THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES AUGENER'S EDITION, No. 9182 HARMONY: ITS THEORY AND PRACTICE. BY EBENEZER PROUT, B.A. Lond Hon. Mus Doc. Trin. Coll. Dublin and Edinburgh, and Professor of Music in the University of Dublin REVISED AND LARGELY REWRITTEN FORTY-FOURTH IMPRESSION AUGENER LTD. LONDON UNTWISTING ALL THE CHAINS THAT TIE THE HIDDEN SOUL OF HARMONY." MILTON: L'^iiegro. b* Autfenet A Cn Printed in England by AUGENEB LTD., 287 Acton Lane, London, W. 4. Maria Library PREFACE TO THE FIRST EDITION. 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 prooably 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 hum- ble 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 com- poser 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. 1900361 iv PREFACE. TO THE FIRST EDITION. It is from this point of view that the present volume has been written. The rules herein given, though in no degree in- consistent 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 sup- ported 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 neeful 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 phys- ical basis being entirely abandoned. It will be seen in Chapter II. ( 44) that by rejecting altogether the eleventh and thir- teenth notes of the harmonic series, and taking in their place other notes produced among the secondary harmonics, the chief objection made by the opponents 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, Helm- holtz is followed to a considerable extent ; but Ouseley's ex- planation 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 in- telligible, perfectly consistent with itself, and sufficiently com- prehensive to explain the pjogressions of the advanced modern school of composers. 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 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, PREFACE TO THE FIRST EDITION. v 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 simplified. It is of the utmost importance that students who wish to master the 'Subject should proceed steadily and deliber- ately. 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 pre- ceded 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 stim- ulate 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 harmonized 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, con- siderably 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 exceeding reasonable limits to illustrate all the points mentioned, it is hoped that at least no rule of importance has been given without quoting some recognized author in its sup- port. It may at all 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 Harmonizing Melodies. vi PREFACE TO THE FIRST EDITION. The volume has, however, extended to so much larger dimen- sions 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 re- ceived 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 sug- gestions, 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. PREFACE TO THE SIXTEENTH EDITION. It is now more than twelve years since the first edition of Harmony : Its Theory and Practice was published ; and the great success with which the work has met has no less surprised than gratified its author. At the same time he must say that, after so many years' experience in teaching from it, he would have been either hopelessly ignorant or incurably conceited had he not become fully aware of its numerous defects and short- comings. He has felt that he could best show his appreciation of its generous reception by the musical public by improving it as far as lay in his power. For some years past it has been his intention to do this as soon as the pressure of work allowed ; but he has thought it best to complete the series of which this forms the first volume before undertaking so serious a task as remodelling this treatise. Though called a new edition, it would be hardly too much to describe the present as a new book. Considerably more than half the text is either additional matter, or has been entirely rewritten. A short account of the modifications introduced is necessary, that the reason for the numerous changes made may be understood. First and foremost among these is the virtual abandonment of the harmonic series as the basis on which the system is founded. Further investigation and thought have convinced the author that the practical objections to the derivation of the higher dis- cords-: the ninths, elevenths, and thirteenths from the natural series of upper partials were far greater than he had realized in first writing the volume. That the acoustical side of the subject has nevertheless an important bearing on harmony he still holds ; and this matter is dealt with in Appendix B, which replaces Chapter II of previous editions. But the modern key, whether major or minor, is so largely the result of aesthetic, rather than of scientific considerations that it is far better for the student that it should be dealt with from the former point of view. It is obvious that this change has necessitated an entirely new treatment of the question of the chromatic constituents cf y, viii PREFACE TO THE SIXTEENTH EDITION. key. The plan now adopted will, it is believed, be found much simpler and easier, especially for self-instruction, than its prede- cessor. The chromatic element is regarded as subordinate to the diatonic, and chromatic chords are considered as being bor- rowed from neighbouring keys. This view it is believed, was first propounded by the author's son, Louis B. Prout, in his Harmonic Analysis, to which little work the author acknowledges his obli- gations for many valuable suggestions. The new treatment of the subject has involved the rearrange- ment of a great part of the contents of the volume. The whole of the diatonic material of the key, up to and including the chord of the dominant thirteenth, is dealt with before the chro- matic chords are introduced. This has necessitated the remodel- ling, and in some cases the entire rewriting of the exercises. To avoid the inconvenience arising from the use of two books, the Additional Exercises have been incorporated in the volume, and the exercises on each chapter have been graduated, as far as practicable, in the order of difficulty. This, however, does not apply to the Chants and Hymn Tunes, which it has been thought advisable to place by themselves at the end of each chapter. A new feature of the present edition is that, from the very beginning, the harmonizing of simple melodies is taught simulta- neously with the harmonizing of figured basses. It is often found that pupils who can do the latter with ease are hopelessly at sea when they attempt the former. The author's own experi- ence is that, if the two are taken together, the harmonization of a melody presents no very great difficulty even to beginners of average ability, and that each branch of the study throws light upon the other. Another most important addition to this volume is the full analysis of the harmony given throughout. The system adopted, though with considerable modification, is that of E. F. Richter : the author has extended the idea of his predecessor by making a difference between "inversions" and "derivatives" of chords (See 252). Though the method looks at first sight complex, it is in practice extremely simple, if systematically pursued from the beginning ; and the insight into the harmonic structure of a composition which is obtained by its means will be found by the earnest student invaluable. For his guidance, and more partic- ularly for teachers who may use this book, special attention is called to the new Key to the Exercises. In this, every chord in each exercise is analyzed on the system here taught. Though the Key should in no case be used as a " cram," it will be most useful to consult it after the exercises have been worked, and the analyses written beneath them. PREFACE TO THE SIXTEENTH EDITION. ix Of the two Appendices, the second has been already spoken of. The first contains a necessarily brief and incomplete account of the Ecclesiastical Modes, a subject of which most students know little or nothing, but a slight acquaintance with which will be found of great use in aiding their comprehension of much of the music of the seventeenth and eighteenth centuries. The sketch for it professes to be nothing more has been compiled from many sources, chief among which should be named Marx's Composition and Charles Child Spencer's Brief Account of the Church Modes. The author would acknowledge his obligations to many who have kindly assisted him with advice and suggestions in the preparation of this new edition. From his son he has received much valuable help ; and he also desires especially to thank his friends Dr. A. J. Greenish and Mr. R. Orlando Morgan for many useful practical hints. He is also indebted to several kind friends for their help in reading the proofs a more than usually laborious task, especially as regards the Key, owing to the multi- plicity of detail. li is hoped, not without some confidence, that the present edition will be found not only more complete, but far simpler for teaching purposes than the work in its earlier form ; if it smooths over the difficulties in the path of the student, the author will feel himself well repaid for the year's hard work spent in its preparation. London : December, 1901. NOTE. Tn consequence of the rearrangement of the subject matter in the present edition, both Chapters and Sections have been renumbered. As there are many references to Harmony in the following volumes of the series, a table is here given of the Chapters and Sections referred to, with the corresponding numbers in the new edition. As the other volumes of the series are reprinted, the necessary alterations of references will be made in the text. In two cases (\\ 423,434), they will not be found, because the passages in question have been replaced in the new edition by others. Previous New Previous New Editions. Edition Editions. Edition. Chapter III Chapter II Section 229 Section 278 IV " III ' 243 " 294 " X IX ' 248 " 307 " XI " X ' 253 " 313 XII XV ' 254 " 3H XVIII ' xvi (w 444-45) ' 2 55 ' 257 " 316 " 3i8 XIX " XI ' 258 " 321 XX " XX " 262 " 324 263 " 325 Section 26 Section 25 " 280 " 475 40 " 289 470 " 70 " 303 " 240 <( >T T " 232, 483, 321 474 7 1 484 " 325 370, 381 " 75 418 " 344 (d) " 585 " 103-105 ' 75-77 " 358-360 542-544 113 ' 104 367 " 392 " 126-128 ' ' 93-95 " 38l " 411 " 137 133 394 " 552 " 156 " 169 404 " 433 " 159 ' 173 " 410 " 428 162 ' 183 " 413 " 552 " 164-166 188, 189 " 426 " 563 " 170 ' 208 " 427 " 564 ' 171 208 " 432 " 440 " 181 ' 215 " 433 " 574 " 190 " 224 439 " 577 < 192 " 202 " 228 " 239 " 460 (d) " 595 (Ex. 523) " 207 " 251 " 49 " 445 " 211 " 258 ' 499 " 450 " 216 " 266 ' 504 " 336 ' ; 219 " 241 ' 517 " 341 223 " 272 ' 546 " 630 " 225 " 273 ' 562-564 " 645-647 TABLE OF CONTENTS. . B. The numbers refer in every instance to the sections, not to I he pages.] CHAPTER I. INTRODUCTION ................................................. page i Amount of knowledge presupposed, I A musical sound ; pitch, 2 Melody and harmony defined, 3 Interval defined, 4 Semitone, 5 Enharmonic interval, 5 (note) Diatonic and chromatic semitones, 6 A Tone 7 Scales, 8 Diatonic scales, 9 The chromatic scale, 10 Names of the degrees of the diatonic scale, 11-13 Consonance and dissonance defined, 14-16 The Resolution of a Dissonance, 16 Discords, 17 How intervals are reckoned, 18 Compound intervals, 19 Different kinds of intervals, 20 Perfect and major intervals, 21 Minor intervals, 22 Augmented and diminished intervals, 23, 24 Inversion of intervals, 25, 26 Inver- sion of compound intervals, 27 Consonant and dissonant intervals, 28 Perfect and imperfect consonances, 29 Table of intervals, 30. CHAPTER II. KEY, OR TONALITY page 13 Definition of Key, 31-33 Development of key from its tonic, 34 The diatonic and chromatic elements in a key, 35 A Chord : major and minor com- mon chords, 36 Major and minor keys, 37 The " primary notes " of a key, 38 The diatonic material of a major key, 39, 40 How to mark the roots, 41 The diatonic material of a minor key, 42 The difference be- tween major and minor keys, 43 Position of the semitones, 44 Other keys than C, 45 Tetrachords, 46 Keys with sharps, 47-49 Keys with flats, 50, 51 How to find the tonic of any major key, 52 Table of key- signatures; enharmonic keys, 53 How to find the signature of a key containing more than seven sharps or flats, 54-56. CHAPTER III. THE GENERAL LAWS OF PART-WRITING page 22 A Part defined, 57 Rules of melodic progression; "conjunct" and "dis- junct motion," 58 Diminished intervals, how treated, 59 Augmented intervals, 60 Large intervals, 61 The leap of a seventh with one inter- mediate note, 62. Leaping to an accented note, 63 Harmonic pro- gression; similar, oblique, and contrary motion, 64 Four-part harmony; names of the voices, 65 Rules of part-writing, 66 Consecutive unisons and octaves, 67-70 Consecutive fifths, 71-73 Hidden octaves and fifths, 74-77 Consecutive fourths, 78 Consecutive seconds, sevenths, and ninths, 79, 80 The progression from a second to a unison. 81 Approaching and leaving the unison by similar motion, 82-84 Overlapping and crossing of parts, 85 An unprepared discord best approached by contrary motion, 86 The chief difficulties of harmony found in the earlier stages, 87. CHAPTER IV. THE DIATONIC TRIADS OF THE MAJOR KEY page 34 A Triad, 88 The diatonic triads of a major key, 89 The Diminished Triad, 90 "Doubling" a note, 91 Compass of the voice;, 92 Close and extended position, 93 The best position of harmony, 94-96. The best notes to double, 97-99 Treatment of the leading note, 100, 101 Omission of one note of a chord, 102 Three rules for part-writing, 104-109 Short score, no Open score, ill The three primary triads xii CONTENTS define the key, H2 The connection of the primary triads with one another, 113 The position of the first chord, 114 Exercises with bass given, 115 Harmonizing a melody with primary triads only; the import- ance of hearing the music mentally, 116 The Cadence : Authentic and Plagal Cadences, 117 The position of the cadence, 118 The choice of chords, 119 A melody harmonized, 120-126 The stiff effect of this harmony: the reason, 127 Melodies given for harmonization, 128. CHAPTER V. THE DIATONIC TRIADS OF THE MAJOR KEY (CONTINUED). SEQUENCES page 49 The " secondary triads," 129 Which notes to double, 130 The intermixing of primary and secondary triads, 131 Three-part harmony with one part doubled, 132 Seqtience defined, 133 Tonal sequence, 134 Length of pattern, 135 Interval of imitation, 136 Irregular sequence ; real sequence, 137 Licenses permitted in repetitions of a sequence, 138 Changing the position of chords in a sequence, 139, 140 Directions for working, 141 Exercises with treble and bass given, 142 Ditto with bass only given, 143 Harmonizing melodies: choice of chords, 144 How to begin and end, 145 Root progressions : the leap of a fourth, 146 The leap of a third, 147 Progressions by step, 148, 149 General principles, 150-152 A melody harmonized, I53-J58 Melodies for harmonization, 159. CHAPTER VI. THE INVERSIONS OF THE TRIADS OF A MAJOR KEY page 61 Inversion in general, 160 Inversion of a chord defined; the number of possi- ble inversions, 161 The first inversion, 162 Figured Bass, 163 Figuring of the first inversion, 164 How to show the position of chords when marking the roots, 165 Which notes to double, 166-168 Treatment of a series of first inversions, 169-171 The second inversion, 172 Its in- complete effect, 1 73 How to find the root of a chord from the figured bass, 174, 175 Second in versions of primary triads, 176 The " cadential | "of the tonic chord, 177, 178 Its position in the bar, 179 The same chord used non-cadentially, 180 The second inversion of the subdominant chord, 181 Progression of the voices in a cadential j, 182. The second inversion of the dominant chord, 183 Second inversions of secondary triads, 184-187 Rules for approaching a second inverson, 188 Rules for quitting a second inversion, 189 The best note to double, 190 Special exercises for the treatment of second inversions, 191 Exercises on a figured bass, 192 Enlarged meaning of the term "progression of roots," 193 Additional rules for root progression, 194-199 A melody harmonized, 200-203 Melodies given for harmonization, 204 The com- position of four-bar phrases, 205. CHAPTER VII. THE MINOR KEY: ITS DIATONIC TRIADS AND THEIR INVERSIONS page 80 The Harmonic Minor Scale, 206 Older forms of scale : the Aeolian, 207 The Dorian scale, 208 Melodic minor scale, 209 The leading-note of a minor key never written in the signature, 210 Relative major and minor keys, 211 Tonic major and minor keys, 212 The diatonic triads of the minor key, 213 The augmented triad on the mediant, 214 The diminished triad on the supertonic, 215 The "Dorian sixth": its employment in the subdominant chord, 217, 218 A modern example of its use, 219 The Dorian sixth in the supertonic chord, 220 The CONTENTS. xiii progression between dominant and submediant in the minor key, 221 The doubling of the submediant, when to be avoided, 222 First in- versions : how to avoid the augmented second, 223 Chords containing the minor seventh of the key, 224-226 How to indicate the roots of these chords, 227 Second inversions in the minor key, 228 The Tierce de Picardie, 229 Exercises, 230. CHAPTER VIII. THE CHORD OF THE DOMINANT SEVENTH page 94 Effect of tonic and dominant harmony, 231 The chord of the dominant seventh: a "fundamental discord" denned, 232 The seventh a disso- nance, requiring resolution, 233 Treatment of the third and seventh of the chord, 234 Omission of the fifth, 235 Resolution on the tonic chord, 236 Ornamental resolutions of the seventh, 237-239 Resolution on theyfrj/ inversion of the tonic chord, 240 Reason of the importance of the chord of the dominant seventh, 241 Resolution on the submedi- ant chord, 242 Resolution on the subdominant chord, 243 The in- versions of the chord of the dominant seventh, 244 Their figuring, 245- 247 The first inversion : its various resolutions, 248-250 Omission of the generator in this inversion, 251 Distinction between inversions and derivatives, 2$2 How to mark the roots of derivatives, 253 Examples of the first inversion and its derivative, 254-256 The second inversion : its most usual resolution, 257 Its resolution on they?/ inversion of the tonic chord, 258, 259 Other resolutions, 260 The derivative of the second inversion, 261 Examples by Handel, 262 The third inversion : its usual resolution, 263 Resolution on the second inversion of the sub- mediant chord : the only satisfactory position, 264 The derivative of the third inversion, 265 Changes in the position of a chord of the seventh before resolving, 266 Treatment of the dominant seventh in a full cadence, 267 Exceptional resolutions, 268 Exercises, 269. CHAPTER IX. KEY RELATIONSHIP MODULATION TO NEARLY RELATED KEYS FALSE RELATION. page 115 Modulation and Transition defined, 270 Key Relationship, 271 Related major keys, 272 Unrelated keys, 273 Chords common to nearly related keys, 274 Table of nearly related keys, 275, 276 How their key signa- tures differ, 277 How to effect modulation, 278 Figuring the bass in modulations, 279 Immediate and gradual modulation, 280 How to regard ambiguous chords, 281 Change of the primary notes, 282 The modulation from a major key to its supertonic minor, 283 Modulations from minor keys, 284 Examples of modulations analyzed, 286-291 General rules for modulating to nearly related keys, 292 Modulation by means of irregular resolutions of the dominant seventh, 293 The choice of modulations, 294 Transitional Dominants, 295 False Relation, 296 Exercises, 297 Harmonization of Chorals : the Cadences, 298 The Invtrted Cadence, 299 The Interrupted Cadence, 300 Position of the cadences, 301 Implied modulations, 302 Chorals to harmonize, 303. CHAPTER X. UNESSENTIAL DISCORDS (I) AUXILIARY NOTES, PASSISC. NOTES, AND ANTICIPATIONS page 131 Notes unessential to the harmony, 304 Auxiliaty notes defined, 305 How taken and left, 306 \Yhen a tone and when a semitone from the harmony note, 307 Chromatic auxiliary notes, 308 Auxiliary notes in mop- than one part, 309 Passing Chords, 310 Examples referred to, 312 Changing xlv CONTENTS notes, 313 Single changing notes, 314 Auxiliary notes taken by leap, 315 Passing notes, 316 Ditto in a minor key, 317 Two successive passing notes, 318 Chromatic passing notes, 319 Passing notes in several parts by contrary motion, 320 Passing notes quitted by leap, 321 Examples referred to, 322 Auxiliary notes cannot make false relation, 323, 324 Anticipations, 325^ 326 Summary of rules for employment of auxiliary and passing notes, 327 Exceptional treatment of auxiliary notes, 328 Their introduction, 329 A choral harmonized with auxiliary notes, 330 Notes of a melody treated as accented auxiliary and passing notes, 331 Directions for work, 332. CHAPTER XI. UNESSENTIAL DISCORDS (II) SUSPENSIONS fage 145 The different kinds of unessential discords, 333 Suspension defined, 334 Which notes can be suspended, 336 Suspensions not to be marked in indicating roots, 337 Their preparation, 338 Their position in the bar, 339 Length of preparation, 340 When it may be sounded with the note of its resolution, 341 Incorrect progressions, 342 Always resolved on the chord over which it is suspended, 343 Practical limitations to suspension, 344, 345 How to know a suspension from the figuring, 346, 347 The inversions of suspensions, 348-350 Ornamental resolutions, 351, 352 Examples from the great masters, 353~356 Suspensions re- solving upwards, 357-363 Double suspensions, 364. 365 Suspensions of complete chords, 366 Directions for work, 367. CHAPTER XII. THE CHORD OF THE DOMINANT NINTH page 161 The chord of the dominant ninth different in major and minor keys, 368 Which note to omit, 370 Figuring, 371 The chord resolving on its own root; the ninth proceeding to the root, 372,373 The ninth proceeding to the third, 374, 375 Irregular resolutions on fifth or seventh of chord, 376 Position of the major ninth, 377, 378 The chord resolved on a different root, 379 Treatment of the fifth, 380 The inversions of the chord rare, 381 Derivatives of the dominant ninth ; how figured, 382 Inversions of the derivatives, 383 The Leading and Diminished Sevenths, 384 Resolution of these derivatives, 385 Examples from the great masters, 386-389 Changing the position of the chord before its resolu- tion, 390 Further derivatives ; the supertonic chords of major and minor keys, 391, 392 Exercises, 393. CHAPTER XIII. THE CHORD OF THE DOMINANT ELEVENTH... page 172 The chord of the eleventh, 394 The difference between the eleventh and the fourth, 395 Which notes mostly omitted, 396 Figuring, 397 Gradual resolution of the higher discords, 398 Resolution, 399 Inversions, 400 First inversion, 401 Second inversion, 402 More than one analysis sometimes possible, 403 The other inversions, 404 Derivative of the first inversion (rare), 405 Ditto of the second inversion, 406 Its resolu- tion on a dominant chord, 407 Ditto on a tonic chord, 408 How to distinguish between chords that are identical in appearance, 409 Other derivatives, 410 The Chord of the Added Stx,'/t, 411 Resolved on a dominant chord. 412 Resolved on a tonic chord, ^.13 Used in approach- ing a cadence, 414 The remaining derivatives, 415, 416 The figuring of these chords; how to tell their real nature, 417 The subdominant chord a derivative of the dominant eleventh, 418. CONTENTS. xv CHAPTER XIV. THE CHORD OF THE DOMINANT THIRTEENTH page 185 The thirteenth completes the series of dominant chords, 419 It differs in major and minor keys, 420 Contains every note of the diatonic scale, 421 The thirteenth a consonance with the root, 422 The inversions, 423 The lower notes of the chord, how treated, 424 Resolutions, 425 The thirteenth and the ninth, 426 Forms in use, 427 Root, third, and thirteenth ; how to distinguish from the mediant chord, 428 Which notes to double; figuring, 429 Resolution on dominant seventh, 430 Ditto on the tonic chord, 43 1 The first inversion, 432 The last inversion, 433 Root, third, seventh, and thirteenth, 434 Resolution on the domi- nant seventh, 435 Alternative explanation, 436 Resolution on the tonic chord, 437 Root, third, fifth, and thirteeenth, 438 Other forms of the chord, 439 Derivatives ; the seventh on the subdominant, 440 Ditto resolved on a tonic chord, 441 The triad on the submediant, 442 The chord of the thirteenth in its complete form, 443 Secondary Discords, 444 Secondary sevenths; how they differ from fundamental sevenths, 445 Their harmonic origin to be disregarded, 446 Rules for their treat- ment, 447 Examples by Handel, 448 A series of secondary sevenths, 449 Secondary ninths, 450 Exercises, 451. CHAPTER XV. CHROMATIC TRIADS THE CHROMATIC SCALE page 198 Chromatic notes in a key, 452 Chromatic chords defined, 453 Illustrations, 454, 455 Chromatic chords are borrowed chords, 456 The keys from which they are borrowed, 457 The chromatic scale, 458 How formed from the diatonic ; its harmonic form, 459 The melodic chromatic scale, 460, 461 Why only nearly related keys are used for borrowing from, 462 In a minor key only the neighbouring minor keys borrowed from, 463 Three methods of averting the modulations suggested by chromatic notes, 464-466 How to mark the roots of chromatic chords, 467 The chromatic common chords of a minor key, 468 The tonic major chord, 469 Seldom used except as a passing chord, 470 The chord on the flattened supertonic, 471 Its first inversion, the " Neapolitan sixth," 472 Its second inversion, 473 The major chord on the supertonic, 474 Treatment of the third of the chord, 475 Examples analyzed, 476 Rarer chromatic chords, 477, 478 List of chromatic common chords in a major key, 479 Which chords are restricted in their progression, 4So Examples of chromatic chords in a major key, 481 Modulation by means of chromatic chords, 482. CHAPTER XVI. CHROMATIC CHORDS OF THE SEVENTH page 215 The real nature of chromatic chords, 483 The chromatic sevenths are the dominant sevenths of nearly related keys, borrowed, 484 The Supertonic Seventh : its chromatic notes, 485 Treatment of the third. 486 Treat- ment of the seventh, 487 The inversions of the chord, 488 Examples of its employment, 489 The seventh rising to the fifth of the dominant chord, 490 The seventh leaping, 491 A seldom used progression, 492 Examples of the inversions, 493, 494 Derivatives of the chord, 495 The Tonic Seventh : its resolution, 496 Progression of the third, 497 Progression of the seventh, 498 The inversions of the chord. 499 Ex- amples of the tonic seventh in root position. 500 Ditto in the first and second inversions, 501 Ditto in the last inversion, 502 The derivatives; how to indicate the roots, 503 Examples, 504. 505 Modulation by means of chromatic sevenths, 506 Sequences of modulating sevenths, 507. xvi CONTENTS. CHAPTER XVII CHROMATIC CHORDS OF THE NINTH FALSE NOTA- TION ENHARMONIC MODULATION page 231 The chromatic ninths, 508 Their various forms, 509 Tk: dominant minor ninth in a major key, 510 Its derivatives, 511, 512 The dominant major ninth in. the minor key, when possible, 513 Resolutions of the chromatic dominant ninth, 514 The supertonic ninth, 515 Its resolu- tion upon its own root, 516 Resolution upon a different root, 517 Pro- gression of the ninth, 518 The supertonic minor ninth taken in the major key; False Notation, 519 Law of the Sharpest Note, 520-522 How to detect false notation, 523 When false notation is to be met with, 524 Examples of derivatives of the supertonic ninth, 525-527 The tonic ninth: its resolution, 528 Progression of the ninth, 529 Double false notation, 530 Resolution of the chord upon a supertonic discord, 531 Examples of derivatives of the tonic minor ninth, 532-535 Ditto of major ninth, 536 A series of diminished sevenths, 537 Rarer deriva- tives, 538 Modulation by means of the diminished seventh, 539 Ex- treme keys; enharmonic change of notation, 540, 541 Enharmonic modulation, 542, 543 Can be effected between any two keys by the diminished seventh, 544 Example from Bach's Chromatic Fantasia analyzed, 545 Example by Beethoven, 546 Caution to the student, 547. CHAPTER XVIII. CHROMATIC CHORDS OF THE ELEVENTH AND THIR- TEENTH page 250 These chords comparatively rare, 548 The chromatic chord of the dominant eleventh in a major key, 549, 550 The tonic eleventh, 551 The super- tonic eleventh, 552 The chromatic dominant minor thirteenth, 553 Its resolution: false notation, 554 Examples analyzed, 555, 556 Tonic and supertonic thirteenth, 557 Derivatives of chromatic thirteenths : rules for their identification, 558 Necessity of ascertaining the key, 559 The derivatives : I. Third, ninth, and thirteenth, 560 Chords that are only partially chromatic, 561 False notation : " False Triads.'' 562 Examples, 563-56511. Third, fifth, ninth and thirteenth ; a " False Tetrad," 566 Example by Mozart, 567 III. Third, seventh, ninth and thirteenth, 568 IV. Fifth, seventh, ninth and thirteenth, 569 V. Third, fifth, seventh, ninth and thirteenth, 570 VI. Fifth, ninth, eleventh and thirteenth, 571 VII. Seventh, ninth, eleventh anil thirteenth used chromatically, 572-575 Enharmonic modulation by means of chromatic chords of the thirteenth, 576-579 The difficulty of these cords ; importance of careful analysis, 580. CHAPTER XIX THE CHORD OF THE AUGMENTED SIXTH page 269 The chord of the Augmented Sixth, 581 Derived from two tonics, 582 On which degrees of the scale found, 583 It has a " double generator," 584 Resolutions of the interval of the augmented sixth, 585 Peculiar construction of the chord, 586 Its usual forms, 587 Its figuring, 588 Distinctive names of the three forms, 589 How the chords are indicated in analysis, 590 The inversions, 591 The chord on the sixth degree t>f the scale the commonest, 592 The Italian Sixth; its resolutions, 593 Its inversions, 594 Examples, 595 The French Sixth, 596 Its resolu- tions, 597 Its inversions, 598 Examples, 599-601 The German Sixth ; its resolutions, 602 Its inversions, 603 Examples, 604-609 Rare forms of the chord of the augmented sixth, 610 Modulation by means of the chord, 6ll Enharmonic modulation only possible with the German sixth, 612 When employed, 613 Examples, 614-618. CONTENTS. xvli CHAPTER XX--PEDALS page 289 A Pedal defined, 620 Which notes can be used as Pedals, 621, 622 Tieat- ment of the harmony when the Pedal is not a note of the chord, 623 How to mark the analysis of a " Pedal point," 624 Examples of dominant pedals, 625 A tonic pedal ; introduction of additional parts, 626 An " Inverted Pedal," 627 Pedal above and below, 628 A pedal in a middle voice, 629 A pedal point generally ends with a chord of which the pedal note forms a part ; exceptions, 630 Modulation on a pedal, 631 Examples, 632-634 Ornamentation of a pedal note, 635 A double pedal, 636. CHAPTER XXI HARMONY IN FEWER AND MORE THAN P'OUR PARTS page 300 Harmony continuously in four parts rare, 637 Three part harmony, 638 Characteristic notes to be retained, 639 Broken chords, 640 Position of chords in three-part harmony, 641 The cadence, 642 Motion of the separate parts, 643 Examples, 644 Two-part harmony, 645 Examples referred to, 646 Examples given, 647-650 Harmony in more than four parts, 651 Greater freedom of part-writing allowed, 652 Which notes to double, 653 Five-part harmony, 654, 655 Six-part harmony, 656, 657 Seven-part harmony, 658 rEight -part harmony, 659-661 Direc- tions for work, 662 Conclusion, 663. APPENDIX A THE ECCLESIASTICAL MODES page 311 Difference between the ancient modes and modern keys, 665 How the modes were formed, 666 The Authentic Modes, 667 The Dominant, 668 The Plagal Modes, 669 How they differed from the Authentic Modes, 670 Table of Plagal Modes, 671 Transposition of the modes, 672 Major and minor modes, 673 Characteristic notes unalterable, 674 Modulation, 675 Alteration of non-characteristic notes, 676 The Dorian Mode, 677,678 The Phrygian Mode, 679-681 The Lydian Mode, 682, 683 The Mixolydian Mode, 684, 685 The Aeolian Mode, 686-68877;^ Ionian Mode, 689 Difficulties of the subject, 690 Nature of its interest, 691. APPENDIX B THE HARMONIC SERIES page 322 Practical use of the study, 692 How harmonics are produced, 693, 694 Pitch and vibration, 695 "Upper partials," 696 The Harmonic Series from C. 697 Ratios of intervals, 698 Compound tones, 700 How far modern scales are derived from nature, 701 The use of the harmonic series in determining key-relationship, 702, 703 The scientific explana- tion of consonance and dissonance, 704 Books recommended for studj HARMONY: ITS THE OR Y 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 meanings of the various musical signs (accidentals, etc.), 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 ///*/* of a sound (that is, its being what is called a high or a low note), depends upon the rapidity of the vibration. (See Appendix B. ) 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 pro- ceed that the question of melody is often so closely connected with that of harmony, that it is impossible to treat of one with- out 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 the two sounds is called the Interval between them. This difference may be so small as to be hardly recogni/.able by the ear : or it may be as great as between the lowest and highest notes of a * This is only a very general definition ; difference of pitch alone is not sufficient to make a good melody ; but for the present purpose the ideas of melody as a succession of sounds and of harmony as a combination of sounds will suffice. For a more complete definition of melody, see Musical /-oi m, Chapter I. 2 HARMON* [Chap. i. 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." 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^f (or D?) on the other side (above). From B to C, and from C to Cjf 'or Db) are therefore both semitones. Similarly from F to 1 Jj 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 At?), comes between them. 6. There are two kinds of semitone. If we look at the two here given, one above, and the other below C, Ex. l. 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 mean- ing which is attached to the word will be explained later. 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 CjJ) the semitone is called chromatic, a word literally meaning " coloured." This use of the word will be further ex- plained later. 7. The word "semitone " means half a tone. A TONE is an interval, the two notes of which are on adjacent degrees of the staff, and which contains two semitones. But if we take two diatonic semitones one above another, * In one sense this statement is not strictly accurate, as the "enharmonic diesis" (i.e., the very small interval between two notes represented by the same sound on the piano, such as Ff and Gfe, or Ctt and B8), is sometimes used in modulation. For ordinary purposes, however, the statement in the text is correct. chap, i.] INTRODUCTION. 3 the resulting interval will be from B to D [? ; which is not a tone as the two notes are not on the next degrees of the staff to one another. And if we take two chromatic semitones, Ex. 3. it is equally clear that they will not make a tone; for now the resulting notes C {? and C $ are both on the same degree of the staff. We see therefore that of the two semitones which make a tone, one must be diatonic and the other chromatic. It matters not which of the two is the lower.* 8. A SCALE is a succession of notes 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 suc- cession 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 diatonic scales are constructed will be ex- explained later (see Chapter II) ; at present we simply give the forms of them. There are two varieties of the diatonic scale, known as the 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. > MINOR SCALE. r- 9 Ex. 5. Pafc= Other forms of the minor scale frequently to be met with will be explained later. It will be seen that each of these scales contains 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 recommences. * The two semitones composing a tone are not of exactly the same size. A diatonic semitone is larger than a chromatic; 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. 4 HARMONY. I chap. i. 10. A chromatic scale is a scale consisting entirely of semi- tones, and it is called chromatic because some of its notes re- quire accidentals (flats or sharps) before them (6). Ex.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. 11. 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 occurrence. 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 (the next in importance to the dominant), is there- fore called the SUBDOMINANT, or lower dominant. We have now got appropriate names for the three chief notes in the key. 12. 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, /. 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 ac- count called the LEADING NOTE. It is sometimes, though rarely, called the " Subtonic," from its position as the next note below the tonic. Chap, i.] INTRODUCTION. 5 13. Having shown the origin and meaning of these different names, we will now tabulate them. First Degree of the Scale = Tonic (Key-note). Second " " = Supertonic. Third " " = Mediant. Fourth " " r= Subdominant. Fifth " " ' = Dominant. Sixth " " = Submediant (Superdominant). Seventh " " = Leading Note (Subtonic). 14. 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. 15. A consonant interval, or CONSONANCE, is a combination of two sounds, which by itself produces a more or less complete and 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, Ex, 7. 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. Ex. 8. pgg II \'" I 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. When he has learned, later in this chapter, which are the consonant intervals, he will see that no others have been used in these chords. 16. A dissonant interval, or DISSONANCE, is a combination of two notes which by itself produces an impression of incom- pleteness, 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. 5 r -f> -- n =?- -S- 3: I I I I 5?* *X" rt* Z& The compound intervals have the same prefixes as the simple ones ; thus C to D Q will be a major ninth, C to A 1? a minor thirteenth, and so on. 22. 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 or lower- ing the upper one a chromatic semitone. Thus from C to E is a major third. If we raise the lower note to C $, the interval C to E is a minor third. Or if we leave the C alone, and lower the E to El?, we also get a minor third from C to Et?. 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 D t? instead of to C $, the interval from Di? 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 D % instead of E (7, C to D $ is a second ; for the two notes are on adjacent degrees of the staff. I Minor 2nds] j Minor 3rds. | Minor 6ths. | Minor yths. | Minor gths. | i- I ^ I I u. I In. * * i chap i.] INTRODUCTION. q 23. 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 Cb to Fty will be an augmented 4th. Again C to A is a major 6th ; and C to A $ or C 1? to A t} is an augmented 6th. The augmented 3rds and yths are not used in harmony ; augmented 2nds, 4ths, and 6ths are frequently, and augmented 5ths some- times to be met with. 24. 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 22. We obviously cannot diminish the minor 2nd, for if we lower Dj? to D^, or raise C to C$, we shall in either case get an interval smaller than a semitone what is called an "enharmonic" interval ( 5, note} and it has been already said (5) that the semi- tone is the smallest interval used in music. The same objection will apply to a diminished gth. But diminished 3rds, 4ths, 5ths and 7ths, especially the last, are of very frequent occurrence. 25. When the relative position of two notes is changed by placing one of them an octave lower or higher than before, the lower one thus becoming the upper, and the upper the lower ; the interval is said to be inverted. Ex. 15. 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 (95 = 4) ; in the same way a 3rd becomes a 6th, a 2nd a yth, etc. 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 dimin- ished, and diminished augmented. 26. The reason of the rule just given will become clear to the student if he observes that the inversion of any simple in- terval is the difference between that interval and an octave. :o HARMONY. [Chap. i. 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 (major) 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. 27. Asa compound interval is larger than an octave ( 19), it is clear that raising or lowering either note an octave will not change their relative positions, and will produce no inversion. It will therefore be necessary to raise or lower one note two octaves, or (which produces the same result), to raise one note an octave and at the same time to lower the other an octave. We will take a major thirteenth, and invert it in each of these ways : (a) (6) ( f ) (d) . Ex.17. At (0) is a major thirteenth ; at () the lower note is raised two octaves ; at (^) the upper note is lowered two octaves, and at ( ; a minor third of G$ and Cl 7 ; a major third of AJ7, Ft}, and D$; a diminished fourth of F$, A fe, and D$; a perfect fourth of B!?, Gty, and A$; an augmented fourth of D>, Fjf, and C#. (3) Write the diminished fifth of Bb, Ft}, G#, and C>; the perfect fifth of Fj}, Bjj, E>, Djf, Fx, B[r>, E>>; the aug- mented fifth of Ejj, Ab, Cjt, G$, and Bb; the minor sixth of D>, Gj, El?, and G>; the major sixth of At>, B^, Eft, DJ(, and C 1? ; the augmented sixth of E ?, G , and A ; the dimin- ished seventh of E$, FX, A$, and Cj} ; the minor seventh of B$, A>, and F$; the major seventh of C^, Fjj, GJ, EP, and BJ} ; and the diminished octave of DJ3, Bi?, and Fx. (4) Write the minor ninth of F)J, B >, I)jf, (J i, and E$, the major ninth of Ft}, A 1?, E^j|, G|{, and Di? ; the eleventh of Et>, A^, Cjf, and F|; the minor thirteenth of E ?, Gjf, Di?, G i', F tj, and D ^ ; and the major thirteenth of G jj, C t*, A jj, E b, and B tj. (5) Write the inversions of all the intervals (a) to (//) in Exercise I, and name each, adding (C) or (D), according to whether they are consonant or dissonant. HARMONY. Chap 6 a CS L i tJ Perfect |J * I Perfect -5 ^ Major 1 1* 1 1 Minor T in g Minor 1 ! $ ' | Major V ^ Diminished i i 4 i Augmented ^ . i . i i * . ! Augmented ui & Diminished g Major c/5 3 Minor .b p Minor u t J Major 1 J Augmented i* t I ) Diminished H 01 | Perfect u ft r JU 1J1 H. \\ Perfect g 11 11 o Diminished !.4 3 i 1 Augmented . Augmented L ; J Diminished Perfect ^ : J Perfect ;3 O fe Diminished J* J Augmented Major i i \\ Minor ^ Minor ,s ) Major <% ^ Augmented B i T : \> Diminished . MR ^ ^ Perfect : A Perfect jl V TN rS O 7i- 3 1 > 3 3 g I Chap, ii.i KEY. OR TONALITY. r^ CHAPTER II. KEY, OR TONALITY. 31. One of the first things which it is necessary that the student should understand is what is meant when we speak of the Key of a piece of music. In order that music should pro- duce 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 recognizable relation to one another. For example, if the first half of " God save the King ' ' be played on the piano everyone can hear what is commonly called the tune 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. 32. 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 TONIC, or KE Y-No TE, to which note the other eleven hear a fixed and definite relationship. 33. 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. Thus in Ex. 6, all the notes of the key lie between the two C's. r^ HARMONY. [Chap. n. 34. 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 composition must be developed from this tonic, and must finally return to it." 35. If the student will compare, the two scales (Exs. 4 and 6), given in our last chapter ( 9 > TO), he will see that the latter (the chromatic scale), contains five notes more than the former (the diatonic), and that each of the additional five notes has an accidental before it. Both these scales are in the key of C major, which will be thus seen to contain two elements, the diatonic and the chromatic. The former includes all those notes which are in conformity with the key- signature, and the latter all those which are inflected by an accidental. When, later in this volume, we come to speak of the chromatic notes and chords in a key, we shall see that they are borrowed from neighbouring keys. They therefore occupy quite a different and subordinate position in the key to the diatonic notes and chords ; it is only of the latter that we have to speak in the present chapter. 36. A CHORD is a combination of not fewer than three notes, placed each at the distance of either a major or a minor third above the note next below it. The lowest note, upon which the chord is built, is called its Root* The most important, and the most frequently used chords are those called COMMON CHORDS, which are made by placing either a major or minor third and a perfect fifth above the root. If the third be major, the chord is called a major chord; if the third be minor, it is called a minor chord. 37. Every key has two " modes," the major and the minor, j> so called from the interval between the tonic and the mediant the third next above it ( 9). A key which has a major third above the tonic is called a " major key," and one that has a minor third above the tonic is called a " minor key." The * Much trouble is sometimes caused to students from the word Root 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 com- bination 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. f These are often spoken of as two distinct keys; but it is better, and more accurate, to regard them as two modes of the same key, as their three chief, or " primary" notes the tonic, dominant, and subdominant. are identical. This will be seen by comparing the two scales given in Exs. 4 and 5- chap, ii.] KEY, OR TONALITY. . 15 scales given in Examples 4 and 5 are therefore respectively the scales of " C major " and " C minor." These names are more convenient and less cumbrous than ' ' the major ' ' and ' ' the minor mode of C." 38. It is implied in what is said in 32, 34, that the tonic is the most important note in every key. Most pieces of music begin, and every piece should end, with a chord upon the tonic. Next in importance to the tonic are those notes in the key which are the most nearly related to it, that is, those which make per- fect consonances with it.* If the student will look at the scales in Exs. 4, 5, he will see that the only notes which make perfect consonances with C are the dominant G (a fifth above), and the subdominant, F (a fifth below). The tonic, dominant, and subdominant are therefore called the three PRIMARY NOTES of every key. 39. Let us first take the major mode of C, which, for the future we shall always call by its usual name, the key of C major. We select this key, because it is what is known as the " natural " key, that is, its diatonic notes require neither sharps nor flats. To obtain the diatonic material of the key, we take the three primary notes, placing the tonic in the middle, with the dominant above, and the subdominant below, and make each of these notes the root of a common chord. In a major key, the three primary chords are all major. Ex. 20, C: IV I 40. If the student will compare these chords with the major scale given in Ex. 4, he will see that every note of that scale is to be found in one or other of these three primary chords, though some of them (F, A, and D), are not in the same octave. But it is clear that the entire diatonic contents of the key are derived from these chords. 41. It is very desirable that the student should from the very commencement accustom himself to think of all chords in their tonal relation to the key to which they belong. In order to do this with more certainty, he should indicate the root be- neath every chord. For this purpose the plan adopted in the example just given (first introduced, we believe, by Gottfried Weber), should be followed. A letter followed by a colon shows the key ; if this be major, the letter is a capital, as above : C : =C major. For C minor a small letter (c :) will be used. * See Appendix B for the reason why the perfect consonances are the most nearly related notes. C 1 6 HAF.KTONY. [Chap. n. The Roman numerals under each chord show the degree of the scale which is the root of the chord. If, as here, the thirds of the chords are major, the numerals are capital letters ; if the thirds are minor, small numerals are employed, as will be seen wnen we come to speak of the minor key. Thus, in Ex. 20, IV, I, and V show that the roots are the fourth, first, and fifth degrees of the scale, and, as the numerals are all capitals, that the chords are all major. Modifications of, and additions to these signs will be dealt with as the necessity arises. The system is perfectly simple, and we strongly advise all students to take the trouble to master it from the first. 42. If the student will look at the scale of C minor (Ex. 5), he will see that its three primary notes are the same as those of C major. To obtain the diatonic material of the minor key, we build common chords on these three notes; but with the tonic and subdominant we shall now have minor chords above the roots, while for the dominant we still have a major chord. Ex. 21. c: iv i V After what was said in 41, the student should have no difficulty in understanding the way in which the key and roots are marked. The reason a major chord is taken upon the dominant is, that if a minor chord were taken, its third (BJ?), would be a tone, instead of a semitone, below the tonic, and the key would have no "leading-note." It will be seen later that a leading-note is equally necessary with major and with minor keys. 43. Looking for a moment at the chords given in Ex. 21, it will be seen that all consist of a major and a minor third placed one above another, and that the third which gives its name to the chord is always the lower of the two. It will further be noticed that the only notes which differ in the two keys are the third and sixth of the scale, which are a semitone lower in the minor than in the major key. To change a major key into its " tonic minor " (i.e., the minor with the same tonic), it is only necessary to flatten the third and sixth notes. The converse process will evidently change a minor key into its tonic major. 44. Let us now turn back to the two scales given in Ex. 4 and 5, which contain all the diatonic notes of C major and C minor.* It will be seen that in the major key the semitones * This statement is correct as regards the harmonies of the key ; but when we come to treat of the minor key (Chap. VII) it will be seen that there are two other notes which can be employed melodically as diatonic notes. chap, ii.] KEY, OR TONALITY 17 come between the third and fourth, and the seventh and eighth degrees of the scale ; while in a minor scale there are semitones between the second and third, fifth and sixth, and seventh and eighth degrees. Between the sixth and seventh degrees of a minor scale is the interval of the augmented second the only interval greater than a tone to be met with in any scale. 45. In older music many other forms of diatonic scale were in use besides the two that we have given (See Appendix A). These various forms were known as "modes." All the modes contained the same notes ; but each began on a different part of the scale, and consequently had the semitones between different degrees. At present only two modes, the major and minor, are employed *; the difference between one major or minor key and another is solely a difference of pitch. We have hitherto spoken only of the key of C ; we shall now show that in keys with any other tonic than C sharps or flats become necessary. For the present we speak only of major keys. 46. If we examine the 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." Lower tetrachord. jx; _, E o 3 = tetrachord. It is important to notice that the lower tetrachord begins with the tonic, and the upper with the dominant the next most important note to the tonic. They may therefore also be called the tonic and dominant tetrachords. 47. If we now take G as a tonic, or in other words make the upper tetrachord of C the lower tetrachord of a new scale, * Occasionally, even in modern music, the older modes are used, if a special archaic or ecclesiastical effect is desired. 1 8 HARMONY. [Chap. \\ we shall find that if we leave all the notes unaltered the upper tetrachord will have its semitone in the wrong place Ex. 23, tf 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 (the subdominant of C), must be raised to Fjt ; and we now have a semitone between the "leading note" (12) 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. 48. If we continue to take the upper tetrachord of each key as we obtain it as the lower tetrachord of a new key /. 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. G F$: D . . F$, Cfc A ... . .. . . . Fft, Cft, Gft. E Fft, C, G, Dft. B . F&C&-Gf,Dft,A|L Fft. ...... F, Cft, Gft, DJ, Aft, Eft. Cft. . V-. Y . . Fft, Cft, GJ, Dft, Aft, Eft, Eft. 49. It would be possible to continue this series, of which the next tonic would be G $ ; but as this would involve the use of a double-sharp, it is more convenient instead of G$ to take its "enharmonic" ( 5 note} AJP, which for all practical purposes is the same note. We never therefore find a piece of music written with the signature of the key of G $ major, though the key is occasionally used incidentally in the course of a piece. If we continue this series to its extreme limit, the next tonic will be D$, then will follow AJf, E$, and B$. We can go no further than this, because B$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 fifths. The incon- convenience of writing in these extreme keys is that they necessitate double-sharps in the signature. Thus the signature of A ft major would be written and of B major chap, ii.] KEY, OR TONALITY. 19 We shall show later in this chapter how to find the key-sig- nature of these "extreme keys," as they are called. Observe that in the series we have just given the tonics always rise by perfect fifths. 50. All these sharp keys have been obtained by making the upper tetrachord 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. Ex.24.f 51. If we examine the lower tetrachord here, we see that it has no semitone ; we also see that there is only a semitone be- tween 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 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), 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 process 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 b B b, E b, A b- Ab Bb, Eb, Ab, Db. Db Bb, Eb, Ab, Db, Gb- Gb Bb, Eb, Ab, Db, G>, Cb. cb Bb, Eb, Ab, nb, Gb, 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 Fb,*that of the E$ is taken, and so on with the others. 52. It should be noticed that in passing to the sharper key - ;'./., taking the dominant as a new tonic it is always the sub- dominant of the old key which is sharpened to become he new 20 HARMON*. [Chap, n. leading note ; and conversely in passing to a flatter key /. t. , taking the tonic as a new dominant, it is always the leading note of the old key which is flattened to become the new sub- dominant. Hence we obtain the easy iiile for finding the tonic of any major key from the signature. In sharp keys, the last sharp is always the leading note, and in flat keys, the last flat is always the subdominant. When we know the leading note or the subdominant of any key, it is a matter of very simple calcu- lation to find the tonic. 53. We defer for the present the discussion of the signatures of minor keys ; these will be more easily explained, and better un- derstood later. We now give the signatures of all the major keys, commencing with C major, and marking the tonic of each key. Keys with sharps. ( Tonics rising by fifths. ) - H^ H^jt Ex, 25. t Keys with flats. (Tonics falling by fifths.) fe"tr-^-nTfr sFF-^^fSji It will be seen that the last three sharp keys are enharmonics of the last three flat keys ; B , with five sharps, being the enhar- monic of Cb, with, seven flats; while F$ (six sharps), and Gb (six flats), likewise C$ (seven sharps), and Db (five flats), are also enharmonics of one another. Let it be particularly noticed that the number of sharps in any one of the sharp keys added to the number of flats in its enharmonic key always amounts to the same number, 12. 54. By bearing this in mind, we shall be able to find the signature of any of the extreme keys referred to in 49. All that is needful is to notice the number of flats or sharps' in the enharmonic of the key whose signature we wish to ascertain, and to subtract that number from 1 2 . The remainder gives the required signature, and it must be remembered that all numbers above seven will represent double sharps or flats,' as the case may be. 55. To make this quite clear, we will find the signatures of the two keys spoken of in 49 and 51 G$ and Fb major. G $ is the enharmonic 01 A b, which has four flats ; 1 2 4 = 8 ; therefore G $ has eight sharps i.e. , one double-sharp. Similarly, F b is the enharmonic of E tj, with four sharps ; it therefore will have eight flats, or one double-flat. Of two enharmonic keys, one will always be a sharp and the other a flat key. 56. There is another method of calculating the signatures of these extreme keys, which some studentc may perhaps find easier than that just given. If we compare the signatures of C % chap, ii.] KEY, OR TONALITY. 21 and Ci? in Ex. 25 with that of Cjf, we shall see that the former has seven sharps more, and the latter seven flats more than the "natural" key. This must obviously be so, because, if we put a sharp or flat before the tonic, it is evident we must put one before every other degree of the scale ; otherwise the semi- tones will not remain in the same places. Applying this reason- ing to other keys, it is clear that the key of A $ for example, must have ten (34. 7) sharps, and that every note which in the key of A was a natural will now be a sharp, while every note which before was a sharp will now be a double-sharp. Similarly, as B 1? has two flats, B #?, must have nine ( 2 -j- 7 ) , with two double- flats ; and so on in every case. The order of double-sharps and double-flats will evidently be the same as that of sharps and flats, beginning with Fx on the one side and Bbl? on the other. We have already said ( 49), that no entire piece of music is ever written in these extreme keys ; but their incidental employ- ment in modern music is frequent enough to render it advisable for the student to be acquainted with them. EXERCISES TO CHAPTER II. (i.) Write the three primary chords in the keys of D, E^, F |t, G [>, A tj, and B V. Prefix the key-signature in each case. (2.) Write major scales (one octave) from the following notes, putting no key -signature, but inserting a flat or sharp be- fore each note that requires one E, A j?, F $, F b, G $, B, B I?, Aj}, 23 HARMONY. fChap .in CHAPTER III. THE GENERAL LAWS OF PART-WRITING. 57. 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, etc. The progression of these parts may be considered in two aspects ; either as melodic pro- gression, 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 ex- plained. 58. The rules for melodic progression are few and simple. A good melody is one that flows naturally and easily ; it is therefore best either to proceed by step of a second (called "conjunct motion ") that is, to the next note above or below ; or by leap ("disjunct motion"), of a consonant interval ( 15). 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 : ^jj~i^^ T-H is better than [~^K~J -j j f| , though either is possible. The former is a diminished fifth, and the latter an augmented fourth. 59. If a part move by a diminished inverval it ought to return to a note within the interval, and not continue in the same direction. The best progression for any dissonant in- terval is, that the second of the two notes forming the interval should proceed to that note which is the resolution of the disso- nance ( 17) made, if the two notes are sounded together. For instance, the student will learn later (Chap. VIII.), t^at jl the diminished fifth just given will resolve thus : \(L p ={] Therefore F, coming after B, moves to the E, just as it would do were it sounded with B (Ex. 26 a~). Had F been the first note and B the second, B would, for the same reason, have gone to C (Ex. 26 b~). chap, ni.j THE GENERAL LAWS OF PART- WRITING. 23 60. An augmented interval should seldom be used in melody unless both the notes belong to the same harmony. But the interval of the augmented second, which we find in the minor scale (Ex. 5) between the sixth and seventh degrees may be used more freely. 61. A large interval in the melody is best approached and quitted in the opposite direction to that in which it leaps. Good. Bad. At (#) 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 they should be approached and left in the same directions, as at (^). 62. It is seldom good to introduce a leap of a seventh in the melody, with one intermediate note, unless all three notes form part of the same chord, the leaps be upwards, and the last note fall one degree.* f Good. Good. Good. Bad. Bad. Good. Good. At (a) () (V) the first three notes all belong to the same harmony a chord of the seventh ; at (V ) (H J-* ^=p ^Jg ^-p=L-p-*-Lg=:=g=a I I I '- II HAYDN. Seasons. KULLAK. Fleurs Animees, No. x. At (a) will be found in the third bar consecutive fifths by contrary 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 (<) are fifths be- tween alto and tenor ; at (V) are seen fifths by contrary motion between tenor and bass, and at ( ' f i *T T chap, ni.] THE GENERAL LAWS OF PART- WRITING. In the second of these examples, is also seen, in the tenor, an exception to the rule given in 61. These passages are not given for 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 haidly 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. 81. RULE VII. It is forbidden for two parts to go from a second into a unison. Ex. 54. ?; This progression is sometimes used when the second is a passing note as at ( # ) ; but the student is advised to avoid it even in this case. 82. RULE VIII. It is generally bad to approach or leave a unison by similar motion. Ei.55 This rule should be carefully observed by beginners ; but it should be added that in the works of the great masters instances of its violation are sometimes to be found. We give a few ex- amples by Mendelssohn, whose part-writing is remarkable for purity and correctness. MENDELSSOHN. Part-Song, "Auf dem See." Op. 41, No. 6. El. 56. MENDELSSOHN. Jagdlied. Op. 59, No. 6. El. 57. ( In both these passages the unison is between the tenor and bass, and the progression is from the dominant to the tonic chord This is the case in which it is most frequently met with. D HARMONY rchap. in. 83. We now give one example of the unison quitted by similar motion. MENDELSSOHN. Hirtenlied, Op. 88, No 3. Ex.58. \ In the first and second bars of this example will be seen consecu- tive unisons ( 67). Their excuse here is found in the fact that they occur between the last note of one phrase and the first of the following ; had both formed part of the same phrase, they would have been objectionable. 84. Though the three passages last quoted show that similar motion to or from a unison is not absolutely prohibited, it will be advisable for beginners to abstain from its employment till they have learned by experience when it can be used effectively. There are many things done by composers which it would be un- wise for the student in the earlier stages of his work to imitate. He will best and most easily acquire the power of correct and fluent part-writing by submitting himself in the first instance to a course of strict discipline. 85. 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 pro- ceed to a note above that previously sounded in a higher part. W Ex.59. 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 () the lower part leaps to C, which is higher than the A of the upper part in the first chord. Such progressions are sometimes necessary, but it is better to avoid them if possible. The crossing of two parts Ex. 60.1 though by no means infrequent in actual composition, especially in older music, should be avoided altogether by the student, in four-part harmony, for the reasons given in the last paragraph. When, later, he has to write for a large number of parts, he will find that crossing is sometimer expedient, and even necessary. Chp. ni.] THE GENERAL LAWS OF PART- WRITING. ^3 86. When two notes making a dissonance with one another (such as a second, seventh, or ninth) are taken without prepara- tion 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. Ei, 61. 1 Not good. Good. Not good. Good. 87. The student must not allow himself to be discouraged by supposing that it ie needful for him to commit to memory the whole of the rules given in this chapter, before he can make further progress. This is not the case. The rules are all tabu- lated here for future use, that they may be referred to when required. Some of them for example, that given in 86 will not be wanted at all at first. If the beginner will proceed steadily, one step at a time, making sure that he thoroughly grasps each point before he goes on to the next one, he will soon find that he is making satisfactory progress. All the chief difficulties of the study of harmony lie in the earlier stages ; and the student who thoroughly masters the first eight chapters of the present volume may rest assured that, when he can hon- estly feel that he has done this, the worst of his troubles are past. 34 HARMONY. ichap. iv. CHAPTER IV. THE DIATONIC TRIADS OF THE MAJOR KEY. 88. It was said in Chapter II ( 36) that a Chord was a combination of not fewer than three notes placed one above another, each note being at. the distance of a third, either major or minor, above the note next below it. A Common Chord was denned as consisting of three notes, the highest of which was a perfect fifth above the root. A chord containing only three notes is called a TRIAD. Every common chord is therefore a triad, but we shall see presently that every triad is not necessarily a common chord. 89. In Ex. 4 (Chap. I) we gave the scale of C major, and in Ex. 20 ( 39) we put a diatonic triad above each of the primary notes of that key, and saw that each of these triads was a major common chord. We will now take each note of the scale as the root ( 36) of a triad, using only diatonic notes : it will be seen that no other notes of the key than the three primary notes will have major common chords above them. ii iii IV V vi vii I 90. In the above example the triads on the second, third, sixth, and seventh degrees all contain a minor third above the root, which is therefore indicated by a small (not by a capital,) Roman numeral ( 41). Each degree of the scale except the leading note has a perfect fifth, and therefore a common chord, above it. But the fifth above the leading note is a diminished fifth, and a chord containing a minor third and a diminished fifth above its root is called a DIMINISHED TRIAD. When in marking the roots nothing is added to the Roman numerals, whether capital or small (I. ii. etc.), a perfect fifth is always im- plied, and the chord will be a common chord ; if the fifth be diminished, a small is added after the numeral. The triad on the leading note is'therefore marked vii, as above. All triads on other than the three primary notes of the key are called "secondary triads." In the present chapter we shall deal only with the treatment of primary triads. 91. It was said in the last chapter (65) 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 evi chm P . iv.] THE DIATONIC TRIADS OF THE MAJOR KEY. 35 dently be necessary to double one of the notes, that is to put the same note in two of the parts, either in unison, or at the distance of an octave, or even two or three octaves. It must be remem- bered that this doubling of a note does not alter the nature of the chord. Though the word ''triad" literally means a com- bination 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 in- troduced into the chord. We shall see presently which are the best notes to double. 92. In writing four-part harmony, the four "voice-parts," soprano, alto, tenor, and bass ( 65), are kept best as far as possible within the compass of the voices after which they are named. The general compass of each voice is about the fol- lowing : SOPRANO. ALTO. TENOK. *>L BASS. Ex. 63. : 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. 93. 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, Ex. 64. the harmony is said to be in close position. If the parts lie at more equal distances, and the tenor is more than an octave from the treble, Ex. 65. 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. HARMONY. [Chap IV 94. The best position of harmony is mostly that which allows the parts to lie at approximately equal distances, when this is possible. At Ex. 64 in the last paragraph, in the first chord there is a tenth between bass and tenor, a third be- tween tenor and alto, and a fourth between alto and soprano. This position is quite correct ; but the position of the same chord at Ex. 65 is generally preferable ; 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. 95. 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. | Very bad. Ex. It should be mentioned that the reason why the position of the chord at (0) is marked "not so good" is not, as the student may perhaps suppose, because of the interval of the twelfth be- tween the tenor and bass, but only because both alto and tenor are lying so high. Had the bass been lower, the same relative position would have been quite good : e.g. Ex. 67. 96. It will be seen that in the examples just given, the rela- tive positions of the chords vary widely ; sometimes the root, at other times the third or fifth is at the top. It must be clearly understood 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 Chap, iv.] THE DIATONIC TRIADS OF THE MAJOR KEY. 37 have inversions of the chord. These will be explained in Chapter VI. 97. With one exception, to be mentioned directly, it is possible to double any of the notes of a triad ; but they are by no means all equally good to double. In the majority of cases, it is better to double a primary than a secondary note of a key. If the student refer to Ex. 62, he will see that each triad con- tains at least one of the primary notes, and that the chords I and IV contain two. It must not be understood that it is com- pulsory to double the primary note ; we sometimes find pro- gressions in which it is quite as good, or even better, to double one of the other notes ; but for general purposes it will be found a good working rule, especially for begirjners, Double a primary, rather than a secondary note. 98. It has just been seen that the chords I and IV contain each two primary notes ; and the question will naturally suggest itself, Which of the two should be doubled? The answer is that this depends upon the position of the chord. If the chord is in root position, and we double the fifth, it will sometimes be difficult, either in approaching or in quitting the chord, to avoid consecutive fifths ( 71). The root is therefore almost always the better of the two notes to double, and our second rule for doubling is, In the root position of a chord, it is seldom good to double the fifth. We shall see later that this rule does not apply to inversions ( 166). 99. It will be noticed that in the chord vii the fifth is the only primary note. Here, however, it cannot be doubled in root position because the fifth of this chord is not a perfect, but a diminished fifth and we shall learn later that it is not generally good to double a dissonant note. Besides this, a diminished triad is very seldom found in root position, except in a sequence ( 138)- 100. The exception referred to above ( 97) in speaking of the doubling of a note was that of the leading note. This note is the third of 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. ) m Ex. 68. Let the student play the chords marked (