times and in different parts of the world ; in 
 modern European music only two are employed, which are called 
 the diatonic and the chromatic scale. 
 
 9. The word " diatonic " has been already explained in 6 as 
 meaning " through the degrees." A diatonic scale is a succession 
 of notes in which there is one note, neither more nor less, on each 
 degree of the staff that is to say, on each line and space. The 
 way in which the scales are constructed will be explained later 
 (see Chapter III.) ; at present we simply give the forms of them. 
 There are two varieties of the diatonic scale, known as major (or 
 greater) and minor (or less) scale from the nature of the interval 
 between the first and third notes of the scale. 
 
 MAJOR SCALE. 
 
 S 
 
 o 
 
 MINOR SCALE. 
 
 Other forms of the minor scale frequently to be met with will 
 be explained later. It will be seen that each of these scales con- 
 tains only seven different notes. This is because the eighth note, 
 or octave (Latin, " octavus "= eighth), is a repetition of the first 
 note at a different pitch ; and from this note the series re- 
 commences. 
 
 10. The scientific reason why octaves resemble each other so much more 
 closely than two notes at any other interval is that the upper of two notes at a 
 distance of an octave from one another is the first "upper-partial " tone ( 36) 
 of the lower, and that all its harmonics are also harmonics of the lower note ; 
 therefore, the "compound tone" (Chapter II., 40) of the higher note 
 contains no new sound, which is not also in the compound tone of the lower 
 note. The higher note is merely the reinforcement of certain upper-partials ot' 
 the lower note ; but this is not the case with two notes at any other interval 
 than the octave, or, of course, the double octave, &c. 
 
 ii. A chromatic scale is a scale consisting entirely of 
 semitones, and it is called chromatic because some of its notes 
 require accidentals (flats or sharps) before them ( 6). 
 
 
 As will be explained later, the chromatic scale is frequently 
 written in a different way from that here given ; but, however 
 written, it equally consists of semitones.
 
 1 6 HARMONY : [Chap. i. 
 
 12. The different degrees of the diatonic scale ( 9) are 
 known by different names, with which it is necessary that the 
 student should be perfectly familiar, as they are of constant occur- 
 rence. The first note of the scale is called the TONIC, or KEY- 
 NOTE. This is the note which gives its name to the scale and 
 key. The scales in 9, for instance, are the scales of C major 
 and C minor, and it will be seen that they both begin with the 
 note C. The term " tonic " is used in harmony much more 
 frequently than " key-note." The most important note in a key 
 after the tonic is the fifth note of the scale. For this reason it is 
 called the DOMINANT, or ruling note of the key. The fourth note 
 of the scale lies at the same distance below the tonic that the fifth 
 note lies above it. This will be seen at once by beginning at the 
 top of the scale and descending. This fourth note is therefore 
 called the SUBDOMINANT, or lower dominant. We have now got 
 appropriate names for three of the chief notes in the key. 
 
 13. About midway between tonic and dominant lies the third 
 note of the scale. We shall see presently that in the major scale 
 it is rather nearer to the dominant, and in the minor rather nearer 
 to the tonic ; but, roughly speaking, it is in the middle between 
 the two. It is therefore called the MEDIANT, that is, the middle 
 note. The sixth degree of the scale lies midway between the 
 tonic and subdominant, just as the third lies between tonic and 
 dominant. We therefore call this sixth note the SUBMEDIANT, or 
 lower mediant. Some writers on harmony call this note the 
 " Superdominant," or note above the dominant ; but the name 
 Submediant is much more usual, and in every way preferable. 
 The second note of the scale is called the SUPERTONIC, 
 i.e., the note above the tonic ; and the seventh note of the scale, 
 which, it will be seen later, has a very strong tendency to lead up, 
 or rise to the tonic is on that account called the LEADING NOTE. 
 It is sometimes, though rarely, called the 'Subtonic,' from its 
 position as the next note below the tonic. 
 
 14. Having shown the origin and meaning of these different 
 names, we will now tabulate them. 
 
 First Degree of the Scale = Ton'c (Key-note). 
 Second = Supertonic. 
 
 Third 
 
 Fourth 
 
 Fifth 
 
 Sixth 
 
 Seventh 
 
 Mediant. 
 
 Subdominant. 
 
 Dominant. 
 
 Submediant (Superdominant). 
 
 Leading Note (Subtonic). 
 
 15. Before proceeding to treat of the names and classification 
 of Intervals, it will be needful to define and explain two terms 
 which we shall very frequently have to use in speaking of them. 
 These are the terms CONSONANCE and DISSONANCE. 
 
 1 6. A consonant interval, or CONSONANCE, is a combination of 
 two sounds, which by itself produces a more or less complete and
 
 Chap. I.] 
 
 Irs THEORY A.VD PRACTICE. 
 
 satisfactory effect, i.e., which does not necessarily require to be 
 followed by some other combination. For example, if the student 
 will strike on the piano any of the following pairs of notes, pausing 
 between each, 
 
 he will find that each is more or less satisfactory. A consonant 
 chord is a chord of which all the notes make consonant intervals 
 with one another. 
 
 Let the student play each of these chords separately on the piano 
 they are not intended to be connected and he will find that 
 each by itself produces a satisfactory effect. W hen he has learned, 
 later in this chapter, which are the consonant intervals, he will see 
 that no others have been used in these chords. 
 
 17. A dissonant interval, or DISSONANCE, is a combination of 
 two notes which by itself produces an impression of incomplete- 
 ness, so that the mind urgently feels the need of something else to 
 follow. Let the student strike on the piano the following pairs 
 of notes, pausing, as before, after each. 
 
 4567 
 
 H 
 
 - 
 
 Everyone will feel the incomplete effect of these combinations, 
 and that they require to be followed by something else to be 
 satisfactory. Let us try. We will put after each of these disso- 
 nances a consonance, and it will be at once felt that the complete- 
 ness which was before wanting has now been obtained. 
 
 r U = H * H ^ H H ^ -r 
 
 The consonance which follows the dissonance is called the 
 RESOLUTION of the dissonance. The laws according to which 
 dissonances are resolved will be learned later. 
 
 18. A dissonant chord, or DISCORD,* is a chord which con- 
 tains at least one dissonance among the intervals made between 
 the various notes. Like a dissonant interval, a dissonant chord 
 
 * The term "Discord" is also sometimes applied merely to the dissonant 
 note itself.
 
 i8 
 
 HARMONY : 
 
 [Chap. I. 
 
 has by itself an incomplete effect. Let the student play the 
 following dissonant chords, and he will feel this. 
 
 HZ U ' II -~? U 
 
 7 -s>- 
 
 3456 
 g- H ^ K-k~5 \\- := & 
 
 E^F^fe^ 
 
 Jfco. 
 
 Now, as before with the dissonant intervals, let us put after 
 each chord a consonant chord for its resolution. The satisfactory 
 effect is felt at once. In general, it may be said that consonance 
 is a position of rest, and dissonance a position of unrest. 
 
 19. Intervals are always reckoned upwards, unless the contrary 
 be expressly stated. Thus " the third of C " always means the 
 third .above C ; if the third below is intended, it must be so 
 described. The number of an interval is always computed accord- 
 ing to the number of degrees of the scale that it contains, including 
 both the notes forming the interval. Thus from C to E is called 
 a third, because it contains three degrees of the scale, C, D, E. 
 Beginners are apt to get confused on this point, and to think of I ) 
 as the first note above C, and E as the second. But the note C 
 is itself counted as the first note of the interval. Similarly, from 
 G to D is a fifth, from F to D a sixth, and so on in all other 
 cases. The same reckoning, but in the reverse direction, applies 
 to the intervals below. Thus A is the third below C, D is the 
 fourth below G, &c. Let the student examine the major scale of 
 C in 9, and he will find within the compass of the octave there 
 given two 7ths, three 6ths, four 5ths, five 4ths, six 3rds, and 
 seven 2nds. It will be a useful exercise for him to discover 
 them for himself. 
 
 20. An interval larger than an octave is called a compound 
 
 interval. Thus the interval 
 
 is compounded of the 
 
 octave, C to C, and the third, C to E. (The octave is 
 printed here as a small note.) Obviously, in addition to the 
 third at the top, the interval contains the seven notes of the 
 lower octave from C to B. The upper C is already counted as part 
 of the third. Thus the number of a compound interval is always 
 7 more than that of the simple interval to which the octave is 
 added. Therefore,
 
 Chap i.] Irs THEORY AXD PRACTICE. 19 
 
 A Compound 2nd = A Qth. 
 
 3rd = A loth. 
 
 4th = An nth. 
 
 5th = A I2th. 
 
 6th = A 13111. 
 
 7th == A 1 4th. 
 
 We never speak of a "compound octave." Such an interval 
 would be called a "double octave." Excepting the 9th, nth, and 
 1 3th, with which we shall presently make closer acquaintance, all 
 compound intervals are, for purposes of harmony, identical with 
 the simple intervals to which they correspond. There is no 
 difference in the treatment of a 3rd and a loth, a 5th and a i2th, 
 or a 1 4th and a yth. We, therefore, never use the names of these 
 compound intervals in harmony. 
 
 21. If the student will examine the intervals contained among 
 the notes of the diatonic scale (as we recommended in 19), he 
 will see that those which have the same name are not always of the 
 same size. From C to E, for instance, is a third, and so is from D 
 to F ; but the former contains two tones (C to D and D to E), and 
 the latter only one tone (D to E) and a semitone (E to F). Similarly 
 the fourth from C to F is smaller than the fourth from F to B, and 
 the sixth from C to A is larger than the sixth from E to C. And if 
 we put accidentals before some of the notes, we shall get still 
 further differences. It is clear, therefore, that the general 
 description of an interval as a second, third, fourth, &c., is not 
 sufficiently precise to show its exact nature. In order to obtain 
 greater accuracy, intervals are described by one or other of the 
 following adjectives : -perfect \ major, minor, augmented, and 
 diminished. These terms we shall now explain. 
 
 22. As a basis for our classification, we take the major scale, 
 and first reckon all the intervals upwards from the tonic, that is, 
 taking the tonic in each case'as the lower note of the interval. It 
 is evident that we shall obtain in succession a 2nd, a 3rd, a 4th, 
 a 5th, a 6th, a yth, and an octave. To these may be added the 
 unison, which, though not strictly speaking an interval, is reckoned 
 as such. Of these intervals, the unison, 4th, 5th, and 8th, are 
 termed Perfect, and the 2nd, 3rd, 6th, and yth major. 
 
 I I I I I I I 
 
 g J *tf *ti 5* g *XJ 
 
 . . S, S, . . 3,. 
 
 The reason why some of these intervals are called perfect, and 
 others major, will be explained presently ( 30). The compound
 
 20 HARMONY : [Chap. i. 
 
 intervals have the same prefixes as the simple ones ; thus C to Di3 
 will be a major ninth, C to F a perfect nth, and so on. 
 
 23. An interval which is a chromatic semitone less than a 
 major, is called a minor interval. A major interval can be changed 
 into a minor, either by raising the lower note a semitone by an 
 accidental, or lowering the upper one, also by an accidental. 
 Thus from C to E is a major third. If we raise the lower note 
 to Cjt, the interval C$ to E is a minor third. Or if we leave the 
 C alone, and lower the E to E7, we also get a minor third from C 
 to E?. But if we alter either note a diatonic semitone, we change 
 the name of the note, and therefore of the interval. Thus, C to 
 E being a major 3rd, if we raise C to Db instead of to C|, the 
 interval from D? to E is no longer a third at all, but a second, of 
 a kind which we shall explain directly. Similarly if we lower E to 
 Di instead of Eb, C to D$ is a second ; for the two notes are 
 on adjacent degrees of the staff. 
 
 Minor znds. \ Minor srds. \ Minor 6ths. \ Minor yths. \ Minor gths. 
 
 ~-jjs& Pig. "^fg- ' -e>- ' 5>- - 
 
 &C. 
 
 24. An interval which is a chromatic semitone larger than a 
 perfect or a major interval is called augmented. Here we reverse 
 the process of making the minor intervals, and we either raise the 
 upper note, or lower the lower note, by means of an accidental. 
 Thus C to F being a perfect 4th, C to F$ or Ci? to Ffl will be an 
 augmented 4th. Again C to A is a major 6th ; and C to A$ or 
 C? to At] is an augmented 6th. The augmented 3rds and 7ths 
 are not used in harmony ; augmented 2nds, 4ths, and 6ths are 
 frequently, and augmented 5ths occasionally to be met with. 
 
 25. An interval which is a chromatic semitone less than a 
 perfect or a minor interval is called diminished. As in the cases 
 just spoken of, it is immaterial to the nature of the interval which 
 of the two notes composing it be altered. Let the student refer 
 to the table of minor intervals in 23. We obviously cannot 
 diminish the minor 2nd, for if we lower Db to Dbi?, or raise C to 
 C$, we shall in either case get an interval smaller than a semitone 
 what is called an "enharmonic" interval ( 52) and it has 
 been already said ( 5) that the semitone is the smallest interval 
 used in music. The same objection will apply to a diminished 
 9th. But diminished 3rds, 4ths, 5ths and 7ths, especially the last, 
 are of very frequent occurrence. 
 
 26. When the two notes composing an interval change their 
 relative position, the lower one becoming the upper, and the upper 
 the lower, the interval is said to be inverted.
 
 -|| <y y 
 JI & B 
 
 Chap, i.] Irs THEORY AND PRACTICE. 21 
 
 Here the first interval is a perfect fifth ; if C be placed above G, 
 the interval is inverted, and its inversion is a perfect fourth. The 
 number of the inversion of an interval can always be found by 
 subtracting the number of the interval from 9. In the above 
 example it is seen that an inverted 5th becomes a 4th (9-5 =4) ; 
 in the same way a 3rd becomes a 6th, a 2nd a 7th, &c. A 
 unison cannot strictly speaking be inverted, as it has no higher or 
 lower note ; but it is said to be inverted when one of the two notes 
 is put an octave higher or lower. Similarly, an octave reduced to a 
 unison is generally said to be inverted. Perfect intervals remain 
 perfect when inverted ; major intervals become minor, and minor 
 major ; augmented intervals become diminished, and diminished 
 augmented. 
 
 27. The reason of the rule just given will become clear to the student if 
 he observes that the inversion of any simple interval is the difference between 
 that interval and an octave. Thus a major 3rd, C to E, and its inversion, 
 a minor 6th, E to C, will together make an octave, either C to C, or E to E, 
 
 according to the note of the 3rd of which the position 
 is changed. A third of any kind taken from an 
 
 octave must leave a sixth ; and if a larger (mnjor) third be taken out, a smaller 
 (minor) sixth will be left ; and conversely, if a smaller (minor) third be taken 
 from the octave, a larger (major) sixth will be left. Evidently the same 
 reasoning will apply to augmented and diminished intervals. 
 
 28. Intervals are divided into two classes, consonant and 
 dissonant. These terms have been already explained in S 16, 17. 
 The consonant intervals are the unison, octave, perfect fifth, 
 perfect fourth, major and minor third, and major and minor sixth. 
 All other intervals of every kind all seconds, sevenths, ninths, 
 elevenths, and thirteenths,* and all augmented and diminished 
 intervals are dissonant. 
 
 29. The consonant intervals are further subdivided into perfect 
 and imperfect consonances. The unison, octave, perfect fourth, 
 and perfect fifth are the perfect, and the major and minor thirds 
 and major and minor sixths are the imperfect consonances. 
 Neither note of a perfect consonance can be altered by an acci- 
 dental that is, raised or lowered a chromatic semitone ( 23) - 
 without changing the interval into a dissonance. But a major 
 third or sixth can be changed to a minor, or vice versa a minor 
 into a major and still remain a consonance. This is one differ- 
 ence between perfect and imperfect consonances. 
 
 30. The researches of Helmholtz have proved that the distinction between 
 consonant and dissonant intervals is not merely arbitrary, but is the result of 
 the nature of the intervals themselves. In the next chapter we shall see that a 
 musical tone is mostly a compound tone, containing besides its principal tone 
 other tones bearing fixed relations to the lowest note, and called harmonics, 
 or "upper partials." Helmholtz has shown that when two of the earlier- 
 
 * The thirteenth, being the octave of the sixth, is itself consonant ( 400), but 
 it is always treated as a dissonance in the chords in which it is found. (See 
 Chapter XVI.)
 
 22 HARMONY : [Chap i. 
 
 produced and stronger of these upper partial tones coincide in two notes 
 sounded together, the resulting tone is pure and free from the inequalities 
 technically known as beats. When this is not the case, the presence of thes-e 
 beats gives a roughness to the tone which is known as dissonance. The most 
 perfect consonances are those in which the upper partials soonest coincide. 
 This is the case with the octave, fourth, and fifth of the major scale. With 
 the unison of course the upper partials are absolutely identical. The explana- 
 tion of the harmonic series given in the next chapter will, it is hoped, make 
 this clear to the student. In a practical work like the present it is impossible 
 to enter fully into the science of acoustics ; we can only touch upon it as far as 
 it is needed in its connection with harmony. The student who wishes for full 
 information on this subject should consult Helmholtz's book, "The Sensations 
 of Tone." 
 
 31. We shall conclude this chapter by giving a table of all the 
 intervals and their inversions within an octave from the note C. 
 (See next page.) The student is advised to make similar tables for 
 himself from other notes. Intervals above an octave cannot be 
 inverted. It will be seen that the inversion of a consonance is 
 always a consonance, and of a dissonance always a dissonance. 
 
 EXERCISES ON CHAPTER I. 
 
 (i) Write the names of the following intervals, indicating those 
 that are consonant by (C) and those that are dissonant by (D) : 
 
 (a) (b] (c) (d) (e) (/) (g) (A) (0 (j) (k) 
 
 (!) (m) () (o) (f) ( f ) (r) (s) (/) () 
 
 (2) Write the following intervals : A minor second of Bb and 
 1)$; a major second of Bfl and GP ; a minor third of G$ and Cb ; 
 a major third of Ab, Ffl, and Df ; a perfect fourth of Bb, GC|, and 
 AJ; an augmented fourth of Db ? F, and C$. 
 
 " (3) Write the perfect fifth of FB, Bfl, EP, D|,Fx , Bbb, Ebt?; the 
 augmented fifth of Eb, Ab, Cf, G$, and B? ; the minor sixth of 
 D?, G, Eb, and G? ; the major sixth of Ab, BD,, EB,, D, and 
 C7 ; the augmented sixth of Eb, GB,, and At) : the minor seventh 
 of BH, AP, and F? ; the major seventh of C2, FO, G#, E7, and 
 Bt| ; and the diminished octave of Db, fib, and Fx. 
 
 (4) Write the minor ninth of F, Bb, D, Gtl, and Eft ; the major 
 ninth of Ft], A?, E#, G$, and D? ; the eleventh of Eb, A#, C#, and 
 F$ ; the minor thirteenth of Eb, GjT, Db, G?, Ffl, and D7 ; and 
 the major thirteenth of Gfl, C?, Ab, Eb, and Bt|. 
 
 (5) Write the inversions of all the intervals (a) to (it] in 
 Exercise I, and name each, adding (C) or (D), according to 
 whether they are consonant or dissonant.
 
 Chap. I.] 
 
 Irs THEORY AND PRACTICE. 
 
 a; 
 
 
 d 
 
 > 
 
 
 o 
 
 Perfect [ 
 
 ^ 
 
 v l 
 
 Perfect "f 
 
 
 
 
 P 
 
 Major I 
 
 > > .! 
 
 . Minor 
 
 
 * 
 
 
 Minor ( 
 > -U 
 
 > ^ 1 
 
 K 
 
 . Major c 
 
 a S 
 
 C/D Diminished i 
 ;' 
 
 ) <k r* 
 
 ' ' ii 
 
 . Augmented ^ 
 
 |H 
 
 1! 
 
 Augmented 
 
 c <h Is 
 
 1 Diminished 
 
 
 
 
 i 
 
 "^ Major 
 
 i! ^ C 
 
 )( Minor 
 
 C/2 
 
 ci ~ L 
 
 t 
 
 Minor 
 
 . 
 
 ^ ^ t 
 
 i E 
 
 1 Major 
 
 1 
 
 Augmented 
 
 it ^ 
 
 (j Diminished 
 
 j 
 
 n 
 
 x\ ^ 
 
 Perfect 
 
 <!$ ] 
 
 i) Perfect ^ 
 d ' 
 
 
 
 11 . 
 
 
 Diminished 
 
 1 
 
 H : 
 fi 
 
 () Augmented 1 "" 1 
 
 ^ 
 
 Augmented 
 
 ^ 
 
 ] Diminished 
 
 J= 
 
 ''i 
 
 : 'i oo 
 
 = Perfect 
 
 :* 
 
 J Perfect ^ 
 
 3 
 
 ^y L 
 
 
 ,0 
 
 j:f 
 
 ~ ' r-r 
 
 Diminished 
 
 1 
 
 
 ] Augmented 
 
 Major 
 
 L : 
 
 |i Minor 
 
 ui 
 
 . K r 
 
 ^ r [/ 
 
 "C 
 
 
 
 .i Minor 
 
 h F 
 
 (\ Major -^ 
 
 
 Uh 
 
 
 H 
 Diminished 
 
 1 
 
 Augmented 
 , 1 1 
 
 Augmented 
 
 It 
 
 ft Diminished 
 
 U5 
 
 -o 
 
 1 5 
 
 f - 
 
 Major 
 
 T 
 
 ft Minor 
 
 o 
 
 SJ 
 
 -^ Minor 
 
 1 
 
 f 5 
 
 Major & 
 
 
 f 
 
 
 
 ik 
 
 I i 
 
 Ji Augmented 
 
 ? 
 
 V) Diminished if 
 
 'a Perfect 
 
 i 
 
 Perfect t5 
 
 "^ 
 
 i ? 
 
 |i
 
 24 HARMONY : [Chap. n. 
 
 CHAPTER II. 
 
 THE HARMONIC SERIES. 
 
 32. Although it would not be correct to say that modern harmony rests 
 solely on natural laws, it is' nevertheless true that a certain amount of know- 
 ledge of the phenomena of the production of musical sounds is of the utmost 
 assistance to the student in enabling him to understand the derivation of the 
 various chords which form the material of which music is constructed. We 
 shall therefore proceed to explain those laws in so far as it will be necessary to 
 apply them subsequently. What is now to be stated is not in the nature of 
 theory, but of physical fact, and is perfectly familiar to students of natural 
 philosophy. 
 
 33. The general law regulating the production of Harmonics may be thus 
 stated : A sonorous body, such as the string of a piano or violin, vibrates 
 not only throughout its whole length, but in aliquot parts of that length, 
 e.g. , in halves, thirds, fourths, fifths, &C ; and the musical tones produced by 
 the vibration of the different aliquot parts will always bear the same relation 
 to one another, and to the note produced by the vibration of the whole string. 
 
 Thus, if the whole string sound the note igl =, each half will sound the 
 
 octave above, viz : & j=^. Similarly, if the whole string sounded this last 
 C, the half of this string, (and therefore the quarter of the longer string first 
 taken), will give ^= ^ ; the eighth will give the C in the third space of the 
 
 treble staff, and so on. If the who'e string sounds a G, the half will give G 
 the octave above, the quarter the G two octaves above, &c. The division of 
 any string into halves, quarters, eighths, or sixteenths, gives the various upper 
 octaves of the "generator," or " fundamental tone," that is the note produced 
 by the vibration of the whole length of the string. 
 
 34. But any string will vibrate not only in halves, quarters, &c., but in any 
 other aliquot parts. If a string vibrate in thirds, it produces the twelfth of the 
 fundamental tone, i.e., the perfect fifth in the second octave. Taking, as 
 
 before, ^= ^ for the fundamental tone, it will be found that one third of 
 
 .jt ^ 
 
 the string gives ^' |" . Obviously one sixth is half of one third, and, 
 
 as it has been already seen that the half of a string always gives the octave 
 above the whole, it is clear that the sixth part of the string will give -^ 
 
 and the twelfth 
 
 35. Here it must be said that \hz pitch of a note depends entirely upon the 
 rapidity of its vibration. What we call low notes are produced by much 
 slower vibrations than high notes. An important law of nature as bearing on 
 this point is that rapidity of vibration varies inversely as the length of the string ; 
 that is to say, that exactly as the length of the string (^creases, the rapidity of 
 vibration zwcreases. Vibration is a periodic oscillation, as with a pendulum, 
 the difference between a pendulum and a musical string being that the former 
 is free at one end, while the string is fixed at both, and the middle of the string, 
 where the deviation from a position of rest is greatest, corresponds to the bob
 
 Chap, ii.] Irs THEORY AND PRACTICE. 25 
 
 of the pendulum. The law of vibration given above can be shown by a very 
 simple experiment. If a ball be fastened to one end of a string, and the string 
 be held by ihe other end, and the ball allowed to swing like a pendulum, it will 
 move at a rate which will depend on the length of string. Now if the string 
 be held in the middle, everyone knows that the ball will swing much faster 
 the shorter the string, the mere rapid the vibration. And this relation of 
 length to vibration is invariable. Thus in the long string C, that we have 
 spoken of above, for every vibration of the whole string there will be in the 
 same time two vibrations of each half, three of each third, four of each fourth, 
 &c. &c. 
 
 36. As the sounds we are explaining are all formed by parts of the whole 
 string, and therefore have more rapid vibrations, and a higher pitch than the 
 generator, they are sometimes called "upper partial tones" or "over tones " 
 A more common name for them, though a less strictly accurate one, is " Har- 
 monics," because those first produced in the series that is by the vibration of 
 the larger aliquot parts of the string belong to the harmonies of the generator. 
 Among the higher " upper partials " are many sounds which are unavailable 
 for harmonic purposes; but as the word "harmonics" is convenient, and 
 generally understood, we shall retain it in speaking of these partial tones. 
 When these notes are tabulated in the order in which they are produced, 
 we get what is called the " harmonic series." We shall now give the first part 
 of this series from C. It will be seen that all the even numbers of the series 
 are octaves of some lower number, because (as has been shown above) the half 
 of a string always sounds the octave of a whole string ; thus the tenth harmonic 
 will be the octave of the filth, the fourteenth of the seventh, and so on. (See 
 Diagram on p. 26. ) 
 
 37. In this diagram the notes produced by the vibration of the aliquot 
 parts of a string are given. The fractions below the notes show the aliquot 
 part of the whole string which produces any given sound, and the " vibration 
 ratios " underneath give the proportion of vibrations in the same time of the 
 fractional parts of the whole string. It will be seen that the harmonics first 
 produced are all consonant to the fundamental tone ; but from the seventh note 
 of the series, inclusive, all the neiv notes produced (i.e., all the uneven num- 
 bers) are dissonant. The yth, nth, I3th, and I4th notes of the series are 
 enclosed in brackets because these harmonics are not exactly in tune in the 
 key of the generator ; the 7th (and of course its octave, the Hth), as well as 
 the I3th, being somewhat too flat, and the nth decidedly too sharp. The 9th 
 and I5th are not "prime numbers," that is to say, they are obtained by 
 multiplying smaller numbers together, 9=3x3 and 15 = 3x5. These are 
 therefore called " secondary harmonics ;" and we shall have more to say about 
 them directly. 
 
 38. A very important use of this series is that it enables us to calculate the 
 ratios of vibration of different musical intervals with accuracy. Thus between 
 numbers I and 2 of the series is an interval of an octave ; therefore in every 
 octave there are two vibrations of the upper note to one of the lower in the 
 same time. Similarly we get the interval of a perfect fifth between the second 
 and third notes, of a perfect fourth between the third and fourth, &c. Hence 
 we get the following ratios for the principal musical intervals : 
 
 INTERVAL. NOTES. RATIO. 
 
 Octave C C ... i 
 
 Perfect fifth C G ... 2 
 
 Perfect fourth G C ... 3 
 
 Major third ... ... ... C E ... 4 
 
 Minor third ... ... ... K -G ... 5 
 
 Major sixth G E ... 3 
 
 Minor sixth ... E C ... 5 
 
 Major tone ... ... ... C D ... 8 
 
 Minor tone ... ... ... D E ... 9 
 
 Major diatonic semitone ... B C ... 15 
 
 Minor diatonic semitone , ... C D7 ... 16 
 
 2. 
 
 3- 
 4- 
 
 6. 
 
 8. 
 
 9- 
 10. 
 1 6. 
 17-
 
 26 
 
 HARMONY : 
 
 [Chap. II. 
 
 * r 
 
 o3 
 
 
 
 iii 
 
 -ua 
 
 jo 
 
 o 
 
 
 1" 
 
 -2 
 
 OS . 
 
 - Minor third. 
 
 1 
 
 E 
 
 co 
 
 
 \ 
 
 1C- 
 
 -o 
 
 - Minor ninth. 
 
 1 
 
 -: 
 
 3 
 
 
 ? 
 
 tr x^- 
 
 S 1 
 
 d 
 
 w 
 
 -|2 
 
 2 
 
 
 u -Tj 
 
 _- 
 
 ^H 
 
 
 X ' 
 
 o 
 
 I 3 
 
 H-" 
 
 ^ 
 
 rt 
 
 
 a ! 
 
 H 
 
 f 35 
 
 Ci 
 
 co 
 
 
 .! 
 
 J * 
 
 -- 
 
 - Minor seventh. 
 
 
 1 
 
 (K -< 
 
 
 
 
 
 Q ^* 
 
 >n - 
 
 - Major third. 
 
 CM ( 
 
 i^> 
 
 
 
 ^> 
 
 1 
 
 >* 
 
 
 1 
 
 
 CO - 
 
 - Perfect fifth. 
 
 
 1 FH|M 
 
 01 
 
 
 Generator 
 
 lA 
 
 
 
 (Fundamental tone). 
 
 lip 
 
 11 
 
 
 I 
 
 i 
 
 
 
 V __, 
 
 _o 
 
 
 C K> 
 
 3 D. Sf- 
 
 o .S 
 
 in O 'S 
 0) 3 
 
 S 
 
 
 
 Interval from the generator 
 (or its upper octaves). 
 
 _p .2" 
 
 1 

 
 chap. ii. i Irs THEORY AND PRACTICE. 27 
 
 The distinction between the major and minor tones and semitones, 
 though it exists, is of no practical importance in connection with harmony ; it 
 is given in the above list for the sake of completeness. 
 
 39. The student must not suppose that the harmonic series terminates at 
 the 2oth note. It might be carried much further ; but no prime numbers 
 above 20 are available for harmonic purposes. 
 
 40. It has been shown that upper partial tones can be obtained by causing 
 a string to vibrate in aliquot pans. As a matter of fact many of these tones 
 are so produced, together with the fundamental tone when a string is struck. If 
 the student will strike one of the bass notes of a piano, and listen carefully, he 
 will hear first the fundamental tone, then, more faintly, the octave, twelfth, 
 double octave, and, under favourable circumstances, even the major third above 
 this. What takes place is that the string, as soon as it begins to vibrate, divides 
 of itself into its aliquot parts, each giving the note proper to itself, and related, 
 as shown above, to the fundamental tone. But the higher harmonics are either 
 not produced at all, or are so faint that the ear cannot distinguish them without 
 artificial aid. It has been shown by the researches of Helmholtz that a good 
 musical tone is a compound tone that is, one containing some upper partials 
 together with its fundamental tone and" that the different quality of various 
 instruments depends on the presence in varying strengths of these upper 
 partials. 
 
 41. Now we proceed one step further. It will be evident that just as the 
 fundamental note C generates its series of harmonics, each of these harmonics 
 can itself become the generator of a series. If we take the third note (G) in 
 the series of C the first new note produced as a generator, it will give a 
 series of upper partials bearing the same ratio to their fundamental tone, and 
 being at the same distance from it, as the series with which we are already 
 acquainted. It will not be needful to give the whole series up to the 2Oth 
 note, as we did from C ; it will suffice to give the first few notes, and the 
 student can add more if he desires. 
 
 &c. 
 
 1 2 S 4 5 6 
 
 By comparing this series with the series of harmonics of C, it will be seen 
 that the six notes here given are all included in that series, where they will 
 be found as numbers 3, 6, 9, 12, 15, and 18 that is, numbers I to 6 multiplied 
 by 3. For G is the third harmonic of C, and all the harmonics of G are 
 therefore harmonics of a harmonic of C. This is what we term "secondary 
 harmonics. 1 ' Thus I) is the third harmonic of the third harmonic of C, and 
 3x3 = 9. Similarly B is the I5th harmonic of C, being the 5th of G; and 
 this note might also be derived as the third harmonic of E, the fifth note in 
 the series of C. Secondary harmonics play a very important part in the 
 formation of many chords, and we shall meet with many others besides those 
 here given as we proceed. For the present an explanation of their nature will 
 suffice. 
 
 42. Let us turn back once more to the harmonic series given in 36. 
 The student will see that as we ascend, the harmonics fall continually 
 closer and closer together. It has been shown that the octaves of the funda- 
 mental tone are numbers 2, 4, 8, 16, 32, &c., of the series. In the lowest 
 octave, I 2, there is no intermediate harmonic, in the second octave, 2 4, 
 there is one, in the octave 4 8 there are three, and in the octave 8 16 seven 
 intermediate harmonics ; and these not at equal distances, but constantly drawing 
 nearer and nearer. Evidently in the filth octave, 16 32, there must be 
 fifteen intermediate harmonics, or more notes than can be used in modern 
 music, which contains only twelve notes to the octave.* For this reason it is 
 
 * The notes of the series next following 20 will illustrate this. No. pi of the series will 
 evidently be the twelfth above No. 7, because 21=3 x 7. Therefore 21 will be F, slightly too
 
 28 HARMONY : [Chap. n. 
 
 necessary to make a selection from the harmonics offered us by nature. Asa 
 matter of fact different selections have been made at different periods, and 
 even at the present time the music of less artistically advanced nations contains 
 fewer notes in a key, scales of only five notes being found among the Chinese, 
 Malays, and other Oriental races ; while, on the other hand, in Arabic and 
 Persian music the octave is divided into 16 intervals. From this we see, 
 in the words of Helmholtz, that " the system of Scales, Modes, and Harmonic 
 Tissues does not rest solely upon unalterable natural laws, but is at least partly 
 also the result of sesthetical principles, which have already changed, and will 
 i^till further change with the progressive development of humanity " (Sensations 
 of Tune, p. 358). 
 
 43. The next question that presents itself is, what considerations are to 
 guide us in making our selections from the harmonic series? In the lower 
 octaves, the partial tones produced are so few in number that we have no 
 choice ; we take them all up to the 8th harmonic. We thus obtain all the 
 consonant intervals (as will be seen by referring to the ratios in 38) and one 
 dissonance, the minor seventh, formed by the seventh note of the series. This 
 seventh, often called the " harmonic minor seventh " is, as already said, 
 slightly flat, being a little more than a true major tone below the eighth note 
 of the series ; for its ratio is 7 : 8, while the major tone is 8 : 9. The differ- 
 ence, however, is so small, being only ^, that it may be disregarded. On 
 this point we are supported by the authority of Helmholtz, who says "the 
 minor seventh approaches so nearly to the ratio 7 : 4 [which is, of course, 
 the same as 7 : 8, transposing the upper note an octave], that it may in any 
 case pass as the seventh partial tone of the compound." (Sensations of Tone, 
 P- 539-* 
 
 44. Thus far then, our path is clear ; but in the next octave, containing the 
 upper partials between 8 and 16, we meet with two notes, II and 13, which 
 are perceptibly out of tune with the other notes which we already have. Our 
 object is to get as many consonances as possible into the key for the sake of 
 making our chords, as will be shown in the next chapter. Now the nth 
 harmonic, F, is distinctly too sharp to be used as a perfect fourth to C ; for the 
 ratio is 8 : n, while that of the perfect fourth ( 38) is 3 14. Here the 
 difference in pitch is much more considerable than in the case of the Jlh 
 harmonic, being -Jg- ; in fact, the nth harmonic is slightly nearer to F;jt 
 than to Ffcj. For this reason this harmonic is not available for the 
 harmonies of the key. But here the secondary harmonics spoken of above 
 come to our aid. In the foot-note to 42, it was shown that in the fifth 
 octave there were two F's, Nos. 21 and 22. The latter, being the octave of 
 II, will evidently be equally out of tune, and therefore unavailable ; but No. 
 21 being the twelfth (or, which is harmonically the same thing ( 20), the 
 perfect fifth of No. 7, a note which, as we have already seen is admissible, 
 
 will equally be admissible, and we here get an F differing from a perfect fourth 
 only by Jy, a fraction which we can disregard. f Similarly A, the 1 3th 
 harmonic, has with C the ratio 8:13, which is --$ flatter than the true major 
 6th, 3 : 5 (\ 38). But by taking the secondary harmonic, 27, of the next octave 
 instead of 26, (see foot-note, 42) we obtain the ratio 16 : 27, which differs 
 from 3 : 5 only by -gV 1 ms note ' s obviously much better in tune, therefore 
 in making our selection we reject II and 13, because we can find better notes 
 
 flat, because Et> is slightly too flat. But 22 will be the octave of n (n x 2=22), which gives 
 another F, somewhat too sharp. Similarly 26 and 27 will give two different pitches lor A, as 
 the student can easily work out for himself. In these cases it is clear that both notes cannot 
 be available in the same key. 
 
 * Helmholtz is speaking of the chord of the dominant 7th, containing the interval G 
 to F. As the law is of general application the specific reference to these two notes is omitted 
 in the quotation. 
 
 t The reason why we do not take F as, the perfect fifth below C will be explained in the 
 next chapter (5 70).
 
 Chap, it.] ITS THEORY AND PRACTICE. 29 
 
 in their place. These secondary harmonics, it will be noticed, are compounded 
 of primes already in the series. 
 
 21 = 3 x 7 
 
 27 = 3 x 3 * 3- 
 
 The latter is, strictly speaking, a tertiary harmonic ; but it will be seen that it 
 is ultimately derived from the same fundamental tone as the other notes. 
 
 45. We have therefore only two more prime numbers, 17 and 19 to con- 
 sider. The former of these, giving the minor ninth to the generator, is one 
 of the most important notes of the series for its employment in harmony ; the 
 
 19th harmonic, which gives the minor third, is only used in one combination 
 which will be dealt with later. (Set Chapter VJI.) When in the next 
 chapter we build up the chords of a key, the only primary upper partial tones 
 we shall require to use will be 
 
 No. 3 giving the perfect fifth to the generator. 
 
 ,, 5 ,, ,, major third ,, ,, 
 
 ,, 7 ,, ,, minor seventh ,, ,, 
 
 ,, 17 ,, ,, minor ninth ,, ,, 
 
 The notes that we may require which are not among these primary harmonics 
 will be furnished by secondary harmonics. 
 
 NOTE TO CHAPTER II. 
 
 The calculation of harmonic ratios. s 
 
 46. It has not been thought expedient to perplex the student of this chapter 
 by all the calculations proving the correctness of the ratios given in $ 43, 44 ; 
 but, as the ability to calculate such ratios with accuracy will be found very 
 valuable to those who wish to understand the subject thoroughly, a few general 
 rules will here be given. 
 
 47. I. To find the difference of the ratios of two intervals. Every ratio 
 can be expressed by a vulgar fraction, the vibration number of the higher of the 
 two notes being the numerator, and that of the lower the denominator. Thus 
 the ratios given in 38 might be written as -f, -f, -f, -f , &c. Two ratios can be 
 compared by reducing them to a common numerator, if the pitch of the higher 
 note of each is to be the same, and to a common denominator if the pitch of 
 the lower note of each is to be the same. To illustrate this, let us take the 
 ratios of the 7th, nth and I3th harmonics mentioned above as being not 
 strictly in tune in the key, and compare them with the correct intervals to 
 which they approximate. The 'harmonic 7th C : B|?, as will be seen from the 
 table of the harmonic series in 36, has the ratio -. But a true minor 7th 
 will be found between D and the C above it, the ratio being \ 6 . To compare 
 these two ratios 
 
 i and V 6 
 
 we reduce them to a common denominator, by multiplying the two denominators 
 together for a new denominator, and each denominator by the opposite numerator 
 for a new numerator, 
 
 4 x 9 = 36 
 
 9 x 7 63 
 
 4 x 16 = 64 
 The fractions will therefore be 
 
 -6.1 nri J 6 
 3 ana 3 6 
 
 that is to say, for every 63 vibrations of the harmonic 7th there will be in the 
 same time 64 of the true minor 7th, proving what was said in 43, that the 
 harmonic 7th is -^ too flat. 
 
 48. As further illustrations of this rule we will take the nth and I3th. 
 The former has to the tonic the ratio V, while the perfect 4th has ^
 
 30 HARMONY: [Chap. n. 
 
 Therefore the nth harmonic is larger (or sharper) than the perfect 4th. 
 The 1 3th again has the ratio ^, while the major sixth has the ratio -f. 
 Reducing as before, to a common denominator, 
 
 i_s _ a 9 A JLP 
 
 8 " 2 3 " " 24 
 
 Therefore the I3th is -$ too flat for a true major sixth. Lastly, we stated 
 that the 27th harmonic was only -y 1 ^ too sharp for a true major 6th. This can 
 be proved in the same way. The 27th harmonic of any series must lie between 
 the two octaves of the fundamental tone 16 and 32 ; above the former and 
 below the latter. Its ratio to the fundamental tone below must be therefore 
 yj, and, as we know, the major 6th has the ratio %. 
 
 2.7. _ K 1_ _5_ 5.O. 
 
 'l 6 " " 4 8 . 3 " " 4 8 
 
 Therefore the 27th harmonic is -^ s sharper than a true major 6th. 
 
 49. The above examples will, it is hoped, sufficiently explain the method 
 of calculating these ratios. A still simpler method of comparing these is to 
 tmiltiply each numerator by the other denominator. This, however, is not 
 sufficient by itself to enable the student to tell by how much one of the two 
 intervals is the flatter or sharper. 
 
 50. II. To find the number in the harmonic series of any note reckoned from 
 one generator. It may often happen that it is desired to find the ratio to the 
 generator of some note not given in the harmonic series up to 20, as in 36. 
 As it has been already said that none of the higher primary harmonics are 
 available for music in any key, it follows that any other notes that we can 
 use must be at least secondary, if not tertiary harmonics from the fundamental 
 tone. In that case, they will be part of the harmonic series of one of the upper 
 partials, and a little practice will soon enable the student who is familiar with 
 the series as far as it has been given to see from which note it can be derived. 
 For example, let us take the note Gft from the fundamental tone C, a note which 
 makes with G^ the interval of the " chromatic semitone." A moment's thought 
 will tell us that G4 is the major 3rd of E. Our harmonic series shows that 
 the major 3rd is the 5th harmonic from the generator ; therefore Gjl is 
 the 5th harmonic of E. But E is itself the 5th harmonic of C ; therefore 
 G$ is the 25th (5 x 5) harmonic of C, and this note must lie between the 
 two Cs which are 16 and 32 of the series. But the G in the same octave will 
 evidently be 24, therefore the ratio of the chromatic semitone G : Gj! is f^, 
 the diatonic semitone being }f or }f. 
 
 51. To illustrate this still further, let us take the two notes FJ and A?. 
 It is clear that F4 is the major 3rd, (and therefore the 5th harmonic) of D, and 
 D being the gth harmonic of C, F$ will be the 45th harmonic, and the interval 
 of the augmented 4th C : Fj is |4. Again, Aj? is the minor gth (therefore 
 the I7th harmonic,) of G, which is itself the third harmonic of C ; A? is con- 
 sequently the 5lst harmonic of C. As a matter of fact, these very high 
 harmonics are not produced in nature together with their generator, or, at 
 least, not to any appreciable extent ; this is why it was said in 32 that modern 
 harmony does not rest solely on natural laws (see also 42) ; but a knowledge 
 of these harmonics is very useful in enabling us to calculate the ratios of dis- 
 sonant intervals. 
 
 52. There is one interval yet to be mentioned, which, as will be seen in 
 the next chapter, is of considerable importance. This is the extremely 
 small interval between two notes which are " enharmonics " of one another, 
 that is to say two notes represented by the same sound on instruments having 
 only twelve sounds in the octave, e.g., Gj and Aj?. It is often supposed that 
 G$ is the higher of these two notes ; but as a matter of fact it is the lower. 
 This will be easily seen by deriving both notes from the generator C. We 
 showed just now that Gj was the 25th harmonic of C ( 50), and also that A|? 
 was the 5 1st harmonic ( 51). Obviously the octave above the 25th harmonic 
 will be the 5oth ; therefore in the series from C, G=5O, and Aj? =51, or the
 
 Chap, ii.] ITS THEORY AND PRACTICE. 31 
 
 next higher harmonic. These two notes are never used in the same key, for no 
 interval smaller than a semitone is used in musical composition ; but, as will 
 be seen later, the two notes are often treated as identical, and the one is 
 substituted for the other for the purposes of modulation. The small interval 
 between the two notes is called the "enharmonic diesis," from a Greek word 
 meaning division.* 
 
 53. For the sake of uniformity we have in this chapter reckoned all the 
 harmonics from the same generator, C. The student must not forget that a 
 similar series can be formed from any other note ; and it will be well for him 
 to accustom himself to work out harmonic calculations such as those here given 
 from other generators. The exercises now to be given will be of material 
 assistance to him in this respect. 
 
 EXERCISES TO CHAPTER II. 
 
 (i.) Write the Harmonic Series up to No. 20 from the follow- 
 ing generators, distinguishing the secondary harmonics by a x , 
 and enclosing in brackets those harmonics which are out of tune. 
 
 fcs "-f=- 
 
 -& 
 
 (2) Find the ratio of the interval of the augmented sixth 
 
 deriving both the notes of the interval from D as a generator. 
 
 (3.) Ft| being the ninth harmonic of E7, compare the pitch of 
 the note B7, when derived as the perfect fifth of E7, with the 
 pitch of the same note derived through F, (a) as a primary 
 harmonic, (b) as a secondary harmonic. 
 
 (4.) Give the number of each of the following notes in the 
 harmonic series of the generator named below it : 
 
 (J 
 
 . c 
 
 HARMONIC, 
 
 GENERATOR. Da Ai> Ej] B!> C$ E!> 
 
 (NOTE. Exercises 2 to 4, being somewhat difficult, should not be attempted 
 by the student until the contents of this chapter are thoroughly mastered.) 
 
 * The enharmonic diesis is usually calculated in rather a different way from that given 
 above ; viz. : by reckoning three major thirds one above another, e.g., 
 
 E G> B| 
 
 C E G$ 
 
 5 x i x i' = V- s 
 
 C " B$ 
 
 As the octave will clearly be W. it follows that the ratio will be iff. 
 
 This is the ratio usually given, and it will be found to be almost identical with the simpler 
 ratio |f given in the text.
 
 32 HARMONY: [Chap. in. 
 
 CHAPTER III. 
 
 KEY, OR TONALITY. 
 
 54. One of the first things which it is necessary that the student should under- 
 stand is what is meant when we speak of the Key of a piece of music. In 
 order that music should produce a satisfactory effect, it is necessary that the 
 notes, whether taken singly, as in a melody, or combined, as in harmony, 
 should have some definite and clearly recognisable relation to one another. 
 For example, if the first half of "God save the Queen" be played on the 
 piano 
 
 s 
 
 everyone can hear what is commonly called the titne that is, can feel that 
 the notes following each other have some definite relation to the first and last 
 note, and to one another. But if we take the very same notes on the staff, 
 and alter several of them by the addition of flats and sharps, thus 
 
 we not only distort the melody beyond recognition, but it ceases to be music 
 at all ; for the notes as they follow one another have no connection, no common 
 bond of union, so to speak. In other words, they are in no key. 
 
 55- From the very infancy of music, the necessity for the relationship of 
 notes to one another has always been felt, though the degree of relationship 
 and its nature have differed as the art has progressed. In its modern sense 
 Key may be thus defined : 
 
 A collection of twelve notes within the compass of an octave, of which the first 
 is called the TON 1C, or KEY-NOTE, to which note the other eleven bear a fixed 
 and definite relationship. 
 
 56. The student must remember that this definition does not imply that 
 all music in one key must lie within the compass of an octave, but only that all 
 the notes used in one key can be found within that compass. In the key of C, 
 for instance, the note C is the tonic, but it is equally the tonic whether it be the 
 lowest or the highest C on the keyboard, or any C lying between the two. 
 
 57- The fundamental principle for the development of modern music 
 cannot be better stated than in the words of Helmholtz (Sensations of Tone, 
 p. 383) : "The whole mass of tones and the connection of harmonies must 
 stand in a close and always distinctly perceptible relationship to some 
 arbitrarily selected tonic, and the mass of tone which forms the whole com- 
 position must be developed from this tonic, and must finally return to it." 
 
 58. A combination of more than two notes in the same key, arranged 
 on certain fixed principles now to be explained, is called a CHORD. Every 
 chord is made by placing not fewer than three notes one above another at an 
 interval of a third, either major or minor. The lowest note, upon which the 
 chord is built, is called its ROOT.* If only two thirds are placed one above 
 
 * Much trouble is sometimes caused to students from the word J\ oot being used in two senses 
 by theorists as the lowest note of any combination of thirds, and also as the fundamental tone 
 in the key from which the combination is harmonically derived. In order to avoid confusion, 
 the word Root will in this book always be employed in the former sense, and the note from 
 which the combination is ultimately derived will be called its Generator. This distinction 
 will become quite clear as we proceed.
 
 chap, in.] ITS THEORY AXD PRACTICE. 33 
 
 another, making in all three notes, the chord is called a Triad. It has been said 
 that the whole contents of a key are developed from the tonic, or key-note. 
 We therefore begin by making a chord on the tonic, and shall choose the key 
 of C major, as requiring no flats or sharps. 
 
 59. If we take C as a root, and put a third (major or minor) above it, that 
 note will be either EJj or E?. Another third above EJ|j wjll give either Gfc| or 
 G, and above E? we shall get cither G? or Gj. How are we to know 
 which of these various notes to select ? Here nature herself is our guide. \Ye 
 have already seen ( 36) that the perfect fifth in this case Gft and the major 
 third (Etj) are the first new sounds generated from C, the octaves being 
 merely repetitions of the same sound at a different pitch. In nature the filth 
 is produced first, and then the third 
 
 but as a chord is made by placing thirds one above another, we re-arrange these 
 notes. We can place them at any pitch, so long as their relative positions to 
 the root are not disturbed, e.g. : 
 
 i/ ^^-H fj II 
 
 P g ""I! 8 - 
 
 
 ig^-^^rrzr. II ===:E 
 
 
 At (a) will be seen the chord of C in its closest position. The student 
 will notice that the chord marked * contains the 4th, 5th, and 6th notes of the 
 harmonic series given in 36. All these positions (as will be shown later) are 
 not equally good ; but the chord is the same in each instance. .At (b) will be 
 seen wider positions of the same chord. It will be remembered that the 
 interval from C to G is always a perfect fifth, and from C to E a major third, 
 no matter how many octaves apart these notes may be ( 20). So long there- 
 fore as G and E are both placed above C, the chord is still a common chord, 
 and is called, after its root, the chord of C. But if either E or G were placed 
 belmv C, we should get a different -combination, which will be spoken of later. 
 
 60. Now let us examine this chord of C major a little closer. 
 
 It will be seen that the two 3rds ot which it is composed are not of the 
 same size. This is because the fifth of every common chord must be a perfect 
 fifth from the root. If we place two major thirds one above another we shall 
 
 get the combination SEa^iEE: ) an d if we take, instead of two major, 
 
 two minor thirds we get -/L L . Both these chords will be met with 
 
 later, but neither of them is a common chord. Hence we get another rule to 
 help us : a common chord is composed of a major and a minor third placed one 
 above another. If, as in the present case, the major third is the lower, i.e., 
 next above the root, the chord is called a major chord. If the minor third is 
 
 the lower, fy ^-^~ =I the chord is called a minor chord. The terms "major" 
 
 and "minor," as applied to keys indicate the nature of the chord on the tonic. 
 C
 
 34 HARMONY : [Chap. in. 
 
 Formerly the keys that we now call C major and C minor were described as 
 "the key of C with the greater third" and "the key of C with the lesser 
 third." At present we are concerned only with the major key, in which the 
 chord of the tonic is always major. 
 
 61. The student will notice that in this chord of C major, the note C is not 
 only the root that is, the lowest of the three notes arranged in thirds but 
 also the generator that is, the note of which the other two, E and G, are 
 upper partials. As the whole of our key is developed from the upper partials 
 (or harmonics) of the tonic, we have to look to our harmonic series ( 36) to 
 find the next third above G. Shall we take Bp or Bfcj ? We select B? 
 because it is the next new tone generated from C, as its 7th harmonic ; for this 
 reason it takes precedence of B|, which is not found till the I5th note of the 
 harmonic series, where, as was seen in the last chapter, it is the fifth harmonic 
 of G, and is therefore only a secondary harmonic of C. Thus far our 
 "fundamental chord " that is, a chord composed of the harmonics of its funda- 
 mental tone, or generator is always the same as to the intervals it contains, 
 these being invariably a major tliird, perfect fifth, and minor seventh from the 
 fundamental tone. These intervals cannot be too clearly and strongly 
 impressed on the mind of the student. 
 
 62. When we come to add another third above the 7th, a choice offers 
 itself; We can either take a minor 3rd (D?, the I7th harmonic) or a major 
 3rd (Dtl, the ninth harmonic). If both these notes \\ereflrimar_y harmonics, 
 we should have no hesitation in choosing DfJ, because the earlier the harmonics 
 are produced, the stronger they naturally are, and therefore the more claim 
 they have to a place in the key. But we saw in the last chapter ( 37) that 
 Dq was a secondary harmonic, being the third harmonic of G, and its claims 
 may therefore be regarded as balanced by the fact that D?, though produced 
 much higher up the series, is a primary harmonic, and springs out of its funda- 
 mental tone without the intervention of any other note. We can therefore take 
 either D? or Pt) the minor or the major ninth from C alternatively ; but we 
 cannot take both in the same chord. This would produce a discord technically 
 known as " false relation," by which is here meant that two notes a chromatic 
 semitone distant from one another cannot be used in the same chord. 
 
 63. Let us now look at this chord as far as we have made it : 
 
 The first chord (a) is the chord of the minor ninth, consisting of the 5th, 
 3rd, 7th, and I7th harmonics (all primary] of the generator. The second 
 chord (b] is the chord of the major ninth, composed, like that of the minor 
 ninth, of the 5th, 3rd, and 7th harmonics of the generator (the tonic], but also 
 containing a secondary harmonic Db, the third harmonic of the dominant. The 
 reason why a harmonic of the dominant, rather than of any other note, is taken, 
 is because the dominant is the first new note generated from the tonic. In the 
 chord (b} the primary harmonics of the generator are printed as semibreves, 
 and a crotchet head ( ) is used to indicate a secondary harmonic.* 
 
 64. It is evident that if we continue to build up our chord by thirds, the 
 next note above D will be an F, and the next note above F will be an A. 
 But it was seen in the last chapter ( 44) that the primary harmonics, F and 
 and A (the nth and I3th upper partials of C), were both too much out of tune 
 to be available in the chords of the key. It was also shown that the secondary 
 harmonics, derived as primes from G, were sufficiently in tune to answer our 
 purpose. Therefore, now that we have once begun (with the major Qth) to 
 employ secondary harmonics, we will continue with them. For the interval of 
 
 * It has been a'ready said in ? 40 that these higher harmonics car, only be heard, if at all, 
 by artificial aid. It must therefore be clearly understood that these ninths are used by com- 
 posers not for physical but for aesthetic reasons (see the quotation from Helmholtz in \ 42). 
 The explanation of their harmonic origin is not on that account the less useful, as showing the 
 relation to one another of the various notes in a key.
 
 Chap, in.] ITS THEORY AND PRACTICE. 35 
 
 the nth we take FJ3, the 7th harmonic of G, and not F$, the iSth, for the 
 same reason for which we took B? and not B3 for the seventh of C. The 
 thirteenth of the tonic, A, is evidently the ninth of the dominant, and, like the 
 interval of the gth from the tonic, it can be either minor or major, the former 
 (A?) being the I7th harmonic of the dominant, and the latter (At]) being a 
 secondary harmonic of the dominant, and therefore a tertiary harmonic of the 
 tonic. 
 
 65. The next third above A will clearly be C. It is impossible to have Gj, 
 because it would make " false relation " with the generator. The note must 
 therefore be C natural, and the series of thirds will begin over again. For this 
 reason, the thirteenth is the highest interval that can be placed over any funda- 
 mental tone. 
 
 66. We have now completed the tonic chord by building up over its 
 fundamental tone as many thirds as can be put one on another. We now give 
 the complete chord, putting the alternative major and minor gths and I3ths 
 side by side, and distinguishing primary and secondary harmonics by white and 
 black heads. 
 
 It will be well to remember that every interval of this chord up to and including 
 the minor ninth is made by a primary harmonic of the fundamental tone, and 
 that all the intervals above the minor ninth are formed by secondary harmonics. 
 In every fundamental chord (see 61), the primary harmonics come in the 
 same order 5, 3, 7, i 7 ; and the secondary harmonics do the same, excepting 
 that the first of these notes the 5th harmonic of the upper fundamental tone 
 cannot be used, because it would make "false relation" with the 7th primary 
 harmonic, which is already in the chord. It was shown in 6l that in this 
 chord of C which we have been building up B7, the 7th harmonic of C, took 
 precedence of Bfcj, the 5th harmonic of G ; and B? being already in the chord 
 Bq is permanently excluded from it. 
 
 67. Having exhausted the available harmonic resources of C as a fundamental 
 tone, we must look elsewhere for the materials to complete our key. It has 
 been already said that the whole key springs out of its tonic ; therefore we con- 
 tinue with the harmonics of the dominant, G, this being the first new note 
 generated from the fundamental tone C, and therefore the nearest related to C. 
 Evidently by taking G as a generator we shall obtain a chord containing 
 precisely the same intervals from G 'as those contained in the chord of C, which 
 we have already constructed. It is needless to go again step by step through 
 the process of building up this chord, as we did that of C, because all the 
 reasonings and all the results will be precisely similar. We shall find, as before, 
 that all the intervals up to the minor Qth will be primary harmonics of G, and 
 all above the minor Qth will be secondary harmonics, that is to say, harmonics 
 of D, the fifth of G. The whole of the dominant chord will therefore he this 
 
 The alternative Qths and I3ths are, as before, put side by s ; de. 
 
 68. It will be seen that several of the notes of this chord have been already 
 met with in the tonic chord as secondary harmonics. There are, in fact, only 
 two notes, Eft and E?, which are absolutely new ; but it will be shown later, 
 when the subject of the treatment of these chords is dealt with, that the effect 
 of notes which look the same on paper may differ entirely according to their 
 surroundings. 
 
 69. In 55, a key was defined as "a collection of twelve notes within the 
 compass of an octave/' If the student will examine the complete chords of the 
 tonic and dominant which we have been constructing, he will see that we have
 
 36 HARMONY : [Chap. in. 
 
 as yet only eleven. We shall have to take some other fundamental tone to 
 obtain the additional note. Just as we took the dominant after the tonic, as 
 the first new note springing from that generator, so we now take \h& supertonic, 
 the fifth of the dominant, for the same reason. Some of the primary harmonics 
 of the supertonic have already been met with as secondary harmonics in the 
 dominant chord, just as the harmonics of the dominant were seen in the upper 
 part of the tonic chord. As the supertonic chord will be constructed in pre- 
 cisely the same manner as the tonic and dominant chords, it is clear that i: will 
 contain exactly the same intervals. 
 
 Here we find one note (F) which was not in either of the previously constructed 
 chords, and which completes the twelve notes required for the material of the 
 key. We cannot take any other fundamental tone than these three in a key 
 without getting mure than twelve notes in the key. For since we have only 
 twelve semitones in the octave, and these are already given from one or other 
 of the three generators we have taken, it is clear that any new note introduced 
 into the key must be a note distant by less than a semitone from some note 
 already in the key. If, for example, we take A, the 5th of D, as a generator, 
 its major third will be Ct But we have already DJ7 in the key, as the minor 
 ninth of C; and between CJJ! and D? is the "enharmonic diesis " ( 52). 
 This interval, being smaller than a semitone, cannot be used. The general rule 
 may be thus stated : No two notes which are enharmonics of one another can both 
 beusedinthe chords of the same key.* Hence, though we can and do employ some of 
 the harmonics of A as secondary harmonics of the supertonic, D, A cannot be 
 employed as a generator in the key ; for it would be impossible to build a 
 fundamental chord upon it without introducing the major 3rd, QJJ!, which is 
 excluded by the D? already in the key. In the same way E, as a generator, 
 would give Gj{, the enharmonic of A[7, one of the primary harmonics of G. 
 This is the reason why we take D as the next generator after G, although E, as 
 the 5th harmonic, is derived from the tonic much earlier than D, the 9th har- 
 monic. On the other hand, if we take F, the subdominant, as a generator, its 
 minor ninth, G?, is the enharmonic of Fit, the major third of D. It is there- 
 fore obviously impossible to continue the series of generators by fifths any 
 further, either upwards or downwards. 
 
 70. Here it may naturally be asked, Why not take the fifth below the tonic 
 as one of the fundamental tones in the key, instead of the fifth above the 
 dominant, thus having the tonic as a centre, with the dominant a fifth above it, 
 and the subdominant a fifth below it ? The reason will be more clearly seen 
 when we come later to consider the different character of tonic and dominant 
 harmony (see Chapter IX. ) ; for the present it will suffice to say that the sub- 
 dominant would be the fundamental tone out of which the tonic springs as its third 
 harmonic ; that is to say, the tonic has the same relation to the subdominant as 
 the dominant has to the tonic ; and if the subdominant be taken as a generator 
 in the key, the tonic at once sinks into a subordinate position, as a note 
 generated out of one of the other notes in the key, instead of being the source 
 whence the whole material of the key is derived. 
 
 71. Let us now collect and endeavour to arrange in order the materials of 
 our key of C major, as we have obtained it from the three generators C, G, 
 and D. For this purpose we exhibit the three fundamental chords side by side. 
 
 * It will be seen later that the apparent violations of this rule by composers aie merely 
 examples of "expedient notation' (see 5 341).
 
 Chap. in. ITS THEORY AND PRACTICE. 37 
 
 It will he noticed that the dominant chord is written a fourth below the tonic, 
 instead of a fifth above. This is simply for convenience ; the octave in which 
 a chord is taken makes no difference in its origin. It must also be observed 
 that the higher notes, C, E, G, and B? of the supertonic chord all diffc-r 
 slightly in pitch from the corresponding notes of the tonic chord, when taken 
 in the same octave. By the help of the rules given in 47, the student can 
 easily ascertain the amount of the difference, which is in every case so small 
 that we can afford to disregard it, the two C's, E's, &c., being for all practical 
 purposes identical. 
 
 72. If we examine the twelve notes which form the three chords given 
 above, it will be seen that seven of them have no accidentals before them, 
 while, of the other five, four are lowered a semitone by a flat, and the fifth is 
 raised a semitone by a sharp. The seven notes having no accidentals are called 
 diatonic notes, by which is here meant notes in accordance with the signature 
 of the key : the five which have accidentals are called chromatic notes, i.e., 
 colouring notes, which give the colour, the light and shade to the key. As 
 every key is generated out of its tonic in the same way as the key of C, it 
 follows that every key contiins seven diatonic and five chromatic notes. 
 
 73. We will now arrange the twelve notes which we have obtained in the 
 key of C in their order of pitch, beginning with the tonic itself C. The note 
 nearest above C is evidently D?, and we have more than once said that the 
 o:tave in which a note is placed does not affect its nature. So that although 
 Dj> is the minor ninth and DJ the major ninth of C in chord-building, yet, as 
 these notes can be found within the limit of an octave from C (see 56), we 
 place them at a distance of a semitone and a tone respectively from the tonic. 
 We now proceed to give the twelve notes in their regular order, using black 
 heads for the chromatic notes, and plating under each note the explanation of 
 its origin in the key. 
 
 
 
 
 
 h 1 
 
 ~fr fr5~ 
 
 
 s- 
 
 -\r tf^ ^* N"- 
 
 -*= i~ 
 
 
 
 
 I 
 
 1 1 1 1 
 
 1 1 
 
 1 1 1 
 
 1 1 
 
 1 
 
 t 
 
 g CO g w 
 
 1 f ff : I- 
 
 w 
 
 B g <ft 
 
 i F * 
 
 S- 3. 
 
 H 
 
 
 
 M K. & n M> - 1 
 
 If p I. p 
 
 
 g S 
 
 It will be seen that all these notes are either fundamental tones, or primary 
 harmonics of a fundamental tone, as stated in 4$. The scale just given is 
 called the chromatic scale of C. It consists entirely of semitones ; and, just as 
 the key itself contains s'even diatonic and five chromatic notes, so the chromatic 
 scale contains seven diatonic and five chromatic semitones. 
 
 74. If we omit from the above scale all the chromatic notes, we get the 
 diatonic scale ( 9) of C major. 
 
 Semitone. Semitone. 
 
 This scale consists of a succession of tones and diatonic semitones five of the 
 former and two of the latter. The semitones are found between the 3rd and 
 4th and the 7th and 8th degrees of the scale ; and the position of the semi- 
 tones, as will be seen later when we come to the minor key, determines th~ 
 nature of the scale and of the key.* 
 
 * As every tone consists of a diatonic and a chromatic semitone, (? 7), it U clear that the 
 five tones toge'her contain five diatonic and five chromat'C semitones. Adding the two 
 diaton'C semitones of the major bcale, we get a total of seven diatonic and five chromat.c 
 semitones, as stated above.
 
 38 HARMONY : [Chap. in. 
 
 75- We will now turn back to the three fundamental chords given in Ji. 
 It is extremely rare to find these chords in their complete shape, partly because 
 some of the notes form harsh dissonances with one another e.g., the nth 
 with the 3rd, or the minor I3th with the fifth but still more because most 
 music is written in harmony of four parts, and as a complete chord of the I3th 
 contains seven notes it is clear that at least three of these must be omitted. 
 But just as we have already shown that all the notes of the key are to be found 
 in these chords, we will now show that all the harmonies of the key are 
 contained in them. As the nature of chromatic chords in the key (that is to 
 say, chords containing some of the chromatic notes) has not yet been explained, 
 we will speak at present only of the diatonic chords. The student will see that 
 without using any of the five chromatic notes in the key, we shall find among 
 the fundamental harmonies a common chord on every degree of the scale 
 excepting the leading note, which bears a triad which we will explain directly. 
 It will be found that each common chord is derived from the harmonics of one 
 of the three generators. 
 
 The tonic chord (C, E, G) consists of the fundamental tone, major third, 
 and fifth of the tonic. 
 
 The supertonic chord (D, F, A) is the fifth, seventh, and major ninth of the 
 dominant. 
 
 The mediant chord (E, G, B) is the major thirteenth, fundamental tone, 
 and major third of the dominant. 
 
 The subdominani chord (F, A, C) is the seventh, major ninth, and eleventh 
 of the dominant. 
 
 The dominant chord (G, B, D) is the fundamental tone, major third, and 
 fifth of the dominant. 
 
 The stibmediant chord (A, C, E) is the fifth, seventh, and major ninth of 
 the supertonic. 
 
 76. The chord on the leading note (B, D, F) differs from all the other 
 chords of this series inasmuch as it does not contain a perfect fifth from the 
 root.* It is therefore not a common chord. ( 60.) The interval from B to F 
 being a diminished fifth ( 31), the chord B, D, F is called a diminished triad, 
 and the notes of which it is composed are the major third, fifth, and seventh of 
 the dominant. 
 
 77- It would be easy to show that all the chromatic concords in the key, as 
 well as all the discords, whether diatonic or chromatic, are to be found among 
 the fundamental chords of which we have been speaking. It will, however, 
 be more convenient to defer this until we come to deal with these subjects. 
 From what has been already said, it will be readily understood that as no key 
 can contain more than twelve notes, from which all the. harmonies of that key 
 are made, and as all these twelve notes are derived from one of the three 
 generators of the key, the tonic, dominant, or supertonic, all the harmonies 
 must also be derived from one of these three generators. It has further been 
 seen that both the dominant and supertonic are ultimately derived from the 
 tonic. It is therefore from the tonic that the whole material of the key is 
 developed ; and we have thus established the correctness of the definition of 
 Key given in 55, and justified the principle which we quoted from Helmholtz 
 in 57- 
 
 78. Hitherto we have in this chapter spoken only of one key that of C ; 
 but as the selection of the tonic is entirely arbitrary, any other note than C may 
 be taken as a key-note. The other notes of the key will bear the same 
 relation to their tonic, whatever that tonic may be, which the notes in the key 
 
 * As we have carefully distinguished between the terms "root" and "generator" (5 58, 
 note) there can be no objection to applying the word '' root " here to the note B, though it 
 aoe^ not bear a perfect fi'th above it.
 
 chap. in. Irs THEORY AND PRACTICE. 39 
 
 of C bear to the tonic C. We selected C to work with at first because in this 
 alone, of all the major keys, there are no flats or sharps in the diatonic scale, 
 and therefore in the key-signature. We now proceed to show how in keys 
 with any other tonic than C sharps or flats become necessary. 
 
 79. In 74 we deduced our diatonic scale from the chords of the key. If 
 we take another tonic, we shall evidently get a similar scale as regards the 
 order of intervals, that is to say, the semitones will come between the 3rd and 
 4th and the 7th and 8th degrees of the scale. When all the notes are naturals, 
 the semitones, as we have seen, lie between E and F and between B and C. 
 But if any other tonic than C is taken, E and F will no longer be the 3rd and 4th, 
 and B and C no longer the 7th and 8th degrees of the scale ; therefore, if the 
 notes be left unaltered, the semitones will come in the wrong places. As a 
 matter of (act, in the older music such scales were used ; and the difference 
 between one key and another was a difference in the place of the semitones. 
 Each " mode," as it was termed, had a different order of tones and semitones.* 
 In modern music only two " modes," the major and minor, are employed ; and 
 the difference between one major key and another or one minor key and 
 another is solely a difference of pitch. 
 
 80. It we examine tbe diatonic major scale of eight notes beginning from 
 the tonic, we shall see that it can be divided into two sections of four notes 
 each, and that these two sections are in their construction precisely similar, 
 each containing the interval of a semitone between the two upper notes, and a 
 tone between the other notes. A series of four notes thus arranged is called 
 a Tetrachord a Greek word signifying four strings ; and the scale consists of 
 two such tetrachords placed one above the other with the interval of a tone 
 between the highest note of the lower tetrachord, and the lowest note of the 
 upper. This tone, separating the two tetrachords, is called the " tone of 
 disjunction." 
 
 "~T r T~^ o. Upper tetrachord. 
 
 Lower tetrachord. 9: 
 
 CO 
 
 = -j. 
 II 
 
 It is important to notice that the lower tetrachord begins with the tonic, 
 and the upper with the dominant the first new note, as we have several 
 times said, springing out of the tonic. They may therefore also be called the 
 tonic and dominant tetrachords. 
 
 81. If we now take G as a tonic, or in other words make the upper tetra- 
 chord of C the lower tetrachord of a new scale, we shall find that if we leave 
 all the notes unaltered the upper tetrachord will have its semitone in the wrong 
 place 
 
 I 
 
 Here the semitone is between the second and third notes of the upper 
 tetrachord, instead of between the third and fourth. To correct this F must 
 be raised to Fi ; and we now have a semitone between the " leading-note " 
 
 * If the student will play a scale on the piano, using only -white keys, and commencing on 
 other note than C, he will obtain the various forms of the old ecclesiastical scales.
 
 40 HARMONY : [Chap. m. 
 
 ( 13) and the tonic, as in the key of C.* It is important to notice that the 
 sharpened note is the leading-note of the new key, and that the sharp, because 
 it belongs to the key, is marked once for all in the key-signature. 
 
 82. If we continue to take the upper tetrachord of each key as we obtain 
 it as the lower tetrachord of a new key i.e., if we make each dominant into 
 a new tonic, we shall clearly have to introduce a fresh sharp for each new 
 leading-note. The student is recommended to work out all the scales rising by 
 fifths after the pattern given above. The result will be the following : 
 
 Tonic. Signature. 
 
 C ......... None. 
 
 A ........ .. Fj, C J, Gf . 
 
 E ......... Ff, C|,G|,Df. 
 
 Ff,.Cf,G*,D&Af,Ef. 
 
 83. It would be possible to continue this series, of which the next tonic 
 would be Gi ; but as this would involve the use of a double-sharp, it is more 
 convenient instead of G4 to take its " enharmonic '' ( 5 2 )> At>, which for all 
 practical purposes is the same note. We never therefore find a piece of music 
 written with the key of Git major as a signature, though the key is occasionally 
 used incidentally in the course of a piece, t 
 
 84. All these sharp keys have been obtained by making the upper tetra- 
 chord of one key the lower tetrachord of the next, or, in other words, by 
 making the dominant of one key the tonic of the following. If we now reverse 
 the process, and make the lower tetrachord into an upper one of a new key, we 
 get a different series. As before we begin with the key of C. 
 
 
 85. If we examine the lower tetrachord here, we see that it has no semi- 
 tone ; we also see that there is only a semitone between the two tetrachords, 
 instead of the "tone of disjunction." In fact the highest note of the lower 
 tetrachord is a semitone too high. We therefore lower this note with a flat 
 making it B?, and the scale of F is now correct. Just as the scale of G, the 
 fifth above C, requires a sharp, so the scale of F, the fifth below C, requires a 
 flat ; and just as each dominant when taken as a tonic, required one additional 
 sharp, it will be evident that each new subdominant (the fifth below the tonic), 
 
 * Of course if we had taken G as a tonic and developed the material of the key, as we have 
 done with C, we should have obtained, as the major 3rd of the dominant D, F4JJ and not FJI, 
 Our object in the text is to show the connection of the scales of C and G, and the order in 
 which the sharps are introduced in the series ot keys we are now considering. 
 
 t If w? continue this series to its extreme limit) the next tonic will be D4fc, then will follow 
 A 2J, EJ, and Bi. We can go no further than this, because Bjjj is the enharmonic of C, and 
 
 we have now completed the " circle of fifths," as it is termed, going through the sharp keys 
 with ascending fi f ths. The inconvenience of writing in these extreme keys is that they necessi- 
 tate double-sharps in the signature. Thus the signature of Aft major would be written 
 
 : ; and of B $ major
 
 Chap, in Irs THEORY AND PRACTICE. 41 
 
 when treated as a tonic will require an additional flat. If the student has fully 
 understood the explanations given above, it will be needless to repeat the pro- 
 cess of forming the scales from the tetrachords. The series of descending fifths 
 with their signatures will be as follows : 
 
 Tonic. Signature. 
 
 C None. 
 
 F Bb. 
 
 Bb Bb, Eb. 
 
 E? Bb, E>, Ab. 
 
 A? Bb, Eb, Ab, Db. 
 
 Db Bb, Eb, Ab, Db, Gb. 
 
 cb Bb, EP, Ab, Db, cb, cb. 
 
 cb Bb, Eb Ab, Db, cb, cb, Fb. 
 
 It is of course possible to continue the series further, as with the sharp keys ; 
 but as so doing would involve the use of double-flats in the signature, it is more 
 convenient to use the enharmonic, keys which contain sharps. For instance, 
 instead of the key of F?, that of Efl is taken, and so on with the others. 
 
 86. It should be noticed that in passing to the sharper key i.e., taking the 
 dominant as a new tonic it is always the subdominant of the old key which is 
 sharpened to become the new leading note ; and conversely in passing to a 
 flatter key i.e., taking the tonic as a new dominant, it is always the leading 
 note of the old key which is flattened to become the new subdominant. Hence 
 we obtain the easy rule for finding the tonic of any major key the minor keys 
 will be explained later from the signature. In sharp keys, the last sharp is 
 always the leading note, and in flat keys, the last flat is always the sub- 
 dominant. When we know the leading-note or the subdominant of any key, 
 it is a matter of very simple calculation to find the tonic. 
 
 87. One point still remains to be noticed in conclusion. As every tonic 
 generates its dominant as its third harmonic, and as the whole key springs out 
 of the tonic, it will be evident that any kev is the generator of keys havingmore 
 sharps than itself, for these must be derived through the dominant. And con- 
 versely, as evt-ry tonic springs (as a third harmonic) out of its own sub- 
 dominant, it will be equally clear that every key must be generated out of keys 
 having more flats than itself. This is a point that will be seen to be of great 
 importance when we come to deal with key-relationship and modulation. 
 
 EXERCISES TO CHAPTER III. 
 
 (i.) Write the three fundamental chords complete in the keys 
 of D, Eb, F, G?, Ati, and B?. 
 
 (2.) Write major scales from the following notes, putting no 
 key-signature, but inserting a flat or sharp before each note that 
 requires one E, A?, F$, F?, G|, B, B7.
 
 42 HARMONY : [Chap. iv. 
 
 CHAPTER IV. 
 
 THE GENERAL LAWS OF PART-WRITING. 
 
 88. In Harmony any number of notes, from two upwards, may 
 be sounded at one time. If each chord contain four notes, the 
 harmony is said to be in four parts, if each contain three notes, 
 the harmony is in three parts, and so on. Each part in the 
 harmony has generally the same relative position to all the other 
 parts ; that is to say, all the upper notes of the harmony form one 
 part, all the lowest notes another part, all the notes next above the 
 lowest another, &c. The progression of these parts may be con- 
 sidered in two aspects ; either as melodic progression, that is, the 
 motion of each part regarded singly ; or as harmonic progression, 
 that is, the motion of each part with relation to all the other parts. 
 Both these kinds of progression are governed by certain general 
 laws, which will now be explained. 
 
 89. The rules for melodic progression are very few and simple. 
 A good melody is one that flows naturally and easily ; it is there- 
 fore best either to proceed by step of a second (called " conjunct 
 motion ") that is, to the next note above or below ; or by leap 
 of a consonant interval ( 16). If, as sometimes happens, it is 
 necessary to leap by a dissonant interval, a diminished interval is 
 
 to be preferred to an augmented one. Thus : ^ = =z i | jj is 
 
 better than jj)~y^'J~~ |j , though either is possible. The former 
 is a diminished fifth, and the latter an augmented fourth. 
 
 90. If a part moves by a diminished interval it ought to return 
 to a note within the interval, and not continue in the same 
 direction. The best progression for any dissonant interval is, 
 that the second of the two notes forming the interval should 
 proceed to that note which is the resolution of the dissonance 
 (17) made, if the two notes are sounded together. For instance, 
 
 the student will learn later (Chap, ix., 199), that the diminished 
 
 i | 
 
 fifth just given will resolve thus : ^jpr^JEl Therefore F, 
 
 coming after B, moves to the E, just as it would do were it 
 sounded with B. Had F been the first note and B the second, 
 
 B would, for the same reason, have gone to C : 
 
 91. An augmented interval should seldom be used in melody 
 unless both the notes belong to the same harmony. But the
 
 Chap, iv.] Irs THEORY AND PRACTICE. 4. 
 
 interval of the augmented second, which we shall, find later in the 
 minor scale ( 171) between the sixth and seventh degrees may be 
 used more freely. 
 
 92. A large interval in the melody is best approached and 
 quitted in the opposite direction to that in which it leaps. 
 
 Good. Bad. 
 
 At (a) will be seen the leap of an octave upwards between the 
 second and third notes It is therefore much better that the first 
 of the two notes should be approached downwards, and the second 
 E left downwards, than that they should be approached and left 
 in the same direction, as at (b). 
 
 93. By harmonic progression, as has been said above, is meant 
 the way in which the parts move in their relation to one another. 
 There are three kinds of motion ; similar (sometimes, though less 
 frequently, called "parallel ") when two or more parts move in the 
 same direction up, or down; oblique, when one part moves up 
 or down while another remains stationary ; and contrary, when one 
 part ascends while another descends. 
 
 *7 
 
 r ~ " i IT '-f-r~r r r r^p* 
 
 Similar. Oblique. Contrary. 
 
 If the music is in more than two parts, it is evident that at 
 least two of these different kinds of motion must be combined, 
 except when all are moving in similar motion, eg., 
 
 % In this passage in the first bar all three parts move in similar 
 motion. Between the last chord of the first bar and the first 
 chord of the second, there is contrary motion between the two 
 extreme parts, and oblique motion between each of the extreme 
 parts and the middle part. Between the first and second chords 
 of the second bar there is contrary motion between the upper part 
 and each of the others, and therefore the two lower parts move 
 by similar motion. The student can analyse the rest of the 
 passage for himself in the same way. 
 
 94. Most music is written in four-part harmony, and the parts 
 are generally named after the four varieties of the human voice, 
 being, in fact, often called " voices." The highest part is called the 
 treble, or soprano, the next below this, the alto, the third part, 
 counting downwards, the tenor, and the lowest part the bass. 
 This refers to their relative rather than their actual positions ; and 
 it is important to remember that the lowest part of the harmony is
 
 44 HARMONY: [Chap. iv. 
 
 called the bass, even where (as in the example irt the last paragraph) 
 it is written in the treble staff. 
 
 95. We shall now give the rules which the student must 
 observe in part-writing, it is only right to say that these rules are 
 not in all cases strictly adhered to by the great masters ; the 
 student will learn by experience, as his knowledge increases, when 
 it is safe to relax them ; but it may be laid down as a general 
 principle that nobody can break rules with good effect till he knows 
 how to keep them. For the present, therefore, no licences can be 
 allowed. 
 
 96. RULE i. No two parts in harmony may move in unison, 
 or in octaves with one another. 
 
 At (a) the alto and tenor are moving in unison in the second, 
 third, and fourth crotchets. At (b) the tenor and bass are 
 moving in octaves between the first and second crotchets ; the 
 soprano and tenor between the third and fourth ; and the 
 alto and bass between the fifth and sixth. Such passages are 
 called consecutive unisons and octaves. It is important to 
 remember that the repetition of the same octave or unison by 
 two parts does not make consecutives, as the parts are not then 
 moving in octaves or unisons. The same remark applies to the 
 consecutive fifths forbidden in Rule 2. Neither is it considered 
 
 to make consecutives if one part leaps an octave, e.g. : 
 
 because the repetition of the same note at the distance of an 
 octave does not chanere the hnrmonv 
 
 97. It must be said here that octaves will be found in most 
 chords in four-part harmony ; but they are not consecutive unless 
 they occur between the same parts. 
 
 In this example each chord contains an octave of the bass 
 note ; but in the first chord the octave is between the tenor and 
 bass, in the second between the alto and bass, in the third between 
 the soprano and bass, and in the fourth between the tenor and bass 
 again. No two of these octaves are therefore consecutive, and the 
 passage is quite correct.
 
 Chap. IV.] 
 
 ITS THEORY A. YD PRACTICE. 
 
 45 
 
 98. The rule just given does not apply to the doubling of a 
 whole passage in octaves, such as frequently found in pianoforte 
 and orchestral music, nor to passages in which all the parts move 
 in unisons and octaves. For instance, the familiar passage in 
 Handel's "Hallelujah" chorus, to the words " For the Lord God 
 omnipotent reigneth " is not considered as "consecutive octaves." 
 Again, in Mendelssohn's "St. Paul," the soprano and alto sing 
 the choral "To Thee, O Lord, I yield my spirit" in unison, the 
 harmony being in three parts throughout. But this is not called 
 consecutive unisons. 
 
 99. RULE II. Consecutive perfect fifths are not allowed 
 between any two parts. They are, however, much less objection- 
 able when taken by contrary than by similar motion, especially if 
 one of the parts be a middle part, and the progression be between 
 dominant and tonic harmony. 
 
 Very bad. Not so bad. 
 
 zoo. This rule is much more frequently broken by great 
 composers than the rule prohibiting consecutive octaves. Con- 
 secutive fifths between the tonic and dominant chords are not 
 infrequently met with, as in the first and third of the following 
 examples : 
 
 BEETHOVEN. Pastoral Symphony. 
 
 BEETHOVEN. Sonata, Op. 14, No. i. 
 
 'j J '| . J^J* 'l 
 
 KULLAK. " Fleurs Anim^es," No. i. 
 
 5 
 
 At (a) will be found in the third bar consecutive fifths by con- 
 trary motion between the tenor and bass; and from the third to 
 the fourth bar, consecutive fifths between the extreme .parts by 
 similar motion. At the second bar of (b) are fifths between alto 
 and tenor ; at (r) are seen fifths by contrary motion between tenor 
 and bass, and at (d) four consecutive fifths between extreme 
 parts. These examples are not given for the student s imitation ;
 
 HARMONY : 
 
 [Chap. IV. 
 
 experience is required to understand when they may be properly 
 introduced ; but it is needful to mention them here, for the sake 
 of completeness. By beginners the prohibition of consecutive 
 fifths must be strictly attended to. 
 
 1 01. If one of the two fifths is diminished, the rule does not 
 apply, provided that the perfect fifth comes first. 
 
 This progression is quite correct. But a diminished fifth 
 followed by a perfect fifth is forbidden between the bass and an 
 upper part, but allowed between two upper or middle parts. 
 
 Bad. Good. Good. 
 
 102. If two parts go by similar motion to octaves or perfect 
 fifths, such progressions are called " hidden " octaves or fifths. 
 
 (a) (b) (c) (aT) 
 
 ' -J-frj-^-t^^jEE* 
 
 I 
 
 m- P i ^~f * -- f 
 
 I ""I 
 
 At (a) are two parts moving to an octave by similar motion. 
 The lower part in leaping from G to Classes ovtr the intermediate 
 notes F, E, D, as shown at (<:). If these notes are introduced, 
 
 D C 
 
 there will be consecutive octaves -p. to n . Similarly at (in the two 
 
 J_y V_/ 
 
 parts leap to a fifth by similar motion. If the intermediate notes 
 
 C 1 1~) F 
 
 are inserted, as at (</), we see the fifths -p . . These octaves 
 
 and fifths, being passed over, instead of sounded, are said to be 
 hidden. 
 
 103. RULE III. Hidden octaves are forbidden between 
 extreme parts, except, ist when the bass rises a 4th or falls a 5th, 
 either from dominant to tonic, or from tonic to subdominant, and 
 at the same time the upper part moves by step,
 
 Chap. IV ] 
 
 Irs THEORY AND PRACTICE. 
 
 47 
 
 2nd, when the second of the two chords is a second inversion 
 ( 
 
 and 3rd, when the second chord is another position of the first. 
 
 In other cases they are not allowed. Let the student examine the 
 three bad examples of hidden octaves here given, 
 
 and he will see that they do not come under any of the exceptions 
 just mentioned. 
 
 104. Hidden octaves are, however, allowed between any other 
 of the parts than the two extreme parts, with one important 
 exception. It is strictly forbidden to move from a yth or pth, to 
 an octave by similar motion, when one part moves a second, and 
 the other a third. 
 
 This is the very worst kind of hidden octaves, and must be most 
 carefully avoided. 
 
 105. RULE IV. Hidden fifths are forbidden between extreme 
 parts,
 
 HARMONY : 
 
 IV. 
 
 except, ist, in a progression from the tonic to the dominant chord, 
 or from the subdominant to the tonic chord, in both which cases 
 the upper part must move by step of a second, 
 
 2nd, from the chord of the supertonic, with the third in the upper 
 part, to the chord of the dominant, when the bass falls a fifth, and 
 the upper part falls a third, 
 
 and 3rd, from one to another position of the same chord, exactly 
 as with hidden octaves : 
 
 .Except between extreme parts, hidden fifths are not prohibited. 
 
 1 06. RULE V. Consecutive fourths between the bass and an 
 upper part are forbidden, except when the second of the two is an 
 augmented fourth. 
 
 Bad. 
 
 Good 
 
 Between any of the upper parts consecutive fourths are not 
 prohibited. They are sometimes to be found between the bass 
 and a middle part , but even these are not advisable. 
 
 107. RULE VI. Consecutive seconds, sevenths, and ninths are 
 forbidden between any two parts, unless one of the notes be a 
 passing note, i.e , a note which does not belong to the harmony 
 (set 255). Even then it will be better for the student to 
 avoid them. 
 
 108. There is one somewhat important exception to this rule
 
 Chap. IV.] 
 
 Irs THEORY AND PRACTICE. 
 
 49 
 
 to be found in the works of the old masters. Corelli, Handel, 
 and others sometimes followed a dominant seventh ( 197) by 
 another seventh on the bass note next below. We give two 
 examples from Handel's works. 
 
 HANDEL'S "Saul." 
 
 . (). k N S ! ! . J I / 
 
 ^-1 * L*4*==3 \ 
 *r * + \ u, i 
 
 HANDEL'S "Joshua " 
 
 In the second of these examples is also seen, in the tenor, an 
 exception to the rule given in 92. These passages are not given 
 fof the student's imitation, but because if no mention were made 
 of such exceptions he might naturally infer, if he met with similar 
 passages in the works of the great masters, that the rule here given 
 was wrong. We have already said that hardly any of the rules 
 in this chapter are strictly adhered to by great composers ; but 
 they are none the less useful, and even necessary for beginners. 
 
 109. RULE VII. It is forbidden for two parts to go from a 
 second into a unison. 
 
 This progression is sometimes used when the second is a passing 
 note ( 255,) as at (*); but the student is advised to avoid it even 
 in this case. 
 
 no. In addition to the above rules, the following recommend- 
 ations will be found serviceable to beginners. They need not be 
 so stringently enforced as the rules just given ; but attention to 
 them will in many cases make the harmonies much smoother and 
 the parts more flowing. 
 
 in. Recommendation I. It is not desirable to allow two parts 
 to overlap, that is, to let a higher part proceed to a note below 
 that previously sounded in a lower part, or, conversely, to let a 
 lower part proceed to a note above that previously sounded in a 
 higher p;.rt. 
 
 At (a) the upper part leaps from E to B, a lower note than C, 
 taken in the first chord by the lower part. At (b) the lower 
 part leaps to C, which is higher than the A of the upper part in
 
 HARMONY : 
 
 [Chap. IV. 
 
 the first chord. Such progressions are sometimes necessary, but it 
 is, better to avoid them if possible. The crossing of two parts 
 
 i 
 
 should be avoided altogether in four-part writing. In writing for 
 a large number of parts (see Chapter XXL), it is sometimes 
 necessary. 
 
 112. Recommendation II. When two notes making a dis- 
 sonance with one another (such as a second, seventh, or ninth) are 
 taken without preparation that is, if neither of them has been 
 sounded in the same voice in the preceding chord it is better 
 that they should enter by contrary than by similar motion, especially 
 in the extreme parts. 
 
 Not good. 
 
 r r ' 
 
 Good. Not good. 
 
 Good. 
 
 1 1 3. Recommendation III. When the same note occurs in two 
 successive chords, it is generally (though not invariably) better to 
 keep it in the same voice. 
 
 I 
 
 J ..fry 
 
 1 1 
 
 J. 
 
 A 
 
 
 _ 1 _ u-p- 
 
 At (a) the two chords have the note C in common. We therefore 
 keep it in the tenor voice. The hidden octaves between extreme 
 parts here are among those that are permitted ( 103). But if at 
 (b] we keep the D in the tenor, we shall get objection'ible hidden 
 octaves between the extreme parts. In this case therefore the 
 position of the second chord will be better as at (c). 
 
 114. Recommendation IV. Each voice should generally go to 
 the nearest note in the following chord : 
 
 (*) 
 
 r " i 
 
 Ji 
 
 In this example the treble note C can move equally well to D or 
 B, each being the next note of the scale. But if C moves to D
 
 chap, iv] ITS THEORY AND PRACTICE. 51 
 
 as at (b], the tenor E will have to go to B, which is not its nearest 
 note in the next chord. It is therefore better to let C go to B, as 
 at (a), which allows the tenor also to go to its nearest note. 
 
 115. Recommendation V. When the bass moves by step, 
 upwards or downwards, it is generally best to let the other parts 
 move in contrary motion to it. In some cases, as will be seen in 
 the next chapter, this is absolutely necessary to avoid consecutive 
 fifths and octaves. 
 
 1 1 6. The student is advised to thoroughly master the rules and 
 recommendations given in this chapter. If he does so, he will 
 find comparatively little difficulty in working the exercises which he 
 will shortly meet with.
 
 52 HARMONY : [Chap. v. 
 
 CHAPTER V. 
 
 THE DIATONIC TRIADS OF THE MAJOR KEY SEQUENCES. 
 
 [NOTE. As Chapters II. and III. are not intended to be studied 
 by beginners, it will be necessary in this chapter to repeat 
 some of the explanations already given.] 
 
 117. By the word "Key," as has been already explained in 
 Chapter III., is meant a collection of twelve notes within the 
 compass of an octave, all of which have a definite relationship to 
 the first note of the series, which is called the TONIC, or KEY- 
 NOTE. As there are only seven degrees of the staff within the 
 two notes of the octave, it follows that five out of these seven 
 degrees must have two notes upon them, if we are to obtain 
 twelve notes. In every key, when two notes are found in the 
 same degree of the staff, one of these will be in accordance with 
 the key-signature, and the other will have an accidental before it. 
 There will be in all seven notes in accordance with the key- 
 signature, one upon each degree of the staff, and therefore called 
 diatonic notes ( 9) ; and there will be five which will bear acci- 
 dentals, and which are therefore called chromatic notes ( 6). For 
 the present we concern ourselves only with the diatonic notes, 
 reserving till later the treatment of the chromatic. 
 
 1 1 8. A diatonic scale ( 9) is one which contains seven notes 
 within the octave, one on each degree of the staff. If the 
 interval between the tonic (12) and the mediant ( 13) is a major 
 third ( 22), the scale is called a major scale; if it be a minor 
 third, it is called a minor scale. Let the student compare the 
 intervals between the first and third notes of the major and minor 
 scales given in 9. Except with two notes of the minor scale, as 
 will be seen later, a diatonic scale never requires accidentals, but 
 conforms strictly to the key signature. 
 
 119. A CHORD is a combination of more than two notes 
 placed at the interval of a third one above another ( 58). The 
 lowest note on which a chord is built is called its root. A chord 
 may consist of any number of notes from three to seven, though it 
 is no't necessary that all should be present at once. We shall see 
 shortly that even in chords containing only three notes one is 
 often omitted ( 134). 
 
 120. A chord made by placing two thirds one above another, 
 and therefore containing only three notes, is called a TRIAD. If, 
 as is generally though not always the case, the upper note of a
 
 chap, v.] ITS THEORY AND PRACTICE. 
 
 53 
 
 triad is a perfect fifth from the root, the chord is called a COMMON 
 CHORD. Every common chord is a triad, but every triad is not a 
 common chord. If the third next above the root of a common 
 chord be a major third, the chord is called a major chord; if it be 
 a minor third above the root, it will be a minor chord. 
 
 121. If we take the diatonic scale of C major, 
 
 > -<& =! ^^^ 
 
 we shall see that the semitones fall between the third and fourth 
 and the seventh and eighth degrees of the scale. The position of 
 the semitones makes the difference between a major and a minor 
 scale, as will be seen when we come to speak of the latter. 
 
 122. If we take each note of the above scale as a root, and 
 place two thirds upon it as explained in 120, we shall get a triad 
 on each degree of the scale. As we shall use no notes except 
 those of the diatonic scale itself, we shall have only diatonic triads. 
 
 It will be seen that all these triads, except that upon the leading 
 note, contain a perfect fifth from the root, and are therefore 
 common chords. The chord upon the tonic is a major chord, and 
 the nature of this chord determines the nature of the key. A key 
 which has a major chord on its tonic is called a MAJOR KEY, and a 
 key which has a minor chord on its tonic is called a MINOR KEY. 
 The note taken as the tonic gives its name to the key; the chords 
 given above are therefore in the key of C major. Any note may 
 be chosen as a tonic ; but the relation of the other notes of the 
 scale to the tonic selected must always be the same, that is, they 
 must be at the same distance from the tonic. The intervals 
 between the tonic and the other notes of a major scale have been 
 given in 22. We have taken C as a tonic because its major 
 scale requires neither sharps nor flats. 
 
 123. By examining the chords given above, it will be found 
 that three degrees of the major scale the tonic, subdominant, and 
 dominant bear major chords above them, and three others the 
 supertonic, mediant, and submediant bear minor chords. The 
 fifth above the leading-note, instead of being, like all the others, a 
 perfect fifth, is a diminished fifth ( 25). There is therefore no 
 common chord ( 120) on the leading-note, and the chord which 
 we find instead is called the DIMINISHED TRIAD. This chord is 
 very rarely used with its root in the bass. 
 
 124. It was said in the last chapter ( 94) that most music is 
 written in four-part harmony. We shall therefore in future give 
 our examples in four parts. But to write a common chord, or a 
 triad, containing only three notes, in four parts it will evidently be 
 necessary to double one of the notes, that is to put the same note
 
 54 
 
 HARMONY : 
 
 [Chap. V. 
 
 in two of the parts, either in unison, or at the distance of an 
 octave, or even two or three octaves. It must be remembered 
 that this doubling of a note does not alter the nature of the chord. 
 Though the word " triad " literally means a combination of three 
 notes, a triad does not cease to be such, in however many parts 
 the harmony may be, unless some additional note, and not a mere 
 doubling of notes already present, be introduced into the chord. 
 We shall see presently which are the best notes to double. 
 
 125. In writing four-part harmony, the four "voice-parts," 
 soprano, alto, tenor, and bass ( 94), are kept best as far as possi- 
 ble within the compass of the voices after which they are named. 
 The general compass of each voice is about the following : 
 
 SorRANO. 
 
 ALTO. TENOR. ~^" BASS. 
 
 to - 
 
 P 
 
 These limits should be very rarely, if ever, exceeded ; and even 
 within them it is best to keep near the middle of the compass, 
 and not to use more of the extreme notes, either high or low, 
 than are absolutely needful for a good progression of the parts. 
 
 126. If in four-part harmony the three upper parts lie close 
 together, and at a distance from the bass in other words, if the 
 soprano and tenor are within an octave of one another, 
 
 the harmony is said to be in' close positi, n. If the parts lie at 
 more equal distances, and the tenor is more than an octave from 
 the treble, 
 
 the harmony is said to be in extended position. In most com- 
 positions a mixture of both positions will- be found. If the treble 
 part lies low, close position will most likely be needful, to prevent 
 the tenor part from going below its compass ; but if the treble is 
 high, extended position will generally be advisable. 
 
 127. The best position of harmony is mostly that which allows 
 the parts to lie at approximately equal distances, when this is 
 possible. At example (a) in the last paragraph, in the first chord
 
 chap, v.] ITS THEORY AND PRACTICE. 55 
 
 there is a tenth between bass and tenor, a third between tenor and 
 alto, and a fourth between alto and soprano. This position is 
 quite correct ; but the position of the same chord at (k) is prefer- 
 able ; for here there is a fifth between bass and tenor, a sixth 
 between tenor and alto, and a sixth between alto and soprano. 
 The intervals between the voices are much more equal. 
 
 128. It will sometimes happen that it is impossible to keep the 
 voices at approximately equal distances without breaking some 
 rule. If there must be a large interval between two voices, it 
 should, with very rare exceptions, be between the two lowest, the 
 tenor and the bass. Excepting occasionally for a single chord, there 
 should never be a larger interval than an octave between soprano 
 and alto, or alto and tenor. We give examples of good and bad 
 positions of the chord of C major ; the student can easily find out 
 from what has been said why each is good or bad. 
 
 Good. i ^ d so | Bad. | Very bad. 
 
 m 
 
 ~^&: 
 
 -f- 
 
 129. It will be seen that in the examples just given, the relative 
 positions of the chords vary widely ; sometimes the root, at other 
 times the third or fifth is at the top. It must be clearly under- 
 stood that the relative positions of the upper notes of a chord 
 make no difference to its nature, provided the same note of the 
 chord is in the bass. Here the root is in the bass in each instance, 
 and the chord is said to be " in its root position." But if the 
 third or fifth of the chord were in the bass, we should have 
 inversions of the chord. These will be treated of in Chapter VI. 
 
 130. With one important exception to be mentioned presently, 
 it is possible to double any of the notes of a common chord. In 
 general the root is the best note to double, though cases will be 
 frequently met with in which, for the sake of a good progression of 
 the harmony, it is expedient, or even necessary, to double one of 
 the other notes. Of the other two notes, if the chord be major, it 
 is mostly better to double the fifth than the third ; but in a minor 
 chord it is just as good, and sometimes better, to double the 
 third. 
 
 131. The reason why it is often unadvisable to double the major third of a 
 chord is that its third "upper partial " ( 36) is only a semitone below the root 
 of the chord of course an upper octave of the root. As this third upper partial 
 is usually rather strong in the compound tone of the note, its power is so increased 
 by doubling as to produce in some positions a distinctly harsh effect against the 
 root. It is nevertheless often necessary to double it, as we shall see later. 
 
 132. The exception referred to above in speaking of the doubling 
 of a note was that of the leading note. This note is the third of
 
 HARMONY 
 
 [Chap. V. 
 
 the dominant chord, and is a semitone below the tonic, toward 
 which it has the strong upward tendency which gives it its name. 
 If the dominant chord be followed by the tonic chord, the leading 
 note, especially if in the upper part, must rise. 
 
 Let the student play the chords marked (a) and (b}. He will feel 
 that the progression (a) satisfies the ear, while (b) does not do so. 
 The effect of the leading note falling is less unsatisfactory when it 
 is in a middle voice, as at (c) ; but this progression, though 
 frequently used by Bach, is not to be recommended to beginners. 
 Excepting in the repetition of a sequence ( 137), in the very rare 
 second inversion of the chord on the mediant ( 162), or when the 
 notes of the dominant chord are taken in arpeggio, that is in suc- 
 cession while the harmony remains the same, 
 
 the leading note must never be doubled ; and when the dominant 
 chord is followed by the tonic chord, the note must always rise a 
 semitone. It is evident that if the leading note is in two voices, 
 and both rise a semitone, consecutive octaves ( 96) will result. 
 
 133. The leading note is, however, free to fall when the 
 dominant chord merely changes its position, 
 
 or when it is followed by some other chord than the tonic, in 
 which case the leading note may either rise or fall.
 
 Chap. V.] 
 
 ITS THEORY AND PRACTICE. 
 
 57 
 
 At (/>) (r) the dominant chord is followed by the chord of the 
 submediant, and the leading note may either rise to the third or 
 fall to the root of the next chord, the former being slightly prefer- 
 able. At (d) (e) the dominant chord is followed by the supertonic ; 
 and either progression is possible, though (d) is much better, not 
 only because contrary motion is mostly to be preferred to similar, 
 but also because at (e) we have objectionable hidden fifths ( 105) 
 between the extreme parts. But if the bass rose from the domi- 
 nant to the supertonic, as at (/) (g), 
 
 00 , , Or) . 
 
 A 
 
 r* r^ 
 
 it would be needful for the leading note to fall, as at (g) ; for if it 
 rises, as at (/) we shall get bad " hidden octaves" ( 103) between 
 extreme parts. 
 
 134. It was said in 119 that one note of a triad was 
 sometimes omitted. This is mostly the fifth of the chord 
 very rarely the third, because the latter note shows whether the 
 chord is major or minor. But it not infrequently becomes 
 necesoary to omit the fifth, in order to secure a correct pro- 
 gression of the parts. For exa.i.ple, if we are harmonising a 
 
 melody in the key of C, . which ends 
 
 EEdl, the two 
 
 last chords must be dominant and tonic in their root positions. 
 The only correct way of harmonising these chords will be 
 
 Here the melody descends to the tonic ; the bass also goes to the 
 tonic ; and the leading note must rise to the tonic ( 132) ; there 
 is therefore only one part, the alto, remaining ; and as the third 
 is needed to fix the nature of the chord, this note is taken, and the 
 fifth omitted.* Occasionally also, though much more rarely than 
 the fifth, the root of the chord is the note omitted in inversions. 
 
 * In old music, especially at the close of a piece in a minor key, the third is 
 occasionally om.tted, as at the end of the " Kyrie " in Mozart's " Requiem " ; but 
 this is exceptional, as we shall see when treating of the minor key.
 
 58 HARMONY : [Chap. v. 
 
 135. Common chords taken in their root position have an 
 effect of greater firmness and strength than when taken in the 
 inversions to be treated of in the next chapter. But if only root 
 positions were used, there would be little variety in the harmony. 
 It is therefore somewhat rare to find an entire passage containing 
 these alone. We give a short one from Mozart. 
 
 MOZART. " Die Zauberflote." 
 
 The voice parts are omitted, as they are in unison with the extreme 
 parts of the harmony. 
 
 136. This short passage is full of instructiveness for the student, 
 and should be carefully examined. Let it be first noticed that 
 within the space of two bars we find all the diatonic common 
 chords of the major key, in each case with the root in the bass. 
 Next let it be observed that the tenor doubles the soprano in the 
 octave through the whole passage. This illustrates what was said 
 in the last chapter ( 98). The law against consecutive octaves is 
 not broken by such a progression as this, though it would be if 
 only two or three notes of the tenor, instead of all, moved in 
 octaves with the soprano. Though there are four parts moving 
 here, the passage is not in four-part harmony, but in three-part 
 harmony with one part doubled. It will be seen further that in 
 every alternate chord the fifth is omitted. This is because the 
 second and third pairs of chords are exact copies of the first pair, 
 but on different parts of the scale. 
 
 137. A passage constructed after a regular pattern of this 
 kind is called a SEQUENCE. When any progression of the bass is 
 repeated on higher or lower degrees of the scale, it should be 
 accompanied by harmony which is also exactly repeated at cor- 
 responding higher or lower degrees. In the passage we are now 
 examining, the bass of the third and fourth chords is the same 
 as that of the first and second, a third lower, and each of the 
 other parts is a third lower than it was at first. The chords are in 
 corresponding positions, in every respect. In the first chord the 
 third is at the top, and in the second the fifth ; the third and 
 fourth chords have precisely the same arrangement, and so have 
 the fifth and sixth. 
 
 138. The pattern set for a sequence may be imitated at any 
 interval above or below. Practically, however, the limit of a third 
 is seldom if ever exceeded, as imitation at a larger interval would 
 soon carry us beyond the range of the voices, e.g., 
 
 
 
 
 n 
 
 
 -(2- 
 
 -\ 
 
 
 
 
 
 
 

 
 Chap V.] 
 
 ITS THEORY AND PRACTICE. 
 
 59 
 
 A pattern may consist of any number of notes ; generally there 
 are two, or at most three. We will now give a simple sequence, 
 and the student will have no difficulty in understanding it. 
 
 ^ 'j. A A 4 t 4 ^ tj 
 
 = 
 
 Here it will be seen that all the parts in the second bar are 
 one note higher than in the first, in the third bar one note higher 
 than in the second, and so on to the end. All the intervals in 
 the chords remain the same as to their names, but differ in quality 
 according to their position in the scale. Thus the first chord of 
 bar i is major, of bars 2 and 3 minor, and of 4 and 5 major 
 again. This is because the music remains in one key. Such a 
 sequence is termed a tonal sequence. 
 
 139. If on the other hand the quality of the intervals is 
 exactly the same in the imitations as in the pattern, the sequence 
 will be real, that is exact ; but the music will not remain in the 
 same key. 
 
 A real sequence is much rarer than a tonal one. 
 
 140. At the fourth bar of the sequence given in 138, the 
 second chord, marked with (*), requires special notice. It will be 
 seen that we have here the diminished triad in its root position 
 ( 123). It will also be noticed that the leading note is doubled 
 ( 132), and that between the two notes of the bass in this bar is 
 seen the unmelodic interval of the augmented fourth ( 91). 
 These departures from the "rules already given are only justified 
 by the fact that they occur in one of the repetitions of a sequence. 
 Had the fourth bar of the example been given as a pattern it 
 would have been incorrect, and the doubled leading note would 
 not have been allowed.* 
 
 141. The sequence we are examining might have been 
 
 * It would have been possible in this passage to have had B? instead of B|j in 
 the fourth bar, making a transition for a moment into the key of F. The pro- 
 gression would then have been analogous to the well-known sequence in Handel s 
 chorus, "The horse and his rider," in " Israel in Egypt," quoted in Macfarren's 
 " Six Lectures on Harmony," p. 60.
 
 6o 
 
 HARMONY : 
 
 [Ch?p. V. 
 
 harmonised differently. We might, for instanc'e, have put the 
 first chord in a different position, still retaining the second as we 
 have given it. 
 
 We preferred to give at first a position which allowed strict 
 observance of the recommendations given in Chapter IV. at 
 113, 114. But the effect of the sequence will be better if 
 carried out after the pattern last indicated ; because as a general 
 rule contrary motion is preferable to similar. 
 
 142. It will in many sequences be found impossible to attend 
 to the recommendations just referred to. For example, suppose 
 we work a sequence on this pattern, 
 
 if after the first bar we attempt to keep the two notes F and A of 
 the chord of D in the same voices in which they were in the 
 preceding chord, as at (#) below, it is clear that the second bar 
 will not correspond to the pattern set by the first, and there will 
 be no sequence. It will therefore be necessary to arrange the 
 upper parts as at (b). 
 
 | , | -, , 
 
 143. The student can now begin to write simple exercises or 
 common chords. He is advised to write them in what is called 
 " short score " that is, on two staves : the treble and alto on 
 the upper, and the tenor and bass on the lower, staff. In order 
 to distinguish clearly the progression of the parts, all the notes of 
 the treble and tenor voices should have their stems turned 
 upwards, and the alto and bass should have their stems down- 
 wards. If two parts written on the same staff are in unison, this 
 is shown by the note having two stems, one upward and the 
 
 other downward 
 two notes side by 
 
 or, in case of a semibreve, by putting 
 side (o-z>). If, however, the student is 
 acquainted with the C clefs (which it is most desirable that he 
 should be), we strongly advise him as soon as possible to bepin
 
 Chap. V.] 
 
 ITS THEORY AND PRACTICE. 
 
 61 
 
 writing his exercises in open score, giving a separate staff to 
 each voice thus : 
 
 This is the first bar of the exercise worked in 145 written in 
 open score. 
 
 144. That the manner in which the exercises are to be done 
 may be clearly understood, we will work a short passage as a 
 model for the student, explaining the progression of every note in 
 each chord, and giving the reasons for the way in which one 
 follows another. Instead of quoting the rules, we shall, to save 
 space, simply refer to the paragraphs in which they are given. 
 We will choose the following simple -bass, numbering each note for 
 the convenience of reference. 
 
 m 
 
 
 12 34 56 7 
 
 145. The position of the first chord is optional ; we may have 
 either the third, the fifth, or the octave of the root at the top. 
 The position of the following chords will depend upon the position 
 of the first. Let us take the chord in the most usual position, 
 with the octave of the root at the top. It has been said ( 130) 
 that the root is generally the best note to double. The first 
 chord will then be in this position 
 
 (> i 
 
 Chord 2 will contain the notes G, B, D. Of these, G was in the 
 first chord; we therefore keep it in the alto ( 113), and put B in 
 the treble and D in the tenor, these being the nearest notes 
 to those which they had in the first chord ( 114). The pro- 
 gression will therefore be 
 
 Between chords 2 and 3 the bass moves by step. The third
 
 62 
 
 HARMONY: 
 
 [Chap. V. 
 
 chord contains the notes A, C, E. Of these, the nearest note 
 to the G of the alto in chord 2 is A ; but if we take A in the 
 alto, C in the treble, and E in the tenor, we shall have consecutive 
 perfect fifths and octaves with the bass. 
 
 Here therefore the recommendation given in 115 comes into 
 force. We can either make all the upper parts move downwards 
 (i.e. in contrary motion to the bass) as at (d\ or take only the 
 octave and fifth of the second chord down, and let the leading- 
 note rise, as at (<?). 
 
 23 23 
 
 Either progression is correct ; but as it is generally best to let the 
 leading note rise when it can, we shall choose (e) ; in this chord it 
 will be seen we have doubled the third instead of the root : note 
 that it is a minor third ( 130). 
 
 146. The chord of F (No. 4) consists of the notes F, A, C. 
 )f these, C is seen both in the treble and tenor of chord 3. In 
 which of the two voices shall we retain it ? If we keep it in the 
 treble, as at (a), it will be seen that the alto and tenor voices over- 
 lap^ ( in) ; we shall therefore keep it in the tenor ; the alto will 
 mo\e to its nearest note, F ; and the treble will fall to A. 
 () i i (*> i . 
 
 34 34 
 
 The fourth and fifth chords have two notes, A and F, in common ; 
 here therefore there is no difficulty ; both remain in the same 
 voices in which they were ; and C in the tenor goes to D, in con- 
 trary motion to the bass. If it came down to A, doubling the fifth 
 of the chord, we should evidently get consecutive fifths. 
 
 4 5 
 
 We can put the sixth chord (G, B, D) in one of two positions.
 
 Chap. V.] 
 
 Irs THEORY AND PRACTICE. 
 
 We can either take the three upper parts in contrary motion to the 
 bass, as at (d}, or we can keep the D of the chord in the same 
 voice (the tenor), which had it in the last chord, as at (e) 
 
 (d) (e) , 
 
 56 56 
 
 In this case the latter is preferable, because it is best, if possible, to 
 finish a piece with the octave of the root in the highest voice ; and 
 if we take position (d} the leading note in the tenor must rise to C 
 (Ji 132), the alto must rise to E, and the treble continue on G. 
 But after position (e) the progression of the last two chords will be 
 as at 00. 
 
 t gF 
 
 147- We now give the whole passage together ; and the student 
 is advised to take the bass in 144, and try to work the exercise in 
 different positions. If he has followed the explanations just given, 
 he will find little difficulty in varying the positions of the chords. 
 
 ' - 1 1 I 1 , i 1- 
 
 F 
 
 ( 2 
 
 
 J_A 
 
 1 2.8 4 5 6 7 
 
 148. All the examples hitherto given have been in the key of 
 C. This has been intentional, as affording a simpler means of 
 comparing one chord with another ; but it is important that the 
 student should be able to write with equal ease in any key. The 
 exercises now to be given will therefore be in various keys. In 
 the earlier ones, the treble as well as the bass will be given, in 
 order to simplify the labour of the beginner; but in the later 
 exercises the student must supply all the upper parts. 
 
 EXERCISES TO CHAPTER V. 
 
 Treble and bass given. Add alto and tenor parts. 
 _(i.) 
 
 rffra J j i ! i \-f~ i i Ji 
 
 i
 
 6 4 
 
 HARMONY : 
 
 [Chap. V. 
 
 (II.) 
 
 (III.) 
 
 '_ J , J ' 
 
 Bass only given. Add the three upper parts. 
 
 (V.) 
 
 (VI.) 
 
 =fr|=g 
 
 (VIL) 
 
 Etz 
 
 | ^ |'-gy T 
 
 B^EB 
 
 e* 
 
 (VIII.) 
 
 (IX.) 
 
 
 IT 1-4 
 
 [We give, as the last exercise, the bass for a Double Chant. The 
 student must endeavour to make the melody in the upper part as 
 interesting as he can.] 
 
 (XI.)
 
 chap, vi.] Irs THEORY AND PRACTICE. 65 
 
 CHAPTER VI. 
 
 THE INVERSIONS OF THE TRIADS OF A MAJOR KEY. 
 
 149. Hitherto all the chords we have used have been in their 
 root positions, but in order to obtain greater variety of harmony, 
 they can be, and frequently are, inverted. The inversion of an 
 interval was defined ( 26) as placing the lower note above the 
 upper, or the upper below the lower. But it is possible to invert 
 some of the intervals of a chord without inverting the chord itself. 
 Thus if we compare the two positions 
 
 (a) and (b) of the chord of C, we shall see that at (a) E is a third 
 below G, and at (b) the relative position of the two notes is 
 changed, and E is a sixth above G the inversion of a third. 
 But the chord itself is not inverted, for the root, C, is in the bass 
 in both cases. 
 
 150. A chord is said to be inverted when any other note than the 
 root is placed in the bass. A moment's thought will show the 
 student that the number of inversions of which any chord is 
 susceptible must be one less than the number of notes which it 
 contains. Thus a triad, which consists of three notes, has two 
 inversions, because it contains two notes besides its root, and 
 either of these notes can be placed in the bass. Similarly a chord 
 of the seventh, because it contains four notes, has three inversions, 
 and so on. If the third of the chord is in the bass, we have the 
 first inversion, if the fifth is in the bass, the second inversion, &c. 
 the inversions being always reckoned according to the number in 
 the series of thirds above the root of the note which is in the 
 bass. 
 
 151. The first inversion of a chord, as we have just said, has 
 the third in the bass. As in the root position, it makes no differ- 
 ence to the nature of the chord which of the other notes stands 
 next above the third. 
 
 Both (a) and (b) are equally first inversions of the chord of C. 
 The root, which was before a third below E, is now inverted with 
 respect to that note, and is a sixth above it ; but the fifth of the 
 
 E
 
 66 HARMONY : [Chap. vi. 
 
 chord, G, which in the root position was a third above E, is still a 
 third above it. The first inversion of a triad is therefore called 
 the chord of the sixth and third, or, more usually, the chord of the 
 sixth, the third being always implied when the shorter name is 
 used. 
 
 152. In order that it may be known, when only the bass is 
 given, what harmony is to be placed above it that is, whether the 
 bass note is the root of a chord, or one of the other notes, and there- 
 fore whether we have an inversion, we use what was formerly called 
 Thorough Bass, but is now more usually spoken of as Figured Bass. 
 This is a kind of musical short-hand, which indicates, by one or 
 more figures placed under (or sometimes over) the bass note, the 
 nature of the chord of which the bass is given. A common chord 
 in its root position, as it contains the fifth and third of the bass 
 
 note would be figured 3 the larger figure, when there is more 
 
 O 
 
 than one, being almost invariably placed at the top. But, as a 
 matter of actual practice, the common chord in its root position is 
 not figured at all, except when one of its notes requires an acci- 
 dental before it, or when two or more chords one of which is a 
 common chord, are to follow one another on the same bass note. 
 (Seel 165.) 
 
 153. As a first inversion has the intervals of a third and sixth 
 
 above the bass note, its full figuring would be ~- But just as we 
 
 speak of it merely as a "chord of the sixth," we mostly figure it only 
 with a 6. In a chord of the sixth the third is always implied, but it is 
 not specially marked, excepting when it requires an accidental, and 
 in that case the accidental 4 or I? is written without the figure 3. 
 Att accidental without a figure at its side always refers to the tJiirtJ 
 of the bass note. But if the sixth required an accidental this would 
 have to be written thus $6 or I? 6, sometimes also 6$ or 6."'. A 
 sharpened note is frequently indicated by a stroke drawn through 
 the figure thus "6. The following examples will make this clear 
 
 Figured. 6 J?6 (or "-&-.) JG 6 6 b6 
 
 3 8 b E 
 
 154. All the triads given in 122 can be used in their first 
 inversion ; and we shall thus have a chord of the sixth on every 
 degree of the scale. 
 
 * 
 
 In four-part harmony one of these notes must be doubled, just as 
 in the root position ; the general rules with regard to doubling 
 given in the last chapter ( 130) apply also here. 
 
 155. At * in the last example will be seen the first inversion
 
 Chap. VI.] 
 
 Irs THEORY AND PRACTICE. 
 
 of the diminished triad on the leading note. Though this chord 
 is very rarely used in its root position ($ 123, 140} it is very often 
 to be met with in its first inversion. Here, though the F and B 
 are still dissonant to one another, they produce a much less harsh 
 effect than in their root position, because now both the notes are 
 consonant to the bass. Greater freedom is therefore allowed in 
 their use and in their progression. 
 
 156. Provided that the general laws of part-writing given in 
 Chapter IV. are observed, there are no special restrictions as to the 
 treatment of first inversions when they occur singly, that is between 
 other positions of chords. But when two or more first inversions 
 follow one another, a little care is needed in their management. 
 If, for instance, in the series of sixths given in 154 we place the 
 fifth of each chord in the upper part, 
 
 it is clear that we shall have a series of consecutive fifths with the 
 roots. Hence we get the following important rule : When two or 
 more first inversions follow one another, the sixth from the bass 
 should be in the upper part if the bass mores by stet>* The fifth 
 will then be a fourth below the root, and consecutive fourths are 
 not forbidden between upper parts ( 106.) 
 
 157. Another po : nt to be attended to in a series of sixths is 
 the doubling of one of the notes of the chords. If either the root, 
 the third, or the fifth be doubled in each chord, 
 
 ! i <*> i 
 
 -g ---- 
 
 IMP 
 
 
 it is clear that there will be consecutive octaves, sometimes also, as 
 at (a) consecutive fifths. From this we deduce another rule : 
 The same note of the chord must not be doubled in two successive first 
 inversions. The usual plan adopted is to double the root and the 
 third of the chord alternately, as in the following example from 
 Mozart, 
 
 MOZART. Mass, No. 15. 
 
 4 J_^LJ!!--B^BI5_* 
 
 
 r 
 
 #-#- 
 
 care being of course taken not to double the leading note ( 13? 
 but the fifth may also sometimes be doubled.
 
 68 HARMONY : [chap, vi. 
 
 158. The second inversion of a chord has the fifth in the bass. 
 
 In this chord both root and third are above the fifth, and are 
 therefore inverted with respect to it. The root is now a fourth 
 and the. third is a sixth above the bass note. This inversion 
 is therefore called the chord of the sixth and fourth, or more 
 commonly "the chord of six~four." It is figured J. The lower 
 figure cannot be omitted, or there would be no means of dis- 
 tinguishing between this chord and the first inversion. 
 
 159. If the student will play this chord on the piano, he will 
 notice that its effect is much more unsatisfying than that of either 
 the root position or the first inversion ; it requires, in fact, to be 
 followed by some other chord, such as a common chord in its 
 root position on the same bass note. This is because a fourth 
 with the bass (though not between two upper parts) produces the 
 effect of a dissonance. For this reason the second inversions of 
 chords require special treatment, as will be seen directly. 
 
 1 60. Any triad may be taken in its second inversion. Some 
 theorists state that only the tonic, dominant, and subdominant 
 chords may be so used ; but this is erroneous, as will appear from 
 the following examples from the works of the great masters, in 
 which the second inversion of a chord on every degree of the 
 scale is given : 
 
 HAYDN. "Creation.
 
 Chap. VI.] 
 
 ITS THEORY AND PRACTICE. 
 
 MENDELSSOHN. March, " Athalie.' 
 (<O * 
 
 <*) 
 
 BACH. " Dazu ist erschienen. ' 
 
 JLLA ^_-i 
 
 In each example the . chord is marked with an asterisk. 
 
 161. All these extracts deserve a little study; and an examina- 
 tion of them will help the student to understand the rules we 
 shall presently give for the treatment of the chord. At (a) is seen 
 the second inversion of the tonic chord followed by another chord 
 on the same bass note. This is by far the commonest way of 
 using a ". and as it mostly occurs at the end of a phrase it may be 
 
 called a " cadential ^ " At (b} we find the same chord, differently 
 followed, viz., by a chord on the next degree of the scale. 
 
 162. The second inversions of chords on the supertonic and 
 mediant are very rare. At (c) is the supertonic chord in the
 
 jo HARMONY : [Chap. vi. 
 
 second inversion ; the bass, as usual, moves by step, and the chord 
 is approached from another position of the same harmony. The 
 second inversion of the mediant chord, as at (d), is probably 
 even rarer. The subdominant chord is tolerably common, 
 especially when used as at (e). Here it is cadential, like the tonic 
 chord at (a). The dominant in the second inversion is also 
 frequent, as at (/) ; but this chord cannot be used like those of 
 the tonic and subdominant cadentially. 
 
 163. The second inversion of the submediant chord .(g) is 
 another which is seldom met with j but the student can find 
 another example at the ninth bar of the first movement of 
 Beethoven's sonata in E minor, Op. 90. In the passage given from 
 
 Mendelssohn there will also be seen a cadential ^ of the tonic 
 chord. The second inversion of the diminished triad on the 
 leading note (//) is only very rarely to be found in four-part 
 harmony. Our extract is from a Church Cantata by Bach, and it 
 gives an illustration of the freedom of his part-writing. In the 
 second bar will be seen consecutive fifths between the treble and 
 tenor. But here the treble note ( A? ) is not a note of the harmony, 
 but an anticipation of a note of the next chord, as will be 
 explained later ( 263). The student is advised not to imitate 
 such licences until he has acquired the skill in part-writing which 
 Bach possessed. The progression of the alto at the end of this 
 example illustrates Bach's partiality (referred to in 132) for 
 taking the leading note down to the fifth of a chord in a cadence. 
 
 164. In consequence of the dissonant effect of the fourth with 
 the bass, it is necessary to be careful in approaching, as well as in 
 
 quitting a " chord. The rules for approaching this chord, which 
 
 are deduced from the practice of the great masters, and which will 
 be all illustrated by the examples given in 160, are the follow- 
 ing: 
 
 I. A second inversion may be approached either by leap, as at 
 (a), or by step, as at (d] (g), from the root position of another 
 chord. 
 
 II. It may be approached by leap, as at (c) (e), from another 
 position of the same chord. 
 
 III. It may be approached by step (but not by leap) from the 
 inversion of another chord, as at (b} (/) (/z). 
 
 IV. It may be preceded by a different chord upon the same 
 bass note, as in the passage from Beethoven's Op. 90, mentioned 
 in 163. See also example from the "Sicilian Mariners' Hymn " 
 in the next paragraph. 
 
 165. No less important than the rules just given are those 
 which regulate the leaving a second inversion. These rules are 
 three in number 
 
 I. The bass of any second inversion may move by step of a tone,
 
 Chap. VI.J 
 
 Irs THEORY AND PRACTICE. 
 
 as at (/>) (f) (/), of a diatonic semitone, as at (g) (//), or of a 
 chromatic semitone, as at (d), upwards or downwards ; and the 
 following chord may be either a chord in root position, as at (t) 
 (/), or an inversion, as at (b) (if) (g) (ti). 
 
 II. The second inversions of the tonic and subdominant 
 chords may be followed by another chord on the same bass note, 
 or its octave, as at (a) (e). In such cases it will be needful to 
 figure the common chord. ( 152.) Thus the last bass note of 
 (a) would be figured 1= ^ . This is what has been 
 
 ~~" 
 
 described as a cadential 6 and in this case the 6 ma y not be on 
 a weaker accent than the chord which follows it. This rule, how- 
 ever, does not apply when the J? chord has itself been preceded by 
 
 another chord on the same bass note, as in the well-known 
 " Sicilian Mariners' Hymn," 
 
 J J 
 
 where the chords are on the second half of the bar and the 
 
 following 5 on the first beat. When the bass moves, as in rule L, 
 
 3 , 
 
 there is no restriction as to the position in the bar of the 9 chord. 
 
 III. The bass of a second inversion may leap to another note 
 of the same chord while the harmony remains unchanged ; but it 
 is not allowed to leap in any other case.* 
 
 1 66. In the second inversion of any chord, the best note to 
 double is not the root, as in the other positions of a chord, because 
 here the root, being a fourth above the bass, is the dissonant note. 
 The fifth of the chord is the best note to double, being of course 
 the octave of the bass note. It will be seen that in all the 
 examples (a) to (f) of 160, the bass note is the one doubled. 
 
 167. The rules just given for the treatment of the ^ chord, 
 
 * An apparent, but not real, exception to this rule is when the bass of a ^ chord 
 rises a seventh to the octave of the note below ; e.g. 
 
 instead of
 
 HARMONY : 
 
 [Chap. VI. 
 
 being necessarily somewhat minute, will probably give beginners 
 a little trouble. It is however strongly recommended that they 
 should be thoroughly mastered, for there is no point on which 
 students are more apt to go wrong, or with regard to which the 
 results of mistakes are more disastrous. 
 
 EXERCISES TO CHAPTER VI. 
 Treble and Bass given. Add alto and tenor parts. 
 
 Bass only given. Add the three upper parts. 
 (in.) 
 
 
 
 
 C 6 
 
 C 6 
 
 (IV.) 
 
 (V.) 
 
 6 65 
 
 4 3 
 
 
 j \ 
 
 (VI.) 
 
 6666 
 4 4 
 
 c 6 5 
 4 8 
 
 6 C 6 6 
 4 
 
 6 6 
 4 
 
 'F^ & es,.. 
 
 6 6 
 
 665 
 4 3 
 
 bass. 
 
 (a) The student must be careful here not to make consecutive 4ths with \\w.
 
 Chap. VI.] 
 (VII.) 
 
 ITS THEORY AND PRACTICE. 
 
 73 
 
 6 5 
 4 3 
 
 6 65 
 
 4 3 
 
 (VIII.) 
 
 6 5 ( 
 
 4 3 
 
 (IX.) Double Chant. 
 
 6 6 
 
 (J 6 6666 
 
 (X.) Hymn Tune (Common Metre). 
 
 6 5 
 4 3 
 
 6 6 
 
 6 6 
 
 6 5 
 4 3
 
 74 HARMONY : [Chap. vn. 
 
 CHAPTER VII. 
 
 THE MINOR KEY. 
 
 168. There is hardly any point connected with harmony the scientific explana- 
 tion of which has given so much trouble to theorists as the nature of the minor 
 key. The difficulty has arisen mainly from the fact that the tonic chord 
 contains a note (the minor third) which is not a part of the compound tone 
 ( 40) of the fundamental. Day in his " Treatise on Harmony " says "The 
 real fact is, that any minor key is an arbitrary, not a natural, change of the 
 major third and sixth of the scale into the minor." Sir Frederick Ouseley 
 takes a different view, and gives the minor third, not as an arbitrary alteration, 
 but as the igth note of the harmonic series (see 36). This certainly appears 
 a more satisfactory view ; but it must be observed with regard to it that (like 
 the I7th harmonic, which gives the interval of the minor ninth) the igth 
 harmonic is produced too high up in the series to be regarded as natural in the 
 same sense in which this word is used with regard to the perfect fifth and major 
 third ; and no generator ever gives a minor third from itself as part of its 
 compound tone instead of a major third. Here therefore the principle of 
 aesthetic selection referred to in 42, 51, comes into play. The only minor 
 third in the compound tone of any note is that to be found between the fifth 
 and sixth partial tones. Helmholtz says that the modern minor mode is a 
 fusion of three of the old ecclesiastical modes, the Dorian, /Eolian, and 
 Phrygian (" Sensations of Tone," p. 381). 
 
 169. In a note to 79 it was pointed out that if we com- 
 mence a scale on some other note than C, and use no flats or 
 sharps, we shall obtain various forms of the old ecclesiastical 
 scales. If we begin with the note A, we shall have the " 
 scale. 
 
 i 
 
 If the student will compare this scale with the minor scale given 
 in Chapter I., 9, and will notice the intervals between each note, 
 he will see that the only note in the JEolian scale which differs 
 from the modern minor is the seventh. In fact the Eolian scale, 
 like several other of the older scales, has no leading note, the 
 seventh being a tone instead of a semitone below the tonic. Such 
 a scale is to be found in some old national airs. As modern music 
 developed, the feeling of the necessity for a leading note became 
 stronger. It was found that a chord on the dominant with a 
 minor third produced a most unpleasant effect before the tonic,
 
 Chap. VII. 
 
 ITS THEORY AND PRACTICE. 
 
 75 
 
 Here, with our modern feeling for music, the effect of the fourth 
 chord is simply detestable. But if the passage is played again 
 substituting GJ for G| the ear is satisfied at once. This alteration 
 of the seventh note changes the ^Eolian scale given above. into 
 our modern minor scale. 
 
 170. As a matter of history, however, this result was not 
 reached by a leap. The old music was mostly written for voices, 
 and augmented intervals were forbidden, as being difficult to sing. 
 But from Ft] to G% in the now altered scale is an interval of 
 an augmented second. This not being allowed, and the G$ being 
 felt indispensable, it became necessary also to sharpen the F. 
 This gave another form of the minor scale, 
 
 which is still in use for melodic, though (as will be seen 
 later) rarely for harmonic purposes. As the necessity of the 
 interval of a semitone between the seventh of the scale and the 
 tonic is not felt in descending, the form of scale last given is 
 chiefly used in ascending passages, the older ^Eolian form given 
 in 169 being generally retained in descending. This will be 
 seen in the two following scales, taken from the finale of Mozart's 
 concerto in C minor. 
 
 Concerto in C minor. 
 
 Occasionally the form of the scale with the sharpened sixth and 
 seventh is found also in descending passages, as in the following 
 passage from Handel's 7th Suite de Pieces. 
 
 HANDEL. Suite de Pieces, No. 7. 
 
 . :| __ 
 
 171. The form of minor scale most used in modern music is 
 that described at the end of 169, which is obtained from the 
 ^Lolian scale by sharpening the seventh note, and which has the 
 interval of an augmented 2nd between its sixth and seventh 
 notes. 
 
 It will be seen that this form of the scale has three semitones ; 
 viz., between the 2nd and 3rd, the 5th and 6th, and the yth and
 
 76 HARMONY : [Chap. vn. 
 
 8th degrees. The point in which all three forms of the minor 
 scale agree is that they all have a semitone between the 2nd and 
 3rd degrees, and therefore a minor third between tonic and 
 mediant. ( 9.) This last given form of scale is very common in 
 instrumental music, and not infrequent in modern vocal music. 
 We give one example 
 
 BEETHOVEN. Sonata, Op. 2, No. i. 
 
 iqs-ii.::-]-- 
 
 SE 
 
 172. As all the diatonic chords of the minor key are made with 
 notes of the scale last given, this form is known as the Harmonic 
 Minor Scale, the other two being called Melodic Minor Scales.* 
 The leading note of every minor scale is always written as an 
 accidental, because in the key-signature (as was shown in 82, 
 85) the sharps or flats are always placed in the same order. If 
 the leading note of A minor were indicated in the signature it 
 
 would be written thus ghrg=EE , and the signature of C minor 
 would be gpFI^^ . This would be extremely confusing, 
 
 especially as the sixth and seventh notes of the minor scale are 
 so frequently met with in two forms, and would in any case often 
 require an accidental whichever way they were written. It is 
 therefore more convenient to write the leading note with an 
 accidental; but it must be particularly noticed that this note is 
 nevertheless considered as a diatonic note in the key, and not as 
 a chromatic note ( 72, 117). 
 
 173. If we compare the harmonic form of the scale of A minor 
 just given with the scale of C major ( 9) it will be seen that six of 
 its seven notes are the same. Only G is different ; and as this 
 note is to be written as an accidental whenever it occurs, it is clear 
 that the keys of C major and A minor will have the same signature. 
 A major and a minor scale which have six notes in common will 
 always have the same key-signature ; for this reason, and also 
 because there is no other minor scale which possesses so many 
 notes in common with a major one, such keys are called RELATIVE 
 major and minor keys. The tonic of the relative minor of any 
 major key is always a minor third Mow the tonic of that major ; 
 and conversely the tonic of the relative major of a minor key is 
 always a minor third above the tonic of that minor. The major 
 tonics have been given in 82, 85 ; their relative minors are the 
 following : 
 
 * Some writers call these forms of the minor scale "arbitrary" an unfor- 
 tunate word, as they are really older forms, and by no means merely arbitrary 
 alterations.
 
 Chap VII.] 
 
 ITS THEORY AND PRACTICE. 
 
 77 
 
 Major 
 keys. 
 
 C 
 
 G 
 D 
 A 
 E 
 B 
 
 Signature. 
 
 Relative 
 minor keys. 
 
 A 
 
 E 
 B 
 
 Ff 
 
 G # 
 
 A* 
 
 Major 
 keys. 
 
 F 
 Bb 
 
 Ab 
 Db 
 
 Gb 
 
 cb 
 
 Signature. 
 
 Relative 
 minor keys. 
 
 D 
 
 C 
 F 
 
 Bb 
 
 Eb 
 
 Ab 
 
 174. It will be seen from the above table that what are called the " extreme 
 keys," that is. the keys with many flats or sharps, have enharmonic equivalent 
 ( 52). Two notes are said to be " enharmonic " of one another when they 
 are represented by the same sound on instruments, such as the piano, which 
 have only twelve notes in the octave. As a matter of fact Gjf and Aj?. if both 
 were played exactly in tune would not be the same ( 52) ; but the difference 
 in "pitch is so small that one note is used for both. Most music is written on 
 what is called the tempered scale, that is, a scale in which every interval 
 excepting the octave is slightly out of tune, but so little that it will serve to 
 represent equally well two or three notes very slightly differing in pitch. Thus 
 the three notes 
 
 are all represented by the same sound on the piano. We cannot discuss at 
 length the question of temperament, the necessity for which arises from the 
 fact that the octave does not contain exactly twelve semitones ; all that the 
 student at present needs to know is the meaning of the term " enharmonic," 
 the practical application of which will be seen later. f It will be seen that the 
 keys of BH and C?, of F and Gb, and of C4 and D? are enharmonics of one 
 another. (See 83, 85.) 
 
 175. A major and a minor scale which begin upon the same 
 key-note are called Tonic Major and Minor scales, and the keys to 
 which these scales belong are called Tonic Major and Minor keys. 
 By examining the table in 172, the student will see that every 
 major scale has three sharps more, or (which comes to the same 
 thing) three flats fewer, than its tonic minor ; and conversely that 
 every minor key has three flats more or three sharps fewer than its 
 tonic major. It is obvious that taking away a sharp and putting
 
 78 HARMONY : [Chap. vn. 
 
 on a flat produce the same result that of lowering a note by a 
 semitone. 
 
 176. As regards the harmonic basis of the keys, there is no connection what- 
 ever between any major key and its relative minor, while there is a very close 
 connection between a major key and its tonic minor. The key of C minor, for 
 instance, is derived from the three generators C, G, D, exactly like C major 
 (. 71), but a different selection of the harmonics is made for the diatonic notes 
 of the scale. As the tonic is the foundation of the key, it would be hardly 
 reasonable to use for the third of its chord so remote an upper-partial tone as 
 the minor ninth of D (the I53rd harmonic of C) ; we therefore take for this 
 chord on lv the igth harmonic , EJ? (see table $ 36) instead of the 5th harmonic, 
 Et|. It has been already said ( 168) that this is a matter of choice, not of 
 necessity. In all the other harmonic combinations of the minor key i.e., in 
 the dominant and supertonic discords to be treated later, we take the same El?, 
 (the 1 7th harmonic of D) as in the major key. As we have the same three 
 generators, it is evident that the chromatic scale ( 73) will be the same as in C 
 major ; but in selecting our notes for the diatonic scale we take a different 
 series. P'rom the table of relative scales we see that the signature of C minor 
 contains the three flats B, E, and A. Of these B flat is raised to Btj for the 
 leading-note; but E|? and Ap are always diatonic in the key. At] and BJ? 
 are also, under certain circumstances to be later explained, treated as diatonic 
 in the key ; but for the present we take only the harmonic form of the scale. 
 E? is, in the chord of the tonic its igth harmonic, in other combinations the 
 1 7th harmonic of the supertonic ; while A? is the iyth harmonic of the 
 dominant. 
 
 177. In 75> 76> we showed that every triad in a major key was found 
 among the fundamental harmonies of the three generators in that key. The 
 same is true of the triads of the minor key, which, as we shall see in the next 
 chapter, differ in several respects from those which we have met with in the 
 major. We conclude this chapter by showing the origin of each diatonic triad. 
 
 The ionic chord (C, E|?, G) consists of the fundamental tone, minor third, 
 harmonic), and fifth of the tonic. 
 
 The stipertonic chord (D, F, A?) consists of the fifth, seventh, and minor 
 ninth of the dominant. 
 
 The mediant chvrd (E?, G, Bh) is the minor thirteenth, fundamental tone, 
 and major third of the dominant.. 
 
 The suMominant chord (F, A?, C) is the seventh, minor ninth, and eleventh 
 of the dominant. 
 
 The domin n' chord (G, BQ, D) is the fundamental tone, major third, and 
 fifth of the dominant. 
 
 The submediant- chord (A)?, C, E|?) is the minor ninth, eleventh, and minor 
 thirteenth of the dominant. 
 
 The diminished triad on the leading note (Bu, D, F) is the major third, fifth, 
 and seventh of the dominant. 
 
 178. It will be seen that of all these chords only two, those on the domi- 
 nant and the leading note, are identical with the diatonic chords of C major 
 given in 75. This is because they are the only two chords which do not 
 contain either the third or the sixth of the scale, the two notes which 
 differentiate the major and minor key.
 
 chap, viii.] Irs THEORY AXD PRACTICE. 79 
 
 CHAPTER VIII. 
 
 THE DIATONIC TRIADS OF A MINOR KEY AND THEIR INVERSIONS. 
 
 179. In the last chapter we showed the various forms of the 
 minor scale, and it was said that the diatonic harmonies of the key 
 were made from that form of the scale given in 171. This 
 statement may be accepted as correct in a general sense, though 
 exceptionally, and especially in older music, chords will be found 
 containing the major sixth of the scale ; and in one case, as will 
 be presently seen, the minor seventh can be used as part of a 
 common chord. But it is none the less true that all the diatonic 
 chords of the minor key in general use are composed of notes of 
 the harmonic minor scale. 
 
 180. As in the major key ( 122), each note of the minor 
 scale can be taken as a root, and a diatonic triad placed above it. 
 But if we do this, we shall find that the series of chords we shall 
 obtain will differ widely from those on the corresponding degrees 
 of the major scale. 
 
 181. If this series be compared with that of the major key 
 ( 122), it will be seen that the only chords which are identical 
 are those of the dominant and leading note. Besides this, while 
 the major key has six common chords, the minor has only four; of 
 which the tonic and subdominant, which were major in the major key, 
 are now minor, while the submediant chord, which was minor in the 
 major key, is now major. It will also be noticed that the supertonic 
 now bears a diminished triad, like the leading note, instead of a 
 common chord ; and the mediant has a chord of a kind which we 
 have not yet met with, containing a major third and an augmented 
 fifth. For this reason the chord on the mediant is called an 
 augmented triad. As all augmented intervals are dissonant ( 28), 
 the augmented triad on the mediant is a discord ; but, unlike the 
 diminished triad, it is as dissonant in its first inversion as in its root 
 position. 
 
 182. The distinctive effect of major and minor keys will be 
 best perceived by the following example of what is known as a 
 " convertible " chant that is, one which can be sung in either a 
 major or minor key by merely altering the character of the 
 intervals.
 
 If these two forms of the chant are compared, it will be seen that 
 every chord is identical, but that the effect produced by the two 
 versions is entirely different. This, being a matter which concerns 
 musical aesthetics rather than harmony, need not be enlarged upon 
 here ; but it is worth passing mention. 
 
 183. Like the diminished triad on the leading note of the 
 major key, the corresponding chord in the minor key is rare in its 
 root position. The diminished triad on the supertonic is however 
 more frequently to be found in this form, as in the following 
 examples, at *. 
 
 BACH. Toccata in F. 
 
 MENDELSSOHN. " St. Paul." 
 ,(*) * 
 
 Though both these extracts are from pieces in major keys, it 
 happens that both are here in the key of A minor. It will be 
 seen that in each case the chord following the dissonance has a 
 root a fourth above that of the dissonance itself. We shall find 
 later, when we have to treat of dissonant chords, that this is a 
 very frequent resolution for a discord. 
 
 184. The augmented triad on the mediant is in reality an 
 inversion of the chord of the dominant minor thirteenth ( 410), 
 and its harmonic origin has been shown in the last chapter ( 177). 
 Its special treatment will therefore be deferred till we come to 
 treat of chords of the thirteenth ; it will be sufficient to say now 
 that when used merely as a mediant triad it is best followed by 
 the chord of the submediant, the root (as in the case of the super- 
 tonic triad mentioned in the last paragraph) rising a fourth to the 
 root of its resolution. 
 
 185. The reason for the treatment of the mediant chord as part of the 
 dominant harmony while the chords of the subdominant and submediant, 
 which are equally derived from the dominant ( 177), are not necessarily con- 
 sidered in their relation to that note, probably arises from the fact that in the
 
 Chap. VI i I 
 
 Irs THEORY AND PRACTICE. 
 
 81 
 
 mediant chord the generator is always present, while the subdominant and 
 submediant chords contain only some of the upper partial tones without the 
 generator. 
 
 1 86. The interval of the augmented second between the sixth 
 and seventh degrees of the minor scale being considered by the 
 older composers to be unmelodic, we frequently find in the music 
 of the last century the major sixth of the scale treated as a 
 harmony note, and used in the chord of the subdominant and, 
 more rarely, in that of the supertonic, instead of the minor sixth. 
 
 (n) i . 3fc HANDEL. " Israel in 
 
 g 
 
 
 
 
 
 
 
 
 
 
 
 5sz 
 
 D 
 
 JP5R 
 
 a 
 
 Sj 
 
 L - i 
 
 ^P 
 
 J J. J. 
 
 mt ~fS . 
 
 -f. 
 
 I ' 
 
 A- 
 
 J 
 
 7 ^ 
 
 * 
 
 
 
 -; 
 
 
 
 
 
 
 
 ^- 
 
 \-Z* 
 
 L_U 
 
 BACH. Cantata " Ich bin ein guter Hirt." 
 I * 
 
 I**! * ' P5*-: j-^~ N I 
 
 At () are two examples of the major chord on the subdominant, 
 the first in three and the second in four part harmony. At (b) is 
 the first inversioaof a minor common chord (with the major sixth of 
 the scale) on the supertonic. The last chord of this example will be 
 explained presently ( 193). These examples are given because 
 of their historical importance, and are not intended for the imita- 
 tion of the student. Our feeling of tonality is so much more 
 definite than that which existed 150 years ago, at the time the 
 passages just quoted were written, that the major sixth of the 
 scale produces a disturbing effect on the key unless very carefully 
 managed. It is therefore better avoided, and it is especially un- 
 pleasant when the subdominant chord comes between the tonic 
 and the dominant, as at (<:). 
 
 187. The general rules for the progression of parts and the 
 doubling of notes already given apply equally to the minor and to 
 the major key ; but there is one very important rule in addition, 
 which concerns the minor key exclusively. We often meet with 
 the progression from dominant to submediant, or the reverse, 
 from submediant to dominant. In 130 it was said that the root 
 was the best note to double; but if we double the root in the
 
 82 
 
 HARMONY : 
 
 [Chap. VIII. 
 
 suhmediant chord of a minor key when it is followed by the 
 dominant chord, we shall find ourselves in difficulties. 
 <) (*) . (0 
 
 We evidently cannot proceed as at (a), for this gives consecutive 
 fifths and octaves ; neither can we avoid these as at (&) or (*r), 
 because an augmented interval in the melody is seldom good 
 unless both the notes belong to the same harmony, which is not 
 the case here. At (</) the treble and alto parts cross, which is 
 bad, and besides this we still have distinctly the effect of fifths 
 and octaves as at (a). The only right way is to double the third 
 instead of the root in the submediant chord, as at (e). In the 
 same way, if the dominant chord comes first, as at (/) (g), 
 ,(/) , , (*) 
 
 it will equally be needful to double the third in the second chord. 
 It will sometimes be found expedient when the first inversion of 
 the subdominant chord (having the submediant in the bass) 
 precedes or follows the dominant chord, also to double the third 
 of the bass note in this case the fifth from the root. The rule, 
 which must be most carefully observed, is this : Whenever in a 
 minor key the chord of the dominant precedes or follows that of the 
 submediant, the third (and not the root} must be doubled in the 
 submediant chord. 
 
 1 88. All the diatonic triads of the minor key, like those of the 
 major, can be used in their first inversion. The general rules 
 given in the sixth chapter ( 156, 157) are still to be observed, 
 and care must be taken to avoid, wherever possible, the interval 
 of the augmented second. Thus the following progression at (a) 
 is faulty, and should be corrected as at (b). 
 
 J. 
 
 A. 
 
 Kd 
 
 -?7_ 
 
 189. The chord of the mediant is probably more common in 
 its first inversion than in its root position, and, as already said 
 ( 184), when not treated as a dominant thirteenth, it is best 
 followed by the chord of the submediant.
 
 Chap. VIII. | 
 
 Irs THEORY AND PRACTICE. 
 
 MENDELSSOHN. Overture " Elijah. 
 
 
 
 At % is the first inversion of the mediant chord in D minor, 
 resolved on the root position of the submediant. 
 
 190. In addition to the first inversions of all the triads given 
 in 1 80, there is one first inversion to be met with under special 
 conditions with which we have not yet made acquaintance. If 
 the bass moves by step from tonic down to submediant it some- 
 times takes the minor seventh of the scale on its way instead of 
 the leading note ; and in this case only (but not in the reverse 
 direction) the minor seventh is allowed to bear a first inversion. 
 This chord in its root position would be a minor chord on the 
 dominant. In such a form it is absolutely forbidden ; and it is 
 only allowed when preceded by the tonic in the bass and also 
 followed by the note below ; either of these notes may, however, 
 bear a chord either in root position or in an inversion. The 
 following is an excellent example of the employment of this chord. 
 It will be seen that on the second occurrence of the chord the bass 
 descends a semitone, instead of a tone, to a supertonic discord in 
 the second inversion. 
 
 MENDELSSOHN. Overture " Ruy Bias." 
 
 191. All the triads of the minor key can be used in their 
 second inversion ; but, as in the major, those of the tonic, sub- 
 dominant and dominant chords are by far the most common. It 
 is needless to give a complete set of examples, as we did with the 
 major key; but a few of the rarer second inversions will be 
 interesting. 
 
 MENDELSSOHN. " St. Paul." 
 * 
 
 '' ' '
 
 HARMONY : 
 
 LChap. VIII. 
 
 (6) 
 
 BACH. " WohUemperirte Clavier," Book 2, Prelude ao. 
 
 * 
 
 At (a) is seen the second inversion of the diminished triad on the 
 supertonic of F sharp minor. The exceptional leap of the bass 
 from this chord to the next is justified by the fact that both 
 supertonic and subdominant chords are really part of the funda- 
 mental harmony of the dominant ( 177), and the bass of a second 
 inversion is allowed to leap to another note of the same harmony 
 ( 165). 
 
 192. The examples at (l>) of the last paragraph will require a 
 little explanation. It must be pointed out that when two or more 
 notes of a chord are taken in arpeggio, as is so often the case in 
 pianoforte music, the harmony they produce is the same as if they 
 had been sounded together. In this passage the G$ and B form 
 the chord above D, and the F and A that above CO, just as if the 
 upper notes were played together as quavers. The chords marked 
 * are therefore the second inversions of the triad on the leading 
 note, and of the common chord on the submediant. It will be 
 seen that the bass of each second inversion moves by step, in 
 conformity to the first rule given in 165. The D$ and CJ in 
 the bass are both chromatic notes in the key, and their use will 
 be explained later. 
 
 193. In old music it is comparatively rare to find a minor 
 chord at the end of a movement, even when that movement is 
 written in a minor key. The effect of such a close was considered 
 unsatisfactory, and a major tonic chord was substituted for a 
 minor to conclude the piece. For some reason which it is im- 
 possible to ascertain with certainty, the major third in the final 
 chord of a minor movement was called the TIERCE DE PICARDIE, 
 or "Picardy third." It is still not infrequently employed, es- 
 pecially in church music ; and, in this connection only, the major 
 third is not considered as a chromatic note in the minor key. 
 The student will see an excellent example of the employment of 
 this chord in the extract from Bach given at (b} in 186. Occa- 
 sionally also the third is omitted altogether, and the piece ends 
 with a bare fifth, as in the following examples : 
 
 MOZART. " Requiem."
 
 Chap. VIII ] 
 
 Irs THEORY AND PRACTICE. 
 
 MENDELSSOHN. "Reformation Symphony. 
 
 194. The real icason of the aversion of the old composers to ending a 
 piece with a minor third was doubtless that, as it is not a part of the compound 
 tone ( 40) of the tonic, it produces beats with the major third, which is often 
 distinctly audible in that compound tone. If on a good piano we play the 
 
 note C in three octaves thus, 
 
 and listen carefully, the E b can be 
 
 distinctly heard. Now, if the minor third above the upper of the three C's be 
 
 also sounded, 
 
 the jarring of the Ej> against the Efcj is quite 
 
 -- 
 
 perceptible to anyone whose ear is at all trained to analysing sounds. Modern 
 music deals so much more freely with dissonances than did ancient, that our 
 ears are far less sensitive to harmonic impurity than the ears of our forefathers ; 
 hence we do not feel the ill effect of a final minor chord to the same extent as 
 they did. 
 
 EXERCISES TO CHAPTER VIII. 
 Treble and Bass given. Add Alto and Tenor. 
 
 (II.) 
 
 
 
 
 
 
 
 
 __ 
 
 
 1 
 
 
 g 
 
 a 
 
 
 
 
 
 
 _a 
 
 
 6 
 
 
 
 
 S 
 
 t , 
 JO 
 
 
 
 
 a 
 
 6 15 
 
 4 5 
 
 Bass only given. Add three upper parts, 
 (in.) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 6 5 
 
 1 
 
 i 
 
 6 
 
 [^ 1 
 6 
 
 6 5 
 
 4 If 
 
 
 (IV.)
 
 86 
 
 HARMONY : 
 
 [Chap. VI II. 
 
 (V.) 
 
 
 
 
 x 6 66 6 5 6 6 X XX5 665 
 
 6 10 6 ff 55 6950 
 
 63 (5 65 
 
 J 6 36 35 6 
 
 (a) In harmonising this progression it will be necessary to take the leading- 
 note in the first chord of the bar down to the fifth, mslead of up to the tonic, or it 
 will be impossible to preserve the sequence (\J 137). The leading-note must, of 
 course, not be in the upper part ( 132). The exceptional fall to the fifth of the 
 tonic chord is here justified by the sequential character of the passage.
 
 Chap, ix.j ITS THEORY AXD PRACTICE. 87 
 
 CHAPTER IX. 
 
 THE CHORD OF THE DOMINANT SEVENTH. 
 
 195. If the student will take a simple progression of the diatonic 
 triads with which he is already acquainted, and will observe the 
 mental effect which they produce, he will find that the character 
 of tonic and dominant harmony is quite different. Let him, for 
 instance, play on the piano the following passage, stopping at the 
 first double bar. 
 
 It will be felt that the effect of the alternation of tonic and 
 dominant chords ending with a tonic chord is one of completeness ; 
 we require nothing to follow. Now let him play the passage 
 again adding the chord at (a). The feeling of finality is gone ; 
 one requires something else to succeed ; but if (b] is played after 
 (a), the reposeful feeling is restored. From this it will be seen 
 that the dominant chord of a key is not fitted for concluding a 
 piece, but that, as compared with the tonic chord, its effect is 
 one of incompleteness. In their relation to one another we may 
 say of these two chords what was said (18) of consonance and 
 dissonance that the tonic- chord is a chord of rest, and the 
 dominant a chord of unrest. 
 
 196. The difference in the character of these two chords no doubt arises 
 from the fact that the dominant chord is generated out of the tonic ( 67), and 
 has therefore a natural tendency to return finally to it. (Compare 57.) 
 
 197. Hitherto all the chords we have used have been triads, 
 that is, combinations of three notes made by placing two thirds 
 one above another. If we place another third on the top of a 
 triad, it is evident that this third will be a seventh from the root. 
 Such a chord will be called a CHORD OF THE SEVENTH. Of all 
 the possible chords of the seventh, that on the dominant is the 
 most frequently used, and the most important. 
 
 In this chord the dominant is not only the root, but also the 
 generator (58, note}. It will be necessary as we proceed to
 
 88 HARMONY : I chap. ix. 
 
 consider it in both these aspects. It is very important that the 
 student should thoroughly grasp the fact that the intervals of this 
 chord are a major third, a perfect fifth, and a minor seventh from 
 the generator. We here meet for the first time with a " funda- 
 mental discord" (611, that is, a discord composed of the 
 harmonics of the fundamental tone or generator. The upper notes 
 of a fundamental discord are in some cases variable ; but every 
 such discord always contains as its thirds next above the generator 
 the notes which make the three intervals just specified. The chord 
 of the seventh is simply figured 7, the fifth and third being im- 
 plied (as in the common chord) unless one of these notes requires 
 an accidental. 
 
 198. If we substitute for one of the dominant chords of the 
 passage in 195 a chord of the dominant seventh (*) 
 
 the feeling of unrest which was before to some extent noticeable 
 becomes intensified. It is no longer possible to end on it, 
 because the interval of a seventh is always a dissonance ( 17), 
 and therefore requires resolution that is, to be followed by a con- 
 sonance, as at (a) above. 
 
 199. If we examine the chord of the seventh given in 197, 
 we shall see that in addition to the seventh from the root there is 
 another dissonance in the chord, namely the diminished fifth 
 between the third and the seventh. 
 
 It is this interval which determines the resolution of the discord, 
 that is, the way in which these two notes shall move with regard 
 to one another. The general rule governing the progression is 
 that two notes forming a diminished interval have a tendency to 
 approach one another. In the present case the third of the chord, 
 being the leading note, rises a second, and the seventh falls a 
 second. 
 
 If the key of the piece were C minor, the seventh would fall a 
 tone, to E7, instead of only a semitone. But if the chord were 
 taken in a different position, and- the seventh were below the 
 third, the interval of the diminished fifth, being now inverted,
 
 Chap. IX.J 
 
 Irs THEORY AND PRACTICE. 
 
 would become an augmented fourth ( 31), and hvo notes forming 
 an augmented interval have a tendency to diverge. This will be 
 seen in the treble and alto parts of the last two chords of the 
 passage in 198. 
 
 200. The chord of the seventh is very frequently found with 
 all its four notes present ; but the fifth is sometimes omitted, and 
 in this case the root is doubled. Neither the third nor the 
 seventh of the chord can be doubled, because the progression of 
 both notes is fixed, and if one of them appear in two voices, we 
 shall either get consecutive octaves or unisons, if both move 
 correctly, or else one of them must move wrongly. 
 
 201. Let us now take the chord of the dominant seventh, and 
 resolve it on the tonic chord its most usual resolution. 
 
 M () M (rf) W (/) 
 
 
 Here the root of the chord, except at (/), falls a fifth (or rises a 
 fourth) to the tonic ; the third rises a semitone ; the seventh falls 
 one degree to the third of the next chord ; and the fifth either 
 rises or falls one degree see (a). When the root falls a fifth, the 
 fifth in the chord of the seventh cannot go to the fifth of the tonic 
 chord, or we shall have consecutive fifths, either by similar 
 motion, as at (b\ or by contrary motion, as at (c). Hence we get 
 a rule that if the fifth is present in the chord of the dominant 
 seventh, it must be omitted in the following tonic chord. . If it is 
 especially desired to have the fifth in the tonic chord, the root 
 must be doubled and the fifth omitted in the chord of the 
 seventh, as at (d} (e}. In . the not infrequent case when the 
 dominant seventh resolves on the second inversion of the tonic 
 chord, as at (/), the fifth can evidently be present in the chord of 
 the seventh without producing consecutives. Here the rule just 
 given does not apply. Examples of the ordinary resolution on 
 the root position of the tonic chord will be found at 100 (c), 
 1 08 (6), and 1 60 (b}. 
 
 202. It is not uncommon for another note of the chord 
 either the root or the fifth to be interposed between the seventh 
 and its resolution, as in the following examples : 
 
 HANDEL. " Messiah."
 
 HARMONY : 
 
 [Chap. IX. 
 
 At (a) the seventh rises to the root, while the harmony remains 
 unchanged, and then descends to the third of the next chord. At 
 (fr) the seventh falls to the fifth, while the root rises to the third of 
 the dominant chord, the fifth rising to the third of the tonic 
 chord the regular resolution of the seventh on the change of 
 harmony. Such a progression is called an ornamental resolu- 
 tion of a dissonance. The example (c) shows at * the same 
 treatment of the seventh seen at (a) ; but it is also interesting as 
 illustrating in the first two quavers of the second bar the resolution 
 of the chord on the second inversion of the tonic chord, referred 
 to in the last section. Notice also that "at the third quaver of this 
 bar the fifth of the chord is doubled, and the third does not 
 appear till the ornamental resolution of the seventh. This is 
 doubtless for the sake of making the alto part more melodious. 
 
 203. In addition to the resolution of the dominant seventh on 
 the tonic chord, several other resolutions are possible. The most 
 common of these is the resolution on the chord of the submediant. 
 In this case the root rises one degree, instead of a fourth, the 
 third and seventh move, as before, by step, the former upwards 
 and the latter downwards ; but the fifth, instead of being as before 
 free to rise or fall one degree ( 201), must fall to the third, as at 
 (a), or. we shall have consecutive fifths with the root, as at (<) below. 
 
 
 If the fifth of the chord of the seventh be omitted and the root 
 doubled, it will still be bad to double the fifth in the submediant 
 chord, as at (c) below ; for this gives the worst kind of hidden 
 octaves between treble and tenor (see 104). It is equally 
 impossible to double the root, for this would make consecutive
 
 Chap. IX.] 
 
 Irs THEORY AND PRACTICE. 
 
 9 1 
 
 octaves with the bass. Hence a general rule : Whenever the 
 dominant seventh is resolved on the submediant chord, the third of 
 the latter chord must be doubled. This, of course, refers, like our 
 other rules, to four-part harmony. Note that the doubled third 
 of the submediant chord is the key-note. 
 
 This resolution is of such frequent occurrence that we need only 
 give one example of it here. 
 
 HAYDN. "Seasons." 
 
 ^. M -N J J* ' j s A~n 
 
 P 
 
 -m-:^- 
 
 204. Another resolution of the chord of the dominant seventh, 
 less common than the one just noticed, but still not so rare as to 
 be exceptional, is on the first inversion of the chord of the sub- 
 dominant. 
 
 Here the seventh, instead of falling, as in the cases hitherto 
 noticed, remains to be the root of the next chord ; the third rises 
 by step, as before, the fifth falls one degree, and the root rises 
 one degree to the third of- the next chord. The following is 
 a good example of this resolution :- 
 
 SCHUMANN. " Paradise and the Peri." 
 . -jt 
 
 4-i 1 n 
 
 -*=< 
 
 r 
 
 r r r " r 
 
 r 
 
 
 r 
 
 , , 
 
 205. As a chord of the seventh contains three notes besides 
 the root, it must of course be susceptible of three inversions. 
 The first inversion has the third in the bass. 
 
 It is evident that the root and the fifth of the chord will now be
 
 92 HARMONY: [Chap ix. 
 
 respectively the sixth and third above the bass, as in the 
 first inversion of a common chord. The seventh will be a fifth 
 above the bass, and the full figuring of the chord will be 5. The 
 figure 3 is however omitted, as in the chord of the sixth ( 153), 
 except when it requires an accidental, and the chord is usually 
 
 figured only \. 
 
 206. In the resolution of this inversion, the dissonant notes 
 (the third and the seventh) obey the rules already given. If the 
 chord be resolved on the tonic chord, the bass (the leading note) 
 will rise a semitone, the seventh will fall one degree, the sixth 
 (the root of the chord) will remain as the fifth of the following 
 chord, and the third (the fifth of the chord) will either rise or fall 
 one degree. 
 
 tJ ^ -<s>- -izzr -<s>- 
 
 If the first inversion of the dominant resolve upon the submediant 
 chord, the latter must also be in the first inversion, because the 
 leading note in the bass must rise. If it falls, as at (a) below, we 
 shall have a diminished fifth followed by a perfect fifth a progression 
 which is forbidden ( 101) between the bass and an upper part. 
 The seventh will fall, as before, the root will rise one degree, and 
 the fifth of the chord, if below the root, as at (^), may either rise 
 or fall one degree. But if the fifth be above the root, it must 
 fall, as at (d] to avoid consecutive fifths as at (c}. 
 
 . () . ( M _ (d) 
 
 This resolution is much rarer than when the chord of the seventh 
 is in its root position ; while the resolution of the first inversion 
 of the seventh on the chord of the subdominant is seldom to be 
 found. 
 
 207. If the root of this chord, which it must be remembered 
 is also the generator, be omitted, which in the first inversion 
 rarely happens, we shall have the root position of the diminished 
 triad on the leading note ( 123). 
 
 This triad is therefore in reality the chord of the dominant 
 seventh, without the generator ; and it must be remembered that 
 it is still impossible to double either the root or the fifth, the
 
 Chap. IX.] 
 
 ITS THEORY AND PRACTICE. 
 
 93 
 
 former because it is the leading note of the key, and the latter 
 because, being dissonant with the bass of the chord, it has a fixed 
 progression, which was shown in 199. The rules for the treat- 
 ment of this dissonance apply equally whether the generator be 
 present or absent. The only note which can be doubled in this 
 position will be the third above the leading note the fifth of the 
 original chord. 
 
 208. We now give a few examples from the great masters of 
 the use of the first inversion of the dominant seventh, both with 
 and without the generator. 
 
 The first two of these examples (a) and (), show the usual 
 form and treatment of the chord, and need no further explanation. 
 At (<r) is a rarer use, the generator being omitted, and the chord, 
 as said above, appearing as the diminished triad on the leading 
 note. It will be noticed that both here and at (d) the third above 
 the bass is the note doubled. Example (d} is further interesting 
 as containing in its last chord a good example of the " Tierce de 
 Picardie" ( 193), and also as illustrating Bach's partiality, referred 
 to in 132, to making the leading note, when in a middle voice, 
 fall to the fifth of the final chord, no doubt for the sake of having 
 the last chord in a complete form. (Compare 134.)
 
 94 HARMONY: [Chap. ix. 
 
 209. The second inversion of the chord of the dominant 
 seventh has the fifth in the bass. 
 
 The root and third of the chord are now (as in the second inver- 
 sion of a common chord) the fourth and sixth respectively above 
 the bass ; and the seventh is now the third above the bass. The 
 chord will therefore be figured *. Evidently the figure 3 (the 
 seventh of the chord) cannot be omitted, or we shall have no 
 means of distinguishing the chord from a ; but the 6 can be dis- 
 pensed with, and the chord is often figured ^, the sixth being im- 
 plied. In a minor key, in which the leading note bears an acci- 
 dental, the sixth must always be marked, and is written $6 or t|6, 
 according to the key ; sometimes also 0. 
 
 210. The second inversion, like the first, is rarely resolved 
 otherwise than on the tonic chord. It is most often resolved on 
 the root position of that chord ; and in this case the seventh (the 
 third of the chord) falls one degree, the leading note (the sixth of 
 the chord) rises a semitone, the root remains stationary, becoming 
 the fifth of the next chord, and the bass falls one degree. 
 
 2ii. There is another resolution of this inversion on the tonic 
 chord. The bass, instead of falling to the root of the tonic chord, 
 may rise to its third, and it is important to remember that in this 
 case the seventh (the third from the bass) may either fall to the 
 third, or rise to the fifth of the tonic. The following examples 
 will illustrate these resolutions. 
 
 CRAMER. Studio, No. 33.
 
 Chap. IX.] 
 
 Irs THEORY AXD PRACTICE 
 
 HAYDN. 4th Mass. 
 * 
 A- 
 
 95 
 
 At (a) both treatments of the inversion are shown. The first 
 (*) is in the key of F, the bass rising to the third of the tonic, and 
 the seventh to the fifth. The ornamentation of the C in the tenor 
 by a kind of shake with BIj does not affect the harmony. B^ is 
 what is called an "auxiliary note," as will be explained in a subse- 
 quent chapter. At ** is the second inversion of the dominant 
 seventh in G minor. Here the bass descends to the root of the 
 tonic chord ; the seventh therefore also descends to the third of 
 the chord. At (b) and (c) are two passages chosen from the same 
 work, to show the different treatment of this inversion according 
 to the position of the notes. In both cases the bass rises to the 
 third of the tonic chord ; at (b} the seventh rises to the fifth of the 
 chord, but at (c} it falls to the third. If it rose here, we should 
 have the objectionable progression from the second to the unison 
 between alto and treble ( 109). It is true that this progression 
 is not infrequently met with in this resolution of the second 
 inversion ; but the student will do well not to let the seventh rise 
 except when it is at the distance of at least a seventh from the root 
 
 212. Unlike the first inversion, the second inversion of the 
 dominant seventh is very frequently found without the generator. 
 It would indeed be hardly too much to say that the generator is 
 more often omitted than not. In this form the chord becomes 
 the first inversion of the diminished triad on the leading note. 
 
 It is then merely figured 6, like any other first inversion. It was 
 said in 155 that though the F and B are still dissonant to one 
 another, they are both consonant to the bass note. As the genera- 
 tor of the chord (G), toward which the seventh must fall, is no 
 longer present, the seventh is not restricted in its progression, and 
 may either rise or fall. It may also be doubled, in which case 
 one of the sevenths rises, and the other falls. The leading note 
 cannot, of course, be doubled.
 
 HARMONY : 
 
 [Chap. IX. 
 
 HANDEL. " Messiah. 
 (f) 
 
 \ U I- 
 
 f r^v &- k r- 
 
 This short extract contains three second inversions, in each case 
 without the generator, which Handel very rarely introduces in this 
 position of the chord. At (a) and (c} the bass is doubled, and the 
 seventh rises in each case. At (b) the seventh is doubled, one 
 rising and the other falling. One of the sevenths may also leap 
 to another note of the next chord, as in the following. The L) 
 here is only a passing note, and does not change the harmony. 
 
 213. The third, and last, inversion of the dominant seventh 
 has the seventh in the bass. 
 
 The root, third, and fifth of the original chord are now respectively 
 the second, fourth, and sixth above the bass ; and this inversion is 
 therefore figured *, more frequently 4 ? and occasionally 2 only. 
 
 The seventh, as in other cases, descends one degree, and the third 
 ascends. 
 
 214. By far the most frequent resolution of this inversion is 
 on the first inversion of the tonic chord ; but it is also occasionally 
 resolved on the second inversion of the submediant chord. 
 
 HANDEL. " Messiah." 
 
 =^2 J^ZZl * i. ~f^\ & ~m- ~m 1 
 
 -,s ^ i y | -^'"F | ^p^ ' F ' i 
 
 ?-: ^-: 1 ai 1 (^ -i^ . ) 1
 
 Chap IX.] 
 
 ITS THEORY AND PRACTICE. 
 
 97 
 
 At (a) will be seen the resolution on the tonic chord, a rest being 
 introduced between the discord and its resolution. The mental 
 effect of rests must never be forgotten by the student ; the last 
 note or last chord preceding a rest is retained by the mind until 
 the following note of the harmony. At (b] this inversion of the 
 seventh is followed by the second inversion of the submediant, 
 which, it will be seen, obeys the law for the treatment of second 
 inversions given in 165, Rule I. 
 
 215. It is very rare in four-part harmony to find the generator 
 omitted in this inversion. This will give us the second inversion 
 of the diminished triad on the leading note ( 163). An example 
 of this form of the chord has already been given at 1 60 (h}. 
 
 216. A very important point to be noticed with regard to the 
 treatment of fundamental discords is that the dissonant notes may 
 be transferred from one part of the harmony to another, provided 
 that they are properly resolved in the part in which they last 
 appear. In this case the seventh is not infrequently doubled even 
 when the generator is present, one of the sevenths leaping to 
 another note of the chord, and the other receiving its regular 
 resolution. (See (a) (b} below.) But it is best when one of the 
 dissonant notes (the third or the seventh) has been transferred to 
 the bass, not to change its position again, but to resolve it in that 
 voice, as at (<:) (</). below. (See also in illustration of this the first 
 bar of the extract from Schumann in 204.) 
 
 BEETHOVEN. Andante in F. 
 
 BEETHOVEN. Symphony in D. 
 
 BEETHOVEN. Sonata, Op. 30, No. 3. 
 )
 
 HARMONY: 
 
 [Chap. IX. 
 
 217. A few exceptional resolutions of the chord of the dominant seventh 
 are to be found, especially in the works of Handel. One of these, in which 
 the chord is followed by a chord of the seventh on the subdominant, has been 
 already noticed ( 108) and examples given of it. We add a few others here 
 not as models for the student's imitation, but for the sake of completeness. 
 
 HANDEL. " Messiah. 
 
 At (a) the mediant chord, with the fifth omitted, is interposed between the 
 chord of the seventh and its regular resolution on the submediant. This may 
 be regarded as akin to the ornamental resolutions spoken of in 202. At (b) 
 the third inversion of the seventh is resolved on the root position of the tonic 
 chord instead of on its first inversion ; and at (c) the seventh is irregularly re- 
 solved by rising to the root. The progressions (b) and (c) are rarely if ever 
 met with excepting in the accompaniment of recitative. 
 
 2 1 8. As we are at present treating only of diatonic chords, 
 we defer till later the explanation of the various chromatic resolu- 
 tions of the dominant seventh, in which the seventh rises a 
 chromatic semitone, or the third falls the same interval. Some of 
 these progressions will be dealt with when, in the next chapter, we 
 have to speak of modulation ; others will be more suitably ex- 
 amined in connection with the chromatic chords of a key. 
 
 219. The great importance of the chord of the dominant 
 seventh is largely due to the fact that when followed by the tonic 
 chord it absolutely defines the key. For instance, the two chords 
 
 might be either dominant and tonic in C, or (if they were not the
 
 Chap. IX.J 
 
 ITS THEORY AND PRACTICE. 
 
 99 
 
 final chords of a piece) tonic and subdominant in G. But if we 
 add the seventh to the first chord, 
 
 the passage can only be in C ; for though the first chord might be 
 a chromatic chord in G, it would in that case, as will be seen 
 later, have a different resolution ; and the fact that it is followed 
 by the chord of C proves that it is a dominant seventh resolved 
 on the tonic. We shall learn more of this in a later chapter. 
 
 EXERCISES TO CHAPTER IX. 
 
 [We shall in future give only the bass of the exercises. The 
 pupil will by this time have gained sufficient experience to be able 
 to dispense with the additional aid afforded by our giving the 
 treble also.] 
 
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 Chap, x.] ITS THEORY AND PRACTICE. 101 
 
 CHAPTER X. 
 
 KEY RELATIONSHIP MODULATION TO NEARLY RELATED KEYS 
 
 FALSE RELATION. 
 
 2 20. If all music remained in one key the resources at the 
 disposal of the composer would be very limited, and the effect 
 would soon become extremely monotonous. In all pieces there- 
 fore of 'more than a few bars' length, changes of key are introduced, 
 a new note being taken as the tonic for a time, and a return being 
 subsequently made to the original key. Every piece must end 
 with the same note as a tonic with which it begins, though the 
 mode of the key may be, and often is, changed from major 
 to minor, or vice versd. A change of key is called a MODULATION. 
 Some writers use the term " Transition," when the modulation is 
 to a remote or unrelated key ( 225), but the distinction is not 
 always made, and the word " Modulation " will suffice for all 
 purposes. 
 
 221. The first question to be answered is, which keys are the 
 best to modulate into, and why do some modulations sound more 
 natural than others? The answer to this question depends on 
 what is known as Key Relationship, which we shall now proceed 
 to explain. 
 
 222. It was said in 30 that two notes are consonant to one another when 
 two of their earlier upper partial tones coincide. The earlier the coincidence 
 of these partials, the more perfect the consonance. In the same way, two 
 common chords are said to be related to one another when they have one note 
 in common. But a moment's thought will show that this is only possible when 
 the roots of the two chords are consonant. For if we take two notes which are 
 dissonant to one another (for instance C and D, or C and B), it is quite clear 
 thit their common chords must (consist of entirely different notes. On the 
 other hand, any two notes which are consonant with one another will have a 
 note of their chords (either root, third, or filth) in common. Thus C and E 
 flat have G as part of their common chords, C and A flat have C, and C and 
 G have G in common. 
 
 223. Two major keys are said to be related to one another 
 when their tonics are consonant, and the more perfect the 
 consonance, the nearer the relationship. Thus F and G are both 
 perfect fifths from C, the former below and the latter above. 
 These therefore are the nearest related major keys to C. The 
 imperfect consonances, the major and minor thirds and sixths, are 
 also related, but more distantly : they stand in the second degree 
 
 of relationship. For the present we deal only with the most 
 nearly related keys.
 
 io2 HARMONY : [Chap. x. 
 
 224. It was shown in Chapters IT. and III. that G was the first new note 
 generated from C, and that all its harmonics were secondary harmonics of C. 
 It is therefore clear that if we take G as a tonic and develop the chords of a 
 key from it as we did from C, we shall have a key derived in the second 
 degree from C. Similarly, C being itself the third harmonic of its fifth below, 
 namely F, the whole key of C must be a derivative from F. Hence these 
 keys are the nearest related to C, the dominant as a generated and the sub- 
 dominant as a generatz'wf key. 
 
 225. Two keys whose tonics are dissonant with one another 
 are said to be unrelated ; and if the tonics are a semitone apart, 
 or are distant an augmented or diminished interval from each 
 other, such keys are often spoken of as "remote." Modulation to 
 unrelated or remote keys will be dealt with later. 
 
 226. If we examine two closely related major keys (e.g., C and 
 G, or C and F), we shall see that, excepting in one note, their 
 diatonic scales are identical. Thus, the only note which differs in 
 the scales of C and G is F, and in the scales of C and F, B 
 ( 8 1, 85). Therefore every common chord in the key of C 
 which does not contain the note F also belongs to the key of G ; 
 and every chord in the key of C which does not contain B belongs 
 also to F. An examination of the table of the triads of C given 
 in 122 will show us that there are four chords common to the 
 keys of C and G, and that there are also four common to the keys 
 of C and F. 
 
 227. If we now compare the major scale of C in 121 with 
 the harmonic form of the relative minor (A minor) given in 171, 
 we shall see that here also all the notes of the scale are identical 
 with one exception G. The key of A minor therefore, like G 
 and F, contains four chords in common with the key of C ; and 
 for this reason (though none of its chords, even though they look 
 identical, are derived from the same generators) a relative minor 
 key is considered to be one of the nearly related keys to its rela- 
 tive major. Similarly, the relative minors of the dominant and 
 subdominant keys, being so closely connected with their relative 
 majors, are regarded as among the nearly related keys to the tonic. 
 We thus obtain the following table of nearly related keys, in which 
 we place the sharper keys to the right and the flatter to the left of 
 the tonic. 
 
 Table of nearly related keys. 
 
 F major. C MAJOR. G major. 
 
 D minor. A minor. E minor. 
 
 228. If we take any minor key as a starting point, we shall get 
 a similar series ; but with this difference, that nearly related minor 
 keys have fewer notes, and therefore fewer chords, in common, 
 than nearly related major keys. But the relations of their tonics 
 to one another will be similar ; and the nearly related keys to A 
 minor will therefore be : 
 
 D minor A MINOR E minor. 
 
 F major. C major. G major.
 
 chap x.] Irs THEORY AND PRACTICE. 103 
 
 Putting the matter concisely and as a general rule, it may be said, 
 that, the nearest related keys to any major key are its dominant and 
 subdominant, and the relative minors of these three keys; and the 
 nearest related keys to any minor key are its dominant and sub- 
 dominant minors, and the relative majors of these three keys. 
 
 229. A modulation to a nearly related key is effected by 
 introducing a chord containing a note belonging to the new key, 
 but foreign to that which we are leaving, and by following that 
 chord by other chords defining and fixing the new key. This last 
 condition is essential, because no single chord can ever define a key. 
 Suppose, for instance, that we wish to modulate from C to G. 
 Naturally we think of using the chord of D major (ggfg^), as 
 being the dominant chord of the new key. But the introduction 
 
 of this chord alone is quite insufficient ; for the same chord may 
 also be the dominant of G minor, the tonic of D major, the sub- 
 dominant of A major, or the submediant of FJ minor ; and it 
 would be quite possible to follow it by harmony in any of these 
 keys. If we take the dominant yth on D (adding Ctj to the above 
 chord), we restrict our choice considerably; but this chord may 
 still be either in G minor or G major, and only the context will 
 show us which. To determine any key, at least two chords, and 
 sometimes more, must be regarded in their relation to one 
 another. 
 
 230. It was said in 152 that a cotnmbn chord in its root 
 position was not figured unless one of the notes required an 
 accidental. In the case of a modulation without a change of key- 
 signature, it would be needful to indicate the chromatic alteration 
 of the third in the chord of the new dominant. Supposing the 
 key of the piece to be C, and we are modulating to G by means 
 of the chord of D major, as in the last paragraph, the D in the 
 
 bass would be figured thus W ^^ . (For the explanation of 
 
 I 
 
 the | without a figure at the side of it, see 153.) If the 
 modulation were effected by means of the dominant seventh of 
 the new key, the bass note would be figured & *? 
 
 7 
 I 
 
 231. A modulation can either be effected immediately that 
 is to say by introducing a characteristic chord of the new key 
 directly after a chord characteristic of that which we are leaving 
 or it can be effected gradually that is by interposing between the 
 characteristic chords of the two keys chords which are common 
 to both. If an abrupt effect is desired, the former plan will be 
 adopted ; but the latter is smoother and sometimes more natural. 
 The larger the number of ambiguous chords (that is chords which 
 belong to both keys, and therefore leave the tonality doubtful)
 
 104 
 
 HARMONY 
 
 introduced, the more gradual will be the modulation, 
 student compare the following passages : 
 
 [Chap. X. 
 
 Let the 
 
 
 The first bar and the last two of each passage are identical ; but 
 at (a) the chord containing F$ follows immediately on one con- 
 taining Ft}, while at (b) six ambiguous chords, belonging to both 
 the keys of C and G, are introduced before the new key is estab- 
 lished at *. The effect of the latter modulation is therefore 
 smoother than that of the former. 
 
 232. In modulating from a major key to the supertonic minor 
 it is necessary to avoid putting the dominant chords of the two 
 keys too near one another. This is because the dominant of the 
 first key is the subdominant of the second, and its chord would be 
 a major chord on the subdominant of the minor key. If these 
 two chords come next to one another we shall have the very 
 unpleasant progression shown at (c) ' 186. It is therefore usually 
 advisable in making this modulation to introduce the minor sixth 
 of the new key before introducing its leading note. 
 
 233. If we compare any minor key with the minor keys of its 
 dominant and subdominant, it will be seen that of all the diatonic 
 triads there is only one in common to each pair of keys. For 
 instance, the only chord common to C minor and G minor is the 
 tonic chord of the former, while the only chord common to C 
 minor and F minor is the tonic chord of the latter. For this 
 reason, while the most frequent modulations for a major key are 
 to the other nearly related major keys, a minor key most often 
 modulates to one of the related major keys, either its relative 
 major, with which it has four chords in common, or its submediant 
 major (the relative major of its subdominant), with which it has 
 three. 
 
 234. The number of chords common to two nearly related 
 keys is largely increased by the addition of chromatic chords, 
 which will be treated of in subsequent chapters. But even with- 
 out using these, considerable variety is possible, since the diatonic 
 chords can be employed in their inversions as well as in their root 
 positions. 
 
 235. We shall now give some examples, from the works of the 
 great masters, of modulation into nearly related keys, with such 
 remarks as may help the student to understand the way in which 
 they are managed.
 
 Chap. X.] 
 
 ITS THEORY AND PRACTICE. 
 
 This passage shows us at (i) a modulation to the dominant effected 
 abruptly, by introducing c| immediately after C3. From D the 
 music modulates to E minor at (2) by taking the first inversion of 
 the chord of E minor as the supertonic of D, and leaving it as the 
 tonic of the new key, following it by a cadence in E minor, with 
 the chord of the dominant seventh fixing the key ( 219). On 
 the repetition of the phrase, the passage is varied from the fifth 
 bar as at (b\ 
 
 I ' _' g i 
 
 - m - J -^. 5=^= 
 
 =EU=W+Th^f 
 
 The music now remains in I), coming to a full close in that key, 
 and the return to G is made by flattening the leading note of D 
 at * and thus making it the subdominant of G ( 86). 
 
 236. The following extract illustrates what was said in 232 
 with regard to the modulation to the supertonic minor key. 
 
 
 HANDEL. " Saul." 
 
 3Efi 
 
 t. 
 
 ir r r 
 
 Here the modulation is effected by taking at the fifth bar the 
 supertonic chord of A and leaving it as the tonic chord of B minor. 
 Let it be noticed that Gt3, the minor sixth of the new key, is intro- 
 duced before A$, its leading note. 
 
 237. In the passage now to be given we find some other 
 modulations. 
 
 BEETHOVEN. Sonata, Op. 14, No. 2. 
 (3) 
 
 At (i) is a modulation to the key of F, the subdominant, by
 
 io6 
 
 HARMONY: 
 
 [Chap. X. 
 
 flattening the leading note of the key of C. At (2) the dominant 
 of F is sharpened to become the leading note of its relative minor. 
 From D minor the music returns to C at (3) by taking the first 
 note of the bar as the tonic of D minor and leaving it as the 
 supertonic of C, thus reversing the process shown 'in 236. The 
 second chord of this bar is a chromatic chord in the key of C, 
 which will be explained later. 
 
 238. A modulation from a major key to its mediant minor is 
 usually made by treating the tonic chord of the first key as the 
 submediant chord of the second, and following it accordingly, as 
 below. 
 
 AUBER. " Masaniello." 
 
 Ill I B~ I ' ' I 
 
 _^_J-^- n ^ A JX-i- r-a-j-^u-kJ , j 
 
 239. Our next illustrations show modulations from a minor 
 tonic to the dominant minor (a) and the subdominant minor (b). 
 The key is changed in each case at the third bar of the extract ; 
 at (a) by treating the previous tonic as a subdominant ; and at (b) 
 by the reverse process, the first inversion of the subdominant 
 being here left as the first inversion of the new tonic. It should 
 be mentioned that the A$ in the third bar of (b) is not a harmony 
 note, but an auxiliary note. 
 
 MENDELSSOHN. "ImHerbst," Op. 9, No. 5. 
 
 .-"o i 1-"' I -i h-*-*-i =j-r a 
 
 tfl A 11 
 
 
 240. Our last example will show a modulation from a tonic
 
 Chap. X.I 
 
 Irs THEORY AXD PRACTICE. 
 
 107 
 
 minor to the relative major of its dominant minor in this case from 
 I) minor to C major. This modulation is not very common, and in 
 the present instance C major is quitted as soon as it is entered. 
 
 SCHUMANN. " Nordisches Lied," Op. 68. 
 
 4- 
 
 The first chord of the second bar is taken as the tonic of D 
 minor and quitted as supertonic of C major. The second chord 
 is a chromatic chord in the key of C, which will be spoken of in 
 the twelfth chapter; the third chord is part of the dominant 
 eleventh of C, with which the student is not yet acquainted, and 
 at the last chord of the bar, the music modulates to A minor. 
 
 241. The general rule for modulating to a nearly related key 
 which may be gathered from the examination of the extracts we 
 have given is that the modulation may be effected either by 
 immediately altering one of the notes of the key we are leaving 
 by means of an accidental (as in 235, 237), or by taking a 
 chord common to both keys, and leaving it as a constituent of the 
 new key, as in 236, 238, 239, 240. All nearly related keys 
 contain at least one chord in common. 
 
 242. Another method of modulating consists of irregularly 
 resolving the chord of the dominant seventh by making its third 
 rise a tone, or fall a semitone or a tone, or its seventh rise a 
 semitone. An example of each procedure will make this clear. 
 
 HANDEL. " Hercules." 
 
 SCHUMANN. " Humoreske," Op. 20.
 
 io8 HARMONY : rchap. x. 
 
 WAGNER. " Die Meistcrsinger." 
 
 At * in (a) the third of the dominant seventh in G falls a 
 chromatic semitone to the seventh of the dominant seventh in the 
 subdominant key, which is followed by the tonic chord, to 
 establish the key. The irregular progression of the bass in the 
 last two chords was spoken of in the last chapter ( 217). At (b] 
 the third in the chord of the dominant seventh in B flat rises a 
 tone to the third of the same chord in the key of C minor (the 
 supertonic minor), and the modulation is confirmed by the tonic 
 chord of that key. At (c) the seventh in the chord of the dominant 
 seventh of C rises a chromatic semitone, and becomes the leading 
 note of the key of G, to which a modulation is made. The chord 
 marked * is a dominant ninth in G, a chord which will be treated 
 of later. 
 
 243. There is yet one more point to be considered in the 
 choice of modulations. As a general rule, a modulation to the 
 dominant side of the tonic that is, to a key containing more 
 sharps, or fewer flats is to be preferred to one to the sub- 
 dominant side that is, a key having more flats, or fewer sharps. 
 The reason of this is that the dominant is a key generated out of 
 the tonic ( 224), as also are all other keys with more sharps than 
 itself (see Chapter III.). When, therefore, we modulate into one 
 of these keys, the tonic still maintains its position as the source 
 whence the whole music springs. But the tonic itself is a deriva- 
 tive of keys having more flats than itself e.g. C is the dominant 
 of F, and the mediant of AP ; and when we modulate into one of 
 these keys the original tonic sinks into a subordinate position as a 
 derived key. If, for instance, we modulate from C to F and make 
 a long stay in the latter key, we shall when we return to C most 
 likely get the mental impression, not of returning to the key of the 
 tonic, but of going into a dominant key. For this reason the 
 feeling of the key is much more readily disturbed and much 
 sooner obliterated by a modulation to the subdominant side of 
 the key than to the dominant side. This applies chiefly to major 
 keys ; in minor keys, owing to their more artificial origin, greater 
 freedom prevails, and the same disturbing effect is not so readily 
 produced. 
 
 244. When in two consecutive chords a note forming part of 
 the first chord is chromatically altered to become a note of the 
 second chord, it is frequently best to keep it in the same voice. 
 This is especially the case when the roots of the two chords are 
 the same.
 
 Chap. 
 
 Irs THEORY AND PRACTICE. 
 
 109 
 
 Bad. 
 
 Good. 
 
 Good. 
 
 The progression at (a) is called a false -Relation. (Another use of 
 this term is explained in 62.) The note to be altered should 
 therefore be kept in the same voice, as at (). If this be done, 
 no false relation occurs by another voice moving to the same 
 note, as at (c}. The false relation is also not considered to exist 
 when the altered note forms part of a fundamental discord ( 197). 
 For instance in example (a) of 235, the first chord of the third 
 bar has CJ| in the bass, and the second has Cl| in the alto. But 
 the latter is not objectionable, because it forms part of a chord of 
 the dominant seventh. Neither do the chromatic chords in a key 
 (to be explained in a later chapter) generally cause false relation, 
 because the essence of false relation is the confusion or obscurity 
 of key which it produces, and chromatic chords, if properly treated, 
 have not this effect. But even in this case, when the chromatic 
 chord has the same root as the preceding or succeeding diatonic 
 chord the altered note should be kept in the same voice. False 
 relation, moreover, has no bad effect when the third of the first 
 chord is either the root or the fifth of the second chord. 
 
 
 At (d) we can modulate from C major to A minor without bad 
 effect because E, the third of the chord of C, is the root of the 
 second chord. At (e) we proceed to D minor, the third of the 
 chord of C being the fifth of the chord of A. It should be added 
 that the interposition of one intermediate chord will not destroy 
 the effect of a bad false relation, e.g., 
 
 J J 
 
 There are few questions on which theorists differ more widely 
 than that of false relation, many passing it over without notice ; 
 and there is probably no rule to which there are so many 
 exceptions. Only experience will enable the student to know 
 with certainty when an apparently false relation is objectionable, 
 and when allowable.
 
 HARMONY : 
 
 [Chap. X. 
 
 EXERCISES TO CHAPTER X. 
 
 [In the exercises henceforth to be given modulations will be 
 introduced, and greater variety and interest obtained. The 
 student must notice what accidentals are employed; this will 
 enable him to determine into what keys the music modulates.] 
 
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 (VII.) Hymn Tune. 
 
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 HARMONY 
 
 [Chap XI. 
 
 CHAPTER XL 
 
 AUXILIARY NOTES, PASSING NOTES, AND ANTICIPATIONS. 
 
 245. In explaining the various examples quoted from the 
 works of the great masters to illustrate the chords already treated 
 of, it has frequently been needful to mention that some of the 
 notes in the extracts were not notes of the harmony, but auxiliary 
 or passing notes. It is now time to show the nature of such notes. 
 
 246. An AUXILIARY NOTE is a note preceding or following a 
 note of the harmony at a distance of a second above or below. 
 As all seconds are discords, the auxiliary note will always be 
 dissonant to at least a part of the chord with which it is sounded ; 
 but, as it is not itself an essential part of the chord, like the dis- 
 sonant sevenths treated of in Chapter IX., it is called an unessential 
 discord. 
 
 247. An auxiliary note is frequently taken, and (excepting in 
 certain cases to be specified presently) should always be left, by 
 step of a second. It may occur on either a stronger or a weaker 
 accent than the harmony note to which it belongs, and it may- be 
 at a distance from that harmony note of either a tone or a diatonic 
 semitone above or below. 
 
 248. When an auxiliary note is above the harmony note, it 
 should be the next note of the diatonic scale of the key in which 
 the music is, whether that note be a tone or a semitone above ; 
 but if it be below the harmony note, it should be a semitone below 
 it, unless such harmony note be the major third of a chord, in 
 which case the auxiliary note may be either a tone or a semitone 
 below the harmony note. 
 
 ' I I I 
 
 Not 
 
 i i [ i 
 
 .* 1* J =4
 
 chap, xi.] ITS THEORY AXD PRACTICE. 113 
 
 At (a) are shown auxiliary notes above the harmony notes, and at 
 (K) they are given below. In this example they are both taken 
 .and left by step ; but we shall see many cases later in the chapter 
 in which they are taken by leap.* 
 
 249. In example (b} of the last paragraph will be seen the note 
 C| in the key of C. A reference to the chromatic scale of C 
 .given in 1 1 will show that this note is not one of the notes of 
 that scale: it was also explained in Chapter III. that the note 
 employed in the chords of the key of C was D? ( 62), and that 
 we cannot also have its enharmonic, C$, without getting more 
 than twelve notes in the key ( 69). But, though C;| cannot be 
 employed as a harmony note in C, it may be used as an auxiliary 
 note a semitone below D. The restriction as to the twelve notes 
 in the key applies only to the notes of the harmony. Similarly 
 1)$, G$, and A$ can be used as auxiliary notes below E, A, and B 
 respectively. 
 
 250. Auxiliary notes may be taken in more than one part of 
 ihe harmony simultaneously, provided that characteristic notes of 
 the harmony remain in some of the parts. 
 
 (/) , (g) W 
 
 =J JJnJJJiiT- iijaUHi 
 
 rrr'Trr 
 
 At (a) is the chord of C major. At (b} an auxiliary note is added 
 in the treble, at (c) in the alto, and at (d) in the bass. In each 
 case the chord remains the chord of C. If auxiliary notes are 
 taken in treble and alto together, as at (e), the C in the bass still 
 preserves the feeling of the chord, as do the E in the alto of (/) 
 and the C in the treble of. (g). But if all three auxiliary notes 
 .are taken together, as at (//), the feeling of the chord of C is gone, 
 and we have instead of it the second inversion of the dominant 
 seventh on G. The notes D, F, and B are no longer auxiliary, but 
 harmony notes. 
 
 * The rule given in this paragraph is almost universally observed by modern 
 composers. Bach, however, frequently adheres to the diatonic scale for an 
 auxiliary note below, as well as above the harmony note, as in the two follow- 
 ing fugue subjects from the " Wohltemperirte Clavier." 
 
 * 
 
 In both these passages a modern composer would have unquestionably written 
 the auxiliary notes a semitone, and not a tone, below the dominant. 
 H
 
 H4 HARMONY: [Chap, xu 
 
 251. It was said in 248 that the auxiliary note below a major 
 third might be either a tone or a semitone below it. If, however, 
 the fifth of the chord have also an auxiliary note below it, which 
 will be at the distance of a semitone, the auxiliary note of the third 
 must also be a semitone below. 
 
 
 Not | 
 
 252. Let the student now examine the auxiliary notes that 
 have been met with in some of our examples. At S 202, in the 
 second bar of (b) the second quaver, C, is an auxiliary note to D 
 taken and left by step. At 2 8 (^) the Et) in bar i is an 
 auxiliary note to F, taken by leap and left by step. In $ 211 (a) 
 the Bfts in the first bar and Cfe in the second are auxiliary notes 
 to C and D respectively. Similarly the upper note of any shake 
 and the upper and lower notes of any turn are always auxiliary 
 notes. In 239 in the third bar of (b) the A$ is an accented 
 auxiliary note, taken by leap of a third and left by step. 
 
 253. There are two cases in which an auxiliary note can be 
 quitted by leap of a third, instead of by step. Such a note, if 
 taken by step of a second from a harmony note, may, instead of 
 returning direct to that note, leap a third to another auxiliary note 
 on the opposite side, provided that this second note return at 
 once to the harmony note from which the first moved. 
 
 * * * * 
 
 J J ^J 
 
 I I 
 
 Two auxiliary notes used in this manner are usually called changing 
 notes. An excellent example of them in two parts will be found 
 in Handel's chorus " For unto us a child is born." 
 
 HANDEL. 
 
 This extract also furnishes an excellent illustration of auxiliary- 
 notes a tone below the third of the chord, and a semitone below 
 the root and fifth. In the third crotchet of the second bar is an 
 auxiliary note a tone, instead of a semitone below the root. Such 
 diatonic progressions were more common in the last century than 
 now. (Compare note to 248.)
 
 Chap. XI 
 
 ITS THEORY AND PRACTICE. 
 
 254. The second case in which an auxiliary note may be 
 quitted by leap is when the harmony notes move by step, and the 
 first note moves a second to an auxiliary note on the opposite side 
 to the next harmony note, to which it then leaps a third. This 
 progression is much more common when the step is upward and 
 the leap downward than in the reverse direction. 
 
 MOZART. Rondo in A minor. 
 
 (*) 
 
 
 HAYDN, and Mass. 
 
 * * * 
 
 &c. 
 
 -9 I ^ 
 
 ! - 
 
 H. LAWES. 
 
 L W * 
 
 ^ f f r 
 
 r 
 ^^^^i 
 
 likji i u L* L 
 
 
 At (a) it is evident that D is not a part of the chord of ^ on E 
 
 which is followed by the common chord on the same bass note ; 
 it is an auxiliary note rising a second to fall a third. At (b) is a 
 series of similar notes in quick time. The very rare reverse case 
 (c), the falling second and rising third, is taken from a song by 
 Henry Lawes given by the late Dr. Hullah in his " Transition 
 Period of Musical History." 
 
 255. When an auxiliary note, proceeding by step from a har- 
 mony note, moves to another note of the harmony, instead of 
 returning to the first one, it is called a PASSING NOTE. These 
 notes, excepting in chromatic passages, are almost invariably taken 
 according to the diatonic scale of the passage in which they occur. 
 For instance, in the chord of C an auxiliary note below G would 
 be F| ( 248). 
 
 But if, instead of returning to G the auxiliary note proceeds to E, 
 it then becomes a passing note ; and, if the music at the time were
 
 HARMONY: 
 
 [Chap. XI. 
 
 in the key of C, and not of G, this passing note must be Ft} and 
 not Fit. 
 
 not F$. 
 
 256. In the minor key, in order to avoid the interval of the 
 augmented second between the sixth and seventh degrees of the 
 scale, it is customary to use for passing notes the two melodic forms 
 of the scale given in $ 169, 170, instead of the harmonic form 
 given in 171. Therefore in passing from the dominant to the 
 leading note of the minor key, or in the reverse direction from 
 leading note to dominant, the major sixth of the key is mostly 
 taken as the passing note. 
 
 ife^-j^^L^p^^a 
 
 jf. ^- J 1 11 U I 1---3-1 U 
 
 Similarly, in passing from submediant to tonic, or from tonic to 
 submediant, the minor seventh of the scale will be used. 
 
 * ~F^" J=^~ r^ H^a~ 
 
 i fe i r 
 
 It \s possible in both these cases to employ the harmonic form of 
 the scale ; but this is very unusual. 
 
 257. If the two harmony notes are a fourth apart, as, for 
 instance, in rising from the fifth to the root of a chord, there will 
 evidently be two passing notes between them. 
 
 * * 
 
 In such a case, it is rot good to return from the second passing 
 note to the first, even though by a change of the harmony the 
 first may have become a harmony note, 
 
 Bad. 
 
 but the second note must proceed in the same direction till 
 reaches the next note of the first chord. 
 
 Good.
 
 Chap. xi. ^ ITS THEORY AND PRACTICE. 117 
 
 If two passing notes follow one another in rising from the fifth to 
 the root of the tonic chord in a minor key, it is usual to employ 
 the melodic forms of the minor scale, as in 256, with the major 
 sixth and seventh in ascending, and the minor seventh and sixth 
 in descending. 
 
 258. The only case in which a passing note may be quitted by 
 leap is when the two harmony notes which it connects are at a 
 distance of a third apart, and the passing note instead of going 
 direct to the second harmony note, leaps a third to the other side 
 of it, and then returns. 
 
 i J I Jiii 
 
 =7 : J * J II J J j^-n 
 
 This is another variety of the ' changing notes ' mentioned 
 in 253.* 
 
 259. Chromatic notes, as well as diatonic, may be used as 
 passing notes ; but if a chromatic note has been introduced it is 
 best to continue the progression by semitones until the next 
 harmony note is reached. 
 
 I I L!__ N^Ll [ _4- 
 
 i 
 
 In this example the chromatic notes are written as Cf and D#, 
 not as \)7 and E? (see 249). This is because they are moving 
 upwards ; if they were descending, it would have been better to 
 write them as flats. This point will be more fully explained later. 
 260. We will now, as with the auxiliary notes, refer to the 
 passing notes in some of the extracts quoted in previous para- 
 graphs. At (d) in 157 we see on the last quaver of the second 
 bar an example of an accented passing note, E. A still more 
 striking example is shown at 160 (g), where the G$ at the 
 beginning of the second bar is not only a passing note on the 
 first beat of the bar, but is of greater value than the harmony 
 notes which precede and follow it. The two scales quoted 
 from Handel in the second example of 170 are instructive, 
 because though they contain exactly the same notes, the passing 
 notes are different, as they are differently accompanied. In the 
 first scale C, B, G, and EC] are passing notes, and in the second 
 C, A, F$, and Efcj. At 183 (<) the second quaver of the bar, 
 
 * Some exceptional progressions of auxiliary notes will be shown later in 
 this chapter ; but these do not invalidate the general rules here given.
 
 HARMONY : 
 
 [Chap. XI. 
 
 C, is an unaccented passing note ; and at ^217 (l>) the quaver 
 B is an accented passing note. 
 
 261. It is important to notice that auxiliary and passing 
 notes cannot make " false relation " (t$ 62, 244), and that even 
 the interval of the diminished octave may be sometimes used 
 with very fine effect. 
 
 BEETHOVEN. Mass in D. 
 
 In the first bar of this example the harmony note of the second 
 crotchet of the treble is B?, the chord being the first inversion of 
 the dominant minor ninth, which will be explained later. The 
 Ct) is an auxiliary note taken by leap of a third ; the Gf is 
 another auxiliary note taken also by leap, but instead of going at 
 once to A it returns to B? first, the latter note thus producing 
 the effect of a changing note. In the second bar, the harmony 
 is that of the tonic chord of D minor ; C is an auxiliary note 
 taken by leap, Bi? is another auxiliary note resembling a passing 
 note, following the first, and therefore moving in the same 
 direction till it reaches the harmony note (S 257), and G| is 
 now a changing note interposed before the harmony note A. 
 It is rare to find a passing note taken by leap, as in this second 
 bar, when at the distance of a third from the harmony note; 
 it is usually a second above it. 
 
 262. Another curious example of an auxiliary note at the 
 distance of a diminished octave from the harmony note is found 
 in one of Bach's Church Cantatas. 
 
 J. S. BACH. " Herr Gott, dicli loben alle wir.' 
 
 Here the effect is much harsher than in the passage from Beet- 
 hoven just quoted, and the extract is not given for the student's 
 imitation, but to show its harmonic possibility. 
 
 263. Besides the two kinds of auxiliary notes already spoken 
 of, there is a third species somewhat resembling them, though 
 possessing features of its own. Sometimes one or more parts 
 of the harmony proceed prematurely to their notes in the next 
 chord, while the others remain. This effect is known as an 
 ANTICIPATION.
 
 Chap. XI. J 
 
 ITS THEORY AXD PRACTICE. 
 
 119 
 
 cfj 
 
 (), 
 
 f=S= 
 
 N 
 
 -^ 
 
 HANDEL. 
 
 5 r^ - 
 
 ^r^^j=i 
 
 " Me 
 
 * 
 
 ssiah." 
 
 F 1 ^^ 
 
 25* 
 W? 
 
 ^r 
 
 1F=* = 
 
 F 
 
 -S- 
 I > - 
 
 I *= 
 
 -n 
 
 r 
 
 1 
 
 E5 1 
 
 r 
 r^fl 
 
 
 l=- i ^ 
 
 
 ]. S. BACH. "Giebdich zufrieden." 
 
 Cfl 
 
 
 SCHUBERT. Sonata in A minor, Op. 42. 
 
 At () is an anticipation of the following harmony note in 
 the treble, and at (b) in both treble and tenor. Such are to be 
 mostly found, as in these examples, in cadences ; but the 
 passages at (c) (d} show that they can be used also in other 
 ways. At (c) each harmony note is anticipated, and at (d) 
 not only harmony notes but passing notes are thus treated. 
 
 264. Occasionally, though rarely, a note is anticipated in one 
 voice, and then taken in another, as in the following examples. 
 
 BACH. " Es ist dir gesagt." 
 f)tt^] * i . I 
 
 g-gg-f:^_^._. =_e*-j -
 
 120 
 
 HARMONY : 
 
 [Chap. XI. 
 
 At (a) the E of the alto is anticipated in the treble, and 
 at (b) the B of the bass is similarly treated. 
 
 265. Exceptional treatment of auxiliary notes is sometimes. 
 found in the works of the great masters. The following passages 
 must be looked upon as licences, as they cannot be explained by 
 any of the rules given in this chapter. They are therefore not- 
 recommended for imitation. 
 
 BACH. " Was Gott thttt." 
 
 At (a) and (b) will be seen auxiliary notes leaping a fourth and 1 
 a diminished fifth, instead of a third, to the harmony note. At 
 (b} we have also an excellent example of the anticipations spoken 
 of in 263. The note D at the end of the first bar of example 
 (c) is clearly an auxiliary note, because the harmony is defined 
 as being that of the common chord of F by the arpeggio in the 
 bass. (&* 192.) Here the auxiliary note leaps a sixth, while 
 at (d) it leaps a fifth. The F in example (d) may also be con- 
 sidered as an anticipation of the implied harmony of the following 
 chord (the dominant seventh), and would thus present some 
 analogy to example (b} of 264. The student will do well to 
 conform to the rules we have given till he has gained sufficient 
 experience to know when they may be safely relaxed.
 
 chap. xii. Irs THEORY AND PRACTICE. 121 
 
 CHAPTER XII. 
 
 THE CHROMATIC SCALE CHROMATIC TRIADS IX A KEY. 
 
 266. A chromatic scale has been defined ( n)as "a scale 
 consisting entirely of semitones." It contains twelve notes within 1 
 the octave, while a diatonic scale contains only seven. It will be 
 remembered that the intervals between the consecutive notes of 
 any major scale are five tones and two semitones. Each tone can 
 be divided into two semitones, thus making twelve semitones 
 in all. 
 
 267. To obtain the correct harmonic form of any chromatic 
 scale from its tonic major ( 175), we divide each tone into its 
 constituent parts, a diatonic and a chromatic semitone ( 7). Let. 
 us do this with the scale of C major. 
 
 Here we have tones between C and D, D and E, F and G, G and 
 A, and A and B. It is clear that each of these tones can be 
 divided in two different ways. Thus, between C and D we can 
 put either Cjjl or D7, in the former case having the chromatic 
 semitone as the lower one, and in the latter the diatonic ; and 
 similarly we can put either a sharp or a flat between each of the 
 other tones. How shall we know which to take ? 
 
 268. If we refer to the three fundamental chords given in 71 
 which contain the whole material of the key, we shall find the 
 necessary guidance. There is no C|l in the key at all ; but we see 
 D? as the minor ninth of the tonic. Similarly we find E? (not 
 D$) as the minor ninth of the supertonic, F$ (not G?) as the third 
 of the supertonic, A? (not Gf) as the minor ninth of the dominant, 
 and B? (not A;|) as the seventh of the tonic. As all the harmonies 
 of the key are made from these three fundamental chords, it follows 
 that the correct harmonic form of the chromatic scale of C will be 
 
 "t< 
 
 as given in n. To write a correct chromatic scale, the upper 
 note of each tone in the major key except that between subdomi- 
 nant and dominant must be flattened for the intermediate semi- 
 tone, and the subdominant of the scale is the only note sharpened. 
 To illustrate this, we give the chromatic scales of A major and A
 
 i22 HARMONY: [ciiap. xn. 
 
 flat major. Let the student compare them with the scale of C 
 given above. 
 
 P 
 
 269. As a matter of convenience, the chromatic scale is often 
 written, especially in ascending passages, with a different notation 
 from that just given. In the chromatic scale of C, for example, 
 Cf, D$, and G$ are often substituted for 1)1?, E? and Ai? ; some- 
 times, also, though more rarely, Aftis written instead of B7. But 
 F$ is almost invariably retained. The scale of C will then appear 
 in the following form. 
 
 This form of the scale is somewhat easier to read, especially in 
 rapid music, as it has fewer accidentals. This is probably the 
 chief reason for its frequent adoption ; because we find that in the 
 descending chromatic scale the correct notation is usually adhered 
 to most likely for the same cause, as it is not uncommon in this 
 form to find the flattened fifth instead of the sharpened fourth, 
 thus saving an accidental. 
 
 270. It is an interesting point, and worthy of passing notice, that the three 
 notes which in the convenient, though inaccurate, notation just given are raised 
 in ascending, are the minor ninths of the three generators in the key. When 
 we come to deal with chords of the minor ninth it will be seen that where that 
 note resolves (as it frequently does) by rising a chromatic semitone, it is usual 
 to adopt the same notation, and to write it, incorrectly but conveniently, as the 
 sharpened octave of its generator. 
 
 271. All the chords of either a major or minor key with which 
 the student is at present acquainted are diatonic chords, that is 
 chords in accordance with the key signature ; and (excepting the 
 leading note of the minor key, which [$ 172] is not considered as 
 a chromatic note) they require no accidentals. But the chromatic 
 notes may also be used in chords of the key under certain con- 
 ditions now to be explained. 
 
 272. When treating of modulation in Chapter X., it was shown 
 that a change of key was effected by putting an accidental before 
 one of the notes of the key we were quitting, the altered note thus 
 becoming a diatonic note of the new key. Thus in the example 
 in 231 the sharp before the F in the third chord of (a] takes the 
 music from C into the key of G, the new key being established by 
 the chords that follow. But it was pointed out in 229 that no 
 single chord can ever define a key. As all music must be in some 
 key, it follows that every chord must be either in the same key as
 
 Chap. XII. J 
 
 Irs THEORY AND PRACTICE. 
 
 123 
 
 the chords that precede it, or in the same key as the chords that 
 follow it. Now if the chords preceding any one chord be in the 
 same key as the chords following that chord, then, if the chord 
 have one or more accidentals before it, it will be a CHROMATIC 
 CHORD in the key. 
 
 273. To make this clearer, we will vary the example given in 
 231 (a). We will take the first three chords as before, but will 
 follow the third chord differently. 
 
 g 
 
 The first two chords are, as before, in the key of C, for there is no 
 H? to indicate the key of F. The third chord (*) looks as if we 
 were about to modulate into the key of G ; but the fourth -and 
 fifth chords absolutely define the key of C ($ 219). The chords 
 preceding and following the chord marked * are therefore all in 
 the key of C ; consequently the chord * must be chromatic in 
 that key. 
 
 274. Let us take another illustration, this time in a minor key. 
 
 
 
 
 
 
 
 ~g- 
 
 
 
 
 
 
 
 
 ^5 
 
 
 ^ 
 
 *T 
 
 
 " F" 
 
 
 =f= 
 
 
 ^ 
 
 
 
 
 
 p 
 
 
 
 
 
 -&- 
 
 ^b-lr*-' = * 
 
 
 -t : 
 
 
 
 
 
 __ ^L-^^ 
 
 
 -i 
 
 
 
 
 
 
 
 ^ 'H 
 
 At * of (a) there is a modulation to A flat ; the first inversion of 
 the chord of U flat is therefore regarded as diatonic in its relation 
 to the new key, and here it is not considered to belong to the key 
 of C minor at all. But the same chord (*) at (b) is followed by 
 chords which define the key of C minor, and in this latter case the 
 fourth chord is a chromatic chord in C minor. Whenever a 
 modulation takes place, the note inflected by an accidental is 
 regarded as belonging to the key in which it is diatonic. But if 
 there be no modulation such note forms part of a chromatic chord. 
 Hence we get the following definition . A chromatic chord i>i a 
 key is one which contains one or more notes foreign to the signature 
 of that key, but which induces no modulation. 
 
 275. For a reason which will become apparent as we proceed, 
 we will first take the chromatic chords of a minor key. These are 
 three in number. Speaking first of those most frequently used, 
 we find a chromatic major chord on the minor second of the 
 scale. 
 
 Here the root is itself the chromatic note, the third and fifth of
 
 124 
 
 [Chap. XII- 
 
 the chord being respectively the subdominant and submediant of 
 the minor scale. This is one of the most important, and in its- 
 first inversion one of the most frequently employed, of the chro- 
 matic chords. There is no restriction as to the chords which 
 shall follow this chord, if only the key remain the same. 
 
 J. S. BACH. Organ Prelude in C minor. 
 
 These two extracts show the chord in its root position. In (a) it 
 is in the key of G minor, and is followed by the first inversion of 
 the dominant chord of that key ; at (b] it is in F minor, and is- 
 followed by the second inversion of the dominant minor ninth. 
 276. The first inversion of this chord 
 
 is generally known as the "Neapolitan sixth," a name for which it 
 is difficult to give a satisfactory reason. This form is much more- 
 frequently met with than the root position. Here follow three 
 examples of this chord, illustrating some of its most usual Jpro- 
 gressions. 
 
 HAYDN. Sonata in CJ minor. 
 fe 
 
 p 
 u 
 
 <*> 
 
 HANDEL. "Joseph." 
 
 . _ 
 
 r=^=r 
 
 i *, ! 
 
 J . *2 . J
 
 Chap. XII. J 
 
 ITS THEORY AXD PRACTICE. 
 
 I2 5 
 
 SCHUBERT. Mass in BiJ, Op. 141. 
 * 
 
 At (a) the ' Neapolitan sixth ' is followed by the tonic chord here 
 in its second inversion, though the bass might also have descended 
 to the first inversion of the tonic ; at (b) the chromatic chord is 
 followed by another chromatic chord (the supertonic minor gih) 
 in the same key ; and at (c) it is followed immediately by the root 
 position of the dominant chord. In the second and third bars 
 from the end the student should notice that the A) in the treble 
 and At] in the tenor make no false relation (>$ 244). 
 
 277. The second inversion of this chord is also sometimes, 
 though more rarely, to be found. 
 
 HANDEL. " Messiah." 
 
 SCHUBERT. " Song of Miriam.' 
 
 At (a) the chord is followed by a dominant seventh, and at (b) by 
 the second inversion of the tonic chord. The consecutive fourths 
 with the bass ( 106) do not here produce a bad effect. 
 
 278. The harmonic derivation of the chromatic chord on the minor second 
 of the key will be seen hy reference to 66. It is composed of the minor gih, 
 nth, and minor i3th of the tonic, the lower note being a primary, and the 
 other two secondary harmonics of the generator. 
 
 279. A chromatic chord by no means inferior in importance 
 to that just treated of is the major common chord on the super- 
 tonic of the minor scale. 
 
 Here both the third and the fifth are chromatic notes, and the 
 chord is itself identical with the dominant chord of G major or 
 minor, from which, however, it is easily to be distinguished by its
 
 126 
 
 HARMONY 
 
 [Chap. XII, 
 
 treatment. The fifth of this chord is free in its progression, but 
 the third is subject to special rules which we must now explain. 
 
 280. It will be noticed that the third of this chord is the lead- 
 ing note of the dominant key ; and if it be treated as such, and 
 rise a semitone to the tonic chord of that key, we shall evidently 
 have a modulation, and the chord will then be not chromatic in C 
 minor, but diatonic in G. In order to keep it in C minor, we 
 must, if the F$ rises a semitone, make G part of the tonic chord of 
 the key, following it by chords which are also clearly in that key. 
 In general terms, if the third of the super tonic chord rises, it should 
 be followed by some form of the tonic chord. 
 
 281. In treating of modulation, it was shown how by lowering 
 the leading note a chromatic semitone we could pass to the sub- 
 dominant key (Example (a) 242). In the chord we are now 
 considering, it is clear that if F$ were the leading note of G, we 
 could modulate to C by changing it to Ft}. Evidently, if, as a 
 constituent of the supertonic chord, we lower it to FC| when the 
 chord changes, we prevent any impression of the chord being the 
 dominant chord of G, for it does not resolve in that key. It 
 matters not what chord does follow, so long as Ft) is a part of that 
 chord. The major third of this chromatic chord, therefore, when 
 it does not rise should fall a semitone, and be followed by some chord 
 of which the subdominant of the key forms a part. It is very im- 
 portant to remember that as the progression of the third of this 
 chord is fixed, it must never be doubled ; for if it be, either one 
 of the thirds must move incorrectly, or there will be consecutive 
 octaves. 
 
 282. The following simple chord progressions will serve to 
 illustrate the rules just given. 
 
 At (a) the chromatic chord on the supertonic (*) has its third 
 rising a semitone ; it is therefore followed by the tonic chord 
 here in its second inversion. Let the student observe the way in 
 which the basses are figured. In the second chord of (a) the 
 second inversion of the dominant seventh requires to be fully 
 figured, because the 6th bears an accidental ($ 209). In the 
 fourth chord it is needful to figure the common chord fully because 
 both 3rd and 5th bear accidentals (S 152). At the last chord but 
 one of (a) will be found a sign which has not yet been explained. 
 A line placed after a figure or an accidental (as here, t} -) indicates 
 that the note represented by the preceding figure is to be con-
 
 Chap. XII.] 
 
 ITS THEORY AND PRACTICE. 
 
 127 
 
 tinned. In the present instance the t^rd (Btj) is to be held on 
 while first the 8th and then the yth of the bass note are sounded. 
 
 283. Though it was said above that this chromatic chord on 
 the supertonic was one of the most important, it is comparatively 
 seldom that it is found in the simple form we are now describing. 
 Generally the yth, frequently also the pth is added to it, as will 
 be seen in the following chapters. We shall give directly a few 
 examples of its employment without these dissonant notes. 
 
 284. When used in its first inversion this chord is subject to 
 the same rules as regards its third as in the root position. 
 Obviously, as its third is now in the bass, if this note rises, it must 
 be followed by the second inversion of the tonic chord. If the 
 third falls there is more choice. The second inversion of the 
 chord is very rare. 
 
 285. We now give a few examples of the use of this chord by 
 the great masters. 
 
 WAGNER. " Der Fliegende Hollander." 
 
 3=3t 
 
 BEETHOVEN*. Sonata, Op. 31, No. 2. 
 
 ic V .b*. & "ig- 
 
 -I | r-J 1 I 
 
 s^PS. 
 
 MENDELSSOHN. Fantasia, Op. 28. 
 
 The first of these extracts illustrates what was said of ambiguous 
 chords in speaking of modulation ( 231). Immediately preceding 
 this quotation is a cadence in the key of F. The first seven chords 
 of (a) may be either in that key, or in B flat (the key of the piece) or
 
 i 2 8 HARMONY: i_chap. xn. 
 
 in D minor ; the chromatic chord at * clearly belongs to this last 
 key as neither F nor B flat contains G$. The chord of E major 
 must therefore be in the key of D minor ; its third (G$) rises a 
 semitone, and the chord is consequently followed by the tonic 
 chord of D minor, leading directly afterwards to a full close in that 
 key. Example (b) gives us some fresh instruction. In the first 
 bar we see the Neapolitan sixth of A minor (S 276) ; it is clear that 
 if the key were D minor, the C of the first chord would be sharp. 
 The fourth bar shows how one chromatic chord in a key (here the 
 Neapolitan sixth again) can be followed by another, the first 
 inversion of the major chord on the supertonic* ; the third of this 
 chord (D$) rises, as before, but now, being in the bass, it is followed 
 by the second inversion of the tonic chord. At (c} the third of 
 the first inversion (B$) falls a semitone to the seventh in the chord 
 of the dominant minor ninth, BCj being an essential note of the 
 new chord. The last example (d} shows the rare second inver- 
 sion, the bass being quitted by step of a chromatic semitone. 
 These examples will, it is hoped, sufficiently explain the treatment 
 of this chord. 
 
 286. The harmonic derivation of this chord is extremely simple. It is 
 merely the generator, major third and fifth of the supertonic ( 69). It has 
 been already said ( 176) that the minor key is derived from the same three 
 generators as its tonic major. 
 
 287. In addition to the two chromatic chords already described 
 there is another, much less frequently used, yet too common to 
 allow of our considering it as an exceptional progression. In $ 193 
 it was said that the tonic major chord was often used to conclude 
 a minor piece, and that in this connection only, the major third 
 was not considered a chromatic note. But the major tonic chord 
 can also be used as a chromatic chord in the course of a move- 
 ment in the minor key. In this case, however, especial care has 
 to be taken to treat the chord in such a manner as not to produce 
 the impression of a modulation. The following examples will 
 show how this is effected. 
 
 3HN. Presto Agitato. 
 
 it -*- 
 
 w 
 
 Compare example (/') of 276.
 
 Chap, xii.] Irs THEORY AND PRACTICE. 
 
 129 
 
 SCHUBERT. Winterreise, No. 15. 
 
 SPOHR. "Calvary. 1 
 
 -^- 
 
 i 
 
 r 
 
 At the third bar of (a) the first chord is the first inversion of the 
 tonic major chord in the key of B minor ; the first chord of the 
 fourth bar is the same chord in its root position. This extract 
 also furnishes in the third bar an example of the supertonic chro- 
 matic chord in root position proceeding to dominant harmony. 
 In the example of the same position quoted in 285 (a) the 
 chord was followed by the tonic chord. The extract (b) shows 
 the first inversion of the tonic major chord ; and also (in the 
 third and fourth bars) the root position and first inversion of the 
 chord on the minor second of the key. The last illustration 
 (<:) gives the second inversion of the tonic major chord. 
 
 288. If we compare these three examples (and many similar 
 ones might be given), we shall see that the treatment of the 
 chromatic note (the major third of the chord) is the same in 
 each instance. It is always approached and quitted by step of a 
 semitone. The D$ in the bass of the third bar of (a) is only 
 apparently, not really, an exception to this rule ; for though it is 
 immediately preceded by F$, it will be noticed that this note 
 is only an ornamental resolution (202) of the E, the seventh of 
 the chord preceding the tonic chord, which really resolves by 
 descending a semitone to the major third of the tonic chord. To 
 understand the progression in example (b) it is important to re- 
 member that the notes of a chord taken in arpeggio are in harmony 
 to be considered as if sounded together ( 192). The first two 
 bars of the lower staff of (b} are therefore equivalent to
 
 130 
 
 HARMONY : 
 
 [Chap. XII. 
 
 where it will be clearly seen that Efc| is approached from E7 and 
 proceeds to F. Here the progression is upwards, while at (a) 
 it was downwards. At (c) the chromatic D$ lies between E and Dt}. 
 
 289. From these and similar examples of the practice of the 
 great masters, we deduce the following rule for the treatment of 
 this chord. A major chord on the tonic may be taken as a chroma- 
 tic chord in a minor key, provided that its third is approached from 
 the semitone on one side of it, and proceeds to the semitone on the 
 other side. 
 
 290. Every triad in a minor key, whether diatonic or chroma- 
 tic, with the exception of the minor tonic chord itself, which 
 would contradict the major key, can be used as a chromatic triad 
 in the tonic major of that minor key.* It will be hardly necessary 
 to remind the student that the triads on the dominant and leading 
 note are the same in both major and minor keys. The major 
 tonic chord, which is chromatic in the minor key ( 287), is of 
 course already in the major key as a diatonic chord. 
 
 291. No new rules need be given for the treatment of the 
 chromatic triads in the major key. The chromatic chord on the 
 supertonic is subject to the same restrictions as regards its third 
 as in the minor key (280, 281) ; but the other chords are free 
 in their progression, provided always that they are followed by 
 chords in the same key. It will be unnecessary to illustrate all 
 the positions of each chord ; but a few examples may be given. 
 
 SCHUMANN. Novellette, Op. 21, No. i. 
 
 m 
 
 * We sometimes meet in the major key with a chord containing the same notes 
 as the tonic minor chord ; but in this case the treatment and progression of the 
 chord prove it to be really the upper portion of the chord of the supertonic eleventh. 
 It will therefore be explained in the chapter on chords of the eleventh ( 394).
 
 chap, xii.] Irs THEORY AND PRACTICE. 
 
 I3 1 
 
 BEETHOVEN. Symphony, No. 8. 
 
 *1 *! | 
 
 r 
 
 r 
 
 3=t 
 
 M 
 
 BEETHOVEN. Mass in D. 
 
 3^* 
 
 JL^JJL 
 
 - 
 
 SCHUBERT. Sonata in B, Op. 147. 
 
 . fa ," 
 
 After the explanations already given, it will not be needful to do 
 more than to remind the student of the nature of each of these 
 chromatic chords. At (a) is shown the root position of the chord 
 on the minor second of the key of F, and at (fr) the first inversion 
 of the same chord (the Neapolitan sixth), also in F. The chroma- 
 tic chord on the supertonic is seen at (<:) in its root position ; 
 while (d) gives the first inversion and root position of the 
 diminished triad on the supertonic ; and (<?) the first inversion of 
 the minor chord on the subdominant. Lastly at (/) are shown 
 the root position and second inversion of the chord on the minor 
 sixth of the scale the submediant chord in the minor key. 
 
 292. We now give for reference a complete table of the 
 chromatic triads in the keys of C minor and C major. 
 
 In C minor. 
 
 n C major. 
 
 It will be noticed that though it was said in 290 that every 
 triad of the minor key except the tonic chord can be used 
 chromatically in the major key, we have not included the aug- 
 mented triad on the mediant ( 184) in the above table. This is 
 because whenever that chord is employed in a major key it is 
 always in its real character, as a chord of the dominant minor 
 thirteenth ; it will therefore be treated of in that relation in a 
 subsequent chapter. 
 
 293. Jn Chapter X. it was shown how modulation was fre- 
 quently effected by means of some chord common to the key 
 which was quitted and that which was entered. By the chords
 
 HARMONY. 
 
 [Chap. XII. 
 
 treated of in this chapter, the means of modulation are very 
 largely increased ; for a chord can be taken as chromatic in one 
 key and quitted as diatonic in another ; or conversely, it may be 
 taken as diatonic in one key and quitted as chromatic in another. 
 We will give one example of each method. 
 
 BEETHOVEN. " Sonata Pathetique." 
 
 Here a modulation is made from C minor to A flat by taking the 
 chord * as the Neapolitan sixth in the former key, and quitting it 
 as the first inversion of the subdominant chord in the latter. 
 
 SCHUMANN. Novellette, Op. 21, No. i. 
 
 In this passage we commence in G flat major. At * the 
 chord of G flat is taken as the tonic chord in that key and quitted 
 as the chromatic chord on the minor second of F, to which key a 
 modulation is thus effected. If the student will accustom himself 
 to analyse the harmonies in the works of the great masters, he 
 will find on nearly every page of their works modulations by 
 means of chromatic chords thus treated. 
 
 EXERCISES TO CHAPTER XII. 
 
 [In these exercises occasional passing and auxiliary notes will 
 be introduced. It is not usual to figure an unaccented passing or 
 auxiliary note. The student may introduce such notes where he 
 can leave them by step. A passing note in the bass is indicated 
 by a line below it which has been continued from the preceding 
 harmony note. If the harmony note has only a line below it, 
 with no figures, it shows that it is the bass of a triad in root 
 position.] 
 
 (i.) 
 
 (II.) 
 
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 Chap. XII.] 
 (III.) 
 
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 134 HARMONY : [Chap. xin. 
 
 CHAPTER XIII. 
 
 THE FUNDAMENTAL CHORDS OF THE SEVENTH ON THE SUPER- 
 TONIC AND TONIC. 
 
 294. If the chord of the dominant seventh shown in Chapter 
 IX. 197 be examined, it will be found to consist of the four 
 lowest notes of the fundamental chord of the dominant given in 
 67. If we in the same manner take the four lowest notes of the 
 fundamental chords of the other two generators in the key, the 
 tonic and supertonic (66, 69), we shall obtain chords of the 
 seventh on those degrees of the scale precisely resembling the 
 chord of the dominant seventh in their intervals, but differing 
 from that chord in the fact that they are both chromatic, the 
 third (and in minor keys the fifth also) of the supertonic chord 
 requiring an accidental, as will also the seventh in the major key, 
 and the third in the minor key, of the chord of the tonic seventh. 
 Here therefore we have chromatic discords in the key. Of these 
 the supertonic seventh is much the more frequently used and the 
 more important ; we will therefore deal with it first. 
 
 THE CHORD OF THE SUPERTONIC SEVENTH. 
 
 295. It has been more than once said (6i, 197) that the 
 distinguishing intervals of any fundamental discord are the major 
 third, perfect fifth, and minor seventh from the generator. If we 
 add a seventh to the diatonic triad on the supertonic, which con- 
 tains both in the major and minor key a minor third from its 
 root, it is clear that we shall have a chord differing in its nature 
 from the chord of the dominant seventh.* In order therefore to 
 obtain a chord identical in its intervals with the chord of the 
 dominant seventh, we add a seventh to the chromatic chord on the 
 supertonic (279). In the keys of C major and C minor the 
 chord appears thus 
 
 It will be seen that the third of this chord is a chromatic note in 
 the major key, and that both third and fifth are chromatic in the 
 minor key. 
 
 296. Just as the chromatic common chord on the supertonic 
 
 * It has been already seen in 75, 177 that the diatonic chord on the super- 
 tonic consists of the fifth, seventh, and major or minor ninth of the dominant.
 
 Chap, xiii.] ITS THEORY AND PRACTICE. 135 
 
 is identical with the dominant chord of the dominant key, the 
 chord we are now considering is identical with the dominant 
 seventh of the dominant key, being distinguished from the latter 
 by its treatment. It is figured like the dominant seventh, but 
 with this difference, that the chromatic notes (the third in the 
 major, and the third and fifth in the minor key) must always be 
 indicated. Thus the two chords shown above would be figured 
 
 7 7 
 
 ' and h 5 respectively. 
 i 
 
 297. If a modulation to the dominant key be effected by 
 means of this chord, it obviously ceases to be a chromatic chord 
 (274). In order to avoid a modulation, the rules for the treat- 
 ment of the third of the chord given in 280, 281 must be 
 strictly adhered to. The fifth of the chord is free in its pro- 
 gression ; the seventh must either fall a semitone to the third of 
 the dominant chord, or remain stationary, or (in two cases to be 
 presently explained) leap to a note of the following chord. 
 
 298. As the chord of the supertonic seventh contains the 
 same notes as the chord of the dominant seventh, it will evidently 
 be susceptible of the same inversions, which will be figured in 
 the same way as the inversions of the dominant seventh, always 
 remembering that the chromatic note or notes of the chord must in 
 all cases be figured, except in the first inversion of the chord in 
 the major key, where the only chromatic note is in the bass, and 
 is of course indicated by an accidental. 
 
 299. In 216 we saw that the seventh in the chord of the 
 dominant seventh could sometimes be doubled, even when the 
 generator was present, one of the sevenths being then free to leap 
 as if it were a concord, while the other receives its regular 
 resolution. The same is the case with the supertonic_seventh, 
 as in the following example. 
 
 HAYDN. 4th Mass. 
 
 *_ 3 3_ 
 
 At * the seventh is doubled and the fifth omitted. The seventh 
 in the treble leaps to the third of the following chord, while the 
 seventh in the tenor remains stationary. 
 
 300. What was said in the last chapter as to the progression 
 of the supertonic chromatic chord without the seventh applies 
 equally when the seventh is present. If the third rises a semi- 
 tone (as in the example from Haydn just quoted), the chord
 
 i 3 6 
 
 HARMONY: 
 
 [Chap. XIII. 
 
 should be followed by some position of the tonic chord ; if the 
 third falls a semitone to the subdominant of the key, that sub- 
 dominant must be a note of the next chord. 
 
 301. Like the seventh in the chord of the dominant seventh, 
 the supertonic seventh may be transferred to another part of the 
 harmony, either with or without a change of chord, provided it is 
 properly resolved in the part in which it last appears. The most 
 usual form of this progression is when the fifth of the chord of 
 the seventh proceeds in the following chord to the note which 
 was the seventh, or to its octave. 
 
 E. PROUT. iooth Psalm. 
 
 As in the case of the dominant seventh ( 212), the second inver- 
 sion of the supertonic seventh is often found without the generator, 
 and in this case the seventh may rise or fall, and may be doubled 
 freely. The prohibition of the doubling of the third ( 278) still 
 holds good. 
 
 302. This chord (like the chromatic common chord on the 
 supertonic 285 (a), 291 (r)) is the same in both major and 
 minor keys, and the rules given for its treatment apply alike to 
 both. It is important to notice that the chord may be resolved 
 direct on the dominant chord without necessarily producing a 
 modulation, provided that the subdominant of the key, contradict- 
 ing the chromatically sharpened third, be introduced immediately 
 after the dominant chord. In this case the three chords together 
 must be taken as defining the key ( 229). 
 
 303. We now give a series of examples of this chord, in all 
 its positions, from the works of the great masters. Its employ- 
 ment is so frequent that the only difficulty is to make the best 
 selection. 
 
 SCHUMANN. "Paradise and the Peri. 1 
 
 EL. t==^
 
 chap, xin.j ITS THEORY AND PRACTICE. 
 
 f* BACH. "Ach, Gott, wie marches Herzeleid." 
 
 
 
 SCHUBERT. Overture. "Rosamunde." 
 
 r ft 
 
 j 
 
 SCHUMANN. " Bunte Blatter," Op. 95. 
 
 
 -4 J _ ^.-[-4 g=*z 
 
 Q* 
 
 I 
 
 ^,BRAHMS. " Deutsches Requiem." 
 
 ' I ^ ' I i 
 
 9S9-- 
 
 
 J- A A 
 
 
 SCHUMANN. Novellette, Op. 21, No. 6. 
 
 
 ^ 
 
 r 'g^ ^ f t 
 * 
 
 HANDEL. " Messiah." 
 
 (A)^. ^ ^..p.- A^..> j J ^^ J J^ J 
 
 MENUEI^SOHM. "Athalie."
 
 138 HARMONY: [Chap. xm. 
 
 These examples are full of instruction in many respects. At (a) 
 is seen the root position of the chord, resolved on the first inver- 
 sion of the dominant minor ninth. It will be noticed that the 
 seventh exceptionally rises, because the bass goes to the note to 
 which in the ordinary course the seventh would fall. At (t>) the 
 chord is shown in two forms, first in its root position, resolved on 
 the first inversion of the chromatic chord of the subdominant, and 
 afterwards in its first inversion, resolved on the third inversion of 
 the dominant seventh. In each case the third of the supertonic 
 chord falls a semitone ( 277). 
 
 304. Examples (c) and (d] show other resolutions of the first 
 inversion of this chord. At (c) is an unusual resolution. When 
 the third of the supertonic chord rises, it is generally to some 
 position of the tonic chord ; here, however, it is to the root posi- 
 tion of the dominant seventh. The C$ in the tenor is a passing 
 note ; and it should be observed that the Dt| in the second crotchet 
 makes no false relation with the D$ in the first, because the latter 
 note is part of a chromatic chord in the key ( 244). The regular 
 resolution of the first inversion when the bass rises a semitone 
 (viz., on the second inversion of the tonic chord) is seen at (d}. 
 
 305. The following examples show the different treatments of 
 the second inversion of this chord. At (e) it is resolved on the 
 first inversion of the dominant seventh. Here the seventh of the 
 chord, D, leaps downwards, instead of stepping, to avoid doubling 
 the leading note. (Compare example (a), where the seventh rises 
 for the same reason.) The A in the left hand part of the present 
 extract is not a part of the supertonic chord, but a " Pedal note," 
 as will be later explained.* Example (/) is interesting because 
 here the supertonic chord is preceded by a chromatic concord in 
 the key the chord on the minor sixth of the major key. The 
 F in the tenor of the second bar is of course a passing note. In 
 this example the third of the supertonic seventh (Et|) rises, and 
 the chord is resolved on the second inversion of the tonic chord. 
 The last bar but one illustrates the change in the position of the 
 dominant seventh referred to in 216. 
 
 306. Our next example (g) illustrates what was said in 302, 
 as to the resolution of the supertonic seventh on the dominant 
 chord without necessarily producing a modulation. The second 
 bar of this extract may, of course, be regarded as a transient 
 modulation to the key of E ; but as this is immediately contra- 
 dicted by the Dtj at the end of the bar, the mental effect of the 
 first chord is rather that of a supertonic chord in A. In the last 
 bar of this example, the second inversion is resolved direct on the 
 root position of the dominant seventh. At (h) is* seen the second 
 
 * A " Pedal " is a sustained note, either tonic or dominant, held on through 
 various chords, of some of which it does, and of others it does not form a part. 
 The rules regulating the employment of pedal notes will be given in Chapter XX.
 
 Chap, xin.] ITS THEORY AND PRACTICE. 139 
 
 inversion without the generator, which, as already remarked ( 212), 
 Handel very seldom introduces in the second inversion of a 
 seventh. Here the fifth remains stationary, and the chord is 
 resolved on the first inversion of the subdominant. The student 
 will also notice the exceptional skip from the auxiliary note B at 
 the end of the first bar to E, which may be regarded as a note in- 
 terposed between the auxiliary note and the harmony note A at 
 the end of the bar. The third inversion of the supertonic seventh 
 is somewhat rarer than the other positions of the chord. At (/") 
 is a good example of its use ; it is here resolved on the first inver- 
 sion of the dominant seventh. 
 
 THE CHORD OF THE TONIC SEVENTH. 
 
 307. The chord of the Tonic Seventh consists of the major 
 common chord on the tonic with the addition of the minor 
 seventh, and is composed of the four lowest notes of the funda- 
 mental chord on the tonic given in 66. Like the dominant and 
 supertonic sevenths, it contains the intervals from its generator 
 characteristic of every fundamental discord, the major third, perfect 
 fifth and minor seventh ( 61), and, like these chords also, it can 
 be used alike in the major and minor key. 
 
 308. The reason why the chord of the tonic seventh in the minor key has 
 not a minor third, like the tonic chord, but a major third, will be understood 
 if it is remembered that the tonic common chord is essentially a chord of rest, 
 ( *95) > ar >d therefore the minor third (the igth harmonic) may be sub- 
 stituted for the major third (the 5th harmonic) because it is still consonant to 
 the notes of the chord above and below it. But the addition of the seventh to 
 the tonic chord changes it at once from a concord to a discord from a chord 
 of rest to a chord of unrest. We saw in 199 that the characteristic interval 
 of a fundamental seventh was the diminished fifth between the third and the 
 seventh of the chord. But if we add the minor seventh to a minor chord 
 instead of to a major one, 
 
 we get a perfect instead of a diminished fifth between the third and the seventh 
 of the chord. The important interval which before was a dissonance is now a 
 consonance, and the chord is no longer a fundamental discord. For this 
 reason (as will be seen from the examples to be presently given) the great 
 masters when they use this chord in a minor key invariably retain the major 
 third. It is probably also because the tonic is naturally a chord of rest, that 
 the great masters use tonic discords so much more rarely than those on the 
 dominant and supertonic. 
 
 309. The chord of the tonic seventh 
 
 is evidently chromatic in both major and minor keys, the seventh 
 in the former and the third in the latter key requiring an accidental. 
 As regards the notes of which it is composed, it is identical 
 with the chord of the dominant seventh in the subdominant
 
 140 HARMONY : [Chap. xni. 
 
 key, just as the supertonic seventh contains the same notes as 
 the dominant seventh of the dominant key. Also, as in the case 
 of the dominant and supertonic sevenths, the two notes of this 
 chord which require special treatment are the third and the 
 seventh. 
 
 310. The usual progression for the third of this chord is the 
 same as that of the third in the supertonic seventh it rises or 
 falls a semitone ; but it is possible for it also to rise or fall a tone,* 
 or even, exceptionally, to remain stationary as a note of the next 
 chord. The seventh must either rise a chromatic semitone, or fall 
 a semitone or a tone. 
 
 311. An important distinction between the tonic seventh and 
 those on the dominant and supertonic is that the latter most 
 frequently resolve on a concord, while a tonic seventh almost 
 invariably resolves on either a dominant or a supertonic discord. 
 If the seventh rises a semitone, to the leading note, a dominant 
 discord will follow t ; if the third rises a tone, to the augmented 
 fourth of the scale, this note forms no part of the fundamental 
 chord of the dominant, and is found in the key only as the third 
 of the supertonic chord. If, on the other hand, the third rises a 
 semitone to the subdominant, this note will be no part of the 
 supertonic chord, and must be treated as the seventh of the 
 dominant, for if made the root of a subdominant chord, we shall 
 have a modulation to the subdominant key, and the preceding 
 chord of the seventh will no longer be chromatic. 
 
 312. What was said in 296, 298 about the figuring and the 
 inversions of the supertonic seventh apply equally to the tonic 
 seventh ; the chromatic notes must in every case be figured. 
 As with the other two chords of the seventh also, the second inver- 
 sion is sometimes to be found without the generator. 
 
 313. As a general rule tonic discords are much less frequently 
 used by composers than either those on the dominant or super- 
 tonic. The following examples will illustrate the use of the 
 various positions of the tonic seventh. It will be observed that 
 no fewer than four are from Bach, whose marvellous genius anti- 
 cipated nearly all the resources of modern harmony. 
 
 BEETHOVEN. Quartet, Op. 59, No. 3. 
 (a) 
 
 cres.% 
 
 Obviously the thirds of the dominant and supertonic sevenths cannot rise a 
 tone, as the note a tone above them would not be in the key. 
 
 t The only supertonic discord containing the leading note is the very rare 
 supertonic major thirteenth ; and it is very doubtful whether any example can be 
 found of the tonic seventh resolving on this chord
 
 chap, xiii.] ITS THEORY AND PRACTICE. 
 
 141 
 
 BACH. "Wohltemperirte Clavier," Book i, Prelude at. 
 
 ~~ J , 
 
 w 
 
 MENDELSSOHN. "Athalie.* 
 
 tr - 
 
 MENDELSSOHN. 3rd Symphony. 
 
 BACH. "Wohltemperirte Clavier," Book i, Fugue 22 
 __() ; | ^ -^ i I.' 
 
 BACH. . " Wohltemperirte Clavier, 
 Book i, Fugue 16. 
 
 oo
 
 142 
 
 HARMONY: [Chap. xin. 
 
 (K) SCHUMANN. "Bunte Blatter," Op. 99. 
 
 
 SCHUMANN. " Er, der herrlichste," Op. 42, No. 2. 
 
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 314. At (a) is seen the root position of the tonic seventh. 
 Here the seventh rises a semitone to the third of the dominant 
 chord, and the third falls a tone to the fifth. The exceptional 
 doubling of the leading note at the third chord is due to the fact 
 that Beethoven here wants a special effect of fulness and power, 
 and therefore uses the "double string." At (b} the root position 
 of the tonic seventh is resolved on a dominant major thirteenth 
 (third inversion), the third remaining to be the thirteenth of the 
 following chord, which chord is in turn resolved on a supertonic 
 minor ninth. At (c) the root position of the tonic seventh is re- 
 solved direct on the third inversion of the supertonic minor ninth. 
 The progression of the parts is not clearly shown in this extract 
 from the pianoforte score, and a quotation from the full score 
 would occupy too much space ; but in the full score the seventh 
 and third each fall a semitone to the fifth and ninth respectively 
 of the supertonic chord. 
 
 315. At (d\ (e), and (/) are seen the first and second inver- 
 sions of the tonic seventh, in each case resolved on a dominant 
 discord. At (d} the first inversion of the tonic seventh resolves 
 on the second inversion of the dominant seventh, the third of the 
 tonic chord falling a tone, and the seventh rising a semitone. In 
 both the other instances the seventh falls a tone and the third 
 rises a semitone. These two examples also illustrate what was 
 said in 308 as to the employment of the major third in this 
 chord when used in a minor key. Note also at (/) the unob- 
 jectionable false relation between the F^ and F| in the fourth and 
 fifth quavers. Example (g) shows the second inversion of the
 
 Chap. XIII.] 
 
 ITS THEORY AND PRACTICE. 
 
 chord resolving on the dominant seventh the same progression 
 as at (a), but in a -different position. 
 
 316. The example (k) is instructive as the first instance yet 
 met with of incorrect notation. If the student will play the 
 passage, he will feel at once that it is in the key of F. But D4 is 
 not a note of that key at all. The minor seventh of the tonic is 
 7, and its enharmonic cannot belong to the key (69). The 
 note is therefore really 7, and the chord marked * is properly 
 
 being the second inversion of the tonic seventh without the 
 generator. The reason Schumann has used the incorrect notation 
 is because the seventh is resolved a semitone upwards. We shall 
 see in the next chapter that this false notation is very common 
 with chords of the minor ninth ; but it is extremely rare with 
 chords of the seventh. 
 
 317. At (/) is seen the last inversion of the chord of the tonic 
 seventh. The key of this extract is B flat minor ; but it was the 
 custom in Handel's time not to write more than three flats in the 
 signature, and to mark the others as accidentals. The chord * is 
 here resolved on the second inversion of the supertonic seventh 
 the third of the chord exceptionally leaping, instead of moving as 
 usual by step. In our last extract (k) the third inversion is re- 
 solved on the second inversion of a supertonic major ninth, the 
 third of the first chord being stationary, as in example (fr). 
 
 318. In 293 it was shown how a chromatic concord could be 
 used for the purposes of modulation. The sevenths treated of in 
 this chapter can also be so employed. Obviously if a supertonic 
 or tonic seventh be quitted as a dominant seventh, it is to be 
 regarded not as a chromatic chord, but as diatonic in the new 
 key ( 274). But a supertonic seventh may be quitted as a tonic 
 seventh, or vice versd, the chord being chromatic in both keys. 
 This method of modulating is not common, as it brings together 
 two unrelated keys ( 225), but it is occasionally to be met with, 
 as in the following example. 
 
 WAGNEK. "Die Meistersinger. 
 
 Here what precedes shows that the first bar is in the key of G 
 major. The chord * is taken as the second inversion of the 
 tonic seventh in that key, and left as the second inversion of the 
 supertonic seventh in F major.
 
 144 
 
 HARMONY 
 
 [Chap. XIII. 
 
 319. We shall now conclude this chapter by an attempt to 
 codify the laws governing all the chords of the seventh. As the 
 chords of the ninth, eleventh, and thirteenth are made by adding 
 new notes to this chord, and the notes thus added do not affect 
 the progression of the notes below them in the chord, but only of 
 those above them, the student will save himself much trouble 
 hereafter if he thoroughly masters the rules for the treatment of 
 the chords of the seventh ; for he will then only have to learn in 
 addition the laws for the treatment of the new notes, which are 
 not in themselves difficult. 
 
 320. I. RESOLUTION OF A CHORD OF THE SEVENTH. No 
 
 chord of the seventh ever resolves on its own root.* The 
 dominant seventh resolves on the tonic, submediant, or sub- 
 dominant chord, or on a supertonic discord. The supertonic 
 seventh resolves on a tonic, subdominant, or dominant chord. 
 The tonic seventh resolves on either a dominant or a supertonic 
 discord. 
 
 321. II. PROGRESSION OF THE THIRD. The third in a chord 
 of the seventh must never be doubled. In a dominant seventh it 
 must either rise a semitone, or fall a semitone or a tone. In a 
 supertonic seventh it must rise or fall a semitone. In a tonic 
 seventh, it may either rise or fall a tone or a semitone, or it 
 may remain to be a note of the next .chord. 
 
 322. III. PROGRESSION OF THE SEVENTH. In the dominant 
 seventh, the seventh may rise or fall a semitone or a tone, or it 
 may remain to be a note of the next chord. In a supertonic 
 seventh, it must fall a semitone, rise a tone, or remain stationary. 
 In a tonic seventh it may either rise a semitone, or fall a tone or 
 semitone. Unlike the sevenths on the dominant and supertonic, 
 the seventh on the tonic may never be doubled. 
 
 323. The root and fifth in a chord of the seventh are both un- 
 restricted in their progression ; and it must not be forgotten that 
 the chord may change its position before being resolved, when the 
 third and seventh must be properly resolved in the position in 
 which they last appear. 
 
 EXERCISES TO CHAPTER XIII. 
 
 (I.) Hymn Tune. 
 
 t*E=* 
 
 ^ 
 
 J4 
 2 
 
 6 
 
 sb 
 
 6 J4 
 
 * The resolution from Handel given in 217 (c) is quite exceptional and 
 irregular.
 
 chap. xiii. j Irs THEORY AXD PRACTICE* 
 
 (ii.) 
 
 46 
 3 
 
 4 6 6 4 
 244 2 
 3 
 
 ti 4 6 
 
 I 
 
 =J =! LJ ~i 
 
 i 
 
 T 
 
 646 464 
 
 52 2 i> 5 4 
 
 p 
 
 s^ 
 
 (III.) 
 
 6 5 6 J4 6 86 6 86 6 
 
 4 8 .* 2 4 
 
 3 
 
 5 6 6 J6 
 8 4 5 f4 
 
 J 2 
 
 6 7 
 I 
 
 86 S 7 8 - 
 
 8 
 
 6 86 66 8" 687 
 
 34 48 
 3 
 
 (IV.) 
 
 
 
 6 6 
 5 
 
 6665 6 
 
 543 5 
 
 6 E6 
 4 
 
 h6 6 
 
 - I * * ~ 
 
 686 
 5 5 
 
 :*: 
 
 665 
 6 
 
 6 to 66 
 
 44 45 
 
 3 2 
 
 687 
 4 
 
 (V.) 
 
 rn 1=^ ^ ] - iv 1 ' h-i 
 
 i>6 8 7 6 6 
 
 I- t) 4 
 
 679 
 4 43 
 
 6 JJ4 
 2 
 
 (VI.) 
 
 176 {6 07 
 
 f 4 h 
 
 A H 
 
 
 6 4 66 6 
 
 4 2 jj5 5 4 
 
 
 .4 
 
 667 
 4 I
 
 140 
 
 HARMONY : 
 
 [Chap. XIII. 
 
 (VII.) 
 
 .__._ __J I 
 
 6 b5 6 
 
 505 
 
 (IX.) 

 
 Chap. xiv. j Irs THEORY AXD PRACTICE. 147 
 
 CHAPTER XIV. 
 
 CHORDS OF THE NINTH. ENHARMONIC MODULATION. 
 
 324. If to the fundamental chords of the seventh on the 
 dominant, supertonic, and tonic, another third, either major or 
 minor, be added above the seventh, we shall have a chord of the 
 major or minor ninth, on these three degrees of the scale.* 
 
 Tonic gths. 
 Major. Minor. 
 
 Dominant gths. | Supertonic gths. 
 Major. Minor. ! Major. Minor. 
 
 The addition of a new note at the top of a chord does not affect 
 the progression of the lower notes, though its own progression 
 will be largely affected by them. The rules for the treatment of 
 chords of the seventh given in 321, 322 are therefore still to be 
 observed, and the only new rules to be learned will refer to the 
 ninth itself. 
 
 325. It will be seen that a chord of the ninth contains five 
 notes. As most music is written in four parts, one note must 
 evidently be omitted. The seventh is almost always either 
 present in the chord, or, if not, it is added when the ninth is 
 resolved. (See examples (b) to 335 and (<?) to 344, later in 
 this chapter.) In the root position, it is generally the fifth of the 
 chord that is absent ; but in the inversions the root (which is also 
 the generator) is mostly omitted, though occasionally it is in- 
 troduced even in the inversions. 
 
 326. As the harmonic derivation of tonic major and minor 
 keys is identical (17 6), it is clear that these fundamental chords 
 of the ninth will be the same in both keys. But, as a matter of 
 actual practice, though all the six chords shown in 324 can be 
 used in the major key, the major ninths of the dominant and 
 supertonic cannot be used in a minor key, because the ninths of 
 these chords are the major sixth and major third of the scale, 
 which notes in modern music are only used in a minor key when 
 the former is part of the supertonic, and the latter is part of the 
 tonic fundamental harmony. 
 
 327. The chord of the ninth in its root position is simply 
 figured 9 when diatonic ; but, as with all chromatic chords, any 
 chromatic notes require accidentals to indicate their presence. 
 
 * The harmonic derivation of these chords from their generator is explained in 
 63.
 
 148 
 
 HARMONY 
 
 [Chap. XIV. 
 
 To make this clear, we give the whole of the chords of the ninth 
 in the keys of C major and C minor, with their correct figuring. 
 
 In C major. 
 
 In C minor. 
 
 We have given these chords in five parts, and have in 
 last chord included the fifth, which is generally omitted, to 
 remind the student of the necessity for figuring that note in the 
 fundamental supertonic chord of the minor key. Though seldom 
 found in the root position, the fifth is frequently present as a 
 chromatic note in the inversions of the supertonic minor ninth. 
 
 328. It was said in 325 that in the inversions of chords of 
 the ninth the root was generally omitted. As the chord contains 
 four notes besides its root it is evidently susceptible of four in- 
 versions. As all chords of the major ninth resemble each other 
 as to intervals, and likewise all chords of the minor ninth, it is 
 needless to show each inversion of all these chords. We will take 
 the inversions of the major and minor ninths on the dominant, 
 and the student will be able to write similar inversions of the 
 supertonic and tonic chords for himself. 
 
 ist Inver. and Inver. 3rd Inver. 4th Inver. 
 
 329. Just as every triad in a minor key, excepting the minor 
 tonic chord, can be used as a chromatic chord in a major key 
 ( 290), every discord of a minor key can be used as chromatic in 
 a major key, though the converse, as mentioned in 326, is not 
 the case. Thus the first four chords of the last example cannot 
 be used in the key of C minor ; but the last four chords can be 
 employed either in C minor or C major. 
 
 330. It will be seen that the figuring of the four inversions of 
 the chord of the ninth is identical with that of the chord of the 
 seventh and its three inversions (Chapter IX.). It is important to 
 remember that figures do not necessarily show the nature of a 
 chord ; they simply indicate the distance of the upper notes of 
 the chord from the bass. The 5 and 3 are always implied with a 
 7, and only marked when an accidental is required. Similarly a 
 
 3 is implied with a , and a 6 with a 4 and 4 , as was seen with 
 5 32' 
 
 the chords of the seventh. But the student need never be in 
 doubt as to whether these figures indicate chords of the seventh 
 or chords of the ninth, if he notices the intervals between the 
 different notes. Thus if the first chord of the example in 328 
 were a chord of the seventh on B, it would have a major third,
 
 Chap. XIV.] 
 
 Irs THEORY AND PRACTICE. 
 
 149 
 
 and perfect fifth ( 197). The fact that it has a minor third and a 
 diminished fifth shows that B is not the generator, but that it is 
 a first inversion of a ninth. 
 
 331. To find the generator of any chord such as that now 
 under consideration, we must remember that all chords are 
 derived either from the tonic, supertonic, or dominant, each of 
 which bears a major third, perfect fifth, and minor seventh as the 
 
 first notes of its fundamental chord. The chord ^yg;-- cannot 
 
 be derived from C, because the seventh in the chord of C is B?, 
 not B C] ; neither can it be derived from D, for the third of the 
 chord of D is F, not Ft). It can therefore only be part of the 
 fundamental chord of G. 
 
 332. Unlike the chords of the seventh, all chords of the ninth 
 can resolve upon their own generator. In this case, while the 
 ninth falls one degree, the seventh usually (though not invariably) 
 remains to form part of a chord of the seventh, and is resolved 
 later. A good instance of this will be seen in 287, example (c\ 
 where at the end the dominant minor ninth in the key of B minor 
 descends a semitone to the root, while the seventh remains, and 
 is resolved in the next bar. 
 
 333. The dominant ninth, whether major or minor, is also 
 allowed to proceed to the third of its own chord, either by rising 
 a second, or falling a seventh, as in the following examples : 
 
 BEETHOVEN. Sonata, Op. 53. 
 
 >Ts 
 
 ^ 
 
 At (a) are seen the root positions of the major ninth in the first 
 bar and the minor ninth in the third, the ninth in each case rising 
 to the third of the chord. The interval of the augmented second
 
 HAR.MOXY: 
 
 [Chap. XII V 
 
 in the last bar ( 91) is not here objectionable. Here the minor 
 ninth is chromatic in the key .of C major. At (b) the same note 
 is diatonic in the key of C minor, and it is resolved by leaping 
 down to the. third of its chord. At (c) is the dominant major 
 ninth in the key of D flat, similarly treated. 
 
 334. It was said in 325 that in the root position of this 
 chord the fifth was generally omitted. But in all the examples 
 just given the third is omitted. This is always the case when 
 the ninth resolves on the third, in accordance with a broad 
 general principle that the note on which a dissonance resolves should 
 not be sounded with that dissonance, excepting only the root with the 
 ninth, ivhich is alwavs allowed. 
 
 335. In the chords of the dominant ninth we sometimes find 
 the ninth resolved freely by leaping to another note of the same 
 harmony. 
 
 SCHUBERT. Overture, " Fierrabras." 
 
 (b) SCHUMANN. " Neujahrslied," Op. 144. 
 
 
 At * of (a) the dominant minor ninth in F minor resolves on the 
 seventh of the same chord ; while at (b} the dominant major ninth 
 in E flat resolves on its fifth. 
 
 336. As the major ninth if placed below the third of the chord 
 will be a major second below that note, it will frequently sound 
 harsh in that position. It is therefore generally better to put the 
 ninth above the third. But to this rule there is one exception of 
 some importance. A dominant major ninth can always be 
 placed immediately below the third, provided that the ninth 
 proceed direct to the generator while the third remains stationary, 
 as in the following example : 
 
 DVORAK " Stabat Mater." 
 *
 
 Chap. XIV. 
 
 ITS THEORY AXD PRACTICE. 
 
 Between the first and second bars of the above will be seen in 
 the soprano and tenor parts hidden octaves of a kind that are 
 specially prohibited ( 104). The infraction of the rule here is 
 palliated, if not justified, by the tenor in the first half of the 
 second bar imitating the first half of the soprano of the previous 
 bar. 
 
 337- ^ e now gi ye a f W examples of various positions of chords 
 of the ninth resolved on their own generators. 
 
 SCHUBERT. Impromptu, Op. 90, No. 4. 
 
 * 
 
 MENDELSSOHN. " First Walpurgis Night." 
 
 * 
 
 ?ggiiElifeffi^f^ll=gg 
 
 ~3 - _ E^l 
 
 BEETHOVEN. "Adelaide." 
 
 J - 
 
 Example (a) shows the first inversion of the dominant minor 
 ninth resolving on its generator. The mental effect of the C$ and 
 B|I in the bass must not be overlooked. Each note is considered 
 as the real bass of the harmony for two bars. Here the generator 
 is exceptionally present in the inversion. In the third bar of (!>} 
 is shown the root position of the dominant major ninth. At (c) 
 are seen the third and fourth inversions of the dominant minor 
 ninth, the seventh rising to the generator when the ninth falls 
 ( 33 2 )- I R the second bar of this example the last inversion of 
 the dominant ninth is resolved on a different chord, as will 
 be explained directly. 
 
 338. The tonic and supertonic ninths are less frequently 
 resolved on their own generators than the dominant ninth. At 
 example (/) of 303, in the fifth bar, the first inversion of the 
 supertonic major ninth is thus treated. We add one example of a 
 tonic minor ninth, also in the first inversion.
 
 152 
 
 HARMONY : 
 
 [Chap. XIV. 
 
 The key of this passage is unquestionably B flat minor. That 
 there is no modulation to E flat minor is proved by the contra- 
 diction in the second chord of the second bar of the chromatic 
 Dt). (Compare 302). 
 
 339. When a chord of the ninth does not resolve on its own 
 generator (or root, which in this case is identical) it always 
 resolves on one of the two other fundamental chords of the same 
 key, unless it is followed by a modulation. Thus a dominant 
 ninth resolves on either a tonic or supertonic chord, a supertonic 
 on a dominant or tonic, and a tonic on a supertonic or dominant 
 chord. The supertonic chromatic triad ( 279) is, however, rarely 
 if ever used, a supertonic discord being almost invariably em- 
 ployed, while the common chord of the dominant can only follow 
 the supertonic ninth if the chromatic third of the latter chord is 
 immediately afterwards contradicted ; otherwise there would be a 
 modulation, the two chords being the dominant ninth and tonic 
 of the dominant key. 
 
 340. In resolving the chord of the ninth on a chord with a 
 different generator, the rules given in the last chapter for the treat- 
 ment of the third and the seventh must be observed. If the fifth 
 is present in a chord of the major ninth, care must be taken that 
 it does not move so as to make consecutive fifths with that note. 
 When resolved on a different generator, the ninth may not only 
 rise or fall, but under some conditions remain to be a note of the 
 next chord. If, for instance, the supertonic ninth resolves on the 
 tonic chord, the minor ninth in a minor key and the major ninth 
 in the major key will remain as the third of the following chord. 
 
 341. When a minor ninth is taken as a chromatic note of a 
 chord in a major key, and resolves by rising a semitone, it is usual 
 to write the ninth as an augmented octave of its generator e.g. 
 
 
 Usually written. 
 
 3*== 
 
 .-= -23- 
 
 This theoretically inaccurate but convenient notation is the
 
 Chap. XIV.] 
 
 ITS THEORY AND PRACTICE. 
 
 '53 
 
 probable explanation of the common method of writing the 
 chromatic scale which was shown in 269.* 
 
 342. We now give some examples of chords of the ninth 
 resolving on other than their own roots. When it is remembered 
 that there are six of these chords, each of which has five positions, 
 and each position several resolutions, it will be seen that it is im- 
 possible within reasonable limits to give an exhaustive series of 
 illustrations All that can be done is to give a few characteristic 
 progressions ; the student, if he will analyse the works of the great 
 masters, will find plenty of others for himself. 
 
 , > | BACH. Organ Fugue in F minor. 
 
 1 
 
 J. 
 
 & 
 
 (6) SCHUMANN. " An der Sonnenschein," Op. 36, No. 4. 
 
 <rf> 
 
 "TjT^ SPOHR. Double Quartet, Op. 136. 
 
 i * JJ-T5- ' 
 
 * The irregularity in the notation of the minor ninth when it is resolved by risin? 
 a chromatic semitone is well shown in the following extract from Beethoven's 
 Sonata in E flat, Op. 31, No. 3 :
 
 '54 
 
 HARMONY : 
 
 [Chap. XIV. 
 
 343. These three extracts show some of the most frequent 
 progressions of the root position of the chord of the ninth. At (a) 
 is seen first the second inversion and then the root position of the 
 dominant minor ninth of C minor, resolved on the root position 
 of the tonic. At (k] is the dominant major ninth also resolved on 
 the root position of the tonic. At (c) is a fine example of the first 
 inversion of the dominant major ninth, resolved on the chord of 
 the tonic. In order to avoid the consecutive fifths which would 
 occur between the fifth and ninth of this chord if both were 
 resolved at once ( 340), the third and fifth of the tonic chord are 
 delayed by auxiliary notes. At (d) we find first the second inver- 
 sion of the dominant minor ninth resolved on the first inversion 
 of the tonic chord, to avoid breaking the rule given in 101. In 
 the second bar of this example is a second inversion of a super- 
 tonic major ninth, followed by the root position of the same chord, 
 and resolved on a dominant thirteenth. Note the progression of 
 the third of the supertonic chord ( 281). 
 
 344. We now give some examples of the first inversions of 
 chords of the ninth. 
 
 HANDEL. "Hercules." 
 
 
 HANDEL. " Israel in Egypt." 
 
 =EEpr=_ E 
 
 BACH. ' Wahrlich ich sage Euch. 
 
 ^i
 
 Chap. XIV.] 
 
 Jrs THEORY AND PRACTICE. 
 
 SCHUBEXT. Sonata, Op. 164. 
 
 At (a) is seen the supertonic minor ninth of E minor resolved on 
 the third inversion of the dominant seventh of the same key. At 
 (A) is a beautiful example of the supertonic major ninth resolved 
 on the tonic chord. Here the seventh and ninth are both 
 stationary. Compare the auxiliary note Dt} in the third bar 
 (making a diminished octave with the D$ of the harmony) with 
 the example from Beethoven in 261. 
 
 345. The quotations (<:) and (d) show how a chord of the 
 ninth may be used for the purpose of modulation. The first bar 
 of () is evidently in the key of E major, and the second bar no 
 less clearly in F$ minor. The chord * in the notation here given 
 does not belong to E major at all, as E$ is not a note in that key. 
 But in effecting a modulation there should always be a connecting 
 link between the two keys. In the present case the chord is taken 
 as the last inversion of the tonic minor ninth of E, the bass note 
 being really FS ; for this FIj its enharmonic, E$, is substituted, 
 whereby the chord is changed to the first inversion of a minor 
 ninth on Cf , the dominant of F$ minor, and it is resolved on the 
 tonic of that key. Here is an instance of enharmonic modulation, 
 of which we shall have more to say later in this chapter. 
 
 346. The example at (d} is somewhat different. Here there 
 is no enharmonic change, but the first chord of the second bar is 
 taken as a dominant ninth in B;?, and quitted as a tonic ninth in 
 F, being resolved on the supertonic ninth of that key. In ex- 
 ample (c} of 337 we see a dominant ninth resolving on a super- 
 tonic ninth. 
 
 347. Our last extract (e) shows a beautiful and somewhat 
 unusual resolution of the supertonic ninth on the third inversion 
 of a dominant eleventh. Many other resolutions could be given 
 did space allow. 
 
 348. The second inversion of chords of the ninth is somewhat 
 less frequently used than the first ; still it can hardly be called 
 rare. The following examples will illustrate its treatment. 
 
 MENDKLSSOHN. Variations, Op. 82. 
 
 () *
 
 HARMONY : 
 
 [Chap. XIV. 
 
 MOZART. Quartett in G. 
 
 SPOHR. " Fall of Babylon. 
 
 -*i ft J Jj -f 
 
 At (a) is the second inversion of a dominant major ninth resolved 
 on the first inversion of tonic harmony. At (^) the second inver- 
 sion of the supertonic minor ninth is resolved on the second in- 
 version of the tonic chord. At (t) the second inversion of a super- 
 tonic major ninth is resolved on the first inversion of the dominant 
 minor ninth ; and lastly at (d) the second inversion of the tonic 
 minor ninth is resolved on the root position of the dominant 
 seventh. Notice in this last example that the minor ninth, Fj?, is 
 written as Etj because it resolves a semitone upwards ( 341). 
 The student should carefully examine the progressions of the third, 
 seventh, and ninth in all these examples. 
 
 349. The examples now to be given of third inversions of 
 ninths will repay careful study. 
 
 BEETHOVEN. Symphony in D. 
 
 ^4 j. A.
 
 chap, xiv.] ITS THEORY AND PRACTICE. 157 
 
 (c) HELLER. " Dans les bois," Op. 86, No. 3. 
 
 MOZART. Quartett in B I?. 
 
 i^~^ 
 
 7 r T-T*- 
 
 LXTti: 
 
 MOZART. Sonata in C minor. 
 
 (/) WAGNER. Das Rheingold." 
 
 SCHUMANN. " Ich grolle nicht." Op. 48, No. 7 
 
 SCHUMANN. " Schone Wiege," Op. 24, No. 5. 
 
 At (a) the note D in the chord * is not a note of the harmony, 
 but a pedal note (see note to 305). In the third inversion of a 
 dominant minor ninth, it is by no means uncommon for the
 
 158 
 
 HARMONY : 
 
 [Chap. XIV. 
 
 seventh of the chord to leap down, as here, to the tonic, instead 
 of descending by step to the third of the chord. At (/>) is seen 
 the usual resolution of the third inversion of the dominant major 
 ninth on the first inversion of the tonic chord, while at (c) is 
 shown the same resolution, but with this difference that (as in 
 example (a) of 337) the root is exceptionally retained in the 
 inversion. Example (d) gives the third inversion of a dominant 
 minor ninth resolved on the first inversion of .a supertonic minor 
 ninth, which in its turn resolves on the tonic chord. 
 
 350. Examples (e) and (/) show other resolutions of the 
 supertonic minor ninth. At () the third inversion is resolved on 
 the first inversion of a dominant seventh, and at (_/) on the root 
 position of the tonic chord. The third inversions of the tonic 
 minor and major ninths are seen at (g) and (h). At (g) the 
 seventh and ninth of the chord, Bj? and Dj?, are written as Af 
 and C| because of their resolving upwards (341) ; the change of 
 notation is much less frequent with the seventh than with the 
 ninth. Here the chord resolves on the first inversion of the 
 dominant seventh. At (h) the third inversion of the tonic major 
 ninth resolves on the second inversion of the supertonic minor 
 ninth, and this chord on the fourth inversion of the dominant 
 eleventh. In this example A*, the third of the supertonic chord, 
 is quite exceptionally written as B(?, because it resolves down- 
 wards. (Compare 269, at the end.) 
 
 351. The fourth inversion of a chord of the ninth will evidently 
 have the ninth in the bass, because that is the fourth note above 
 the generator ( 150). The major ninth is seldom if ever used in 
 this position, because of the harshness of the ninth below the 
 third. We have given an example of the fourth inversion of the 
 dominant minor ninth at (c) 337, we now add a few specimens 
 of supertonic and tonic ninths thus employed. 
 
 MENDELSSOHN. Octett. 
 j* 
 
 * * 
 
 BEETHOVEN. Quartett, Op. 18, No. 3.
 
 Chap. 
 
 ITS THEORY AND PRACTICE. 
 
 '59 
 
 SPOHR. " Fall of Babylon." 
 
 R=it^5 
 
 ufl 
 
 *itaSJ 
 
 itr=:4 
 
 * 
 
 oSa_4 
 
 In the very fine progression at (a) the last inversion of a super- 
 tonic minor ninth is resolved on the second inversion of a 
 dominant major ninth, the latter resolving upon its own generator. 
 At (b) we have the last inversions of a tonic minor ninth in the 
 second, and of a supertonic minor ninth in the third bar. In both 
 chords we find examples of the theoretically inaccurate, but con- 
 venient notation mentioned in 341. Trie tonic ninth in the 
 second bar as wntttn could only be derived from Eh as a 
 generator ; but if it were so derived it could not possibly resolve 
 on a chord of the seventh of F ; for E h and F not only are not 
 generators in the same key, but they are not even generators in 
 related keys. In all chords of the ninth (as well as in those of 
 the eleventh and thirteenth, to be subsequently described), the 
 student must carefully examine the notation before he can decide 
 on the nature of the chord. 
 
 352. Another instance of false notation, exactly similar to 
 that just spoken of, is seen at example (<:). Here both the seventh 
 and ninth of the tonic ninth are written inaccurately, partly 
 because both resolve upwards, partly also, no doubt, because the 
 passage occurs in a chorus, and is far easier to sing with the 
 notation given than if it had been correctly written thus : 
 
 353. Like the chords of the seventh ^216), chords of the 
 ninth can change their position, being resolved in the position in 
 which they last appear. The following example shows all the in- 
 versions of the chord of the dominant minor ninth. 
 
 354 The first inversion of the chord of the minor ninth, the 
 generator being omitted, is commonly known as the chord of the 
 diminished seventh. It is clear that there are three of these 
 chords in every key, and it is by no means uncommon to find a
 
 i6o 
 
 HARMONY : 
 
 [Chap. XIV 
 
 succession of them, each part moving a semitone, without pro- 
 ducing a modulation, as in the following example : 
 
 In order to make the progression of the harmony in the second 
 bar easier to the student, we have given each chord with its 
 correct notation. Meyerbeer has written them in a very promis- 
 cuous manner. The first diminished seventh is the first inversion 
 of the dominant minor ninth ; this resolves on the third inversion 
 of the tonic minor ninth, which again is followed by the second 
 inversion of the supertonic minor ninth ; then the series 
 (dominant, tonic, supertonic) recommences, but each time with a 
 different inversion. The music continues in A minor throughout. 
 
 355. But the chord of the diminished seventh is of great 
 importance from its use in modulation. We have already seen 
 ( 2 35> 236, 318, &c.) how to modulate by taking a chord as 
 belonging to one key, and quitting it as a different chord of 
 another key. Clearly we can apply the same method to the 
 
 chord now under consideration. For instance, the chord jm~ftg~7 
 
 may be the first inversion of a dominant minor ninth in C major 
 or minor, of a supertonic minor ninth in F major or minor, or of 
 a tonic minor ninth in G major or minor. It is evident that it may 
 be taken in any one of these keys, and left in any other of the six. 
 But its utility for the purposes of modulation by no means ends 
 here. A little explanation is necessary to enable us to understand 
 its further use. 
 
 356. It has been already said (52) that two notes which are 
 enharmonics of one another (e.g. Gft and A [7), can never both be 
 used in the same key ; it was also mentioned (83) that, to avoid 
 the use of double sharps and double flats, an enharmonic change 
 of notation was not infrequently employed, especially in " extreme 
 keys," that is, keys with many flats or sharps in the signature. If, 
 for instance, being in the key of F4 major, we wish to modulate 
 to the mediant of that key (Aft) which would have ten sharps
 
 Chap. XIV.] 
 
 Irs THEORY AXD PRACTICE. 
 
 161 
 
 (three double sharps) in the signature, it would make the music 
 difficult to read were it written thus 
 
 Even an experienced player might stumble at first sight over such 
 a passage. It is therefore much more convenient to mark a 
 change of signature, and substitute for the key of A|: its en- 
 harmonic, B7, as follows - 
 
 
 *-=>-*- 
 
 
 The passage is now perfectly easy to read. 
 
 357. It must be especially noticed that here the change of 
 notation makes no difference whatever in the progression of the 
 harmonies. Every chord is exactly the same as before. If the 
 student will transpose the above passage a semitone higher 
 or lower, he will find that no change is required. We have here 
 therefore simply an enharmonic change of notation. 
 
 358. If, however, by enharmonically altering one or more of 
 the notes of a chord we change the harmonic origin of the chord, 
 so that its altered notation induces a different progression of the 
 harmony, we get an enharmonic modulation. We have already 
 met with one case of this in example (<:) of 344 ; and the chord 
 of the minor ninth lends itself to this process more readily than 
 any other. If, for instance, we take the chord given in 355, we 
 can change any of its notes enharmonically ; each change alters 
 the nature of the chord, and gives it a new generator. 
 
 /t \ / \"/ V-/ V"- ,/ 
 
 We have already seen that the chord at (a) is the first inversion of 
 the minor ninth of G, and belongs to the keys of C major, 
 C minor, F major, F minor, G major, and G minor. If now for 
 A? we substitute G| as at (b) the nature of the chord is 
 changed. It is no longer derived from G, but it is the second 
 inversion of the minor ninth of E, and is dominant in A major 
 and A minor, supertonic in D major and D minor, and tonic in 
 E major and E minor. 
 K
 
 162 
 
 HARMONY .- 
 
 [ Chap. XIV. 
 
 359. Now take E| instead of FJ3, as at (c), and the chord 
 becomes the third inversion of the minor ninth of G|, and 
 evidently belongs to the keys of F| major and minor, B major 
 and minor, and C| major and minor. Again changing D^ to C x , 
 the chord is the fourth inversion of the minor ninth of A$, (d}. 
 It now belongs to the keys of D$ major and minor, G| major and 
 minor, and A$ major and minor. As these three major keys are 
 very rarely used, their enharmonics, EP, A?, and B?, being em- 
 ployed instead, the chord is for these keys written as at (e) where 
 it is the last inversion of the minor ninth of B7. It should be 
 noticed that while between each of the chords (a) (b} (c) and (d) 
 there is an enharmonic modidation, there is only an enharmonic 
 change of notation between (d} and (e) as the generator A| of (d) 
 is the enharmonic of the generator B!? of (e). 
 
 360. It will thus be seen that any chord of the diminished 
 seventh can be used, by enharmonic modulation in every key, 
 being taken in its relation to the key quitted, and resolved 
 (as dominant, supertonic, or tonic) in the new key entered. 
 When a modulation is thus effected, it is usual to write the chord 
 in the notation of the key which is being approached, as in 344 
 (f), where the chord was taken as a tonic ninth, and resolved as a 
 dominant ninth ( 345), being written in the notation of F$ minor, 
 and not of E major. In this example it is used to modulate to a 
 nearly related key ( 227), but it is frequently also employed for 
 modulations to more remote keys. A familiar example of this 
 will be seen in the recitative " Thy rebuke hath broken his 
 heart" in the "Messiah." The passage is too long to quote; 
 but if the student will refer to it, he will find in the sixth bar a 
 modulation from G minor to E minor, and at the thirteenth bar a 
 modulation from D minor to B minor by means of a diminished 
 seventh, taken in each case ^s a last inversion, and quitted as a 
 first inversion of a ninth, B!? being in both passages changed to 
 A| 
 
 361. The following magnificent progression is one of the finest 
 existing examples of enharmonic modulation by means of the 
 chord of the diminished seventh, and deserves careful analysis. 
 The bars are numbered, for convenience of reference. 
 
 BACH. "Chromatic Fantasia." 
 
 I 
 
 The first bar gives the first inversions of the dominant seventh 
 and dominant minor ninth in the key of E minor. The first 
 chord of the second bar is taken as the last inversion of a super- 
 tonic minor ninth (with A$) in E minor, and quitted as the third
 
 Chap. XIV.] 
 
 ITS THEORY AND PRACTICE. 
 
 163 
 
 inversion of a tonic ninth in A minor, resolving on the first inver- 
 sion of the dominant ninth in the same key. The third bar con- 
 tinues in A minor, the first chord being the last inversion of a 
 dominant minor thirteenth (as will be explained later), and the 
 second being the first inversion of the tonic minor triad. 
 
 362. At the beginning of the fourth bar there is clearly a 
 modulation, for the notes E? and G? in the first chord are not in 
 the key of A minor at all. This chord as written can only be 
 derived from F ; therefore the chord is taken (with F$ and D$ ) 
 as the second inversion of a supertonic minor ninth in A minor, 
 then changed enharmonically to the last inversion of the minor 
 ninth of F. But whether F is a tonic, supertonic, or dominant 
 can only be seen from the context. The first chord of bar 5 is the 
 dominant seventh of B flat ; we therefore look on bar 4 as being 
 in this key. The first chord is the last inversion of the dominant 
 minor ninth, while the second is the second inversion of the super- 
 tonic minor ninth. The last two chords of our example are 
 dominant minor thirteenth and tonic of D minor ; the second 
 chord of bar 5 is taken (with D?) as the second inversion of a 
 supertonic ninth in B flat (F being a pedal note, and not a part of 
 the harmony), and quitted, as just said, as a dominant thirteenth, 
 in which F is a note of the chord. 
 
 363. The student will probably find this analysis somewhat 
 difficult to follow ; but it is so instructive, as showing the nature 
 of enharmonic modulation, that it is well worth careful study. It 
 ought to be added that Bach's part-writing is here, as in most of 
 his works, somewhat free. We will now give a more modern 
 example of the same method, showing how enharmonic modula- 
 tion can be used to connect two very remote keys. 
 
 BEETHOVFN. Trio, Op. 70, No. i. 
 
 
 This passage begins in B? minor. At * the first chord is 
 taken (with Di?) as the second inversion of the supertonic minor 
 ninth in that key (generator C) ; then by the enharmonic change 
 from D? to Cf the generator changes to A, and the chord 
 becomes the third inversion of the dominant minor ninth of D. 
 The ninth resolves on its root in the following bar, and the
 
 164 HARMONY: [Chap. xiv. 
 
 dominant seventh which remains is resolved, after several changes 
 of position, on the tonic chord. 
 
 364. While the chord under consideration offers an easy 
 means of modulation to any key, it is well to warn young 
 composers not to use it too freely for this purpose. A modulation 
 to a remote key can often be quite as well effected by some other 
 means, and the too frequent use of diminished sevenths soon be- 
 comes monotonous, and palls on the ear. To modulate almost 
 exclusively by this chord would show great poverty of invention. 
 
 365. The first inversion of the dominant major ninth is some- 
 times called the " Chord of the seventh on the leading note," and 
 sometimes simply the "Leading Seventh." It is a far less im- 
 portant chord than the chord of the minor ninth. An example of 
 its use has been given in $ 342 (c}. 
 
 366. If both root and third are omitted in the chord of the 
 dominant major ninth, the remaining notes (the fifth, seventh, 
 and ninth) form the diatonic chord on the supertonic of the 
 major key ( 75). 
 
 This chord is now a concord ; for the root and third are the only 
 notes with which the seventh and ninth are dissonant, and in 
 considering the progression of the upper notes of a fundamental chord, 
 only such notes are to be taken into account as are actually present 
 in the chord. We shall see this principle more fully illustrated 
 when we come to chords of the eleventh and thirteenth. 
 
 367. If the root and third are omitted in the chord of the 
 dominant minor ninth, we shall have the supertonic triad of the 
 minor key (177), 
 
 which, though dissonant in its root position, is consonant in its 
 first inversion. 
 
 368. If the root and third of the supertonic major ninth be 
 omitted, 
 
 we shall have another consonance the chord of the submediant 
 in the major key. W T hat was said in 366 applies equally here. 
 
 369. As with the chords of the seventh in the last chapter, 
 we shall conclude with summarising the rules for the treatment of 
 the chords of the ninth. 
 
 I. RESOLUTION OF THE CHORD. A chord of the ninth may 
 resolve on its own generator, or on one of the other two gene- 
 rators in the same key. The third and seventh of the chord 
 follow the rules relating to chords of the seventh ; the fifth is free.
 
 Chap. XIV.] 
 
 ITS THEORY AND PRACTICE. 
 
 370. II. PROGRESSION OF THE NINTH. When a. chord of the 
 ninth resolves on its own generator, the ninth descends one 
 degree to the qctave of the generator, or, if it be a dominant 
 ninth, may proceed to any other note of the same chord. If the 
 chord resolve on a chord having a different generator, the ninth 
 may either rise or fall a second, or it may remain to be a note of 
 the next chord. 
 
 371. III. In the inversions of chords of the ninth, the root is 
 generally omitted. The first inversion of a chord of the minor 
 ninth is generally called the chord of the Diminished Seventh, 
 and differs from all other chords in the fact that by means of en- 
 harmonically changing one or more of its notes it can be used for 
 modulation between any two keys. 
 
 (1.) 
 
 EXERCISES TO CHAPTER XIV. 
 
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 HARMONY : 
 
 [Chap. XIV. 
 
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 Chap. XIV.j 
 
 ITS THEORY AND PRACTICE. 
 
 167 
 
 [In the following exercises enharmonic modulations by means 
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 (VIII.) 
 
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 1 68 HARMONY: [Chap. xv. 
 
 CHAPTER XV. 
 
 CHORDS OF THE ELEVENTH. 
 
 372. If we proceed still further to develop the fundamental 
 harmonies of a key by adding another third to the chords of the 
 ninth treated of in the last chapter, we shall obtain a series of 
 chords of the eleventh. The new note now to be added is always 
 a perfect eleventh (the octave above the perfect fourth) above the 
 generator ( 64) ; but as the chord of the ninth may be either 
 major or minor, there will be six varieties of the chord of the 
 eleventh, three of which will contain major and three minor 
 ninths. 
 
 Dominant I iths. Supertonic nths. 
 
 ^^s^^rnEQ ^ 
 
 g +^kg H-^t-8-^H-it-g 
 
 All these chords are possible in a major key ; but the prohibition 
 of the dominant and supertonic major ninths in a minor key 
 (326) applies equally to those forms of the chord of the eleventh 
 which contains a major ninth from these two generators. 
 
 373. Some theorists deny the existence, of chords of the eleventh alto- 
 gether; hut there are many combinations, especially in modern music, which 
 it is quite impossible to explain satisfactorily and clearly on any other 
 hypothesis. 
 
 374. Although we have given above six possible chords of the 
 eleventh, only those on the dominant are in frequent use. A few 
 exceptional cases of the employment of tonic and supertonic 
 elevenths will be given later in this chapter. 
 
 375. The chord of the eleventh in its complete form contains 
 six notes ; it will therefore be necessary in four-part writing to 
 omit at least two of these notes. As the eleventh is a dissonance, 
 the usual resolution of which is by descent of a second, the third 
 is mostly omitted in accordance with the general principle given 
 in 334- Either the fifth or ninth of the chord is also generally 
 omitted ; but the seventh is usually present, though occasionally 
 this note is only added when the eleventh is resolved, as in the 
 example to 240, at the last chord of the second bar. 
 
 376. Unlike the ninth, which seldom appears at a less distance 
 than a ninth from its generator never, indeed, excepting occa- 
 sionally when it is resolved on its third the eleventh may, and 
 often does, occur at the interval of a fourth above its generator.
 
 Ch.ip XV.] 
 
 ITS THEORY AND PRACTICE. 
 
 169 
 
 It is then sometimes figured u, but more frequently 4, and the 
 remaining figures of the harmony indicate the true nature of the 
 chord. As an instance of this, let the student refer to example (V) 
 of 337, where the chord in the second bar is a dominant eleventh 
 in its root position with the 7th and gth. The bass would be 
 figured thus : 
 
 It has been already said that in figuring a chord the largest 
 numbers are generally placed at the top. 
 
 377. Like the chords of the ninth, the chord of the eleventh 
 may either resolve on its own generator,* as in the example just 
 referred to, where the eleventh resolves before the rest of the 
 chord, or on one of the other generators in the key. When the 
 generator is present, the chord seldom resolves on a different 
 generator, though even in this case the dominant eleventh occa- 
 sionally resolves on a tonic chord, as in the following example. 
 
 SCHUMANN. Novellette, Op. 21, No. 8. 
 
 tf, * * & fc 
 
 Here the third'chord from the end is a supertonic minor ninth, 
 the A in the bass being a dominant pedal ($ 305, Note) ; the 
 chord * is a dominant eleventh, with the third and ninth omitted. 
 In this case the eleventh remains to be a note of the following 
 chord (compare $ 340). When the eleventh moves, it will go 
 either one degree down, to the third, or one degree up, to the 
 fifth of its own generator. When in the root position of this 
 chord the eleventh rises to the fifth, the ninth is usually also 
 present, and rises to the third as below. (See also Exercise VI., 
 .at_the end of the^chapter, at (a).) 
 
 WEBER. Duet, " Se il mio ben." 
 
 * Of course it is not meant by this that the note itself (the eleventh) falls 
 a fourth to the generator, but only that it is followed by another harmony on the 
 same generator.
 
 170 
 
 HARMONY 
 
 [Chap. XV. 
 
 378. The chord of the eleventh, as it consists of six notes, can 
 of course have five inversions. As these are somewhat more com- 
 plex than the inversions of the ninth, we shall treat them separately. 
 
 379. Owing to the harsh dissonance of the third against the 
 eleventh, the first inversion of this chord is rarely used. We give 
 two examples. 
 
 (*) 
 
 C. P. E. BACH. Fantasia in C minor. 
 ** 
 
 At (a) is the first inversion of the eleventh on F, the eleventh 
 rising to the fifth of the chord. It will be seen that the generator 
 is present ; this is much more frequent with inversions of chords 
 of the eleventh than of the ninth. At (fr) is a peculiarly harsh 
 first inversion of the eleventh of C, the eleventh resolving on the 
 third, and thus breaking the rule given in 334. This is not 
 recommended for imitation ; Wagner has no doubt introduced 
 this very rough dissonance for the sake of the dramatic effect. 
 The El? and D at the beginning of the last bar are chromatic 
 passing notes. 
 
 380. The second inversion of this chord is to be found both 
 with and without its generator. In the following example 
 
 BACH. Organ Prelude in Efe. 
 
 the generator and the seventh are both present, and the eleventh 
 resolves on the third of the chord (377). We have given the 
 figuring below the chord. It need hardly be said that the manner 
 in which the chord is figured will depend on the notes which are 
 present. An example of the second inversion without the gener-
 
 Chap. XV.] 
 
 ITS THLORY AND PRACTICE. 
 
 ator will be seen at 160 example (g) in the fourth quaver of the 
 third bar. Here the chord on the bass note B contains the 
 minor third, perfect fifth, and minor seventh. Like a chord of 
 the seventh, it would be figured 7 only ; but the fact that the third 
 from the root B is minor and not major proves that B is not the 
 generator of the chord, but only the root The second inversion 
 of the eleventh when the generator is absent is like a chord of 
 the seventh with a minor third and in a minor key with a dimin- 
 ished fifth also. 
 
 381. The third inversion of the chord of the eleventh, having 
 the seventh in the bass, is in frequent use. The generator is 
 rarely used with this inversion. In the following example 
 
 SCHUBERT. " Rosamunde." 
 
 the generator is clearly treated as a pedal note above the other 
 parts for if Schubert had looked upon the note as a part of the 
 chord, the B in the bass would have fallen, instead of rising. The 
 more usual form of this inversion contains only the fifth, seventh, 
 ninth, and eleventh of the chord. 
 
 As the chord in this shape resembles the common chord on the 
 subdominant with the sixth added to it, it is generally called the 
 
 " Chord of the Added Sixth." It is figured (?, like the first 
 
 inversion of a chord of the seventh, from which it may be 
 distinguished by the fact that it has a major sixth above the bass 
 note, while the first inversion of a chord of the seventh has always- 
 a minor sixth. 
 
 382. The term "added sixth " is somewhat misleading. Clearly the chord 
 cannot be derived from the subdominant ; because this note is not a generator 
 in the key ( 70), and the subdominant chord itself is merely a portion of the 
 fundamental chord on the dominant ( 75). 
 
 383. The third inversion of the dominant eleventh can resolve 
 either on its own generator or on one of the other two generators 
 in the key. In the form now under consideration (with both 
 generator and third absent), the seventh is no longer a dissonance 
 ( 366) and is free to rise or fall. An example of the chord 
 resolving on its own generator has already been given in 344 at 
 the third bar of the extract (e) ; we now add two short examples 
 of resolutions on the other generators.
 
 172 
 
 HARMONY : 
 
 BEETHOVEN. Sonata, Op. 14, No. 2. 
 
 | Chap. XV. 
 
 At (a) the chord * is the third inversion of the dominant eleventh 
 in G resolved on the first inversion of the supertonic minor 9th ; 
 at (b) the same chord in the key of E is resolved on the second 
 inversion of the tonic chord. 
 
 384. Like the third inversion of the dominant ninth ( 349), 
 the third inversion of the eleventh can be resolved on the root 
 position of the tonic chord, the seventh being free to leap, as in 
 the following example : 
 
 CHOPIN. Nocturne, Op. 32, No. 2. 
 
 385. The fourth inversion of the dominant eleventh, which 
 has the ninth in the bass, is scarcely ever found with the generator 
 present, owing to the harshness of the effect of that note above 
 the ninth. Its most usual form is with the fifth, seventh, and 
 eleventh above the ninth, as in the following example : 
 
 HAYDN, ist Mass. 
 
 
 
 _ . 
 
 
 $*= 
 
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 I 
 
 A 
 
 hi) 
 
 
 
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 =^rp^_ 
 
 s-'-b 
 
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 It will be seen that the figuring of this chord is identical with 
 that of a second inversion of a chord of the seventh, from which 
 it may be distinguished by the nature of the intervals. (Compare 
 ;5 381.) Another example of this inversion may be seen in the 
 last bar of extract (b} in 275.
 
 Chap. xv. j 
 
 ITS THKORY A. YD PRACTICE. 
 
 173 
 
 386. The last inversion of this chord, with the eleventh in the 
 bass, has generally the fifth, seventh, and ninth above it, though 
 occasionally the generator is used instead of the ninth. 
 
 Rarer. 
 
 -s2=- 
 
 This inversion is seldom resolved on any but its own generator. 
 One example of its use will suffice. 
 
 , HANDEL. "Messiah." 
 I i 
 
 
 We shall now, as with the chords of the seventh and ninth, sum 
 up the rules for the treatment of the chord of the dominant 
 eleventh. 
 
 387. I. RESOLUTION. A chord of the dominant eleventh 
 resolves either on its own generator, or on one of the two other 
 generators in the same key. In its root position, and in its fourth 
 and fifth inversions it mostly resolves on its own generator. The 
 first inversion is very rarely met with. 
 
 388. II. PROGRESSION. All the notes of the chord up to and 
 including the ninth follow the rules for their progression given in 
 previous chapters. The seventh and ninth are free in their pro- 
 gression when none of the lower notes of the chord with which 
 they make dissonances are present. If the chord resolve on its 
 own generator, the eleventh either falls to the third or rises to the 
 fifth of the chord. If it resolve on a different generator, the 
 eleventh mostly remains to be a note of the next chord. 
 
 389. If the generator,, third, and fifth of the chord are all 
 absent, the seventh, ninth, and eleventh give the triad on the 
 subdominant ( 75, 177). 
 
 As none of these notes are dissonant with one another, the chord 
 is a concord, and the eleventh (the fifth of the chord) is now free 
 in its progression. 
 
 390. The chords of the eleventh on the tonic and supertonic 
 are very rare, and far less important than that on the dominant. 
 They are seldom, if ever, found with the generator present, and 
 are generally used either with the seventh, ninth, and eleventh 
 only, in which case the chord is a consonance, or with the fifth, 
 seventh, ninth, and eleventh, when the eleventh is a dissonance 
 with the fifth. A few illustrations of their employment are all 
 that will be needed here.
 
 HARMONY : 
 
 [Chap. XV. 
 
 BEETHOVEN. Sonata, Op. 28. 
 
 -f f. u.- 
 
 
 At (a) is shown in the third bar the third inversion of the tonic 
 eleventh with the minor ninth. The context unmistakably shows 
 the key of the passage to be D. But in this key there is no D$ ; 
 this note therefore is clearly the minor ninth of the tonic (E?), 
 written as an augmented octave because its resolution is a semi- 
 tone upwards ( 341). It is also evident that the generator of this 
 chord must be D, since neither of the other generators in the key 
 (E and A) can give El? as one of their harmonics. The chord 
 therefore is a tonic eleventh. 
 
 391. At (fr) is seen the fourth inversion of the tonic eleventh 
 with the major ninth. All that precedes the chord * is evi- 
 dently in the key of C, as also is all that follows it ; therefore the 
 chord * must be in C ($ 272). As every chord in a key must 
 be derived from one of the three generators of that key, it 
 follows that the chord in question must be derived either 
 from C, D, or G, the three generators of the key of C. As 
 each generator produces its major third, it follows that the 
 chord * cannot be derived from D, or it would have Fl|, not Ft). 
 Neither can it be derived from G, for it would then have Btj, not B?. 
 It can therefore only be derived from C, from which generator D 
 is the major ninth, B7 the seventh, and F the eleventh. 
 
 392. The two illustrations now to be given differ in some 
 respects from those already quoted. 
 
 SCHUMANN. Concerto in A minor, Op. 54.
 
 Chap. XV.] 
 
 Irs THEORY AND PRACTICE. 
 
 At (a) is seen the same combination of the tonic eleventh as that 
 shown in 390 (a) ; but in this case it is approached from the 
 second inversion of the supertonic seventh of the same key. At 
 (/>) we find the second inversion of the tonic eleventh resolved on 
 the root position of the dominant seventh. 
 
 393. The chord of the eleventh on the supertonic is even 
 rarer than that on the tonic. It can be so seldom employed that 
 it is needless to give rules for its use. The following examples 
 will suffice as illustrations : 
 
 <*> 
 
 SCHUBERT. Mass in Eb. 
 I I l 
 
 J. S. BACH. "Ach, lieben Christen, seid getrost." 
 
 - 
 
 The chord at (a) is evidently not a dominant chord in C minor, 
 because the major ninth is not allowed in this chord in a minor 
 key ($ 372) ; neither is it a tonic chord, because it contains E7, 
 and all tonic discords have a major third ( 308) ; the chord * 
 must therefore be a supertonic eleventh. 
 
 394. The chord * of example (fy requires a little explanation. 
 It cannot be a minor chord on the tonic ; because if there were a 
 modulation to the minor key the Bt] of the preceding chord 
 would certainly have been B? ; neither can it be derived as a 
 chord of the dominant minor thirteenth ; for a chord of the
 
 i 7 6 
 
 HARMONY : 
 
 ;Chap. XV. 
 
 thirteenth is never found with only the generator, eleventh, and 
 thirteenth present, as will be seen in the next chapter ; the chord 
 in question therefore consists of the seventh, minor ninth, and 
 eleventh of the supertonic, here resolved upon its own generator 
 (see note to 8 290). 
 
 395. In example (c) the form of the chord is the same as at 
 (a) ; but it is here preceded by a dominant ninth, and resolved 
 on the first inversion of the dominant seventh without the 
 generator, i.e., the root position of the diminished triad on the 
 leading note ( 207). These examples will sufficiently show 
 the treatment of this chord. 
 
 EXERCISES TO CHAPTER XV. 
 
 [In the exercises now to be given many new combinations of 
 figures will be met with. These will show which notes of the 
 chord of the eleventh are present. The student must remember 
 what has so often been said that figures only show the intervals 
 made by the upper parts with the bass.] 
 
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 Chap, xv.] ITS THEORY AND PRACTICE. 
 
 (IV.) Hymn Tune. 
 
 177 
 
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 7 
 
 8 
 
 4 6 
 
 2 
 
 ,. , b^ r 
 
 he b5 6 
 
 b D 
 
 i i | .1 - 1 
 656 466 
 4352 5 
 
 6 ? 
 
 4 b 
 
 
 

 
 i 7 8 
 
 HARMONY : 
 
 [Chap. XVI. 
 
 CHAPTER XVI. 
 
 CHORDS OF THE THIRTEENTH. 
 
 396. To the chords of the eleventh treated of in the last 
 chapter another third, either major or minor, can be added. This 
 will give the chords of the major and minor thirteenth on each of 
 the generators in the key. The chords in their complete form 
 will be the following : 
 
 i 
 
 Tonic major 
 I 3 ths. 
 -<&- .-<&- 
 
 Tonic minor 
 
 Dominant 
 major i3ths. 
 
 Dominant 
 minor i3ths. 
 
 Supertonic 
 major i3ths. 
 -ft- , -52- 
 
 bupertofnc 
 minor isths. 
 
 As each major and each minor thirteenth can have either a major 
 or minor ninth with it, it follows that there are twelve varieties of 
 the chord, as shown in the above table. 
 
 397. These formidable looking groups of notes are, as a 
 matter of fact, of the rarest possible occurrence in their complete 
 form. In four-part harmony three at least of their notes must be 
 omitted ; and we shall find several forms of the chord, as we 
 proceed, in which only three of its notes are present. 
 
 398. It has been already said that in a minor key the major 
 third of the scale can only be used in tonic harmony, and the 
 major sixth only in supertonic harmony ( 326). Therefore both 
 forms of the tonic major thirteenth, both forms of the dominant 
 major thirteenth, and the dominant minor thirteenth when it 
 contains a major ninth, are unavailable in a minor key. 
 
 399. Owing to the large number of notes which it contains, of 
 which seldom more than four are used at once, the chord of the 
 thirteenth differs widely from the chords hitherto spoken of; for it 
 appears in so many forms that it is impossible to treat it in the 
 same manner in which we have treated the chords of the seventh, 
 ninth, and eleventh. When we come to examine the various 
 examples to be given, it will be seen that the notes selected 
 largely depend upon the general principle given in 334. Thus, 
 if the thirteenth is resolved on the fifth, the fifth will be absent 
 from the chord ; similarly the third and eleventh are hardly ever 
 used together, and in many cases all the lower part of the chord 
 is wanting.
 
 chap. xvi. j ITS THEORY AND PRACTICE. 179 
 
 400. An important difference between the thirteenth and the 
 other higher notes of a fundamental chord is, that while the 
 seventh, major and minor ninth, and eleventh are all dissonant 
 with their generators, the intervals of the major and minor 
 thirteenth (being the octaves of the major and minor sixth) are 
 themselves consonant to their generators, though in chords of the 
 thirteenth dissonant intervals are usually found between some of 
 the notes. 
 
 401. All the notes of this chord, up to and including the 
 eleventh, are bound by the rules for their treatment given in 
 previous chapters ; it being always remembered that account has 
 only to be. taken of such notes as are actually present in the chord 
 ( 366). It will only be needful now to give the rules regulating 
 the treatment of the thirteenth itself. 
 
 402. Like the chords of the ninth and eleventh, the chord of 
 the thirteenth may resolve either upon its own generator, or on one 
 of the other generators in the key. As a matter of actual practice, 
 however, the chords of the tonic and supertonic thirteenth (which 
 are far less frequently met with than those of the dominant) 
 seldom resolve on their own generator. 
 
 403. If the chords shown in $ 396 are examined, it will be 
 seen that the minor ninth is a perfect fifth below the minor 
 thirteenth, and the major ninth is a perfect fifth below the^ major 
 thirteenth. When therefore these notes are both found in the 
 chord, care must be taken in the resolution to avoid possible 
 consecutive fifths. 
 
 404. When a chord of the thirteenth resolves on its own 
 generator, it usually descends one degree to the fifth ; but it may 
 occasionally rise to the seventh, or even leap to the third of its 
 own chord. It is mostly also resolved sooner than the lower notes 
 of its chord, as we have already seen with the ninth ($ 332) and 
 eleventh (377). 
 
 405. If a chord of the thirteenth is resolved upon one of the 
 other generators in the key, the thirteenth may either remain to be 
 a note of the next chord, or it may fall a second or a third. A 
 minor thirteenth may also (like a minor ninth) be resolved by 
 rising a chromatic semitone ; and in this case it is very often 
 written as an augmented fifth from its generator, just as the minor 
 ninth under the same conditions is written as an augmented 
 octave ( 341). 
 
 406. When it is said that there are about \$>o possible com- 
 binations of four notes in the various chords of the thirteenth, to 
 say nothing of the combinations of three and five notes, and that 
 each combination has at least two or three possible resolutions, it 
 is evident that we cannot give a few simple rules for the treatment 
 of these chords, as with the chords discussed in previous chapters. 
 In truth, however, the matter is far less complex than would 
 appear from the above remark ; for many of the combinations
 
 180 HARMONY : [chap. xvi. 
 
 which are possible are, owing to their harshness, seldom or never 
 employed. The real difficulty with these chords for beginners 
 arises from the fact that they are to be met with in such widely 
 different forms that it is impossible to give any general principle 
 for their figuring, either in their root position, or in any of their 
 inversions. It must never be forgotten that figures alone are not 
 sufficient to indicate the true nature of a chord, but only serve to 
 show the distance of the upper notes from whichever note of the 
 chord may happen to be in the bass. 
 
 407. As a complete chord of the thirteenth contains six notes 
 besides its generator, it can of course have six inversions. Of 
 these the fifth, having the eleventh in the bass, is rare ; all the 
 others are more or less common. We shall give examples of each 
 inversion as we proceed. 
 
 408. In discussing the chords of the tonic and supertonic 
 eleventh, the derivation of these chords was proved by a process 
 of what may be called " destructive criticism." For example in 
 391, 393, 394, it was proved that the chords analysed were 
 only capable of derivation from one of the three generators in the 
 key. A similar method of reasoning will often assist the student 
 to determine the nature of a chord of the thirteenth ; but he must 
 in the first place be sure of the key in which the music is, and 
 then see whether any of the notes of the chord are incorrectly 
 written according to the chromatic scale of that key. If so, 
 he must substitute for them their true enharmonic equivalents 
 before proceeding to reason from them. The notes which can be 
 incorrectly written are the sevenths, the minor ninths, and minor 
 thirteenths. We shall follow this method of analysis in the chords 
 now to be examined ; and if the general principle is grasped, the 
 student will soon gain the needful experience for making similar 
 analyses for himself. 
 
 409. We shall now give examples of various forms of the 
 chord of the thirteenth, classifying them according to the notes 
 which they contain, dealing chiefly with those most commonly 
 used, but including also some of the rarer combinations which 
 possess special features of interest. 
 
 CHORDS OF THE THIRTEENTH INCLUDING THE GENERATOR. 
 
 410. I. Generator, third, and thirteenth. This is the simplest 
 form of the chord, and is rarely found except on the dominant. 
 In its root position it is identical with the first inversion of the 
 triad on the mediant. 
 
 It will obviously have a major thirteenth in the major key, and a 
 minor thirteenth in the minor key, which latter can also be used
 
 Chap. XVI.] 
 
 Irs THEORY AND PRACTICE. 
 
 181 
 
 as a chromatic chord in the major key, though the major 
 thirteenth cannot be used in the minor key. 
 
 411. The following examples show this form of the chord 
 both in major and minor keys. 
 
 SCHUMANN. " Waldscenen," Op. 82, No. 6. 
 
 I* 1 J 1 S ' 
 
 r -i 
 
 At (rt) is shown the dominant major thirteenth with its generator 
 and third, resolving on the root position of the tonic chord. The 
 B flat is of course the real bass note of the last chord. Here the 
 thirteenth leaps a third to the next harmony note, the first semi- 
 quaver, C, being an accented passing note ( 260). Example (b) 
 shows the dominant minor thirteenth, leaping to the generator (C) 
 of the following chord. In both these passages the chord is 
 resolved on a different generator. At (c) it is resolved on its own 
 generator, the thirteenth falling to the fifth, while the octave of 
 the generator falls to the seventh. 
 
 412. Our next illustration is somewhat different. 
 
 T* ' f! " 
 
 GOUNOD. " More 
 
 ^ ^ J- 
 
 t Vita." 
 
 <^= 
 
 - 5 ^ 
 
 jlf Nif 
 
 ^ 
 
 ~ii f 15 ^'" ~1~1 
 
 1 H 
 
 
 
 Here we see the sixth inversion of the dominant major thirteenth. 
 As all the notes present are consonant with one another, it is
 
 182 
 
 HARMONY 
 
 LChap. XVI. 
 
 allowed to double the thirteenth ( 366). Here one of the 
 thirteenths moves down by step to the fifth of the chord, while the 
 other remains. We then have the second inversion of the chord, 
 with the generator, third, fifth, and thirteenth, which is in its turn 
 resolved on the root position of the tonic chord. 
 
 413. The chord of the minor thirteenth is often used as 
 a chromatic chord in the major key, when it mostly resolves on 
 the tonic chord, the thirteenth rising a chromatic semitone to the 
 third of the tonic. In this case the note is mostly written as an 
 augmented fifth ( 405). 
 
 BEETHOVEN. Sonata, Op. 30, No. 2. 
 
 |J5 I 
 
 The key of this extract is clearly A!?; but the note Bt) does not 
 belong to that key ; it must therefore be in reality Ct?, the minor 
 thirteenth of the dominant. 
 
 414. In the following interesting passage 
 
 WAGNER. " Die Meistersinger." 
 
 (frr4-^= 
 
 -I *- 
 
 -I -f- 
 
 K~S "6" 'S ''Sg{g ftete~%rfc~ : &: igS'-T^M 
 
 be 
 
 it 
 
 we see a new resolution of the chord. The context shows the key 
 of the extract to be G major. The chord * is therefore the 
 chromatic dominant minor thirteenth ; on its first and second 
 appearances it is resolved on a chord of the augmented sixth, 
 which will be explained in the next chapter ; but the third time it 
 is resolved on the second inversion of the tonic chord. 
 
 415. The example just quoted illustrates what was said in 
 292 as to the employment of the augmented triad on the 
 mediant of the minor key as a chromatic chord in a major key. 
 
 416. Our next illustration shows three different forms of the 
 chord of the thirteenth. 
 
 WAGNER. "Tristan und Isolde." 
 
 
 7 6 
 4
 
 Chap. XVI.] 
 
 ITS THEORY AND PRACTICE. 
 
 As the passage is clearly in E?, the F$ of the second bar must be 
 G?, and the chord on the second crotchet is the sixth inversion of 
 the dominant minor thirteenth (the generator and third being 
 above the thirteenth) the chord resolving on the first inversion of 
 the tonic. The chord in the third bar looks at first sight like a 
 modulation into C minor ; but as there is nothing in what follows 
 to confirm this view, while the whole mental effect of the passage 
 is that of E?, it is clear that the chord is the first inversion of the 
 supertonic chord, as in the next bar, and that fit] and D are 
 "changing notes" ( 253). The chord at the end of the fourth 
 bar can only be a supertonic chord, because the augmented 
 fourth of the key (At|) can be derived from no other generator; 
 it is therefore the root position of the rather rare supertonic minor 
 thirteenth, resolved on a dominant ninth, the thirteenth rising a 
 chromatic semitone. At the second crotchet of the fifth bar is the 
 root position of the dominant major thirteenth, with a seventh 
 and major ninth. 
 
 417. II. Generator, third, fifth, and thirteenth. This form of 
 the chord is much rarer than that last given. We have seen one 
 instance in 412. It is, however, mostly to be found with a minor 
 thirteenth, and in the last inversion. A very fine example of this 
 form will be seen at (a) in 303, in the first bar. We give a 
 somewhat different illustration. 
 
 BACH. Organ Prelude in D minor. 
 
 4 1 8. III. Generator, third, seventh, and thirteenth. This is one 
 of the commonest, as well as one of the most useful forms of this 
 chord. It is found in both major and minor keys, and it must be 
 remembered that the thirteenth should ahvays be above the seventh, 
 because of the harshness of the dissonance of the second which it 
 will make if below that note. The chord of the major thirteenth 
 in this form usually, though by no means invariably, resolves on 
 its own generator ; the minor thirteenth quite as often resolves on 
 a different generator. 
 
 419. The following passages show the treatment of this form 
 of the major thirteenth. 
 
 SCHUMANN. Bunte Blatter, Op. 99, No. z.
 
 1 84 
 
 HARMONY : 
 
 BEETHOVEN. Sonata, Op. 10, No. i. 
 
 [Chap. XVI. 
 
 At (a) E is the real bass throughout the bar ; we have therefore 
 the root position of the chord resolved on the root position of the 
 tonic chord. In 342 at the second bar of (d) will be seen the 
 same position of the chord resolved on its own generator. Our 
 next examples (fr) and (c) show the first and third inversions of 
 this chord, the thirteenth in each case falling to the fifth, and the 
 chord resolving on its own generator, and becoming a chord of 
 the seventh. 
 
 420. The chord of the minor thirteenth with the third and 
 seventh is often used in the major key, as in the following 
 passages : 
 
 CHOPIN. Valse. Op. 34, No. 2. 
 
 In ; the first example the dominant minor thirteenth (for the 
 notation compare 413) resolves on the tonic chord, while at (b] 
 the chord of the dominant minor thirteenth is held over the bass 
 of the tonic chord on which it resolves.
 
 Chap, xvi.] ITS THEORY AND PRACTICE. 185 
 
 421. Our next example shows the same chord used in a minor 
 
 SCHUMANN. Faschingsschwank aus Wien. 
 
 key. 
 
 &C. 
 
 ^ 
 
 The key of this extract is F minor ; the thirteenth at * leaps to 
 the third of the same chord, instead of (as usual) to the root of 
 the tonic chord when the harmony changes. In the continuation 
 of the passage here given will be found the third inversions both of 
 the major and minor thirteenth, the thirteenth in each case 
 leaping to the third of its own generator. 
 
 422. It has been already remarked that chords of the 
 thirteenth are much more common on the dominant than on 
 either the tonic or the supertonic. We subjoin a very good 
 example of a tonic minor thirteenth. 
 
 SCHUMANN. "Wanderlust," Op. 35, No. 3. 
 
 
 This chord resolves on the chord of E 7 ; for it is clear that the 
 
 A * fl 
 
 P at the fifth quaver of the bar are accented auxiliary notes. 
 
 With this resolution, the chord of the minor thirteenth on B 7 
 would mostly be a dominant chord in E flat ; but that it is not so 
 in the present case is shown by the fact that the auxiliary note of 
 G is At}, not Aj?, and an auxiliary note above a harmony note is 
 always in the diatonic scale of the key in which the music is 
 ( 248). The passage is therefore in 67, and the chord * is a 
 tonic minor thirteenth. 
 
 423. IV. Generator, ninth, eleventh, and thirteenth. This, 
 though not one of the most common forms, is important enough 
 to deserve a word of mention.
 
 i86 
 
 HARMONY : 
 
 [Chap. XVI. 
 
 BEETHOVEN. Mass in D. 
 
 At * is the fourth inversion of the dominant major thirteenth, 
 resolving on its own generator. The major ninth in the bass rises 
 to the third, while the generator falls to the seventh, the eleventh 
 rises to the fifth, and the thirteenth falls to the fifth. 
 
 CHORDS OF THE THIRTEENTH WITHOUT THE GENERATOR. 
 
 424. Those forms of the chord of the thirteenth in which the 
 generator is absent are frequently more difficult to recognise and 
 identify than those in which it is present. The explanation of 
 the examples now to be given will, it is hoped, assist the student 
 in this matter. 
 
 425. V. Third, minor ninth, and thirteenth. This tolerably 
 common form is mostly found, if the thirteenth is major, with a 
 disguised notation, the minor ninth being written as an augmented 
 octave of the generator. In this shape it looks like a major triad 
 on the mediant. 
 
 BEETHOVEN. Sonata, Op. 81 
 
 g 
 
 7^ 
 
 if r r - 
 
 
 
 -,&- 
 
 l_2 j r=gs;g| 
 
 j 
 fji 
 
 
 r^ 1 ^ 
 
 
 
 S-^=^^ 
 
 S 
 
 
 
 9 
 
 -S- 
 
 ~S b 1 ^""^ "^ 
 
 In this passage the resolution of the chord * shows that it is not 
 what at first sight it would appear to be the. first inversion of the 
 dominant triad of C minor, for it resolves on a chord which 
 cannot be in C minor, but which (like the chords preceding) is 
 distinctly in the key of Ej?. The chord * is therefore chromatic 
 in E|?, and its real notation is 
 
 426. The last inversion of this chord is sometimes found in 
 cadences, written as a major chord on the mediant. 
 
 AUBER. " Le Dieu et la Bayadfere."
 
 chap, xvi.] ITS THEORY AND PRACTICE, 
 
 i8 7 
 
 W 
 
 LISZT. "Die Ideale." 
 
 In the second and third bars of (a) there is clearly a temporary 
 modulation to B minor ; but the last chord (which is the close of 
 the overture the key of the whole being D major) is evidently 
 not in B minor at all. No one chord can ever establish a key 
 ( 229) ; therefore in this passage the chord * must be taken as the 
 dominant of B minor, and quitted (with the enharmonic change 
 of A4 to B;?) as the last inversion of a dominant major thirteenth 
 in D with a minor ninth and third. The same reasoning applies 
 to example (b] ; but here no previous modulation into the relative 
 minor has been established ; the chord in the first bar is the tonic 
 of F, and is followed by the dominant major thirteenth written as 
 a mediant triad. Repetitions of the chord of F conclude the 
 movement. 
 
 427. Though the form of the chord now under notice is most 
 common as a dominant, it is also found derived from other 
 generators, as in the following beautiful examples. 
 
 SCHUBERT. Sonata in A minor. Op. 42. 
 
 DVORAK. ' Spectre's Bride. 
 
 r 
 
 The chord * at (a) suggests a modulation to E minor ; but its 
 resolution on the dominant seventh of C shows that the chord 
 itself belongs to that key. D$ not being a note in the key of C 
 must be in reality Ej? ; while FJ! can only be derived from D as a
 
 i88 
 
 HARMONY : 
 
 [Chap. XVI. 
 
 generator ; and the chord is composed of the same intervals as the 
 two shown in $426, but derived from the supertonic instead of the 
 dominant. At (b) is shown the same chord, but with a minor 
 thirteenth, written in its correct notation, with the third in the 
 bass. The passage is evidently in B flat minor, in which key Efcj, 
 A?, and D? are respectively the third, minor thirteenth, and 
 minor ninth of the supertonic. The chord here resolves on a 
 dominant triad, the ninth leaping to the root of the next chord, 
 before descending to its regular note of resolution, the fifth ; and 
 the thirteenth rising a chromatic semitone to the third of the 
 dominant. 
 
 428. VI. Third, fifth, ninth, and thirteenth. Not very common, 
 but striking in its effect, as the following examples show 
 
 WAGNER. " Tristan und Isoiae." 
 
 I 
 
 E^ 
 
 BEETHOVEN. " Sonata Pathtique," Op. 13. 
 
 -b-= 1 <-=. -ft 
 
 The chord at (a) is written in its true notation, and is the fourth 
 inversion of the dominant major thirteenth with a minor ninth, 
 the chord resolving on its own generator. The well-known 
 passage at (b] -can be in no other key than C minor ; the C4 in 
 the chord * must therefore be D?, which can only be derived from 
 the tonic. The chord is therefore the second inversion of a tonic 
 minor thirteenth, with third and minor ninth, resolved on the root 
 position of the dominant minor ninth. It would also be possible 
 here to consider G as a dominant pedal ; in which case the chord 
 belongs to form V. (425). 
 
 429. VII. Fifth, seventh, ninth, and thirteenth. The rare 
 example of this form of the chord given here 
 
 WEKTHOVEN. Sonata, Op. 22.
 
 Chap. XVI.] 
 
 Irs THEORY AND PRACTICE. 
 
 189 
 
 deserves quotation for more than one reason. The Eft in the 
 bass is really Q?, the minor ninth of the tonic, which therefore 
 determines the derivation of the chord, as it can come from no 
 other generator in the key. The major thirteenth (G) leaps to the 
 third of the chord ( 404), then returns to the fifth, whence it goes 
 to the eleventh, and the chord ultimately resolves on the dominant 
 triad. 
 
 430. VIII. Third, fifth, seventh, ninth, and thirteenth. 
 
 BRAHMS. " Deutsches Requiem." 
 
 
 --fey-* 
 
 The key of this passage is D major. The first bar gives the first 
 inversion of the tonic triad. The second bar sounds like the root 
 position of the dominant minor ninth of B minor, but its resolu- 
 tion in the third bar on the chord of D shows that the B? of the 
 tenor is the minor ninth from A, and that the chord is the last 
 inversion of a dominant major thirteenth. The first chord in the 
 third bar is the first inversion of a tonic minor thirteenth in the 
 form spoken of in 410, the thirteenth falling to the fifth at the 
 second minim of the bar. 
 
 431. IX. Fifth, ninth, eleventh and thirteenth. A very rare 
 form of the chord, of a peculiarly poignant effect. 
 
 DVORAK. " Stabat Mater." 
 
 The first two bars indicate the key of G major, as do the fifth and 
 following bars. The discord in the third and fourth bars conse- 
 quently belongs to that key. G4 must therefore be really A), 
 which in this key can only be the minor ninth of the tonic ; and 
 the chord * is a second inversion of a tonic major thirteenth, 
 resolved on the third inversion of the dominant seventh. The 
 minor ninth (as usual when it is written as an augmented octave) 
 rises a chromatic semitone, and the eleventh and thirteenth each 
 rises one degree. The same chord, similarly resolved, may be 
 seen in the third bar of Wagner's prelude to " Die Meistersinger." 
 432. X. Seventh, ninth, eleventh, and thirteenth. A very 
 common form of the chord, usually described by theorists as a
 
 HARMONY : 
 
 [Chap. XVI. 
 
 chord of the seventh on the subdominant. In this form of the 
 chord the seventh, ninth, and eleventh are all free in their pro- 
 gression, being consonant to one another. The thirteenth makes 
 with the seventh the dissonance of the seventh, and requires to be 
 resolved accordingly. If the chord resolves on its own generator, 
 the seventh usually rises one degree, and the thirteenth falls. 
 
 In both these examples the thirteenth, which is minor at (a) and 
 major at (b) falls to the fifth of the next chord when the seventh 
 rises to the generator. 
 
 433. Like the third inversions of the dominant minor ninth 
 ( 349) and eleventh ( 384), this form of the dominant minor 
 thirteenth, when the seventh is in the bass, can resolve on the 
 root position of the tonic chord, as in the following cadence 
 
 E. GRIEG. Lyrische Stiickchen, Op, 43, No. 6. 
 
 434. The next examples show this form of the tonic thirteenth, 
 which (as already said) seldom resolves on its own generator.
 
 Chap. XVI.] 
 
 ITS THEORY AND PRACTICE. 
 
 191 
 
 S. HELLER, Etude, Op. 90, No. i. 
 
 ^m 
 
 ^ J I J- 
 
 3: 5: 
 
 The passage quoted at (a) is in the key of B major ; the GQ shows 
 that the chord * cannot be derived from the supertonic, C|, and 
 the Ah shows that it cannot be derived from the dominant, Fft ; 
 it must therefore be a tonic chord, and is the third inversion of a 
 minor thirteenth with a major ninth, resolved on the first inversion 
 of a dominant seventh. At (b) we have the sixth inversion of 
 the same chord resolved on the root position of the dominant 
 seventh. 
 
 435. XI. Ninth, eleventh, and thirteenth. This form of the 
 chord is by no means uncommon, but, as none of the notes are 
 dissonant to one another when the ninth and thirteenth are both 
 minor, or both major, it is usually treated as a concord. The 
 minor ninth, eleventh, and minor thirteenth of the dominant 
 chord give the triad on the submediant of the minor key ( 177), 
 and the same notes of the tonic chord give the chromatic chord 
 on the minor second of the scale ( 274).* An example of the 
 same combination of the supertonic harmonies in the fifth in- 
 version resolving on the dominant is worth quoting. 
 
 WAGNER. "Tannhauser." 
 
 t, 
 
 ** 
 
 
 
 =^.g if i LT 11- i- J ^L i > =g= 
 
 4.36. The combination of the generator, eleventh and 
 thirteenth (either major or minor) is never used, because these 
 forms would simply give ordinary second inversions of concords. 
 
 * The major ninth, eleventh, and major thirteenth, being all secondary har- 
 monics, (66), these notes of the dominant harmony are the fifth, seventh, and 
 major ninth of the supertonic. already given as the derivation of the chord of the 
 submediant in the major key (75).
 
 192 
 
 HARMONY : 
 
 .[Chap. XVI. 
 
 On the dominant we should get the second inversion of the tonic 
 chord, on the tonic the second inversion of the subdominant, and 
 on the supertonic the second inversion of the dominant. 
 
 437. It was said in 397 that the chord of the thirteenth was 
 scarcely ever found in its complete form. As exceptional in- 
 stances we give two examples. 
 
 At (a) the chord * is the fifth inversion of a dominant minor 
 thirteenth in E minor, with all the notes present except the ninth. 
 At (b) is the last inversion of the dominant minor thirteenth in 
 D minor, every note of the chord being present. We very rarely 
 find the dissonant notes together with the notes of their resolution, 
 as here ; and it is only right to add that both these combinations 
 may also be explained as tonic chords with the notes of the 
 dominant harmony sounded over them as auxiliary notes ' (See 
 Chapter XI.) 
 
 438. As all fundamental discords can be derived from any one 
 of the three generators in the key, it is evident that the chords of 
 the thirteenth can be used for the purposes of modulation ; lor 
 they can be taken as derived from the dominant, and quitted 
 as derived from the tonic or supertonic, and vice versa. But their 
 use may be still further extended by enharmonic modulation, 
 as with the chords of the ninth. For example, the dominant 
 minor thirteenth in the key of C, with only generator, third, and 
 thirteenth present j^ g J may, by changing E b to D #, become 
 the last inversion of a similar chord derived from B 
 if instead of this change we substitute C b for B tj 
 
 the chord will become a first inversion of a minor thirteenth on 
 E[?. In these cases the chord can be quitted in any of the keys
 
 Chap. XVI. J 
 
 ITS THEORY AND PRACTICE. 
 
 in which the new generator is either tonic, supertonic or 
 dominant. 
 
 439. We have already met with one instance of an en- 
 harmonic modulation by means of this chord at example (a) of 
 Ji 426 ; we now add a few curiosities of a similar kind, which will 
 repay careful study. 
 
 MENDULSSOHN. Fantasia, Op. 16, No. i. 
 
 This passage begins in A minor : the chord * is the augmented 
 sixth (see next chapter) in that key ; but it is resolved in the key 
 of C ; the last chord but one must therefore be in the key of C, 
 since in making a modulation there 'must always be a connecting 
 link. The D4 of the chord * is treated as an E?, and the chord 
 itself thus becomes the form X. of the thirteenth (432), the 
 seventh in the bass leaping to the generator of the tonic chord, as 
 in the passage quoted in 433. It will be observed that here the 
 general rule as to writing the chord in which an enharmonic 
 change is made in the notation of the new key (S 360), is not 
 observed. 
 
 440. The two passages from Wagner next to be quoted are 
 even more curious 
 
 WAGNER. " TannhSuser." 
 
 In this well-known progression from the overture to " Tann- 
 hauser," the first bars are clearly in E minor. The first chord of 
 the second bar is the tonic chord of that key ; it is followed by 
 the chord of the sixth on the minor seventh of the scale ( 190), 
 which descends a semitone to the second inversion of the 
 chromatic chord on the supertonic. To effect the modulation 
 into D, the A4 is enharmonically changed to B?, and we get 
 form V. (425) of the dominant thirteenth in D resolved on the 
 tonic chord of that key. The modulation two bars later is pre- 
 cisely similar, the C|; being quitted as a D?. 
 
 441. In the passage just analysed the enharmonic change 
 made the chord into a chord of the thirteenth ; in that now to 
 be given the process is reversed, a chord of the thirteenth being 
 changed into something different.
 
 J94 HARMONY: ; chap. xvi. 
 
 WAGNER. "Die Meistersinger." 
 
 \L_N- 
 
 At the first bar of this passage the chord is the third inversion of 
 the dominant eleventh (the "chord of the added sixth") in B?. 
 At * the fifth rises to the minor thirteenth of the same chord. 
 As it resolves in the next bar on the dominant seventh of G, it is 
 clear that there must be here a double enharmonic change ; ( r? 
 and Dj? being changed into F& and C4, and the chord being 
 quitted as an unusual form of the augmented sixth (476). 
 
 442. To deal exhaustively with a chord so full of varied pos- 
 sibilities as the thirteenth will be seen to be, would require not 
 a chapter but a small volume. It is hoped that the explanations 
 that have been given will assist the student to understand the 
 nature of the chord, and to analyse similar passages for himself. 
 We shall conclude, as in previous chapters, with a general 
 summary. 
 
 443. I. RESOLUTION. Like the chords of the ninth and 
 eleventh, a chord of the thirteenth can either resolve on its own 
 generator or on one of the other generators in the key. The 
 chords of the tonic and supertonic thirteenths, however, are 
 seldom resolved on their own generators. 
 
 444. II. PROGRESSION OF THE NOTES OF THE CHORD. All the 
 notes present in the chord, up to and including the eleventh, follow 
 the rules given in previous chapters. When the chord resolves on 
 its own generator, the thirteenth moves to another note of the 
 chord, mostly to the fifth or seventh. When the chord is resolved 
 on a different generator, the thirteenth remains, falls a second 
 or a third, or rises a chromatic semitone. 
 
 445. III. FORMS OF THE CHORD. Of the many possible forms 
 of this chord, those most frequently met with are (i) generator, 
 third, and thirteenth, (2) generator, third, seventh, and thirteenth, 
 (3) seventh, ninth, eleventh, and thirteenth, and (4) ninth, 
 eleventh, and thirteenth. This last combination is a consonant 
 triad, and is mostly treated as such. In all chords of the 
 thirteenth any note may be treated as a consonance, provided 
 that no lower note of the complete chord with which it would 
 form a dissonance be present. 
 
 446. As the third next above the thirteenth is the double octave of the 
 generator, it is evident that the series of fundamental chords which we have 
 been building up by thirds must end here. The chords to be treated in the 
 next chapter are formed, as will be seen, in a different manner. All the 
 combinations hitherto treated of are found in the three fundamental chords 
 of the key shown in 71.
 
 Chap. XVI. 
 
 (I.) 
 
 Irs THEORY AND PRACTICE. 
 EXERCISES TO CHAPTER XVI. 
 
 195 
 
 
 77 77 6766 J 7 
 
 65 4 3 8 5 5 6 
 
 I 
 
 76 6 b(i 6 7 
 4 456 
 
 2 - 8 
 
 7J6 67 
 
 4 465 
 
 2 3 
 
 (II.) 
 
 6767 6 6 66 
 5 4 84 85 5 54 
 
 2- 8 
 
 7 7 
 I 
 
 n 
 
 6 767 {6 76 76 676 6 767 
 f |4- fas - 4 - 56 8 4- 4 8 
 
 2 b b4 3 2 
 
 (III.) Hymn Tune. 
 
 -Ulr-l- 
 
 4 
 
 82 
 
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 3 
 
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 86 6 37 86 
 4 
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 (IV.) 
 
 7 5 
 
 66 77 
 54 65 
 
 
 4 98 
 
 3 7 
 
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 4 3 
 
 ge e 
 
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 7 
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 8* 7 6 C6 J6 6 8 
 2 J5 458 
 
 
 78 6 7 6 7 6 !)6 7 6 84 7 6 86 
 554 _fl5 4 5 j,3 55 
 
 2 a ' 
 
 (VI.) 
 
 
 85 85 b7 .6 34 6 46666 b7 
 
 & 652 2 5454 5 
 
 [|2 b 4 
 
 
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 6 6 86 35 9 Q 7 86 S6 7 4 7 07 
 
 5554 7 44 J B D
 
 196 
 
 HARMONY : 
 
 [Chap. XVI. 
 
 (VII.) 
 
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 6 7 
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 6 
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 mil 
 
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 6 5 
 
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 6 676 
 
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 76 26 69865 6 
 
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 99 55 
 
 (VIII.) Hymn Tune. 
 
 6 6.47 
 
 $6 
 5 
 
 7 6 
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 56 I; 6 
 
 (IX.) Hymn Tune. 
 
 
 
 6 jf6 
 
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 chap. xvir. i Irs- THEORY AND PRACTICE. 197 
 
 CHAPTER XVII. 
 
 THE CHORD OF THE AUGMENTED SIXTH. 
 
 447. In addition to the series of fundamental chords already ex- 
 plained, there is one chord of great importance and of frequent 
 use still to be noticed, which is formed in a different manner from 
 any of the chords yet shown, and which contains an interval which 
 we have not yet met with. This is the Chord of the Aiigmented 
 Sixth. 
 
 448. If we take the second inversion of a chord of the seventh, 
 or of a chord of the minor ninth on either the dominant or super- 
 tonic, and lower the bass note a chromatic semitone, we shall 
 obtain a new combination. 
 
 (a) \ 2 3 (b) 1 2 3 
 
 p 
 
 r'S '-tr^-J=FS : 
 
 At (a) i, 2, are seen the second inversions of the chord of the 
 supertonic seventh, both with and without the generator, and at 
 (a) 3, the same position of the minor ninth of that chord, but /// 
 each case with the bass note lowered a semitone. At (b) i, 2, 3, are 
 corresponding positions of the dominant chord similarly altered. 
 The chords of the seventh and ninth on the tonic cannot be 
 treated in the same way, because the diminished fifth of the tonic 
 is not a note in the key. 
 
 449. It will be obvious that the alteration of the bass notes 
 changes the nature of the chords. The three chords at (a) are 
 now no longer supertonic chords ; because A? cannot be derived 
 from D as a generator. Neither, for the same reason, are the 
 chords at (b) dominant chords. We must seek another explana- 
 tion. 
 
 450. If the student refers to the derivation of the notes of the 
 chromatic scale given in 73, he will see that A? is the minor 
 ninth of the dominant, and that D? is the minor ninth of the 
 tonic. The three chords at (a) are therefore dominant chords of 
 a kind we have not yet met with, and the chords at (b) are 
 similarly tonic chords. 
 
 451. As only the lower note of each chord has been changed, 
 it would seem at first sight as if the chords were derived from two 
 generators. Thus all the upper notes of the chords at (a) are 
 still part of the supertonic chord, and all the upper notes of the
 
 198 HARMONY: [Cha r . xvn. 
 
 chords at (V) are still part of the dominant chord. As a matter of 
 fact, many theorists speak of these chords as " double-root chords." 
 But it is important to remember that the supertonic is itself de- 
 rived from the dominant, and the dominant is derived from the 
 tonic ; therefore all the harmonics of the supertonic are derived in 
 the second degree of descent (i.e., as secondary harmonics, see 
 41) from the dominant, and all the notes of the dominant chord 
 are secondary harmonics of the tonic. It will be convenient to 
 speak of the notes of this chord as derived from the upper or 
 lower generator ; and it must not be forgotten that the upper 
 generator is itself derived from the lower one. 
 
 452. It was seen in 66 that in all fundamental chords all the notes at a 
 greater distance from the generator than a minor ninth were secondary 
 harmonics ; the chords now under notice are similarly compounded ; the only 
 primary harmonic generally used being the bass note (the minor ninth), and 
 all the other notes of the chord being secondary harmonics. The important 
 distinction between this chord and all others is that in this chord alone the 
 major seventh of the generator (the 1 5th harmonic) is used. Of course it 
 absolutely excludes the minor seventh. 
 
 453. From the nature of the interval between the minor ninth 
 and major seventh of the generator, 
 
 in whatever form it may appear, is called the Chord of the Aug- 
 mented Sixth. It is always chromatic, the major seventh of the 
 dominant, and the minor ninth of the tonic being chromatic both 
 in major and minor keys. The chord is by no means of modern 
 introduction ; it is to be found in the works of composers of the 
 1 7th century, such as Purcell, Pelham Humphreys, and Caris- 
 simi.* 
 
 454. The general tendency of the notes of an augmented in- 
 terval being to diverge ( 199), the most frequent resolutions of 
 the augmented sixth are those in which the two notes move each 
 a semitone in contrary motion, as at (a) below. More rarely each 
 descends a semitone, as at (^) ; or one remains stationary while 
 the other moves a semitone, as at (c) (d). It is even possible, 
 though rare, for both to approach each other by step of a semitone, 
 as at (e). 
 
 455. The only two notes of a key on which the interval of the 
 augmented sixth can be found are the minor second and minor 
 sixth of the scale. It has just been shown that the chord on the 
 minor second of the scale is derived from the tonic, and that on 
 the minor sixth from the dominant. The latter chord is much the 
 
 * Some interesting examples of the chord are to be seen in the specimens from 
 these composers given in Hullah's "Transition Period of Musical History," and 
 " History of Modern Music." Want of space prevents their quotation here.
 
 Chap. xvn. ITS THEORY AND PRACTICE. 199 
 
 more frequently used ; we have seen in earlier chapters that tonic 
 discords are of far rarer occurrence than dominant, and the present 
 is no exception. 
 
 456. The chord of the augmented sixth is most commonly 
 found in one of the three forms shown in 448. We shall speak 
 of these first, commencing with the simplest. 
 
 457- I- Chord of the Augmented Sixth with the Third 
 This form of the chord consists of the 
 
 minor ninth of the lower, and third and seventh of the upper 
 generator, and is generally called the " Italian Sixth." As it 
 contains only three notes, one must be doubled in four-part 
 harmony. As with other discords, neither of the dissonant notes 
 of the augmented sixth can be doubled ; the third from the bass 
 note therefore appears in two parts. The chord is figured like an 
 ordinary first inversion of a triad ; but the sixth being a chromatic 
 note in the dominant chord, will require to be indicated ac- 
 cordingly * 6 or tj 6. We now give this chord with its principal 
 resolutions, putting the commonest progressions first. 
 
 W 
 
 458. At (a) the chord is resolved on the root position of the 
 dominant chord, and at (b) (<:) on the second inversion of the 
 tonic ; these are the most usual resolutions. At (d) the bass 
 remains, and the chord resolves on a dominant discord ; at (e) the 
 sixth remains as the third of a supertonic discord. The resolution 
 at (_/) is very rare. 
 
 459. This form of the chord is susceptible of two inver- 
 sions (jfo \**, \-^t% \. Of these the first is infrequent, and 
 
 / -&- 
 
 bo 5 
 
 54 bs 
 
 the latter extremely rare. 
 
 460. We now give a few examples of the use of this form of 
 the chord. 
 
 HANDEL. " Hercules."
 
 HARMON y : [ Chap, xvi i . 
 
 BEETHOVEN. " Prometheus.' 
 
 m 
 
 SPOHR. " Fall of Babylon. 
 
 l 1 
 
 SCHUMANN. Humoreske. Op. 20. 
 
 At (a) is shown the Italian sixth on the minor sixth of the key of 
 D minor, resolved on the dominant seventh, with the unusual 
 progression of the notes of the augmented sixth by similar motion. 
 At (<) and (c) we see the same chord in the keys of C and F 
 major, with their usual resolution on the dominant chord. At (</) 
 is the chord on the minor second of B?, resolved on the tonic 
 chord. The D at the end of the first bar is, of course, an " anti- 
 cipation." The less usual resolutions given at (d} (e) (/) of 457 
 are not often found with this form of the chord. 
 
 461. II. Chord of the Augmented Sixth with the Third and 
 Fourth. This form of the chord consists of the minor ninth of 
 the lower generator, and the seventh, octave, and third of the 
 upper ; and it is commonly known as the " French Sixth." 
 
 gTj It is figured like the second inversion of a 
 
 chord of the seventh ; but the chromatic note must of course be 
 indicated when not in the bass. 
 
 462. The principal resolutions of this chord are the following.
 
 Chap. XVII.] 
 
 ITS THEORY AND PRACTICE. 
 
 201 
 
 After the explanations already given the student will have no 
 difficulty in analysing these progressions for himself. 
 463. There are three inversions of this chord. 
 
 be 
 8* 
 
 2 
 
 The last inversion is so harsh in effect that, though possible, it is 
 seldom used. It may be said that in general it is better not 
 to invert the interval of the augmented sixth, thereby making it a 
 diminished third, excepting in the third form of the chord next to 
 be considered. It is needless to show all the resolutions of these 
 inversions, as the progression of the dissonant notes is governed 
 by the rules with which the student may be reasonably presumed 
 to be by this time quite familiar. 
 
 464. The following examples of this form of the chord will 
 sufficiently illustrate its use. 
 
 HANDEL. " Susanna. 
 
 In each of these examples the French sixth resolves on a position 
 of the tonic chord. This is a very frequent, though (as we shall 
 see directly) by no means an invariable progression for this form 
 of the augmented sixth. At (a) the root position of the chord on 
 the minor sixth in A minor is resolved on the second inversion of 
 the tonic. At (ti) is the second inversion of the same chord in 
 the key of G minor, resolved on the first inversion of the tonic 
 chord. The G in the bass at the end of the passage is the entry
 
 HARMONY : 
 
 [Chap. XVII. 
 
 of a new voice, and does not affect the progression of the discord. 
 At (c) is the root position of the same chord in the key of G 
 major, resolved like example (a) on the second inversion of the 
 tonic chord. Note that in the major key both the notes forming 
 the augmented sixth in this chord are chromatic. 
 
 465. The examples now to be given illustrate other resolutions. 
 
 MENDELSSOHN. "Athalie. 
 
 grj^nfc r V-i fcrL-j^ .J!L^: J^i J-q-g-^f^-^ 
 
 SCHUMANN. " Paradise and the Peri." 
 
 J . J I \-r-,S- 
 
 (c) WAGNER. " Die Meistersinger.' 
 -tr.. 
 
 Our first extract (a) shows an irregular resolution of the chord. 
 Instead of resolving on the root position of the dominant, the 
 augmented sixth here resolves on the first inversion of the 
 dominant seventh. At (^) the chord on the minor second of 
 B flat minor is resolved on the tonic chord. We then see the 
 same chord on the minor sixth of the scale, the octave of the 
 upper generator falling to the seventh, thus changing the form of 
 the chord to the Italian sixth, which is resolved in the last bar. 
 A chord of the augmented sixth often changes its form in this 
 way before being resolved. The passage at (c} begins in F, in 
 which key the French sixth is resolved, quite regularly, on the 
 dominant. In the second bar there is a modulation to G, the 
 chord of C being quitted as the subdominant of that key. The 
 chord at the end of the second bar is the second inversion of the 
 augmented sixth on the minor sixth of the key resolved irregularly 
 on the first inversion of the dominant. 
 
 466. It is important to notice that as this particular chord is 
 not made, like all which have been previously studied, by placing 
 thirds one above another, the second inversion of a French sixth 
 has the fourth in the bass instead of the fifth.
 
 Chap. XVII.J 
 
 Irs THEORY AND PRACTICE. 
 
 203 
 
 467. The last example to be given of this chord offers some 
 new features. 
 
 VERDI. " Requiem." 
 
 * 
 
 -W- 
 
 Here are shown two positions of the chord. We find first the rare 
 last inversion, with a diminished third above the bass note. The 
 last chord of this bar is clearly an augmented sixth, for C| is not 
 in the key ; the D? is incorrectly written because of its resolving 
 upwards, like the minor ninths and thirteenths which have been 
 already met with. The passage is further interesting as furnishing 
 an example of the rare progression of the notes of the augmented 
 sixth shown at $454 (e), where they approach each other by step 
 of a semitone. The chord in the passage now under notice re- 
 solves on the fourth inversion of a dominant eleventh. 
 
 468. III. Chord of the Augmented Sixth with the Third and 
 Fifth. This form of the chord consists of the minor ninth of the 
 lower generator, and the seventh, minor ninth, and third of the 
 upper ; and it is generally called the " German Sixth." * 
 
 l>5 b5 
 
 The resolutions of this chord are the same as those of the other 
 two forms, excepting that when the bass falls a semitone it usually 
 resolves on a second inversion. 
 
 gfrSa i \& - u, b frgz 
 
 ge 
 
 ft' 
 
 So 
 
 5 
 
 Here the fifth of the chord (the minor ninth of the upper gene- 
 rator) rises a chromatic semitone in the major key, and remains 
 
 * As students are very apt to confuse the names of these three forms of the 
 augmented sixth, the following artificial "aid to memory" may be found 
 useful : The three forms correspond to the character of the music of the three 
 countries Italian music is the simplest ; and the " Italian sixth" is the sim- 
 
 plest form of the chord / h-J^-^H. French music is the most piquant, and 
 Vy u 
 
 so is the "French sixth," with the discord between the upper generator, and 
 
 r\ * ^ 
 
 its seventh ^ b^^j. Lastly, German music is the richest and fullest in 
 character ; and the " German sixth " : /fs "^^^ = ^ is richer in its effect than 
 either of the others.
 
 204 
 
 HARMONY : 
 
 | Chap. XVII. 
 
 stationary in the minor. Evidently, if the chord is resolved direct 
 on the dominant, the fifth will fall a semitone, making consecutive 
 fifths with the bass. Such fifths (as will be seen presently) are 
 not always objectionable. 
 
 469. Like the French sixth, this form of the chord has three 
 inversions. 
 
 -4 jfo 
 
 * , ?*-'- 
 
 B'S !i ' ^ 
 
 (fl) I.? s - 
 
 =^=fl=^- 
 
 "^ bT 
 
 S b 4 
 
 6 U7 
 b4 5 
 52 > 
 
 The last inversion, containing the interval of a diminished third 
 from the bass, is much more frequently used than the last inver- 
 sion of the French sixth, being much less harsh in this form of the 
 chord. 
 
 470. The German sixth is more common than either the 
 Italian or French, and it will be needful to give a larger number 
 of illustrations to show its use. 
 
 HAYDN. " Creation." 
 
 P . b ^ -yf 
 
 MOZART. Quartett in E !>. 
 
 Of 
 
 BEETHOVEN. Sonal.i, Op. 57. 
 
 At (a) is the commonest resolution of the chord on the 
 second inversion of the tonic, though Haydn has first changed 
 the chord to the Italian sixth. (Compare 465.) Example (If) 
 shows the same resolution, but with a freedom not to say licence 
 in the part writing by no means usual with Mozart, and which 
 we do not recommend for the student's imitation. The dotted
 
 Chap. XVII.] 
 
 ITS THEORY AND PRACTICES 
 
 lines show the progression of the separate parts in the score. The 
 next example (<:) shows the resolution of the chord on dominant 
 harmony -in this case on a dominant eleventh ; the two notes of 
 the augmented sixth move in similar motion ; F?, the minor ninth 
 of the upper generator, is written as E3, and moves in consecutive 
 fifths (S 468) with the bass. 
 
 471. In the following passage the chord * is written as if it 
 were a dominant seventh in E flat. 
 
 BEETHOVEN'. Symphony in D. 
 
 Its resolution shows it to be the last inversion of the German sixth 
 on the minor sixth of the key of I). In this key Al? is not a note ; 
 its true notation is G$. 
 
 472. The commencement of Schubert's fine song " Am Meere" 
 gives an excellent example of the first inversion of this chord, 
 resolved on the root position of the tonic. 
 
 SCHUBERT. "Am Meere." 
 
 J- 
 
 
 473. The passages next to be given show some other resolu- 
 tions of this chord. 
 
 (a) MACKENZIE. "Colomb 
 
 ( [) S A = 1 1 ! N h 
 
 /u *fr n * *.. * --^f-^===* 
 w 4 f ^ ij* b* i yy 
 
 o - 
 
 
 (i) VERDI. " Requiem 
 
 --h b-j ! i -I U U J I , i _i 
 
 I^^S^l^^;
 
 2O6 
 
 HARMONY: 
 
 [Chap. XVII. 
 
 BRAHMS. ' Schicksalslied." 
 
 feE 
 
 The first of these passages gives the chord on the minor second of 
 G resolved on the chord of the dominant seventh, the upper note 
 of the augmented sixth remaining to be the third of the following 
 chord. At (b) is the last inversion of the chord (compare 471), 
 resolved on the dominant chord ; and at (c) the chord on the 
 minor sixth of the scale (here written as Btj instead of C?) is 
 resolved on a supertonic seventh. Example (d) is somewhat 
 similar, the augmented sixth being resolved on the supertonic 
 chord ; but here it is taken above a dominant pedal, and the 
 minor ninths from both the generators are written as augmented 
 octaves ( 341). The real notation of the chord * is of course 
 
 474- Schumann's music is full of interesting examples of this 
 chord. We select a few typical specimens. 
 
 (a) SCHUMANN. Novellette, Op. 21, No. 7. 
 
 1 
 
 'rfdn 
 
 -i^S-ft^r- J J 1 J ' SL_ 
 
 7 S-T m ^ ^^ 1* 
 
 
 ^ 8 ^->' ^ bJJ * ! ' =H^' J " =| 
 
 SCHUMANN. Kreisleriana, Op. 16, No. 2. 
 
 (6) /sx ^ -^ 
 
 L r T"^ ~~i i 1 j"i~s ~~; h ' ~n 
 
 
 &^^E^F=&P J ~ J ' ^-^^-^=-8 
 
 -# , 
 
 -r^ ~~~TiT J 'l 1 "u i <gl : , 1 
 
 5SE4 F* ' ^*^ - -J. .>_^- > J-CTI (I 
 
 v I^ZL-i *= L p -^=i " 
 
 .. H V-s>- .
 
 Chap. XVII.] 
 
 ITS THEORY AXD PRACTICE. 
 
 207 
 
 At (a) the chord * is the augmented sixth on the minor second 
 in the key of A. The notation is evidently inaccurate, as B? and 
 Kj cannot possibly belong to the same key. The E$ is really Ffc|, 
 and the resolution of the chord is unusual, being on the first 
 inversion of the submediant triad. Example (b] in which, to make 
 the progression clearer, the distribution of the first chords between 
 the two hands has been altered is another illustration of false 
 notation. C$ and G? cannot belong to the same key. The 
 whole passage is in B?, and the C;jt in the second bar is really D?. 
 The chord * is the second inversion of the augmented sixth on 
 G? (with a diminished third), resolved on the root position of a 
 supertonic seventh. 
 
 475. The following passage 
 
 SCHUMANN. " Bunte Blatter," Op. 99. 
 
 is probably unique, as containing four chords of the augmented sixth 
 in as many consecutive bars. The first two bars are on a " double 
 pedal '' (see Chapter XX.). The first chord of the second bar is the 
 second inversion of the German sixth on the minor second of the 
 key resolved on a minor tonic chord. This is a very rare resolu- 
 tion ; the augmented sixth on the minor second of the key, when 
 resolved on a tonic chord, mostly resolves on a major chord. 
 Another example of its resolution on a minor chord is seen at (b) 
 $ 465. The third bar of our present illustration shows the last in- 
 version of the German sixth, resolved on the dominant of C minor, 
 as in 473, at (b) ; the fourth bar gives the second inversion of the 
 same chord on the minor second of C minor, here resolved on a 
 major chord ; and the fifth bar contains the first inversion of the 
 chord on the minor second of G minor, the minor ninth of the lower 
 generator rising a chromatic semitone, and the chord being thus 
 resolved on the second inversion of a dominant minor ninth. 
 
 476. The three forms of the chord of the augmented sixth 
 already explained are by far the most common, but by no means 
 the only ones to be found. We give a few of the rarer forms. 
 
 HAYDN. "Creation."
 
 208 
 
 HARMONY: 
 
 [Chap. XVII. 
 
 WAGNER. " Tristan und Isolde." 
 
 Jb- 
 
 
 VERDI. " Requiem." 
 
 -g_g_s=s=g=a 
 
 -*-&tf-3^ ^~ 
 
 /) E. PROUT. Overture "Rokeby." 
 
 f TT" 
 
 At () will be seen a chord in which Btf, the third of the lower 
 generator, is substituted for C, the seventh of the upper. The 
 
 chord is the first inversion of gji-fcj^^. Another position of the 
 
 same chord is seen at (ft). At (c) a beautiful effect is obtained by 
 the substitution of the major for the minor ninth of the upper 
 generator. The last example (d} shows the chord on the minor 
 second of the key, with the minor ninth of the lower generator, 
 and the octave, third, and minor thirteenth (instead of the seventh) 
 of the upper. It may be said in general terms that any combina- 
 tion of the harmonics of the two generators is possible for this 
 chord, so long as the minor ninth of the lower and the third of the 
 upper generator are present, and that no false relation is induced 
 between the notes employed. 
 
 477. Like other fundamental chords, the chord of the aug- 
 mented sixth can be .freely used for the purposes of modulation. 
 As it can be taken in either a major or minor key on either the 
 minor sixth or minor second of the scale, it can evidently be taken 
 in any one of four keys and quitted in any other of the same four 
 without an enharmonic change. But it is also largely available 
 for enharmonic modulation. When thus employed, the upper note 
 of the augmented sixth, being enharmonically changed, becomes a 
 minor seventh, of which the lower note of the interval is the 
 generator.
 
 Chap. XVII. 
 
 ITS THEORY AND PRACTICE. 
 
 209 
 
 This change evidently converts the chord into a fundamental 
 seventh. 
 
 478. A little thought will show the student that it is only the 
 German sixth that can be thus converted ; for the French sixth 
 contains a note which will not be a part of a chord of the seventh 
 at all ; while the Italian sixth will be impossible as a seventh in 
 four-part harmony, owing to the doubling of the third. 
 
 479. This enharmonic change of an augmented sixth to a 
 dominant seventh, and vice versa, is mostly used when a modula- 
 tion is desired to a key a semitone up or down. If, for instance, 
 in C the chord of the augmented sixth on Aj? is taken, and the F:$ 
 changed to G?, the chord becomes the dominant seventh in Dj?, 
 and can be resolved in that key. It might also be quitted as the 
 supertonic seventh of G?, or the tonic seventh of Aj?, but these 
 resolutions, the latter especially, are seldom if ever to be met with. 
 Conversely, if we take the dominant seventh in the key of C, and 
 change the F to E|, the chord becomes an augmented sixth in 
 B major or minor possibly, even, in F|! major or minor. We 
 shall see directly that some other enharmonic changes are possible ; 
 but the above are by far the most usual. 
 
 480. We shall conclude this chapter with a few examples of 
 enharmonic modulation by means of this chord. Our first extract 
 
 SCHUBERT. Sonata, Op. 164. 
 
 illustrates what has been said in the last paragraph. At the third 
 and fourth bars is the chord of the dominant seventh in F major. 
 In the fifth bar Bb is changed to A| and the chord becomes the 
 last inversion of the German sixth in E major, in which key it 
 is resolved at the seventh bar. 
 
 481. Our next illustration will be found rather more difficult 
 to follow
 
 210 
 
 HARMONY : 
 
 [Chap. XVII. 
 
 BACH. "Chromatic Fantasia." 
 
 The first bar of this passage is clearly in Bj? minor ; as the note 
 F? in the bass of the second bar is not in that key, it is evident 
 that there is here an enharmonic modulation, the chord being 
 written in the notation of the new key. The chord in B) minor 
 will have Efc|, and will be the last inversion of the German sixth. 
 The modulation, as proved by the last half of the second and the 
 first half of the third bar, is to A;? minor. In this key the chord * 
 cannot be dominant, because of the G?, nor supertonic because of 
 the D;? ; it must therefore be the last inversion of the tonic minor 
 thirteenth. 
 
 482. The progression of the harmony in the fourth and fifth 
 bars is the same as that just analysed. The chord * in the fourth 
 bar is taken as the last inversion of a German sixth in A flat (the 
 notation of every note being enharmonically changed), and quitted 
 as the last inversion of a tonic thirteenth in F sharp minor, which 
 is resolved in the next bar on the dominant harmony of that key. 
 At first sight the chords marked * in the second and fourth 
 bars of this extract look like third inversions of dominant 
 sevenths in Cj? and A ; but they cannot be so regarded, because 
 these keys are never established. The chords in question must 
 therefore be taken in their relation to the keys next following. 
 
 483. One of the most beautiful and novel harmonic progres- 
 sions ever written will be seen in our next example. 
 
 ofcAK. " Spectre's Bride. 

 
 Chap. XVII.] 
 
 ITS THEORY AXD PRACTICE. 
 
 This passage commences in C minor, and passes rapidly to Cb. 
 major, whence by a simple, but most unexpected, enharmonic 
 change, a return is made at once to Cft major. The chord * is 
 taken as the augmented sixth in Q>, its notation in that key being 
 
 ^Wtv 
 I 
 
 If the chords in the third and fourth bars be written in Bfc| major 
 (the enharmonic of Q>), the student will follow the progression 
 more easily, as then only one note will need to be changed for the 
 key of C. It will be a useful exercise for him to do this for him- 
 self; we shall therefore not do it for him. In this passage we see 
 the converse of the modulation shown in $ 480. 
 
 484. Our last example is chosen to show the student how to 
 overcome some of the difficulties arising from incorrect notation 
 in the extreme keys, as well as from incomplete, or merely 
 suggested harmonies. 
 
 SCHUMANN. Romanze, Op. 28, No. 2. 
 
 The first chord, being a chord of the diminished seventh, can be 
 (as we already know, 360), in any key. We look at the next 
 bar to see what key the music goes into, and we find C|; minor 
 clearly indicated by the Afc), -B4, and F x . The fourth quaver of 
 the bar is an outline chord of the augmented sixth, in its last in- 
 version, with the diminished third, for this interval occurs in no 
 other chord. The G X must be of necessity a false notation for 
 Afc[, because of the Efc] in the upper part at the beginning of the 
 last bar ; for no key which contains Efcj as one of its notes can 
 possibly also contain G x . The chord at the end of the bar is 
 therefore the augmented sixth Aft, C|, F x ; the last note is en- 
 harmonically changed to Gft, the change being exceptionally 
 written, and the chord is left as a tonic minor thirteenth in B 
 major, and resolved on the dominant of that key. The en- 
 harmonic change is the same as in the passage from Bach given 
 in 481 ; and the Gx was evidently written by Schumann instead 
 of At] because of the note resolving by rise of a semitone as we 
 have so often seen to be the case with the minor ninths and 
 thirteenths. 
 
 485. The student will no doubt find the analyses just given 
 somewhat difficult to follow. Such passages require, but they will
 
 212 
 
 HARMONY : 
 
 [Chap. XVII. 
 
 certainly repay, careful study. It must not be supposed that we 
 have given all the possible examples of such modulations. This 
 our space will not allow ; we can only give a few representative 
 specimens. Besides this, the resources of art are not yet ex- 
 hausted ; new combinations are constantly being discovered, and 
 it is certain that whenever such combinations are good, they will 
 be capable of a satisfactory theoretical explanation. 
 
 (i.) 
 
 EXERCISES TO CHAPTER XVII. 
 
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 Chap. XVII.] 
 (V.) 
 
 Irs THEORY AND PRACTICE. 
 
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 214 HARMONY: [Chap. xvm. 
 
 CHAPTER XVIII. 
 
 THE SO-CALLED "DIATONIC DISCORDS." 
 
 486. The development of harmony has been very gradual, and 
 many of the progressions which have been met with in preceding 
 chapters are of comparatively recent introduction. The earliest 
 modern music consisted of concords only ; and discords, when 
 first employed (unless they were passing notes), were always con- 
 sidered to require Preparation. By the preparation of a discord 
 is meant the appearance of the dissonant note as a consonance in 
 the preceding chord, .and in the same voice in which it is to be 
 heard as a dissonance. 
 
 487. But about the beginning of the iyth century Claudio 
 Monteverde ventured to introduce the dominant seventh into his 
 music without preparation. The bold innovation raised an outcry 
 in his day, just as the harmonic experiments of Schumann and 
 Wagner have done in our own times ; but it soon began to be 
 realised that the effect was good, and before long, the use of the 
 unprepared dominant seventh became general. It was probably 
 nearly a century later before any one ventured to use any other 
 discord without preparation. Bach, who anticipated nearly all 
 modern harmonic progressions, was one of the first to use other 
 chords of the seventh than that on the dominant in this manner ; 
 and at the present time it is allowed to use any essential discord 
 that is, a discord in which the dissonant note is itself an essential 
 part of the chord without preparation. 
 
 488. In order that the explanations now to be given may be 
 clearly understood, it is very important to remind the student of 
 the distinction made in the foot-note to 58 between the Root 
 and the Generator of a chord. By the latter term is meant that 
 one of the three notes in a key tonic, dominant, or supertonic 
 from which the chord is harmonically derived; by the word " Root " 
 is meant the lowest of the series of thirds which happens at the 
 time to be present in the chord. Sometimes, as for instance in 
 the triad on the dominant, the root and generator are the same ; 
 but very often the two notes are different. Thus, in the key of 
 
 C major, the diatonic triad on the supertonic gfaiipEE is derived 
 
 from the dominant as its fifth, seventh, and major ninth ( 75); 
 here G is the generator of the chord ; but D is its root.
 
 Chap, xviii.] ITS THEORY AX D PRACTICE. 215 
 
 489. By the term " diatonic discords " are meant all discords 
 which are in accordance with the key-signature, and contain any 
 other intervals from their root than a major third, perfect fifth, and 
 minor seventh, which, as we have already seen, are the character- 
 istic intervals of a fundamental discord. For example, if we add 
 a seventh to the supertonic triad given above, it will be a diatonic 
 discord, for, though it contains a perfect fifth and minor seventh, 
 it does not contain a major third. Similarly, if we add B fcj to the 
 tonic chord of C, we shall have a diatonic discord with a major, 
 instead of a minor seventh. Some theorists call such chords 
 "secondary sevenths," to distinguish them from the dominant 
 sevenths. 
 
 490. If we place a seventh, in accordance with the key-signa- 
 ture, above each of the triads of the major key given in 75, we 
 shall obtain a series of diatonic sevenths. 
 
 If the student will examine the fundamental chord of the domi- 
 nant shown in 67, he will see that each of the chords of the 
 seventh given above is a part of that fundamental chord : 
 
 The tonic seventh is the eleventh, major thirteenth, generator, 
 and third. 
 
 The supertonic seventh is the fifth, seventh, major ninth, and 
 eleventh. 
 
 The mediant seventh is the major thirteenth, generator, third, 
 and fifth. 
 
 The subdominant seventh is the seventh, major ninth, eleventh, 
 and major thirteenth. 
 
 The dominant seventh is already known. It is only " diatonic " 
 in the sense of containing no note foreign to the key-signature, 
 and is not included among the diatonic discords. 
 
 The submediant seventh is the major ninth, eleventh, major 
 thirteenth, and generator. 
 
 The seventh on the leading note, sometimes called the leading 
 seventh, is the first inversion of the dominant major ninth, without 
 the generator. 
 
 491. The triads of the minor key ( 177), can be similarly 
 treated, and the diatonic sevenths thus obtained can equally be 
 derived from the fundamental chord on the dominant with the 
 minor ninth and thirteenth. The tonic seventh in the minor key 
 
 is found in two forms (^ t?'b g E *r^\ The forin with the 
 
 ' 
 
 minor seventh can only be used when the seventh descends in the
 
 2l6 
 
 HARMONY 
 
 [Chap. XVIII. 
 
 next chord ; and in this form it is used to avoid an augmented 
 second, and must be considered as an arbitrary alteration of the 
 major third of the dominant to a minor third, such as has been 
 seen in 190. As all the other derivations are precisely the same 
 as in the major key, with the substitution of minor for major ninths 
 and thirteenths, it is needless to give the list in full. 
 
 492. The old rule for the treatment of diatonic discords was 
 that the seventh must be prepared ( 486), and that the chord 
 must resolve on another chord the root of which was a fourth 
 above its own. The extracts given in previous chapters will 
 furnish illustrations of this. At 202 (b} the first chord of the 
 second bar is a diatonic seventh ; the root is B ; the seventh, A, is 
 prepared in the preceding chord, and is resolved on the third of 
 the chord of E a fourth above B. At (d) of 208, is seen on 
 the third crotchet of the bar the same chord in a minor key. The 
 root of the chord is Btj (the generator is E) ; the seventh, A, 
 appears in the tenor of the preceding chord, and the diatonic 
 seventh on the supertonic is resolved on the dominant, the root, 
 as before, rising a fourth. In the example at 301 the same rule 
 is observed. Here the sixth chord in the extract is a diatonic 
 seventh on the submediant; the seventh is prepared, and the 
 chord is resolved on a supertonic seventh, the root of which is the 
 fourth above A. 
 
 493. The following passage 
 
 S5 
 
 HANDEL. " Joshua." 
 
 shows a sequence (137) of diatonic sevenths in their third 
 inversion, each being resolved (like the third inversion of the 
 dominant seventh) on the first inversion of the chord whose root 
 is a fourth above their own. The first chord marked * is the 
 third inversion of the supertonic (the seventh being prepared in 
 the preceding chord) resolved on the first inversion of the
 
 Chap. XVIII.] 
 
 ITS THEORY AND PRACTICE. 
 
 217 
 
 dominant ; then comes the third inversion of the tonic seventh, 
 resolved on the first inversion of the subdominant ; and so on to 
 the end of the passage, the last seventh being the third inversion 
 of the dominant seventh, and therefore a fundamental chord. 
 
 494. In a progression of this kind, a sequence of sevenths 
 is often met with, as in the example now to be given 
 
 MOZART. Litany in Bb. 
 J I 
 
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 * r^-r ' * r^-r 1 ^ * f 
 
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 7 7 7 7 7 7 
 
 Here is a fine sequence of sevenths, commencing at the second 
 bar, the third of each seventh remaining to be the seventh of the 
 next chord, and the seventh of each chord descending a note to 
 be the third of the following. The roots in every instance rise a 
 fourth. 
 
 495. Having given examples of the observance of the old rule 
 referred to in 492, we shall now show cases in which it was 
 disregarded, even by the old masters. In 313 at (b} will be 
 seen a diatonic chord of the seventh on the subdominant. Here 
 the seventh is duly prepared ; but instead of being resolved on a 
 chord of the leading note (which would be the fourth above the 
 subdominant), it is resolved on the first inversion of a supertonic 
 minor ninth. Again, in the example at 361, the first chord of 
 the third bar is a diatonic ninth on the mediant of A minor (in 
 reality, as was there shown, a last inversion of a dominant minor 
 thirteenth), and it is resolved, not, according to the law of diatonic 
 discords, on the submediant, but on the tonic. Again, at 432, 
 example (a) is a subdominant minor seventh, not prepared, and 
 resolved on a dominant chord ; while at 432 (b) is a subdominant 
 major seventh, here prepared, but resolved also on a dominant 
 chord, the root of which is a second, and not a fourth above the 
 root of the discord. 
 
 496. These examples show that in many cases the strict rule 
 was not observed even by the old masters. Where disregarded 
 it appears to have been that the chords were looked at according 
 to their true derivation, which has been seen to be as part of the 
 dominant harmony. A few more examples may be given. 
 
 (a) J.S.BACH. " O Ewigkeit, du Donnerwort. 1 
 
 r~ C^T
 
 2l8 
 
 [Chap. XVIII. 
 
 J. S. BACH. " Johannes Passion." 
 
 L: J. 
 
 1 j j j . j 
 
 p 
 
 V 
 
 T. S. BACH. " Ich armer Mensch, ich Stindenknecht." 
 
 J J J3 
 
 At () is seen the chord of the mediant seventh, with the seventh 
 prepared, resolving on the submediant seventh. Here the root 
 rises a fourth, according to rule ; but the submediant seventh is 
 resolved on the first inversion of the triad on the leading note, the 
 root rising only a second instead of a fourth.* At (&) the super- 
 tonic seventh is taken without preparation, and resolved on the 
 second inversion of the tonic. The chord * at the end of the 
 first bar of (c) is a parallel case to the second at (a) but in the 
 third bar of this passage is a mediant seventh with an ornamental 
 resolution, the seventh going to the fifth of the chord ( 202), 
 before the change of harmony, and the chord itself being resolved 
 on the first inversion of a subdominant triad. 
 
 497. These examples, which might be increased to any extent, 
 show that even in the last century the old law as to the prepara- 
 tion and progression of these chords was often disregarded ; and 
 it would be absurd to fetter ourselves now by any such rule. Let 
 it therefore be clearly said that any essential discord can be taken 
 without preparation. None the less can we deduce from the 
 examples here given some general principles for our guidance in 
 the treatment of these discords. 
 
 498. It has been several times said, in speaking of the funda- 
 mental discords in preceding chapters, that account has only to be 
 taken of such notes as are actually present in the chords. We 
 adopt the same course in dealing with these diatonic discords, 
 which, as has been seen, are parts of fundamental dominant 
 harmony. For example, in the chords shown in 490, the upper 
 note of each chord is the seventh above the lowest note, which is 
 in each case the root ; and the seventh of each chord should fall 
 
 * In this explanation the F is considered as a suspension. If, however, the first 
 quaver of the bar be regarded as a supertonic seventh without the fifth, the old rule 
 is still broken, as this chord resolves on the first inversion of a triad on the leading 
 note, the root falling a third instead of rising a fourth.
 
 Chap. XVIII.] 
 
 ITS THEORY AND PRACTICE. 
 
 219 
 
 one degree. This rule, like all other rules given in this volume, 
 is founded upon the practice of the best masters, and is illustrated 
 in all the examples we have quoted. It is also generally advisable 
 if the seventh of these chords has appeared in the preceding 
 chord, to prepare it, that is to say, to retain it in the same voice 
 in which it first appeared. (Compare 113.) It will also be very 
 often expedient to follow such a chord, according to the old rule, 
 by another chord, the root of which is a fourth above its own. 
 
 499. Diatonic ninths are much rarer than diatonic sevenths, 
 but are treated in the same manner, the ninth falling one degree, 
 and the root generally rising a fourth. We give one example. 
 
 CHERUBINI. Mass in F. 
 i i *~ ^ ^ 
 
 -J 1 U 
 
 j * ** 
 
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 EXERCISES on CHAPTER XVIII. 
 
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 (IV.) 
 
 HARMONY : 
 
 [Chap. XVIII. 
 
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 Chap, xix.] ITS THEORY AND PRACTICE. 
 
 CHAPTER XIX. 
 
 SUSPENSIONS. 
 
 % 
 
 500. With the exception of the auxiliary notes explained in 
 Chapter XL, all the discords hitherto treated of have been 
 essential discords that is to say discords in which the dissonant 
 notes have been essential parts of the harmony. But there is a 
 large and important class of discords in which the dissonant notes 
 are unessential to the chord and in fact do not belong to it at all, 
 but are merely held on from the preceding chord. Such discords 
 are termed SUSPENSIONS. 
 
 501. A suspension may be very simply denned as a note of 
 one chord held over another of which it forms no part. The 
 sounding of such a note as a note of a chord is called the pre- 
 paration of the suspension (compare 486) ; the holding of the 
 note over the following chord, to which it does not belong, is the 
 suspension itself; and the ultimate progression of the suspended 
 note to its place in the chord is called its resolution* 
 
 502. At first sight a suspension would seem to have consider- 
 able resemblance to the prepared diatonic discords spoken of in 
 the last chapter ; but it differs from them in two very important 
 respects. In the first place a prepared diatonic discord is always 
 an essential note of the chord, and a suspension never is ; and, 
 secondly, a diatonic discord may be used on any part of a bar, 
 while a suspension can only be introduced on an accented beat 
 the first or third beat of a bar of common time (or, if the passage 
 be in quavers, on the first half of each crotchet), or on the first or 
 second beat of the bar in triple time. 
 
 503. The general rules governing the treatment of suspensions 
 are extremely simple. It will be convenient first to enunciate 
 them as clearly as possible, and then to give examples enforcing 
 the rules from the works of the great masters. 
 
 504. I. Any note of one chord may be suspended over the 
 following chord, provided that it is able to move by step of a 
 second upward or downward to one of the notes of that chord. 
 For example 
 
 * Evidently a suspension cannot be prepared by a passing-note, for that 
 would not be a note of the preceding chord.
 
 HARMONY : 
 
 [Chap. XIX. 
 
 (a) 
 
 at (a) the U of the first chord is held over the chord of C, and 
 moves down by step to the octave of the root ; and (b] it rises by 
 step to the third of the chord. But at (c) there is no suspension ; 
 because the D is a note of the second chord, and moves by leap 
 of a third ; besides which a suspension cannot form a part of the 
 chord over which it is held. (501.) 
 
 505. II. The preparation of a suspension (that is, the 
 sounding in the preceding chord of a note to be suspended) must 
 be in the same voice, and may not be on a stronger accent than 
 the suspension itself. Thus, if we alter the position of the bars in 
 (a) and (b] of the above example, 
 
 the passage is faulty, because the suspension is prepared on the 
 first beat of the bar, and appears on the second. It is, however, 
 possible for the preparation also to be on a strong beat 
 
 506. III. The suspension must always be on an accented 
 beat of a bar ; the suspended note may be either tied or repeated ; 
 and the resolution of the suspension should be on a less strongly 
 accented beat than the suspension itself. 
 
 507. IV. If the suspension be tied to its preparation, the 
 latter should be of at least equal length with the suspension ; it 
 may be longer, but it should not be shorter. When the sus- 
 pended note is sounded again, this rule is not so strictly observed ; 
 e.g. 
 
 () H 
 
 ANDEL. 
 1 
 
 " Samson." 
 
 -^-* 
 
 i/~^ *~ 
 
 1 
 
 '*' JA 
 
 
 
 Here the B flat in the second chord is not a part of the dominant 
 triad ; neither have we here a chord of the eleventh, for that 
 chord is not used without the seventh (375). The note B is
 
 Chap. XIX.] 
 
 Irs THEORY AM> PRACTICE. 
 
 223 
 
 therefore a suspension of the value of a dotted minim ; but its 
 preparation is only a crotchet. Occasional exceptions from this 
 rule are to be found, as in the following passage : 
 
 BEETHOVEN. Mass in D. 
 
 where a suspended crotchet is prepared by a quaver, but the 
 student is advised to adhere strictly to the rule here given. 
 
 508. V. The note on which a suspension resolves may be 
 sounded at a distance of at least an octave below the suspension ; 
 but should not be heard above it, excepting sometimes in the 
 suspension of a whole chord, as we shall see later, when all the 
 parts move by step of a second ($ 5 1 5 (d}}. This exception is 
 illustrated at example (<) above, where D is heard above its sus- 
 pension E, all the parts moving by step. 
 
 509. VI. As a suspension is only a temporary .substitute for 
 the harmony note which follows it, a progression which would be 
 incorrect without a suspension is not justified thereby. 
 
 For example, 
 
 the suspension (a) involves the same consecutive octaves as are 
 seen at (c) ; and (b) the same consecutive fifths as at (d). This 
 rule, however, is not always strictly observed by the old masters 
 in the case of fifths, where the progression is less unpleasing than 
 with the octaves. 
 
 If the suspensions are omitted here, we have clearly consecutive 
 fifths between the two upper parts ; but the effect of the passage is 
 quite unobjectionable. 
 
 510. VII. A suspension always resolves on the chord over 
 which it is suspended ; and it is by no means unusual for the
 
 224 
 
 HARMONY : 
 
 TChap. XIX. 
 
 chord to change its position on the resolution of the suspension, 
 as in the following example. 
 
 BACH. " Matthiius Passion." 
 
 Here the G in the last bar of the alto is a ninth suspended over 
 the root position of the chord of F minor. At the moment of 
 resolution the bass goes to A flat, and the suspension is resolved 
 on the first inversion of the same chord. The student will hardly 
 need to be reminded that G and B in the bass are passing notes. 
 But if the root of the chord changes at the moment of resolution, 
 the preceding chord is not really a suspension. 
 
 HAYDN. "Creation." 
 
 r~r 
 
 The E in the second bar of example (b) would be a suspension of 
 the octave by the ninth, if the second chord of that bar were the 
 chord of D; the fact that the root changes to B proves that the 
 first chord is a diatonic ninth. Similarly at (c) the B7 in the 
 tenor of the chord * cannot be a suspension of the ninth ; for the 
 root rises a fourth to Db, when the B resolves ; we have here an 
 incomplete chord of the diatonic ninth, with only root, third, and 
 ninth present. A somewhat different example will be seen at 191 
 (a). The E in the alto of the second chord looks like a fourth 
 suspending the third of B, but the resolution on a different root 
 shows that it is really an incomplete chord of the dominant 
 eleventh. When the ninth and eleventh resolve on their own 
 roots, they are very rarely found without the seventh. 
 
 511. It was said just now ( 504) that any note of a chord can 
 be suspended over the next chord, provided it can move by step
 
 Chap. xix. j 
 
 Irs THEORY AND PRACTICE. 
 
 225 
 
 to a note of that chord. But to this general rule there are certain 
 practical limitations. 
 
 (/) 
 
 J 
 
 
 J. 
 
 I s " 
 
 ' 
 
 7 8 
 
 23 
 
 65 
 
 At (a) the root is suspended by the ninth and at (b) by the seventh ; 
 at (c) the third of the chord is suspended by the fourth, and 
 at (d) by the second. At (e) the fifth is suspended by the sixth, 
 and at (/} by the fourth. These last suspensions are very seldom 
 used ; that at (e) is ambiguous, as the first chord in the bar may 
 be regarded as a chord of the thirteenth with only generator, third, 
 and thirteenth present (S 410), resolving upon its own generator. 
 It may be remarked here that, excepting the suspension of the 
 leading note over the tonic chord, as at (I)), the suspensions 
 resolving upwards are much rarer than 'those resolving down- 
 wards. 
 
 512. The practical limitations just referred to are seen as soon 
 as we try to suspend any of the higher notes of a fundamental 
 chord. The seventh cannot be delayed by a suspension, because 
 the note below it is the thirteenth from its generator, and the note 
 above it is the octave ; similarly the notes above and below the 
 ninth, eleventh, and thirteenth are all notes of the same harmony ; 
 and a suspension has been defined as a note held over a chord of 
 which it forms no -part. In practice therefore, the only available 
 suspensions are those given above ; but these may be taken in any 
 part of the harmony, and even, as we shall see presently, in two 
 or three parts at once. 
 
 513. The figuring of suspensions will give the student little 
 trouble, if he will bear in mind the often repeated rule that the 
 figures simply indicate the distance of certain of the upper notes 
 from the bass. But, inasmuch as some suspensions bear the same 
 figuring as some of the chords which have already been studied, it 
 is very important to know how to distinguish between a suspension 
 and a chord which is figured in the same way. 
 
 514. If we look at example (a) of S 511, we shall see that it is 
 figured like a chord of the ninth. But if th : s were really a chord 
 of the ninth resolved on its own generator, the seventh would be 
 present, or if not, would certainly be introduced when the n : nth 
 resolved ( 325). The resolution and the form of the chord 
 therefore prove the 9 to be here a suspension. 
 
 515. If we take the inversions of the same chord, still retaining 
 the suspension,
 
 226 HARMOKY : [Chap, xix. 
 
 A 
 
 7666 4 7 
 
 54 2 4 
 
 2 
 
 \ve shall find that the first inversion (a) is figured like a chord of 
 the seventh. But if it were a fundamental seventh it would have 
 a major, not a minor third, and it could not possibly resolve on 
 the chord of C. If it were a diatonic seventh (see Ch. XVIII.), it 
 would resolve on a root a fourth, or a second above its own. The 
 resolution proves it to be a suspension. Whenever the figure 7 is 
 followed by 6 on the same bass note, 7 indicates a suspension of 
 the sixth, not a chord of the seventh. There is no mistake which 
 students are more apt to make than to confound the two 
 meanings of the figure 7. Whenever it is a suspension of 6 
 
 (which implies ^), the seventh must not be accompanied by the fifth ; 
 
 because the fifth is not a note of the chord of the sixth ?ver which 
 the seventh is suspended. 
 
 516. The same reasoning applies to examples (b} and (c). 
 The figures - and 4 are the same as for the first and third inver- 
 
 J 2 
 
 sions of a chord of a seventh ; but the resolution in each case 
 shows that we have here only suspensions. If the at (b) were 
 
 the first inversion of a seventh, it could not resolve on its own 
 bass note, but would resolve on the chord of A minor ; as would 
 
 also the ^ at (c) were it really the last inversion of a seventh. As 
 2 /- 
 
 the chords are only suspensions, the must contain no third, and 
 the 2 n o sixth ; these notes being no part of the harmony on 
 which the suspension is resolved. 
 
 517. It has been said above (508), that the resolution of a 
 suspension should not generally appear above it. In order to 
 avoid this, the leading note at (c) leaps upwards to the third of the 
 next chord. In this case, however, it would be possible (though 
 less usual) for the resolution to appear above the suspension, as 
 at (cf). Such a progression is allowed when all the parts move, as 
 
 7 
 here, by step. This suspension would be figured 4, and there is 
 
 2 
 no danger here of confounding it with an inversion of a seventh. 
 
 518. For the sake of completeness, we now give the inversions 
 of the suspended fourth shown in (c) 511, with the figuring of
 
 Chap. XIX.] 
 
 Irs THEORY AND PRACTICE. 
 
 227 
 
 each. The suspended fourth is distinguished from a chord of the 
 eleventh by the absence of the seventh ( 375). 
 
 After what has already been said, the student will have no diffi- 
 culty in understanding these inversions. 
 
 519. Like the chord of the seventh ( 202), a suspension can 
 have an ornamental resolution. This is effected by giving the 
 suspended note only half its proper length, and interposing be- 
 tween it and the note of its resolution either a note of the chord 
 taken by step or leap, 
 
 er notes of shorter value, which may be either harmony notes or 
 a mixture of these with auxiliary notes, the latter being of course 
 subject to the rules given in Chapter XL 
 
 
 The resolutions at (a) and (b] are very common ; that at (c) is 
 rarer, and less advisable. In example (b) it must be remarked 
 that although the suspension takes the note of its resolution, 
 C, on the second crotchet of the bar, this is not counted as its 
 actual resolution, because it does not remain there. If after 
 descending to C we had returned to D for the second quaver, 
 
 the progression would have been monotonous and weak. 
 
 520. Occasionally in the case of an ornamental resolution of 
 a suspension, the preparation of the suspension will be only of 
 the same length as the suspension without its ornamentation. 
 Though examples of this are not infrequent, the method is not 
 recommended for the student's imitation. 
 
 521. We shall now give a series of examples illustrating the 
 treatment of suspensions by the great masters. Our first extract 
 will be a well-known passage from the " Messiah."
 
 228 
 
 HARMONY: i.chap xix. 
 
 HANDEL. "Messiah.' 
 
 
 4JJ3767687 4365 
 
 We have here some of the most frequently used suspensions. At 
 the first crotchet of the bar is 4 3, at the second and third, 7 6 
 (the first inversion of the suspension 9 8). There is no sus- 
 pension at the last crotchet of the first bar, because the quaver C 
 is a note of the harmony. The second crotchet of the second 
 bar shows the somewhat rare and always ambiguous (511) 
 suspension 6 5. 
 
 522. Our next illustrations 
 
 (a) 
 
 SCHUBERT. " Rosamunde." 
 
 > ft g=M J=; J~j =^ 
 
 US 
 
 9 8 
 
 BACH. Organ Fugue in C minor. 
 
 show the ninth suspending the octave. At (a) the octave is the 
 root of the chord ; at the last bar of (b) it is the third of the chord ; 
 ,the 9 is therefore accompanied by 6, and we see here, as at 518 
 >(a), the first inversion of the suspended fourth. The second bar 
 tof this extract shows the last inversion of the same suspension 
 ((compare (c), S 5 1 8). 
 
 523. We shall now see the two suspensions 98 and 4 3 in 
 their second inversions.
 
 Chap. XIX.j 
 
 ITS THEORY AND PRACTICE. 
 
 BEETHOVEN. Mass in D. 
 
 229 
 
 r-fl-S. . J_ .-* J ' * -* V 
 
 P^^p3gp^||^|]|l^|^| 
 
 The chord * at (a) is the second inversion of 9 8, as will be seen 
 by reference to (/>) $51 5- At (/;) the third crotchet is the second 
 inversion of 4 3 compare 518 (b}. The last bar of this example 
 also shows the rare first inversion of the suspension 6 5. 
 524. In the following example, 
 
 MENDELSSOHN. " St. Paul." 
 
 i p* ^j i p i^_f^^m 
 
 is seen in the second bar the last inversion of the suspension 9 8, 
 the third and fifth of the chord being at a distance respectively of 
 the second and fourth above the ninth. The root position of the 
 suspension 4 3 will also be seen in the last bar. 
 
 525. We now give examples of suspensions resolving upwards. 
 Some theorists call these " Retardations " ; but there seems no 
 reason for giving a different name to them. By far the most 
 common upward suspension is 7 8 the suspension of the root of a 
 chord by the note below it. 
 
 BACH. Organ Prelude in E minor. 
 
 87 8 -[- 
 BEETHOVEN. Sonata, Op. 10, No. x.
 
 230 
 
 HARMONY : 
 
 [Chap. XIX. 
 
 The first of these passages is too simple to require explanation. 
 Example (b) is interesting not only for the 7 8 in the second bar, 
 but because in the first bar we see the second inversion of a 
 dominant eleventh, looking like a suspended fourth ($518), not, 
 as before in a triad, but in a chord of the dominant seventh. The 
 last bar also contains a very rare inversion 9 8 as the second 
 inversion of 6 5. 
 
 526. We hardly ever find a 7 8 suspension except on the 
 tonic of the key. The following passage, therefore, deserves 
 noting for its rarity. 
 
 WAGNER. " Die Meistersinger." 
 
 3; 
 
 ^m 
 
 Here the submediant of the key is suspended by the dominant. 
 
 527. The two examples next to be given show the first in- 
 version of the suspension 7 8. 
 
 BACH. " Gott der Vater wohn'uns bei." 
 
 BACH. "Wachetauf! ruft uns 
 (6) die Stimme." 
 
 The first of these extracts is very rich in suspensions ; the student 
 has already met with all of them excepting the one marked *. 
 This at first sight looks like the triad on the mediant ; that it is 
 not so is proved by the preceding chord ; for a second inversion 
 of a dominant seventh never resolves on a mediant chord (fee 
 Chapter IX.). The C is therefore the suspension of the following 
 D. 
 
 528. The extract (b} is rather different. Here the root of the 
 dominant chord, G, is suspended by a note a tone, instead of a 
 semitone, below it. If we had here a chord of the seventh with
 
 Chap. XIX.] 
 
 ITS THEORY AND PRACTICE. 
 
 231 
 
 an ornamental resolution ( 202), the F of the alto after rising to 
 G would fall to E. 
 
 529. The following examples show the other inversions of 7 8. 
 
 f \ BACH. " Wohltemperirte Clavier," Book 2, Prelude I. 
 
 ' ^mf^mm^^t^a^ ^MHM^HM 
 
 * * 
 
 BACH. " Wohltemperirte Clavier," Book 2, Prelude n. 
 
 In the last crotchet of (a) the harmony is that of the chord of A 
 minor in the second inversion, the root in the upper part being 
 suspended by the leading note. The chord * looks like the root 
 position of a dominant minor thirteenth, but, as at 527 (a), this 
 would be no proper resolution of the preceding dominant seventh ; 
 the chord is therefore a suspension. The last inversion of the 
 same chord is shown at (b} ; and it will be observed that both 
 the suspensions at (a) and (l>} have ornamental resolutions. 
 
 530. Though the 7 8 is the most common of the upward 
 suspensions, it is by no means the only one. It is not rare to find 
 the third suspended by the second. Many examples of this 
 might be quoted from Bach ; but as all our last extracts have been 
 taken from that composer, we select two more modern passages. 
 
 BEETHOVEN. Quartett, Op. 18, No. 3. 
 
 ~ 
 
 52 3 
 
 DVORAK. "Stabat Mater." 
 
 
 (g;V 
 
 'W' V 
 
 
 7 6
 
 232 
 
 HARMONY : 
 
 [Chap. XIX 
 
 At (a) the third of the supertonic chromatic minor gth is suspended 
 over the first inversion of the tonic chord, and rises a semitone ; 
 at (If) the second of the chord is suspended over the third. The 
 last bar of this passage shows a very curious doubling of a sus- 
 pension ; the C is suspended both in alto and tenor, one part 
 falling and the other rising to the note of the harmony. 
 
 531. The suspension 4 5 is extremely rare, though tolerably 
 
 common in its inversion , an example of which is seen at (a) 
 530. We give an instance of the 4 5 in its less usual form. 
 
 BACH. " Wohltemperlrte Clavier," Book 2, Prelude n. 
 
 * *^. i p^k_ 
 
 532. We have already met with a few cases of ornamental 
 resolution ; we add two more. 
 
 * _ r 
 
 MOZART. Sonata in A minor. 
 
 P3- 
 
 786 
 CHERUBINI. Mass in F, No. i. 
 
 r J 
 
 At (a) the resolution is approached by leap of a third from a 
 harmony note a second from the suspension. In the last bar of 
 
 (a) there is no suspension ; the student will easily see why. At 
 
 (b] is the very common ornamental resolution where the suspension 
 leaps to another note of the harmony before returning to its note 
 of resolution. 
 
 533. Two or more notes of a chord may be suspended at the 
 same time, or a whole chord may be suspended over the following 
 one. If two notes are suspended, this is called a " Double 
 Suspension " ; if three or more are suspended, it is usually called 
 the "Suspension of a complete chcrd."
 
 chap xix. ] Irs THEORY AXD PRACTICE. 
 
 (a) , MENDELSSOHN. "St. Paul." 
 
 2 33 
 
 We see here examples of double suspensions, which will require 
 no further explanation. 
 
 534. The following example is rather more intricate : 
 
 MOZART. Sonata in F. 
 
 ^ 
 
 *-^r& *~ 
 
 Here is a series of double suspensions, in every case with an 
 ornamental resolution which is frequently chromatic. In the first 
 three bars the two lower parts are suspended, and in the fourth 
 and fifth bars the two upper parts. We have not figured the bass 
 of this example, because no ordinary system of figured bass will 
 apply to such unusual and chromatic progressions. It is of course 
 possible to figure them, but the student would be more perplexed 
 than assisted by the strange combinations which would be 
 necessary. ' 
 
 535- We conclude this chapter by a few examples of the 
 suspensions of complete chords. After the explanations already 
 given, the student will have no difficulty in analysing them for 
 himself.
 
 234 
 
 HARMONY : 
 
 [Chap. XIX. 
 HANDEL. " Samson." 
 
 HELLER. "Promenades d'un Solitaire," Op. 78, No. 5. 
 
 (3) f^^ ^^ .^^ ^ 
 
 ii ^ r J F^ d 5 - 
 
 
 (f) s f s f sf sf BEETHOVEN. Sonata, Op. 14, No. 2. 
 
 ?=^- 
 
 SE 
 
 S& 
 
 ^^Sr-3: 
 
 yj= 
 
 EXERCISES TO CHAPTER XIX. 
 
 [In these exercises many essential discords looking like sus- 
 pensions are introduced, that the student may learn to distinguish 
 between the two. See ^512, 525.] 
 
 (i.) 
 
 g 
 
 
 
 76439870 J6436443987656C 43 
 6 5 3 6 54 
 
 (II.) 
 
 E^^^feE5EJ=J=EJ 
 
 ^=3- 
 
 4 b 
 2 2 
 
 43 6 676464365 
 
 42 43 
 
 333 
 
 623 
 4 
 
 (in.) 
 
 
 696 
 
 G 4 {4 6 Jf6 7 
 
 2 44 
 
 32 
 
 7 6 
 
 
 6 98 98 76 76 76 4Jf 
 5 4 3 b<> 4 r, 
 
 4 t
 
 Chap. XIX.] 
 (IV.) 
 
 ITS THEORY AND PRACTICE. 
 
 2 35 
 
 439643 
 
 5 
 
 2 
 
 
 7J6 J7 6 768 7 
 
 9 
 
 4787643 4 
 
 2564 2 
 
 3 
 
 &*=t 
 
 986 5 676656743 
 
 (a) The figures here indicate a suspension of a complete chord, and the order 
 in which they are placed shows on which note each suspension is resolved. 
 Compare Exercise VI. to Chapter XV. 
 
 (VI.) 
 
 f=f f- 
 
 698 J6 76 5 6 J767676 
 
 543 85 2 J4 4 J4 
 
 f2 - - 2- 
 
 766643 987668* 76 7 7 7fc66 85 
 fa 6 5 8 6 4 2 85 85 4 C5 4 4 |3 
 
 4 2 2 
 
 (VII.) f . . 
 
 ~-3^+=Z -"1 - ""^ ~ 
 
 i i i 
 
 6 6 
 
 4 
 
 7656 6 
 
 
 4 3 987666 
 X6 5 
 
 7 _ 
 3 
 
 
 4 
 
 2 
 
 i 6 H4 3 9 8 786 J5 4343435667 
 Q5 6 55 6 5443 
 
 4 
 
 (VIII.) 
 
 676^ 766 48 
 455 
 
 B 
 
 6 j 9 3 C6 fl7 8 57 4 3 
 4B7846- 
 
 b 4 
 
 ^ 
 
 4 676 7 
 
 2 |]4 54 
 
 b 2 
 
 67 
 5 4 J
 
 236 
 
 [Chap XIX. 
 
 (IX.) 
 
 =p TT~~I J I ^ 
 ~f=-+~\?3~ 
 
 4 9898 98656 623 
 3 56767054 23 
 
 54 2 
 
 
 76767678 236 
 456 4 
 
 1 H ' 
 
 6 4 543 
 22 
 
 9364 
 78 2 
 
 43986 fc9 87 
 
 3l>3 6 5 
 
 4 3 
 
 (X.) 
 
 
 fig 
 
 645J576J665 76 743X6 65 J7 
 
 5 8 J5 6 5 4'- {* 
 
 698 86 6- 
 4 
 
 6 76 66057 
 4 5 \ 4 j 
 
 3
 
 Chap, xx.] ITS THEORY AND PRACTICE. 
 
 CHAPTER XX. 
 
 PEDALS. 
 
 536. A Pedal is a note sustained by one part (generally, 
 though not invariably, the bass) through a succession of harmonies 
 of some of which it does, and of others it does not, form a part. 
 The name is no doubt derived from the pedals of the organ, 
 which are used to play the bass of the harmony, as is evident from 
 the fact that what we call a pedal is called in France and Germany 
 an "Organ-point." 
 
 537. There are few points in connection with harmony in which the rules 
 given in old treatises differ more widely from the practice of modern composers. 
 It will therefore be necessary in this chapter to modify very considerably the 
 precepts of the old theorists. The more important of such modifications will 
 be noted as they present themselves. 
 
 538. The note used as a pedal is almost invariably either the 
 Tonic or the Dominant, the latter being the more common. A 
 tonic pedal is mostly found toward the end of a movement, though 
 it is sometimes met with at the beginning. A well-known instance 
 of this is the opening of the Pastoral Symphony in the " Messiah." 
 
 539. It is not unusual for the same piece to contain toward 
 the end both a dominant and a tonic pedal ; when this is the case, 
 the dominant pedal almost always comes first, the tonic pedal 
 being reserved for the close. A very good example of this may be 
 seen in the last ten bars of the great five-part fugue in Cft minor 
 in the first book of Bach's " Wohltemperirte Clavier." 
 
 540. When the pedal note forms no part of the chord above 
 it, the note next above the pedal is considered as the real bass of 
 the harmony for the time being, and is not allowed to move in a 
 way in which it could not move were it a bass note. An example 
 will make this clear. 
 
 CO 
 
 At the third crotchet of (a) is the second inversion of the 
 dominant chord over a tonic pedal. The pedal note is not a part 
 of the chord ; therefore D, the bass for the time being, may not 
 leap to A ( 165). But the progression at (t>) is quite correct.
 
 2 3 8 
 
 HARMONY : 
 
 [Chap. XX, 
 
 54i- The following examples of dominant pedals require little- 
 explanation. 
 
 CHEKUDINI. Mass in F. 
 
 At (a) is a dominant pedal in the key of B flat minor, and at (b) 
 a dominant pedal in the key of D major. Note that at the be- 
 ginning of the second bar of (b} there is a momentary modulation to 
 E minor, and at the end a modulation to A major, the dominant 
 pedal thus becoming for the time a tonic pedal. 
 
 542. The end_ of the second fugue of Bach's "Forty-eight" 
 gives an excellent example of a tonic pedal, ending with a " Tierce 
 de Picardie" ( 193). 
 
 BACH. " Wohltemperirte Clavier," Book i, Fugue 2 
 
 This passage illustrates a point of some importance which must 
 here be noted. The fugue from which it is taken is for three voices 
 (or parts), but the pedal note itself is not reckoned as one ; it 
 will be seen that in the last bar of the extract there are three parts 
 exclusive of the pedal. This is a case of very frequent occurrence, 
 both in three and four part harmony. 
 
 543. Though a pedal note is most often found in the bass, it 
 is by no means unusual to meet with it in an upper part. In 
 such a case it is called an " Inverted Pedal." 
 
 BEETHOVEN. Mass in C. 
 
 I
 
 Chap. XX.] 
 
 ITS THEORY A.VD PRACTICE. 
 
 2 39 
 
 At (a), of which the voice parts only are given, is seen the domi- 
 nant in the upper part of the harmony, sustained as a pedal note, 
 and at (b) is the tonic, also in the upper part, similarly treated. 
 
 544. Sometimes a pedal note is found at the same time above 
 and below, as in the following examples : 
 
 
 MOZART. Sonata in D. 
 J- 
 
 The student will remember what has so often been insisted upon 
 that broken chords (as in the bass of example (a) are harmonically 
 the same as if all the notes were struck together ($ 192). At (a) 
 we see a dominant pedal, and at (&} a tonic pedal in the highest 
 and lowest part of the harmony at the same time ; at (b) the pedal 
 note is also in the middle. 
 
 545. A pedal note in a middle voice is somewhat rarer. Two 
 examples have already been seen in 303 (e) and 349 (a). We 
 add another of a somewhat different character.
 
 240 
 
 HARMONY . 
 
 [Chap. XX. 
 
 te^r^> ^ 
 tj i~ i 
 
 BEETHOVEN. Sonata, Op. 27, No. i. 
 
 The fifth quaver of the second bar is here evidently the first 
 inversion of the chord of F minor, and the B is a dominant 
 pedal. The special interest of the passage arises from the fact of 
 the pedal note being so close to the other notes of the harmony. 
 In the orchestra, with different qualities of tone, such combinations 
 are common enough ; on the piano they are very rare. 
 
 546. A pedal passage often begins, and generally ends, with a 
 chord of which the pedal note itself forms a part. This has been 
 the case in all the examples hitherto given ; * but it is by no 
 means invariable, as the following passages prove. 
 
 
 AUIIKR. " Masaniello." 
 
 " .-^-- 
 
 So much judgment and experience are required to know when it 
 is advisable to quit a pedal in this manner, that the student is 
 advised to adhere to the general practice rather than to imitate 
 t lese somewhat rare exceptions. 
 
 547. The old masters seldom modulated much on a pedal ; 
 beyond the keys in which the pedal was tonic or dominant they 
 rarely ventured, excepting into the key of the supersonic minor, 
 
 * The old theorists give it as an invariable rule that a pedal point must end 
 with a chord of which the pedal is oi<e of the. notes ; but in thf? lij;ht of ir.ortern 
 practice, this must be regarded as a recommendation rather than as a lav/
 
 Chap. XX.] 
 
 ITS THEORY AND PRACTICE. 
 
 241 
 
 as in the passage from Haydn, 541 (t>). Modern composers, 
 however, recognise no such restriction ; and almost any modula- 
 tion may now be used on a pedal. It is, however, very rare to 
 find much modulation on an inverted pedal. 
 
 548. The following examples of free modulation on a pedal 
 will be instructive. 
 
 MENDELSSOHN. Variations, Op. 82. 
 
 Here we see a dominant pedal in E flat ; at the third bar a modu- 
 lation is made to G minor, and the music continues in that key to 
 the end of the extract. 
 
 549. Our next passages are more curious. 
 
 m 
 
 SCHUMANN. " Paradise and the Peri." 
 I k-, n I 1-, 
 
 I I J J 
 
 We have here a dominant pedal in B minor, with a distinct 
 modulation to G major ; in the continuation of the passage, a 
 further modulation is made to E minor. 
 
 SCHUMANN. Humoreske, Op. 20
 
 242 
 
 HARMONY : 
 
 [Chap. XX. 
 
 On this dominant pedal in B flat is a modulation to the very 
 remote key of E minor, the music returning at once to the key of 
 B flat. 
 
 550. Two examples by living composers will conclude our 
 illustrations of this point. 
 
 aJ 
 
 DvcmXK. " St. Ludmila." 
 
 J. J* 
 
 r g 
 
 w 
 
 f -h&y -^tf 
 
 J I Ix- }[ I M 
 
 bJ . * -*- gJ- -^ 
 
 I 7-0 i-^ r E . 
 
 Here is seen a tonic pedal in D minor, on which is a modulation 
 to the keys of C and B flat. 
 
 (g. MACKENZIE. "Jason." 
 
 
 
 In this passage there is a dominant pedal in D, with a modulation 
 in the third bar to the key of F.* 
 
 551. It is possible, though rare, to introduce ornamentation 
 on a pedal note. In his organ fugue in D minor in the Dorian 
 mode, Bach has a shake on a pedal ; and towards the close of the 
 finale of his symphony in A, Beethoven introduces a pedal E 
 alternating with an auxiliary note. 
 
 552. A double pedal, with both dominant and tonic sustained, 
 is sometimes to be met with ; but in this case the tonic must be 
 below the dominant, otherwise the fourth above the bass will 
 
 * Some theorists strictly forbid any modulations on a pedal except those 
 referred to in 547; but, provided the modulations are properly managed, the 
 restriction appears unnecessary.
 
 Chap. XX.] 
 
 ITS THEORY AXD PRACTICE. 
 
 243 
 
 produce an unpleasant effect. An example will be seen in 475. 
 With a double pedal it is rare to find chords of which neither 
 pedal note forms a part. A very fine example of a double pedal, 
 which is unfortunately far too long to quote, will be found at the 
 end of No. 2 of Brahms's " Deutsches Requiem." 
 
 553. Schumann, in the second movement of his symphony in 
 E flat, has tried a novel experiment, which (so far as we are aware) 
 has never been imitated. In the episode in A minor of that 
 movement, he has used the mediant, C, as a pedal note. A few 
 bars would not be sufficient to give a fair idea of the passage ; we 
 must therefore refer our readers -to the work itself. 
 
 EXERCISES TO CHAPTER XX. 
 
 [The figures in these exercises will require careful attention, as 
 numerous combinations will be found not hitherto met with. In 
 a few cases where four figures are placed under a pedal, it indi- 
 cates that there is to be four-part harmony exclusive of the pedal 
 note itself.] 
 
 1 1.) 
 
 
 
 
 37675656567876 7 5 6 4 
 445 3g4 g4 fl4 3 4 g4 5 jf 5 63 3 
 2 4 2 b3 b 2 2 J2 3 4 
 
 6 67 g4 6 g6 7 
 * I fl * 
 
 m i ^j * | ~ | 
 
 5 6 7 $7 8 Jf4 765 
 3 J4 5 g5 ti J2 $543 
 2 b3 3 4 3 
 
 6J7 5 t!6 5 JS 6' 7 8 7 6 7 5 6 6 
 453 G455g553f 4 
 2 3243 2 
 
 
 7 
 
 (II.) 
 
 ^: v|3 : = * i~tvi~ 
 
 1 i* r f 5 ' i* ^ i f 
 
 
 V^b I J \ ^ ^__*L ^ ^ > 
 
 6 6 4 6 D4 IT6 
 
 5 9 B2 t 
 
 i ' i i 1 i r- 
 b7 Q4 5 |j7 6 C7 i 
 1] J2 3 5fa4 54 J 
 2 2 
 
 '"^^"j^r^T 
 
 6 6 
 I 
 
 * p ~f~- 
 
 1 1 1 1 . _ _ 
 
 
 
 5 4 tt 2 2 2 
 () 
 
 2 543 
 398 
 
 5 
 1 J || 
 
 ^^ 
 
 6 8 9 3 8 b7 
 
 6378b65 7 5 
 
 -j-^ B 
 
 * 
 
 8 5 7 b7 
 
 
 
 656 5 
 
 2943 4 
 2 2 
 
 
 (a) From this point there should be four parts above the pedal.
 
 244 
 
 HARMONY : 
 
 (in.) 
 
 J==t 
 
 56 556 6 7 6 5 b7 6 57 8 H6 6 56 
 
 4 5 4 53 D5 4 3 4 4 
 
 3 17 23- 
 
 PfeJ 57 S7 656757 8 -J4 6 
 VG7j6tJ653 4*754 6532 
 483 4 j|3 2 4 5 - 
 
 765 
 4 - 9 
 
 (IV.) 
 
 57575657 5 
 
 34343434 3 
 
 22 2 
 
 6 656 
 44 
 
 7676 
 444 
 2 
 
 57 D7 6 7 94 
 5554 1 
 
 43 ' 
 
 6756 78 
 
 4 4 3 4 176 5 
 
 2 43 
 
 (V.) Hymn Tune. 
 
 6 7 50 
 
 6 6 
 175 
 
 
 7 
 
 fc 
 
 88 88 
 
 5b7D6b6 
 
 11 5 5 ^ ^ 
 
 [The following exercise will be found more difficult than any 
 of the preceding. At the third and fourth bars an inverted pedal 
 (543) is introduced ; while the last two bars have a pedal note 
 both above and below ( 544)-] 
 
 (VI.) 
 
 =t=tai 
 
 LLf LLT 
 
 6 465 
 
 2 66 6 666666 
 
 = f C =fr 
 
 JJ4 
 4 2 
 
 6 6 56 6 
 
 56 
 4 
 3 
 
 677 
 
 4 5 5 
 
 m 
 
 3 57 6 fl7 7607 
 443 5435 
 222 
 
 7 5 7 87 (7 fl7 8 $? 
 
 3344 354 65 
 
 2 2 48 
 
 6 7 56 5,4 6 56 6 BT 8 
 
 !| !J5 4 B5 876567 S078 
 
 3 354342 3451
 
 Chap, xxi.] ITS THEORY AND PRACTICE. 245 
 
 CHAPTER XXI. 
 
 HARMONY IN FEWER AND MORE THAN FOUR PARTS. 
 
 554. Though four-part writing is justly considered the founda- 
 tion of harmony, it is very seldom that a composition of any 
 greater dimensions than a hymn-tune is found in which the 
 harmony is in four parts throughout. Even in a quartett, one 
 part, and sometimes two, often have rests ; while in instrumental 
 compositions we frequently meet with passages in five, six, and 
 even more parts. We shall therefore conclude this book with a 
 chapter explaining the general principles by which a composer is 
 to be guided when writing in fewer or more than the four parts 
 hitherto treated. 
 
 THREE-PART HARMONY. 
 
 555. In writing in three parts, it is clear that if all the notes of 
 a triad are present none can be doubled. It is advisable, where 
 practicable, to introduce all the notes of the triad ; but it frequently 
 becomes needful to omit the fifth, e.g., 
 
 (") i i (fl i i (<3 l 
 
 If the chord of C be followed by the chord of A minor, both 
 being in root position, and with the third of each chord at the 
 top, it is clear that if we introduce the fifth in both chords, as at 
 (a), we shall have consecutives ; it will therefore be necessary to 
 double either the root, as at (), or the third, as at (c), in the 
 second chord, the former being preferable. 
 
 556. In chords containing more than three notes, the most 
 characteristic notes of the chord should be retained. Thus in a 
 chord of the seventh, if the root be present, the fifth should be 
 omitted, so as to keep the third and seventh ; similarly, in a root 
 position of a chord of the ninth, either the third or seventh, and 
 in an inversion of the ninth both these notes should be retained. 
 Not seldom, however, one of the parts takes two notes of the 
 harmony in succession, as in the last bar of the example 425, 
 where the upper part moves from the seventh to the fifth, and the 
 middle part from the fifth to the third of the chord. In this 
 passage, though there are four notes, we have only three-part 
 harmony, with one part doubled in the octave ( 98). 
 
 557. In instrumental music, three-part harmony is often vir- 
 tually in four parts because of one of the parts moving in arpeggio.
 
 246 
 
 HARMONY : 
 
 [Chap. XXI. 
 
 For instance, at example (a) of $419, there are never more 
 than three notes sounded at once ; but the middle part being in 
 arpeggio, there is practically four-part harmony throughout. This 
 is not three-part writing in the sense in which we are treating of it 
 now. 
 
 558. The rules for the position of the chords given in 127, 
 128, apply no less to three-part than to four-part harmony. The 
 parts should either be at approximately equal distances, as at 
 example (<:), 510, or the largest interval should be between the 
 bass and the part next above it, as at 509. But in vocal music 
 the compass of the voices may render it necessary to break this 
 rule. Cherubini's first mass is written for three voices soprano, 
 tenor, and bass ; and at 532 (b) will be seen an extract in which 
 it would be impossible to place the tenor nearer the soprano than 
 the bass. At $ 532 (a) the rule just given is observed. 
 
 559. In a full cadence in three parts it is desirable that the 
 penultimate chord (the dominant in its root position) should be in 
 as complete a form as possible, even if this involves leaving the 
 final chord incomplete. Thus in the example just referred to, 
 510 (c), the last chord but one contains root, third, and seventh ; 
 and the fifth is omitted in the tonic chord. In the final cadence 
 of the same movement, 
 
 MENDELSSOHN. "St. PauL" 
 
 -I * 
 
 the tonic appears without either third or fifth. 
 
 560. The close of the example just given illustrates another 
 important principle that the part-writing should have special 
 regard to the motion of the separate parts. The fewer the parts, 
 the more clearly each is individualised. Had Mendelssohn made 
 his tenor leap to A7 for the sake of having the third in the last 
 chord, the purity of the part-writing would have been impaired. 
 
 561. We now give two examples, one vocal and one instru- 
 mental, to illustrate the principles laid down. 
 
 MEYERBEER. " Robert le Diable."
 
 Chap. XXI.] 
 
 ITS THEORY AND PRACTICE. 
 
 247 
 
 This passage is the opening of the unaccompanied trio in the 
 third act of the opera. Notice how the chords are mostly com- 
 plete, and observe the reasons for the exceptions. 
 
 BEETHOVEN. Trio, Op. 87. 
 
 ( ' " J 
 
 ^~'=^^ 
 
 f^-j 
 
 -S r r r f 
 
 m-^ * * 1 -==-> 
 
 sf 
 
 
 ^ u^- 
 
 Jfc&=?-r f M^^^F 
 
 
 
 p i ^-i 
 
 
 
 
 L _u 1 
 
 Here is a piece of imitation between the outside parts in the first 
 four bars. In the fifth bar it will be seen that for the sake of an 
 easy flow of the parts, the two leading notes, which are implied 
 (F4 in the fourth quaver and D4 in the eighth), are omitted, 
 though they are important notes of the chord. As an excellent 
 specimen of pure three-part harmony, the student is recommended 
 to analyse the trio " Lift thine eyes," in " Elijah." 
 
 TWO-PART HARMONY. 
 
 562. Harmony in only two parts differs in one very essential 
 respect from harmony in three or four. As every chord consists 
 of at least three notes, it is obviously impossible here to have any 
 chord in a complete form. In two-part writing, therefore, the 
 harmony is merely suggested or indicated, and the chords are 
 only outline, or skeleton chords. 
 
 563. Genuine two-part harmony is somewhat rare; but we 
 very often in instrumental music find passages written in only two 
 parts, but of which one, being in arpeggio, indicates in reality 
 three, or even four-part harmony. To refer to some of the 
 extracts already quoted in this volume, at 191 (b\ 265 (d), 291 
 (/), and 333 (b), a three-part harmony is evidently present, though 
 the notes of two of the three parts are sounded in succession, 
 instead of together ; while the passages at 265 (c] and 428 (b) no 
 less clearly indicate four-part harmony. 
 
 564. It is common even in the strictest two-part writing (which 
 is found more frequently in instrumental than in vocal music) to 
 find broken chords, as in the examples above referred to. There 
 is not the least objection to these ; the important point is that the 
 suggested harmony shall be perfectly clear. Some of the most 
 masterly specimens of two-part harmony ever written are Bach's 
 "15 Inventions " ; the opening of the first is given here.
 
 248 
 
 HARMONY : 
 
 [Chap. XXI. 
 
 BACH. Inventio, No. i. 
 
 Let the student examine this passage carefully, and notice how 
 distinct is the indication of every chord. The addition of middle 
 parts here would increase the fulness, but not the clearness of the 
 harmony. 
 
 565. Our next example will be of later date. 
 
 MOZART. Duet for Violin and Viola. 
 
 
 Here, while we have several broken chords, a great part of the 
 harmony is strictly in two parts, an additional part being scarcely 
 suggested. 
 
 566. Two-part harmony continued for any length of time is 
 comparatively rare in vocal music ; even in duets for voices the 
 harmony is mostly completed by the accompaniment. As an 
 example of this, let the student refer to the extract from the duet 
 " The Lord is a man of war " quoted in 344 at (b}. Here the 
 two voice-parts, which are printed on the upper staff, are supple-
 
 Chap. XXI.] 
 
 ITS THEORY AND PRACTICE. 
 
 249 
 
 mented by the orchestra. The rule in such cases as this is that 
 the voice parts must make correct, though not neccessarily com- 
 plete harmony by themselves. A progression of two voices in 
 fourths thus 
 
 
 
 
 would therefore be incorrect, even although an instrument played 
 in thirds below the lower voice. If another voice adds the thirds 
 below, the harmony will be correct. The passage just given is 
 taken from the trio " Lift thine eyes " in " Elijah," omitting the 
 lowest voice part. 
 
 567. We now give two examples of vocal two-part harmony. 
 The first 
 
 (a) HANDEL. " Judas Maccabaeus." 
 
 is the commencement of a double fugue given out by the alto and 
 tenor voices. The second 
 
 (*> 
 
 -^ WAGNER. " Fliegende Hollander.' 
 
 ifTT F-F-~^ 
 
 T*-V- 
 
 
 -V- 
 
 
 
 
 
 1 j n 
 
 *r 
 
 1 
 
 1 1 1 if I . 
 
 
 U fL-J- 
 
 " 
 
 
 
 c F r r ! 
 
 
 
 
 
 is a cadenza for unaccompanied solo voices. The fact of its being 
 a cadenza is indicated in the original by its being printed in small 
 notes ; this also explains the irregular time of the first three bars. 
 It will be seen that the outline of the harmony in this passage is 
 perfectly clear.
 
 250 
 
 HARMONY 
 
 [Chap. XXI. 
 
 HARMONY IN MORE THAN FOUR PARTS. 
 
 568. There is hardly any limit to the possible number oi parts 
 in which harmony may be written. Orazio Benevoli, an Italian 
 musician of the lyth century, composed many masses and 
 anthems for sixteen, and even for twenty-four voice parts. But 
 the most astonishing feats in part-writing are probably Tallis's 
 " Forty-part Song," and the " Kyrie and Gloria " in 48 real parts 
 by Gregorio Ballabene. Such pieces are merely ingenious 
 curiosities ; but harmony in ten, twelve, and more parts is by no 
 means uncommon in the works of Bach. The opening chorus of 
 his cantata " Herr Gott, dich loben alle wir " (founded on the 
 choral known in this country as the xooth Psalm) is mostly in 
 fifteen real parts ; and many similar instances might be given. 
 Among modern composers, Brahms and Wagner are especially 
 distinguished for their skill in "polyphonic" (many part) writing; 
 while of our own countrymen the place of honour in this depart- 
 ment is probably due to the late Rev. Sir Frederick Ouseley. 
 
 569. It will readily be understood that every part above four 
 added to the harmony increases the difficulty of the task, because 
 of the danger of incorrect progressions, consecutives, &c., which it 
 is needful to avoid. But in proportion as the number of parts, 
 and therefore the difficulty, increases, the stringency of the rules 
 relaxes. Thus, hidden fifths and octaves are allowed, even when 
 both voices leap ; consecutive octaves and fifths by contrary 
 motion may be used freely ; we even meet in the works of the 
 great masters with examples of a doubled leading note, though it is 
 better to avoid this, if possible. No new rules need be given for 
 writing in more than four parts ; the practice of the best composers 
 will be most clearly understood by a careful study of the examples 
 now to be given of harmony in five, six, seven, and eight parts. 
 Beyond this number it is not needful to go. 
 
 570. In adding to the four voice parts, it is immaterial which 
 voice is the new one. In five-part harmony, the fifth voice is 
 usually either a second soprano (as in the examples to be given 
 below) or a second tenor, as in the five-part choruses of Handel's 
 " Acis and Galatea ; " but instances may also be found of a second 
 alto or second bass part. 
 
 BACH. " \\'er \vciss uic nahe niir intin Elide.' 
 
 
 This passage, from one of Bach's cantatas, is for two trebles, alto, 
 tenor, and bass. We have placed the alto part on the lower staff,
 
 Chap. XXI. j 
 
 Irs THEORY AND PRACTICE. 
 
 instead of the upper, as usual, because it makes the passage 
 clearer to read. 
 
 571. Our next example is arranged in the usual way, the two 
 trebles and the alto being on the upper staff. 
 
 MENDELSSOHN. "Tu es Petrus." 
 
 F7Pj?~Tt 5 ~^ 5?-^"t~^' 
 
 
 IBII R^^^^ 
 
 fflE=l 
 
 C\) v -^ =E=: 
 
 *; 
 
 ^- ^ : -^' -4 
 
 ^> : V(fr 1~ FT ' 
 
 J _-- 
 
 _J_= [_ 
 
 -K_ . J -^ = 
 
 M^=8 B 
 
 ^=| 
 
 ^ S v 
 
 
 im ' j^i : 
 
 ^ " 
 
 It will be seen that in the last bar but one the A and G on the 
 upper staff are printed in small notes. The third of the chord, 
 G, could not well be omitted, and Mendelssohn has therefore 
 added it in the orchestra, AS he had left no voice available He 
 might, however, have easily managed it, had he arranged the 
 
 As this would have 
 
 upper parts in the third bar thus 
 
 been a more usual position for the chord, it is difficult to see why 
 he did not adopt it. 
 
 572. Our next illustration will be in six parts. 
 
 HANDEL. " Belshazzar." 
 
 
 This passage is written for two trebles, alto, two tenors, and bass. 
 The dotted lines on the lower staff indicate that the tenor parts 
 cross each other. In the fifth and sixth bars the first tenor has B 
 throughout. In more than five parts, and sometimes even with 
 five, it becomes needful to cross the parts in order to avoid con- 
 secutive fifths or octaves. This is not the case in this particular 
 passage, where the crossing seems to be the result of a wish to 
 give melodic interest to the second tenor part ; but in the passages 
 in seven and eight parts to be quoted shortly, such a procedure 
 often becomes absolutely necessary. 
 573. The following extract, 
 
 BRAHMS. "Gesang d^r Parzen."
 
 2 5 2 
 
 HARMONY : 
 
 [Chap. XXI. 
 
 which is written for treble, two altos, tenor, and two basses, affords 
 illustration of some other points. Notice in the first chord the 
 crossing of the tenor below the first bass to avoid consecutive 
 octaves with the first alto. At the third crotchet of the second 
 bar is an instance of a doubled leading note (569) ; and in the 
 last crotchet of the following bar is a doubled eleventh, one F 
 rising to G, and the other falling to E in the following chord. 
 
 574. Seven-part harmony is comparatively rare. In the 
 example we give we have printed the passage in score, as it would 
 have been impossible, had it been condensed on two staves, to 
 show clearly the progression of the different voices. 
 
 HANDEL. 
 
 SOP. I, 2. 
 
 ALTO i, 2. 
 
 FEN. i, 2. 
 
 BASS. 
 
 1 Parnasso in Festa." 
 
 J S J 
 
 
 
 iM 
 
 ^ 
 
 ^ 
 
 ^ 
 
 In this passage will be seen numerous crossings of the parts and 
 the leading note doubled in every bar at the end of the fourth 
 bar it even appears in three parts. Between the last chord of the 
 fourth bar and the first of the fifth are found consecutive perfect 
 fifths between the first tenor and the bass. This is of course a 
 slip of the pen, probably the result of the haste with which it is 
 notorious that Handel composed ; but such slips are by no means 
 uncommon in his seven and eight part writing. 
 
 575. Choruses in eight parts are much more common than in 
 seven. Sometimes they take the form of double choruses, that is 
 for two separate choirs, as in Handel's " Israel in Egypt " and 
 Bach's " Passion according to Matthew ; " at others, as in
 
 chap. xxi. j ITS THEORY AND PRACTICE. 
 
 2 53 
 
 Handel's "Athalia" and Mendelssohn's ii4th Psalm, we find 
 only one choir, but with all the voices divided. We give one 
 example of each arrangement. 
 
 CHERUBINI. Credo & 8 Vocf. 
 
 ^=Lfe 
 
 -A A.^A A 
 
 Szk 
 
 S.A. 
 
 r r g 
 
 - r- g 
 
 j j - ? 
 
 T.B. 
 
 j 
 
 J 
 
 f r< r^ h_l^-J 
 
 ! ! 
 
 
 
 i 
 
 ' ' 
 
 r 
 
 Here we have printed each choir in "short score*' (143), as irt 
 previous examples. The part-writing here is remarkably pure, and 
 will repay close study. 
 
 576. In our second illustration 
 
 SOP. I, 2. 
 
 ALTO i, 2. 
 TEN. i, 2. 
 BASS i, 3. 
 
 MENDELSSOHN. "Am Himmelsfahrtbtage," Op. 79, No. 3. 
 
 I "fc _ I"*! I m \ I I 
 
 ^? 
 
 
 nJ u 
 
 A. 
 
 we have only one choir instead of two ; the voices are therefore
 
 254 HARMONY : Irs THEORY AND PRACTICE. [Chap. xxi. 
 
 arranged differently. It will be seen that after the first two bars 
 the harmony is only in seven parts. 
 
 577. Much of what is called eight-part writing is not really 
 such. For instance in Mendelssohn's Octett for eight stringed 
 instruments, when all are employed at once some of the parts are 
 mostly doubled in the octave or unison. This can be seen in the 
 quotation given from this work at ()35i. Though there are 
 eight notes, the harmony is not in more than five parts, some of 
 which are doubled. Even in the strictest writing it is frequently 
 expedient, for the sake of variety and contrast, to give rests to 
 some of the voices, as great fulness of harmony, if too long con- 
 tinued, becomes tedious. 
 
 578. In concluding this work, we offer one piece of advice to 
 the student. Much, but not" all, can be learned from a text-book ; 
 if the principles underlying the science of harmony are thoroughly 
 grasped, endless instruction is to be gained from the study of the 
 great masters. It is from their works that the rules laid down in 
 this volume have been deduced ; the best theory is that which 
 agrees most closely with their practice. It is impossible to make 
 any text-book absolutely exhaustive ; for art is always progressing. 
 Let the earnest student, therefore, while founding his practice 
 mainly on the example of the acknowledged masters of the past, 
 not neglect to acquaint himself with the more modern develop- 
 ments of music ; let him welcome what is excellent, from whereso- 
 ever it may come ; and let his motto be, " Prove (i.e. test) all 
 things ; hold fast that which is good." 
 
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 E Wrist Studies (9935) 
 
 net 
 
 51. F. Chopin, in G flat 
 A. Henselt, in E flat 
 minor 
 
 L. Kb'hler, in c 
 Th. Dohler, in D flat 
 
 52- 
 
 S3- 
 
 54- 
 
 s. d. 
 
 2 6 
 
 2 O 
 
 2 6 
 
 2 6 
 
 5 
 
 1 O 
 
 2 6 
 
 1 O 
 
 2 O 
 
 F School of Octaves 
 
 (9936) net 
 
 55. F. Hiller, in c 
 
 56. "W. Taubert, in F 
 minor 
 
 57. J.Brahms, in A minor 
 
 58. E. Pauer, in B 
 
 59. J. N. Hummel, in 
 
 G minor . . .10 
 
 60. L.van Beethoven, 
 
 in F. . . .10 
 
 61. C. Mayer, in A minor 2 6 
 
 62. J. Schulhoff, in B 
 
 flat minor . .26 
 
 63. L. Kohler, in B .26 
 
 64. C. Mayer, in F .26 
 
 65. F. Hiller, in B .26 
 
 66. J. C. Kessler, in c 
 
 minor . . .26 
 
 G Studies in Chords 
 
 (9937) net 2 
 
 67. F. Hiller, in c minor 2 
 
 68. J. C. Kessler, in F . 2 
 
 69. C. "V. Alkan, in c $ 3 
 
 H Extensions in Arpeg- 
 gio Chords (9938) net 2 
 
 70. F. Chopin, in c .2 
 
 71. A. Henselt, in D flat 2 
 
 72. H. Seeling, in A flat 2 
 
 I School of the Staccato 
 
 (9939) net 5 
 
 73. C. E. F. Weyse, in c 2 
 
 74. F. Kalkbrenner, in 
 
 B flat minor . I 
 
 School of the Staccato (cont.) s . d . 
 
 75. F. Hiller, in A minor 2 o 
 
 76. C. Mayer, in E flat . 2 o 
 
 77. A. Loeschhorn, in G 2 6 
 
 78. J. C. Kessler, in K . 2 6 
 
 79. "W". Taubert, in A 
 
 minor . . .26 
 
 80. M endelssohn-Bar- 
 
 tholdy, in E . .30 
 
 81. F. Liszt, in E . .20 
 
 82. Th. Dohler, in A minor 2 6 
 
 83. S. Thalberg, in E flat 2 6 
 
 84. C. M. v. Weber, in 
 
 K flat . . .26 
 
 K School of the Legato 
 
 (9940) net 2 6 
 850. Mendelssohn-Bar- 
 
 tholdy, in E minor i o 
 85/>. Mendelssohn-Bar - 
 
 tholdy, in D .10 
 86a. R. Schumann, in c i o 
 86/>. R. Schumann, in D 
 
 minor . . .26 
 86<r. R. Schumann, in E 2 6 
 
 87. J. N. Hummel, in A i o 
 
 88. A. Henselt, in B flat i o 
 
 L School for the Left 
 
 Hand (9941) net 5 
 
 89. A. Loeschhorn, in 
 
 G minor . . .20 
 
 90. C. Mayer, in D minor 2 6 
 
 91. A. Loeschhorn, in 
 
 E flat . . .26 
 
 92. F. Hiller, in G .26 
 
 93. C. Mayer, in c minor 2 6 
 
 94. J. Brahms, in c .40 
 
 95. A. Henselt, in E flat 2 o 
 
 96. F. Chopin, incminor 2 6 
 97* W. Taubert, in D flat 2 6 
 98. Th. Dohler, in B . 2 6 
 99* E. Pauer, in A flat . 2 6 
 
 100* R. Willmers, in E . 2 6 
 
 * For the Lejt Hand only. 
 
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