jpij itJlf ' Z^mJj'mj/jjgii J^ji ^^^^^^^icz?/-^ :^i; \ Vvv t3 „ "A^J'\Si*'l- TBI' //' r '' (E( ar or ',^^r XJii if f ^s^.. >C-*^ ^aw' iff (-( ■^jr(gr etc THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA RIVERSIDE jgy^ rm^ !■»■■»■■ £■* Libris : c. K. OGDEN .jlT /?'■ t}> ' V •\ o- V A TREATISE ON HARMONY \^ BY THE Eev. Sir F^ A. GOEE OUSELEY, Bart., MA., Mus. Doc. PROFESSOR OF MUSIC IX THE UNIVERSITY OF OXFORD AT THE CLARENDON PRESS M.DCCC.LX^^[II MTyo €Iar.entr0iT ^ress Btxm PRmCIPLES OF HARMONY OUSELEY Hoution MACMILLAN AND CO. FUBLISHEBS TO THE UNIVEBSITT OF xioxti PREFACE. The object aimed at in this volume is the combination of true philosophical prin- ciples with simplicity of explanation. It also endeavours to include all necessary- details in the smallest possible compass. So many treatises on Harmony have appeared since the beginning of this century, that some apology is perhaps due for adding yet one more to the number. The author must plead as his apology, the conviction that although the existing treatises on the subject contain much very valuable matter, yet all seem to him to be either founded on erroneous principles, or faulty in arrangement. Some mix up together the elements of Harmony, Counterpoint, and Pianoforte-practice; others start from principles not based upon nature, but too often contradicted by the now better ascertained phenomena of acoustics; others repudiate physical science altogether, and treat of Music as though it were only an emotional art. The present volume tries to avoid these and other similar errors. Wherever existing works explain or illustrate any point with clearness, they have been quoted freely and acknowledged with gratitude. But in other and more frequent cases it has been necessary to take up entirely fresh ground, and to employ new explanations and illustrations. It is confidently hoped that the system thus evolved may prove useful to the student, by giving him natural and rational explanations of the facts of Music, and of the rules deduced from those facts. The author has aimed throughout at a consistent theory, founded in nature, progressively expanded, and involving no purely arbitrary rules. He now lays the results of his endeavour's before the public. viii PREFACE. His warmest thanks are due to many who have aided him in the production of this work, especially to Professor Pole, Mus. Doc, for his admirable illustration of the comparative magnitude of intervals, and the lucid explanations accom- panying it, which invest the work with a scientific value it would not otherwise have possessed. In conclusion, a suggestion is offered as to the best way of using this little work. The student is recommended to study Harmony and Counterpoint con- currently, working at them a little at a time, alternately. He will find that neither can be perfectly mastered without the other. With this view, the author proposes, before long, to bring out a Treatise on Counterpoint, based on those of Fux, Mai'pm'g, Reicha, and Cherubini, which will be so constructed as to run parallel to the present volume, as far as possible ; and he desu*es that the two works may be regarded as pai-ts of one whole, for the instruction of such as desire to grasp the subject in its completeness. St. MiCH-iEL-'s College, Tenbuey, January, 1868. TO THE REV. G. W. KITCHIN, M.A. THIS TREATISE IS AFFECTIONATELY DEDICATED BY HIS OLD FRIEND THE AUTHOR EKRATA. Page 32. First stave of music from top of page, under the last two chords, 6 6 B6 be for 4 4 read 4 4 2 2 b2 b2 Page 34. Second stave of music from top, bar 2, for Page 47. Second stave of music from bottom, bar 5, for ^ :cz — read -^ 221 4^±- Page 53. Second stave from top, bar 3 , for g -.'S>- ^^ read y?W=?^_ -^- read E -- Page 85. Bar 6 of example, bass stave, for V^ ^^j, — read V^ y^^y Page 88. Line 4 from bottom, for t]4 read :5 ; and in line 2 for B5 read D4 3 3 b b Page 89. Second stave from top, bar 5, for - V ^ l rj I'ead 4^ '^Q rJ Page 97. Second stave from top, for read ..88 Page 108. Last bar but one of Exercise on Dissonances, for - read 5 Page 114. Second stave of music, bar 1, for £ 't d^ read / 4 s &- F eJ ^" Page 129. Last stave of music, in bars 4 and 7, for Page 136. First stave, first bar, for ySi rea •q^^ read @ ^-f^ ad ^7~^ ^ Page 139. Second stave from top, in bar 1, for '^ read *'^; in bar 3 for jfs b6 T . , (js be read 4^; and in bar 5 for Jfo read 44- 2 b b Paere 149. Lowest stave, bar S, for \ ^^ read $ bs D 9 D Page 163. In second bar of example, for -if — hn ^gtd V 7 u,*^^ J a^ /m\' I le;. v^ 1 ^ 1 read 5 c Page 169. Page 174. Page 178. Page 192. Last stave, bar S, for ^^ read '^^ In fourth chord of example, for B read A I 2^= Second stave, bar 6, for - /W- -& read fe e/ The Jt in the signature of the example should be omitted Page 20O. Upper stave, bar 2, for ^^^ fj „ Lowest stave, bar 4, for y^-g!— ^ ,ead ^r: p^1=p: I— J— read vfiv- Sl— S *! ^ - r f^^ Page 205. Page 210. Lines 1 and 2, for " subdominant " read ^^ supertonic First bar of exercise, for -tS>- r^-H :?:2 - read rJ rJ Page 227. Second stave from bottom, bar 3, for „ Last stave, bar 2, for ^ read ^ Page 248. Bar 4 should be the same as bar 3 Page 262. Line 10, for "or as the fourth below A'^ read "or as a fifth below A.' TABLE OF INTERVALS. THE SAME COUNTING DOWNWASDS. fttJ I f J j ^^rr 1''' ^ I I jzin a ^^^^^ ^i^- ^kHi s^^s j ff?rrbj:^ ^^ Trjfr^ ^ ^ti=^ *3 p This nomenclature will be strictly adhered to throughout this treatise. A SHORT TREATISE ON THE PEINCIPLES OF HARMONY, CHAPTER I. P^'eliminary Remarks. 1. It is presumed that the student who uses this book is abeady acquainted with what are commonly considered the first elements of music. A very brief summary of these will, therefore, be all that is required here, before we begin to treat of harmony strictly so called. And it will be sufficient for our pui'pose to indicate most of these elementary matters in a tabular form, as our chief object in so doing is merely to set forth the system of nomenclature which will be adhered to in these pages. 2. In the first place, it is important that the student should have clear views as to musical notation. For this purpose he is referred to Hullah's admirable little Treatise on the Stave, than which notliing can be more definite and intelhgible. B THE PRINCIPLES OF HARMONY. He will there learn, that our ordinary stave of five lines is derived from what is called the great stave of eleven — The three lines to which the letters G, C, and F are here appended, represent fixed sounds of definite pitch : from these all others are counted. Formerly, the whole eleven lines were occasionally used, and sometimes six, seven, four, or three. But five lines have been found sufficient for all purposes, and that number is now universally employed. It is obvious that in selecting the five adjacent lines which we shall use for our stave, we must be guided by a consideration of the notes we have to write, and their place in the scale or range of musical sounds. If we are writing for violins, flutes, or other high-toned instruments, or for treble voices, or for the upper portion of keyed instruments, it will be necessary to select the topmost fines of the great stave. These five lines do not include either the fixed note C or the fixed note F, but they do include the fixed note G, which is represented by the figure j[^ called the G clef — or treble clef — or violin clef, and which will be ^ placed so that the principal curl of it shall fall on the second line counting upwards; thus q If we are writing for bass instruments or voices, it will be necessary to select the five lines at the bottom of the great stave. These will not include either the fixed notes G or C, but only the fixed note F, which is represented by the figure ^: or ^ , called the bass clef, and which will THE PRINCIPLES OF HARMONY. 3 be placed so that the two dots shall fall above and below the fourth line counting upwards ; thus ^ ~@' Instruments of large range, such as the organ, pianoforte, or harp, require the combination of these two staves ; thus and on comparing this with the great stave, it will be seen that it contains the whole of it except the middle line, which represents the fixed note C. If we are writing for instruments of medium range, such as the viola, or the alto and tenor trombones, or for contralto or tenor voices, neither of the above staves will be found convenient ; the one being too high and the other too low in the scale : therefore we must select some other set of five liaes, to suit the requirements of the case. Whichever set we select will include always two and sometimes the three fixed notes ui the great stave ; and as we have already appropriated the clefs repre- senting G and F to the highest and lowest staves respectively, it will conduce to clearness and distinction if we use the symbol of the fixed note C, which is thus formed, u , or H , to distinguish our medium staves. Of such staves only three are now in use. The most acute is called the soprano, and is used in instrumental scores of oratorios, and in old Church music, for the soprano or treble voices. It consists of the 6th, 7th, 8th, 9th, and 10th lines of the great stave, countiag upwards, and is expressed thus „ , — ^-G. B 2 THE PRINCIPLES OF HARMONY. The next is the truly medium stave, and consists of the 4th, 5th, 6th, 7th, and 8th lines of the great stave, counting upwards. It is used for the viola, or alto violin, for the alto trombone, and for the alto voice parts in scores. It is expressed thus C- F The last is called the tenor clef (or, more properly, the tenor stave), and is used for the tenor voices in scores, as well as for the tenor trombone, and sometimes also for the higher notes of the violoncello and bassoon. It consists of the 3rd, 4th, 5th, 6th, and 7th lines of the great stave, counting upwards, and is expressed thus For further details the student is referred to Hullah's book above- mentioned, or to Marx's " Theory and Practice of Musical Composition," (translated by Saroni. New York, 1852). 3. It will be sufficient to give a list of the different kinds of notes and rests, an ordinary acquaintance with them being presupposed. A Breve |Io|| is equal to two Semibreves o G> or to four Minims 1 1 p 1 f P 1 • or to eight Crotchets p 1 p 1 p f or to 0":^. r^ cc It ti cr G' tm- -'» sixty-four Demisemiquavers, and so forth. THE PRINCIPLES OF HARMONY. 5 The corresponding 7'ests (to represent intervals of silence) are the breve , the semibreve ^^^^ , the minim ---- , the crotchet [*, the quaver *|, the semiquaver sj, and the demisemiquaver ^. A dot after a note or rest makes it half as long again : thus o*, or a dotted semibreve, is equivalent to three minims, or six crotchets, or twelve quavers : [O*, or a dotted minim, is equivalent to tlnree crotchets, or six quavers, or twelve semiquavers. If a second dot is added, it will lengthen the note or rest half as much again as the first dot: thus i***, or a doubly dotted crotchet, is equi- valent to seven semiquavers, or foiu-teen demisemiquavers ; for it consists of the value of a crotchet, a quaver, and a semiquaver, i. e. two quavers, one quaver, and a semiquaver, i. e. four semiquavers, two semiquavers, and one semiquaver, i. e. 4+2 + 1 = 7 semiquavers. When a note or a rest is to be lengthened indefinitely, this figure ^^, called a ijcmse, is placed \uider or over it. For further details and exercises on this branch of musical knowledge we must refer to more elementary treatises. 4. Supposing the student to be famihar with the use of bars and double bars, with the theory of accent, and the art of beating time, it wiU be sufficient for our purpose to give a time-table, to shew the system and arrangement adopted in this book. THE PRINCIPLES OF HARMONY. a. Common or Duple Time — divided into i. Simple, and ii. Compound. in a bar. (g or Double Common, 4 rzt (1^ or Alia Breve, 2 |C5 in a bar. (3 01* - or Common, 4 |# in a bar. 4 I or 2 # in a bar. r — or 4 |0» in a bar. 4 I ]2 "8" 6 4 _ or 2 (#• in a bar. 8 I or 4 !#• in a bar. I or 2 |0* in a bar. jS. Triple Time — divided into i. Simple, - or 3 o or 1 o* in a bar. 2 I 3 - or 3 |# or 1 fO* in a bar. 4 I I 3 - or 3 |# or 1 !#• in a bar. 8 I I 16 or 3 |# or 1 !#• in a bar : and ii. Compound. 9 4 or 3p 1 • in a bar. 9 8 or 3 p. 1 in a bar. 9 16 or 1/ in a bar. 5. It may be taken for granted tliat the student knows tlie meanings of the figures J|;, b, Q, x, bb, and the use of them, both as accidentals and in the signature. If not, we must again refer Mm back, to the elementary treatises, contenting ourselves with a tabular list of the keys, major and minor, with their several signatures. THE PRINCIPLES OF HARMONY. Major keys with shai-ps. 7 sharps CJf 6 sharps FJf 5 sharps B|^ 4 sharps Ep 3 sharps Ap 2 sharps Dp ] sharp Gp Minor keys with sharps. 7 sharps Aj( 6 sharps • n 5 sharps G|j! 4 sharps C Jf 3 sharps Fjf 2 sharps Bp 1 sharp Ep Major natural key . . Cp Minor natural key AP -d i\Iajor keys with flats. 1 flat Fp 2 flats Bb 3 flats Eb 4 flats Ab 5 flats Db 6 flats Gb 7 flats Cb fm^ Minor keys with flats. 1 flat Dp 2 flats Gp 3 flats cp 4 flats Fp 5 flats Bb 6 flats Eb 7 flats Ab THE PRINCIPLES OF HARMONY. Each of these major keys is supposed to be nearly related to that minor key which has the same signature, and stands on a Hne with it in the above table : hence the terms relative major and relative minor. 6. It will be necessary to give a table of intervals, although it is supposed that the student has already mastered them before using this book, because some of them have been variously named by different writers, and it is essential, as a preliminary, to fix our own nomenclature, which will be strictly adhered to in the following pages. Intervals are divided into consonant and dissonant intervals, (or, as they are sometimes called, concords and discords). Consonant intervals are of two kinds, perfect and imperfect. Imperfect consonant intervals are subdivided again into major and minor. Perfect consonances cannot be so subdivided. Dissonant intervals also, like the imperfect consonances, are either major or minor. All intervals are susceptible likewise of certain alterations, called augmentation and diminution; excepting only that major intervals can- not be diminished, and that minor intervals cannot be augmented. Perfect consonances alone can both be diminished and augmented. Intervals I Consonant Dissonant Perfect Imperfect major mmor h major minor §• B c p g &- T c p p p 'ft-' p. P^ B p p B c-t- eri- g fl> rt> B » 2 p 1-S i-j a n> rt> tjD O' CI- &- ft- ft- Cr" n> fD Q^ ft. ft- THE PRINCIPLES OF HARMONY. 9 The smallest unaltered interval is the minor second, or semitone : and it is convenient to compare and measure all larger intervals by the number of semitones they contain, or to which they are equivalent. The perfect consonant intervals, according to the usual computation, are the octave and the fifth ; to which may be added the fourth also, though only under certain restrictions, to be considered hereafter. The imperfect consonant intervals are the major and minor third, and the major and minor sixth. The dissonant intervals are the second and the seventh, each of which may also be major or minor. If the interval of a semitone be subtracted from any perfect or minor interval, by means of a sharp or flat, to alter the pitch of the lower or upper notes respectively, such interval is said to be diminished. If by the same means a perfect or major interval be enlarged to the same extent, such interval is said to be augmented All augmented or diminished intervals so produced are called chro- matic dissonances, except the two which occur in the diatonic major scale, i. e. the augmented fourth or tritone between the 4th and 7th degree, and the diminished fifth between the 7th degree and the 4th in the next octave. Of all which more will be said hereafter. If an octave be added to any interval, its original character (as re- gards divisibility into major, minor, augmented, or diminished) remains the same ; only that in the case of the 2nd, 3rd, 4th, 5th, and 6th, they are in that case sometimes designated as the 9th, 10th, 11th, 12th, 13th, as will be more fully explained in a future section. We here annex a list of all the intervals, illustrated in the key of C natural. Note. Some writers call augmented intervals " extreme." Others call augmented perfect intervals " pluperfect." Others call diminished fifths " false fifths." Others call augmented fourths " tritones." Others call minor sevenths and seconds " flat." Others call major 10 THE PRINCIPLES OF HARMONY. sevenths and seconds " sharp." Others call diminished fourths " flattened." Others call augmented fifths "sharp." But every one of these terms is unsystematic and unphilosophical, and some of them are absolutely incorrect. 7. There are yet remaining a few technical terms which ought to be explained in this preliminary chapter. (a) A cliord is the simultaneous sounding of several different notes, selected according to certain fixed principles and rules. (j8) A triad is a chord of three notes consisting of a bass with its third and fifth, e.g. Triads are of three kinds : — i. Major; ii. Minor; iii. Imjperfect, or diminished. A major triad consists of a major third and perfect fifth. A minor triad consists of a minor third and perfect fifth. An imperfect triad consists of a minor third and diminished fifth. (7) If the octave of the bass is added above a major or minor triad, it constitutes what is called a common cliord. ^OTE. The imperfect triad being (as its name imports) not formed after the same perfect model as the major and minor triads, cannot be converted into a common chord by the addition of the octave of its lowest note. {Examples) fL ^ * A major triad. § Imperfect triad. -^s- m -^ '^^ t X § 1[ t A minor triad. If Common chord major. X Imperfect triad. II Common chord minor. THE PRINCIPLES OF HARMONY. 11 Besides these triads we occasionally meet with what is called an augmented triad, which consists of a major third and augmented fifth, or of two major thirds superposed ; thus M ^ 8. It may be as well here to explam one or two terms which belong rather to counterpoint than to harmony, but to wliich reference must necessarily be made in treating of the resolution of discords, and in some other cases also. i. Motion is of three kinds : — similar, obhque, and contrary. a. Similar motion is said to exist between any two or more parts, or voices, which ascend or descend simultaneously, but not in unison ; e. g. ^^g: &C. j8. Oblique motion is when one part remains without moving while another ascends or descends ; e. g. ici xz 7. Contrary motion is when two parts, or voices, move in opposite directions ; e. g. ^^ ^^ :jrj=^ Avr r r ii. Consecutive fifths or octaves are produced when two parts move so as to produce the same interval (of a fifth or an octave, as the c 2 12 THE PRINCIPLES OF HARMONY. case may be) between tbem in successive chords. There are several kinds of such consecutive fifths and octaves. a. By similar motion ; e. g. ^ -(S^- :,^=^ a or /8. By contrary motion ; e. g. ^^ -(S>— ' -tS>- :^d or 5^^ r -p^ I "?:^ 7. Hidden fifths or octaves, which occur when an unaccented note intervenes without any change of fundamental harmony ; e. g. --i^i By the laws of strict counterpoint, every consecutive fifth or octave, of any of the above kinds, is altogether forbidden. Consecutive major thirds have been hkewise forbidden by some ancient authorities, though modem composers use them very freely. Still it must be admitted that several unaccompanied major thirds in succession do produce a very disagreeable effect. Any one playing the following notes on a keyed instrument will perceive their badness : — ^ ^ ^ C^ — Consecutive fourths are also strictly forbidden, unless accompanied THE PRINCIPLES OF HARMONY. 13 by thirds below them, when they become perfectly correct and pleasant to the ear ; thus ^^^ :^ -(S»- is bad, while Wff is good. It is unnecessary to go any further now into this matter, as the harmonic principle involved will be duly explained in its proper place. CHAPTER 11. Fii'st Principles of Harmony. 1. The origin of harmony must be sought in natural phsenomena. This involves an acquaintance with the science of acoustics, and is therefore more or less foreign to the peculiar subject of this treatise. Still, it will be necessary to obsei-ve here that the primary chord given us by nature is the following : — root, octave, twelfth, fifteenth, seventeenth, and nineteenth. As an example of such a natural and primary chord, we wiU assume C as our root, or generator, and give the notes which result from it as natural harmonics : — ^ 3 q: root octave 12th 15th -&■ 17th 19th The interval of an octave is so perfect a concord, that it may be regarded as almost equivalent to an unison. It is therefore possible for us to omit for the present the root and its octave ■^^^- 2!2 "3r , as they are sufficiently represented by its double octave (or fifteenth), ^ Similarly, we may for the present omit the twelfth of the root, sufficiently represented by its own octave _ , as it is , which is the nineteenth, or octave- twelfth, or double-octave-fifth, of the original generator. THE PRINCIPLES OF HARMONY. 15 We get then, as our residuum, the major triad harmony of nature. which is the primary 2. If we were to continue the natural series of harmonic sounds as they occur in nature, we should arrive at some which would lead us out of the key in which we began, and which in fact do not belong to that key at all, but to others related to it in a peculiar way, which will be explained when we come to speak of modulation. For instance, starting from the generator C, as before, we should find the following sequence of notes succeeding those given above : — r^^ ^ P - ^^"p- ^ .. ^^ ^2: W^- ^- ^^=^2=^ ^ -j:2. i? ±zz± O X o o Now of these there are four, marked x , which not only are foreign to the key of C, but are out of tune in any key. Four others, marked © , are merely repetitions of the intervals of the major triad in an upper octave. They may therefore be considered as identical with the primary chord we have already obtained. Two only remain, D and B, which we can make use of for our present purpose. On carefully examining these two notes, and combining them with the note G which we already possess, we find that they constitute together a major triad. To shew this more clearly, let us take the highest G in the above figure. the B next above it, ■^ ^2_ written beyond the double bar ^i : and let us take the higher octave of the D, which is and this gives us the major triad «y 16 THE PRINCIPLES OF HARMONY. of which G may be regarded as the generator, just as C is the generator of the triad 3. Seeing that octaves may be neglected in these considerations, it will be convenient to write this new major triad an octave lower, thus ; and setting out the whole harmonic series of G, of which it forms the principal part, we produce this scheme — cz i :& n 'P "- 3^#^ No. i. VI. Vll. VIU. IX. XU. Xlll. XIV. XV. XVI. Here it will be observed, that every note belongs to the key of C till we come to the double bar : and although the note F, marked x , is not perfectly in tune, yet we can sub- stitute a really true F without at all materially disturbing our new series of sounds. Omitting, then, numbers i, ii, iii, viii, x, (and of course all beyond the double bar,) as we did in the case of the harmonic series of the generator C, we get as our residuum the chord , which is called the " dominant chord of nature," being based on the fifth of the key; which fifth is called the, dominant, because it exercises the most powerful influence on the harmony. Reducing our two chords to the same octave, and writing them in juxtaposition, , our ear at once detects a close relationship between them, and on hearing the former of the two, it immediately feels a desire to hear the latter also, and feels relieved and satisfied when it has been sounded. 4. From the preceding section it will have been seen that while the dominant harmony suggests the idea of chcmge or motion, and tends to the primary major triad, that primary major triad suggests no such idea, but rather induces rest, and in fact determines the key in which the music is written ; gives, as it were, the chai'acteristic tone to the music : hence THE PRINCIPLES OF HARMONY. 17 the original generator is called the Tonic, and its triad or common chord is called Tonic harmony. If we were to be confined to tonic harmony alone, we should be like a i)risoner within the four walls of a gaol — we could never get out of one groove. Our fatigue would become unbearable. d: ^ S^ s= i-y=j-^ ^h=^ ^ 'n^^t^rr ^ ^ rrf^==rrf =^ 5S= If this passage be played over several times, it will give a fair idea of the irksomeness of purely tonic harmony. On the other hand, if we were confined to dominant harmony, not only should we have a continually unsatisfied craving after a Tonic chord, but we should be even more wearied than in the above case. An example will sujfficiently prove this : — ^3 -(S>- ^ij=d: -- --^ \r^f-^^ 2=k -m JPL -^ rr '^' d^r^d-^^^ d: S -fS*- -Gh -& _»__ 1^2 S^ '^^• .«• Play over this exclusively dominant piece, and it will remind you of a traveller wandering homeless from place to place, seeking a welcome, and finding none. 18 THE PRINCIPLES OF HARMONY. But an alternation of tonic and dominant chords will always excite and satisfy the ear,' just as alternate activity and rest are salutary and pleasant to the body. 5, It may, then, be taken as proved, that the key-note or tonic, with its third and fifth, satisfies the ear, and leads it to desire no further change. For this reason, the close or " perfect cadence " of a piece of music must always terminate in a tonic chord. It may also be taken as proved, that the fifth of the tonic, which is called the dominant, with its third, fifth, minor seventh, and major ninth, does not satisfy the ear, but leads it to desire a change to the tonic harmony. For this reason the dominant harmony never can end a piece, but should precede the tonic. And it is from this characteristic feature of the dominant harmony that the whole system of the resolution of fundamental discords is derived, of which we shall soon have to speak. This is a most important first principle, and should be thoroughly understood and mastered by the pupil before going any further. For which object it is desirable that he should strike the following dominant chords, pausing after each, and realizing the unsatisfactory impression they leave on the ear : — 22: ~JZZ. M^ b^ -- \i^ "<>> \y -Gh -^ S>- v^ \U \:^ THE PRINCIPLES OF HARMONY. 19 To shew how the ear may be satisfied by a tonic chord succeeding each of these dominant chords, let the student play the following : — jQ_ s ^. ^ i± Z21 -Gh Si^ !2 a T2-- ^ :22: ^ --^ C2: T^ -"Gh X^ ^ -- s>- "^?" M=n=3B =g: iq: ^^ g ;^^ -!?(S^ -^^^ "^7" -iZf:^ 12^:2 -_ !2^ i2^ J^L e 172^ -•S) -ts^ -?»^- t54Si'- ^Is*- -9|S>- -c^- ^s^ and he will perceive that in each case the tonic chord, which here suc- ceeds the dominant, at once satisfies the ear, and produces the sensation of rest. 6. It must next be observed, that it is not every note in the domi- nant harmony which necessarily possesses the peculiarity of leading thus to the tonic. For if we simply take the major triad of the dominant root, there is notliing in it to shew that it is not a tonic triad : for instance. g= ^ may be simply the tonic harmony of the key of G. But the moment we add the next note in order which belongs to the dominant harmony, which in this case would be F, thus ^ ^ we preclude all impressions of the key of G, or of G as a tonic root, for D 2 20 THE PEINCIPLES OF HARMONY. tlie F here is natural, and therefore out of the scale of G, which of course requires the F to be sharpened. Moreover, the intei'val of a minor seventh, from G to F, and of a diminished fifth, from B to F, at once destroy all the rest and jpei^ma- nence of the chord. This minor seventh, then, is clearly the characteristic note which invests the chord with its distinctively dominant character. Hence this chord is generally called the " chord of the dominant seventh." Note. This chord is often called the " added seventh," because it is composed of a seventh added to a major triad. It is also sometimes called the " fundamental seventh," to distin- guish it from other chords of the seventh. Likewise some writers call it the " minor seventh" chord. None of these designations are incorrect, but in this work it will be invariably styled the " chord of the dominant seventh." 7. Tliis chord, as we have just seen, contains discords, and these are made to satisfy the ear by means of the chord of the tonic, which must follow immediately. When the discords have thus been rendered agree- able to the ear, they are said to be resolved ; and this resolution of dis- cords forms the most important element of the science of harmony. The rule for resolving the chord of the dominant seventh is a very- simple one : " Each of the discordant notes leads to and is resolved into that note in the succeeding tonic chord which is nearest to it in pitch, whether that note be above or below it on the scale." Thus, in the chord i ^~"8 — , which is resolved into 7/K ^ > the F goes to E, and the B to C, the bass note G of course goes to the bass note C (root to root), while the D, being equidistant from C and E, may go to either. It appears, then, that the only notes (beside the root) which have a compulsory resolution, are the third, B, and the seventh, F. Of these, THE PRINCIPLES OF HARMONY. 21 the " third of the dominant" always goes to the octave of the tonic root, and is therefore called the " leading note/' while the seventh always falls to the third of the tonic. This is a most important rule, and leads to many essential conse- quences. It should therefore be thoroughly learnt and appreciated at this early stage of the student's progress. We accordingly give a few examples in different keys, by way of illustration : — 8. It would seem that the tendency to a tonic resolution, which, as we have seen, is the characteristic feature of the chord of the dominant seventh, is attributable mainly to the discordant interval which exists between the third and seventh, and which is either a diminished fifth, (if the third be below the seventh,) or an augmented fourth, (if the seventh be below the third,) as may be seen by a careful examination of the preceding examples, where the intervals are purposely placed in various positions. Note. Any interval within an octave is susceptible of what is called " inversion." Which may be thus explained : — 22 THE PRINCIPLES OF HARMONY. If the lower of the tvvo notes forming any interval be changed into its upper octave, e. g. I^ changed to /L ~r^ - J , the interval is said to be inverted, or, in other words. the new interval thus formed is an inversion of the former : thus if the interval ^ be given, then its inversion will be m :^ The same result will be obtained if the upper note be taken down an octave : thus — P It will be seen by the subjoined table that the inversions of 'perfect intervals are perfect; of major, ai*e minor; of minor, are major; of augmented, are diminished; of diminished, are augmented. The student is recommended to copy out this list, and name all the upper and lower intervals according to the table given in Chap. I. sect. 6. Indeed, if a diminished fifth or an augmented fourth be played, alone and unaccompanied, the same craving after resolution will ensue. Thus, if 2^^y be struck, the ear craves the regular resolution f^ ^ -& J and if the inversion the regular resolution THE PRINCIPLES OF HARMONY. 23 be played, a similar craving is felt for This effect, however, is enhanced by the discord which exists between the minor seventh and the root, and more strongly still between the minor seventh and the octave, where the interval of a major second is produced. Thus i fr)—^^ — g— or ^f IQ ^- -is»- 9. The dominant harmony contains yet another note besides those we have been considering, as will be seen by a reference to Section 3 of this Chapter. It is called the major ninth, and it is occasionally added to the chord of the dominant seventh, to strengthen, vary, and enhance its effect, although it is not an essential element of that chord. When thus enhanced, the complete chord is appropriately called the " chord of the added ninth," or the " chord of the fundamental ninth," to distinguish it from certain other chords of the ninth which will be ex- plained hereafter. The addition of this new dominant interval to the chord of the dominant seventh does not alter the resolution of the two essential notes of that chord (i. e. the third and the seventh). The third rises to the octave of the tonic root, and the seventh falls to the third of the tonic, just as they would were the ninth absent. The ninth itself, being as it were a coadjutor and strengthener of the seventh, pursues a similar course to that which is peculiar to that 24 THE PRINCIPLES OF HARMONY. interval. While the seventh falls to the third of the tonic, the ninth falls to its fifth. Thus -& :^3: Here we see by an example the necessary resolution of the dominant root, the third, the seventh, the ninth ; but how about the fifth, D % If the ninth were not there, this note might either rise to E, or fall to 0, being (nearly) equidistant from both. But the addition of the ninth. A, renders it impossible for the D to proceed downwards to C, without violating that rule of counterpoint which forbids consecutive fifths (see Chap. I. sect. 8. no. ii.), for their joint progress would be ^ :P2: Therefore the fifth of the dominant root (here D) is forced in this case to adopt the alternative of rising to the third of the tonic (here E). To shew this more clearly, we will arrange the notes of the chord in the following position : — m :ezz2: 122115" t:?~i -1— (S>- m Note. Logier does not appear to recognize any dominant interval of harmony beyond the dominant minor seventh. He regards the major and minor dominant ninths as no more than substituted notes. THE PRINCIPLES OF HARMONY. In this Fetis agrees with him in his valuable Treatise on Harmony. But this seems to be a very unphilosophical view of the case. For nature supplies us with both these intervals, (the major and minor ninth,) one perfectly, and the other almost perfectly, in tune, as will be shewn hereafter (Chaps. IV and VI), whereas the natural dominant minor seventh is by no means so perfect. Then again, it is not philosophical to account for an interval by the hypothesis of a substituted note, and yet to allow of the coexistence and simultaneous use of that note and of the one for which it is supposed to be substituted : e. g. 8 > Q ^ -3— Sit IBZZZS where the A according to these theorists would be substituted for G, and yet the G is allowed to be sounded in another octave. Surely it is more consistent with analogy, and more agreeable to the phjenomena of nature, to regard the ninth here as an added harmonic, derived from the chord aflforded us by nature, XT- as will be more fully explained hei-eafter (vide Chaps. Ill, IV, VI). E CHAPTER 111. 1. From what has been abeady said, it will have appeared that the intervals of a chord may be inverted amongst themselves, and their order of acuteness interchanged, without thereby altermg either their relations to the root or to each other. This may be illustrated by taking the triad or common chord of C, and the chords of the dominant seventh, and of the added ninth, and arranging them variously. ^ ^ .CL e ■^ ^ "7^r~ ^-1~r^ H— ^ ^ :^ ?2: -^ iS»- -^ :g=U :^--gr ^^^ ^^y --& &c., &c. @: ?2: P^zif^ T^ rJ rj y^-rj rJ ^ \^j ^~r5 g2 ^ rj f j zp r^ r j^ :B :^i ^ S ^- -sf -^ — f^ — s & -cP=g^ -^hT- S=^ :& -^2. -f=^ -^^ -|Q- &C.J &c. P=r S» lis — & — ts- 3^ -&- THE PRINCIPLES OF HARMONY. 27 But we may now go a step further, and omit the root altogether, substituting for it occasionally its octave in an upper part. The effect of this will obviously be that one of the other notes of the chord will be at the bottom of the harmony. But although the root be in such a case unheard, yet it must be always imagined. All the other notes of the chord are equally derived from it, and dependent on it. And the chords thus modified are called inverted chords — or inversions — and their constituent intervals will require the same treatment, for the most part, as though the root were sounded. The effect on the ear, however, will not be the same, especially in the inversions of the tonic harmony, for the sensation of rest and fixity will be absent : and therefore " every piece of music must end with an un- inverted tonic chord." 2. The tonic triad consisting of three notes, and either of the upper ones being capable of being taken as the lowest note in the harmony, by inversion, as above described, it is clear that only two such inversions are possible — first, when the third is in the bass, and secondly, when the fifth is in the bass. When the third is in the bass, it is called the " first inversion of the common chord ;" and as the octave of the root then forms the interval of a sixth with the third which is placed in the bass, this chord is also called the " chord of the sixth." For instance, deriving it from C, / ■i s Cj E 2 28 THE PRINCIPLES OF HARMONY. it will appear iii the following forms, according to the collocation of the upper notes of which it is composed : -f^ -Q- -f^ C^ -^ t^ r^ :^ ^2: q: e :?2: -*s- -- -» -o- -iS r ^ r^ r^ ^ 2~r> q: 'f^ r^ r:t r:^ r^ r n — r^ ^^ ^'^ THE PRINCIPLES OF HARMONY. 29 This second inversion is less satisfactory than the first, and should be more sparingly employed. 3. And perhaps this will be the most fitting place to introduce the subject of what is known as " thorough-bass-figming." This is a kind of musical short-hand, of no great value, but occa- sionally convenient, by which figures, placed under or over the bass notes of a piece of music, are used to mdicate, vaguely but concisely, the kind of harmony to be played with the given bass. A bass note with no figures attached to it, indicates a simple imaltered triad or common chord. If, however, one of the notes of the harmony has to be modified by an accidental sharp or flat, such modification is indicated by an accidental instead of a figure, (in the case of the third,) or by an accidental added to a figure, (in the case of the fifth). Thus the chords ^ -^ ^^'- "-^^ :^ ^&r -<^- :p2: ^± ^- ^-^ may be thus expressed z:± -G^ 3 P «5 -S* — " The chord of the dominant seventh is always expressed by the figure 7, either alone or with any accidental natural or flat which it may require. If the fifth or third in the chord require it, accidentals 30 THE PRINCIPLES OF HARMONY. may be used for them as in the case of the common chord. Thus the chords -^ e=»3 fS>- &- -^ ^^m zd: -& j::z. would be figured as follows m -s^ T^ I C7 3 -- ^-^ = 3— r S rJ jfr^ ^— ^ g^- 1^ The second inversion has the fifth in the bass, and consequently this THE PRINCIPLES OF HARMONY. 33 note may either ascend to the thii'd of the tonic or descend to the tonic itself, according to circumstances, the fifth of the dominant having no comptdsory resohition. The third in this chord, being the seventh of the root, must of course descend to the third of the tonic ; while the sixth, which is the leading note, is compelled to rise to the octave of the tonic root. The tliird inversion has the dominant seventh itself in the bass, which is therefore compelled to fall to the third of the tonic. But the third of the tonic in the bass constitutes the first inversion of the common chord : therefore the third inversion of the chord of the dominant seventh must always be followed by the first inversion of the tonic common chord. The augmented fourth in this chord is the leading note, and goes of course to the octave of the tonic root. In all the inversions, the octave of the dominant root remains without motion, being converted into the fifth of the tonic. The next examples will shew all this more clearly. m '-^^ -&- 2i Z2: f- -- iq: XF'rj-^ g: Z2: "Cr ^ ^ 'IZ2L •^~ r T^ ^G)- 2i: ici: -G^ ^in^i b6 F 34 THE PRINCIPLES OF HARMONY. We will now give a longer example, including all the inversions of the common chord and the chord of the dominant seventh, with their various resolutions and coiTect thorough-bass figuring. H _c^ :d JOL. :g=^ fe -^- jCk- -G>- iS"- Z2: HS"- -s>- g -«s>- :e =P^ 1!S>- q: -Q_ -- £i ^z z::i lS>- gE^ S^ -^ :^ 22: -«s^ Z2: -.C2- -(S*- :s =P2: ^=^^^=0 -«s^ |S>- ?::^ 1^- :z:2 -s>- 1^2: -(S»- z± "C5" -«S>- i ^ -^ -J^2L «=^ -e>- -G^ S^3 -s^ -- _Q „ ^ ZZ -(S- if^ 3^ -!S»- -^s — - -&- The student is recommended to copy this out, and to analyse each chord carefully, putting the letter D over those chords that are dominant, and T over those that are tonic, and drawing Hues from all the discordant notes to those which resolve them, according to the foregoing rules. THE PRINCIPLES OF HARMONY. 35 6. In the chord of the dominant seventh, in its original position, neither the root nor the seventh can be omitted. The leading note may be, on an emergency, but it is always better to avoid this. The fifth may be omitted whenever it is convenient to do so, as it is not an essential note in the chord. In the first inversion of the dominant seventh the leading note is the bass, and of course cannot be omitted. The seventh of the root, which here becomes a diminished fifth, is the characteristic note, and therefore cannot be spared. The octave of the root, which here is a sixth, may occasionally be omitted ; but if this is done, the chord is converted into a dimmished triad, and becomes weak and ambiguous, as we shall see hereafter ; still, when necessary, it may be omitted. The fifth of the root, which here is a third, may be omitted at pleasure, when convenient. In the second inversion, the fifth of the root becomes the bass, and cannot be omitted ; the fundamental seventh, which is here the third, cannot be omitted, for the same reasons as in the former cases. The leading note, which is the sixth in this inversion, ought not to be left out, except when absolutely necessary. But the octave of the root not only may be omitted, but it is generally better that it should be, because of the harsh effect of the interval of a perfect fourth which it makes with the bass. In the third m version, the fundamental seventh being in the bass is essential, and must remain. The octave of the root should hardly ever be omitted. The leading note, which forms the augmented fourth, cannot well be dispensed with ; but the fifth of the root, which is here a sixth, may be omitted, whenever it is convenient. These rules about omissions will be found specially useful when the student begins to write music in less than four-part harmony. F 2 36 THE PRINCIPLES OF HARMONY. Examples of imperfect chords, with the various omissions allowed- -- zz -s>- -- -^- lor. 22: -s>- -- -- 321 -- -- S: -- i^z: -(S>- 22: -s>- 1221 05 -7 1 rzi ^--i n Xl .-^ <"-^ ^-^ r^ >5 C^ 1 1^^ '^-^ r^ 1 ^"i ^ r^ 1 c^ II y>\) ^-^ ^-^ '^-^ 1 r^ rj ^-. w J ^ ^ /'i^• • — - i'-:) ' rj r^ ift)- rj rj r^ 1 V-^ 1 r^ r^ 1 A 7. It is allowable, and indeed often necessary, to donhle a part, i. e. to let the same note be heard at once in two different octaves. Now it is obvious that if we thus double a note whose progression is compulsoiy, we must either break the compelling rule or fall into the great contrapuntal error of consecutive octaves. For instance, if we double the leading note "^ ^- _Q^2. S5^ iS ^^=g: .jQ_ -Gf- rj fj zz: 1^2: rjrj '^ZiL S: ^2: -- -o- -^ ^ ^ &e., &c. 1^21 -(S>- :s2: 3: ~rjr- r\ IZH -- 8. There is yet one case which remains to be noticed, which is when the chord of the dominant seventh is resolved by the second inversion of the tonic triad, instead of the original common chord of the tonic. This occurs only in the course of a piece of music, never at its close ; for the second inversion is no position of rest, but quite the contrary. Example- i The student's ear will at once tell him that something must follow this ere rest can be gained. 9. We must now speak of the chord of the added ninth and its inversions. This chord has five notes in it, and is therefore susceptible of four inversions. It is fovmd, however, in practice, that the fourth of these is seldom available. Still it will be as well to give the whole here, and then to state the cautions and limitations which are required. THE PRINCIPLES OF HARMONY. 39 The first inversion has the third in the bass ; thus I F The second inversion has the fifth in the bass ; thus m fS»- The third inversion has the seventh in the bass ; thus (S^ And the foiu"th inversion has the ninth in the bass ; thus 7=v (S> 1 © — F= 3 Now it will be perceived, on examining these chords, that all the intervals except the ninth (A) are precisely the same as in the inversions 40 THE PRINCIPLES OF HARMONY. of the dominant seventh. Consequently the resohition of these notes is the same as it is in the former case ; except where the new interval of the added ninth might cause consecutive fifths to occur. We have already shewn that such is the case in the original position of the chord of the added ninth : for if the fifth be resolved downwards to the tonic or its octave, it makes consecutive fifths with the ninth ; thus -& :r± Therefore in this case the fifth is of necessity resolved by ascending to the third of the tonic. And this holds good also in the inversions. The first inversion is thus resolved — ^=rjr:^'' -^ or else thus -^± -is> — ei>& in (where the D skips downwards to the G). The latter plan is, on the whole, preferable. In the second inversion, the fifth, being in the bass, cannot go to the tonic root, but is forced to go to the third of the tonic ; thus :z2: THE PRINCIPLES OF HARMONY. 41 In the third inversion of course the same resolution must take place, (i. e. to the first inversion of the tonic common chord,) because the domi- nant seventh is in the bass ; thus l(S>- P^ This is perhaps the pleasantest of these inversions, and the easiest to manipulate. The fourth inversion is crude and harsh, and should be avoided. In it the ninth is in the bass : consequently its natural resolution is into the second inversion of the tonic triad ; thus ^ T2: In this case, the ninth being below the fifth, no consecutive fifths are produced, and therefore the fifths may either ascend or descend. 10. The ninth may go into its resolution before the rest of the chord ; thus -Gh ^± r± e=n =s -- «=^ Ty :^± -o- i^z: -(S>- -^ -o- Z2: -«s^ T^ -^ G 42 THE PRINCIPLES OF HARMONY or thus — -f- S r3 n y ^ r^ rv n ((\ >^ r^ II 11 ; c^ c^ W tJ — 1^— iP^ -^— zp:_ _^^ — H V^ ' 1 II II This renders the fourth inversion rather less unwieldy, but still it is very harsh and awkward. 11. Probably this harshness results simply from the fact that the interval of the ninth is not susceptible of regular inversion, as it is beyond the limits of the octave. In the chord of the added ninth, the added note is essentially a ninth and not a second, seeing that it is originally added on above the seventh. Therefore it must always be kept at the distance of a ninth from the root, or the octave of the root, in order to preserve its essential character. But in the fourth inversion of the chord of the added ninth this feature is destroyed by the position of the ninth in the bass. Hence the harshness and awkwardness of this ugly inversion. 12. The thorough-bass figuring of the added ninth is §. The first 766 7 inversion is figured 6; the second 5; the third |; and the fourth §; ^32 2 together with any accidental flats or sharps which may be required. 13. The chord of the added nmth is so full of notes, and consequently of dissonances, that it is greatly improved by omissions and curtailments. These must now be considered in order. When this chord is uninverted, the best note to omit is the fifth. THE PRINCIPLES OF HARMONY. 43 both because it is not essential to the character of the chord, and also because by this omission all danger of consecutive fifths with the ninth is avoided ; and in the resolution the third of the tonic triad need not be doubled, as it otherwise must be. The leading note may also be omitted, as the seventh and ninth are sujBB.cient without it to give a distinctively dominant character to the chord. But neither the root, the seventh, nor the ninth can be omitted. In the first inversion it is always desirable to omit the octave of the root, as this note forms very harsh discords both with the seventh and ninth. The fifth may also be omitted freely, for the reasons given before. The leading note, being in the bass, is essential, and of coiu"se cannot be omitted. Neither the seventh nor the ninth of the root can be ever left out. Note. This rule may perhaps be occasionally relaxed in tlie case of tlie seventh, when it could not be introduced without contravening the rules of counterpoint. But such licence is not recommended. In the second inversion, the fifth, being in the bass, cannot be dis- pensed with. The octave of the root, however, is better away. The leading note, seventh, and ninth of the root cannot be omitted. Note. Here again some relaxation of the nile is sometimes necessitated in the case of the seventh. But the chord when thus weakened loses much of its dominant character. In the tliird inversion, which has the seventh in the bass, (and therefore to be retained as essential,) the octave of the root may be omitted, and so may the fifth, although the latter omission renders the chord somewhat bare. The leading note and the ninth can on no account be dispensed with. In the fourth inversion, which has the ninth in the bass, it is almost always necessary to leave out the octave of the root ; indeed the chord G 2 44 THE PRINCIPLES OF HARMONY. is hardly ever seen in its complete form, on account of its extreme harshness. The Jifth may also be omitted, and even the seventh and leading note, though these two last omissions almost divest the chord of its dominant character : whichever of these two is omitted, therefore, the other must always be retained. The following are the usual forms of this chord, with the omission of intervals, and with resolution and figuring : — The chord of the added ninth with the fifth omitted. All the resolutions in this case are compulsory. m ts>- T^ -^ B=z^ S- 2± The first inversion, with the octave of the root omitted. The fifth resolved by a skip, to avoid consecutive fifths. Note. The leading note is in French called "la note sensible;" and the first inversion of the added ninth with the octave of the root omitted is therefore called " la septieme de sen- sible." But inasmuch as this designation ignores the true derivation of the chord, it has not been adopted in this work. The first inversion, with the fifth of the root omitted. r:? -&- :a -^±. THE PRINCIPLES OF HARMONY. 45 The first inversion, with the octave and fifth of the root both omitted. -Gi- Bl -s^ zz The first inversion, with the dominant seventh omit- ted. This should generally be avoided. fefe HS>- -L 1^2: The second inversion, with the octave of the root omitted. I m -s>- -fS>- ~X2. The second inversion, with the octave and seventh of the root both omitted. This, however, is not recommended. 16 THE PRINCIPLES OF HARMONY. :^==^ The third inversion, with the octave to the root omitted. ^— f^^ |S>- T^- ^ '^- The third inversion, with the fifth of the root omitted. -Q- :b ^~- -P rj The third inversion, with the octave and fifth of the root both omitted. da ?^ gd -^^ The fourth inversion, with the octave of the root omitted. Even thus it is too harsh to be used, ^ ^^ ' except in very rare cases. '-^'- THE PRINCIPLES OF HARMONY. 47 -Gh ZZ S Z2: The fourtli inversion, with the octave and fifth of the root both omitted. This is just tolerable, and may be occasionally used, with caution. We can also in every case allow the ninth to fall to its resolution before the other notes of the chord. This very much facilitates the use of the fourth inversion. Examples — S J J J ,1 J j ?2 ^Bgp ^ J^ u ^ A ^ ^- T2L T^ -4^ -<2. is>- (S>- -^ ^- 9 8 7 - 7 6 5 - I '- 6-6 5 4 6 - 5 4 fc^iLd: J=^ _C^_ (S»- ^^ f^ s is>- &c., &c. PSEE^ '¥=^ ^ :q -^- -1^- ^. 6 7 6 7 3 14. Some of the above chords have a strong resemblance, on j^9aj9er, though not when sounded, to the dominant seventh and its inversions. The thorough-bass figuring is also very often similar or identical. It 48 THE PRINCIPLES OF HARMONY. will be as well, therefore, to shew how to distingLiish the chords belonging to one series from those which belong to the other. I which is the first inversion of the chord of the added # _C2_ ninth, with the octave of its root, G, omitted, might at first sight be mistaken for a chord of the dominant seventh on the root B. But the chord of the dominant seventh on B would require D and F to be sharp, not natural, as here, and would be written thus -^2- f For in the chord of the dominant seventh the tliird, or leading note, is always major, and the fifth perfect : whereas in the first inversion of the chord of the added ninth, the third is minor and the fifth diminished. Again, which is the second inversion of the chord of the added ninth with the octave of its root, G, omitted, might at first THE PEINCIPLES OF HARMONY. 49 sight be mistaken for the first inversion of a dominant seventh on the dominant root B flat. But such a chord would require both the B and the A to be flattened, and not natural, as here ; thus * ^-- ^^S For in the first inversion of the chord of the dominant seventh, the sixth is minor and the fifth diminished : whereas in the second inversion of the chord of the added ninth, the sixth is major and the fifth perfect. Again, which is the third inversion of the chord of (S» — the added ninth, with its root, G, omitted, might at first sight be mis- taken for the second inversion of a chord of the dominant seventh on the root B flat. But such a chord would require both the B and the A to be flattened, and not natural, as here ; thus m -o- ^l 50 THE PRINCIPLES OF HARMONY. For in the second inversion of the chord of the dominant seventh, the fourth is perfect and the third minor : whereas in the third inversion of the chord of the added ninth, the fourth is augmented and the third major. Again, which is the fourth inversion of the chord of m the added ninth, with its root, G, omitted, might at first sight be mis- taken for the third inversion of a chord of the dominant seventh on the root B natural. But such a chord would require both the D and the F to be sharp, and not natural, as here ; thus m ife --^ For in the third inversion of the chord of the dominant seventh, the sixth is major and the fourth augmented : whereas in the fourth inver- sion of the chord of the added ninth, the sixth is minor and the fourth perfect. Before going any further, we would strongly recommend the student to transpose all the examples m this Chapter into several other keys. In no other way can he impress all the intervals, chords, and resolutions THE PRINCIPLES OF HARMONY. 51 so firmly on his memory. We also subjoin a longer example, wliich he is advised first to analyse and then to transpose — zt £Eg :2^ S :^=£ - -j:^ ^ ^ f^^9- r^^ ^ =2=25 l6 b5 i-^t 23 :i^3 22 t22: rJ rJ J^- -(S^- r±:^E=^ 3 S ^21 fci T2L 21: ^±=^ - 'i -o ^^m -oi ^ zi: 22 22- 3: 47 — r2- 22: :^ 22t -fS'- f 22: :^ to: 22: 22_ -(S>- jdzz M=^ ^ 5^ :^ 3^ S I o - rJ r J i -O- -^- H 2 -fS"- Z2] S=° -- ^ -^ is: -- 122: 22: :zz: Z2: zz: 1^21 :^z: Z2: Nay, more, the seventh may skip sometimes to the fifth on the same bass, instead of being resolved regularly; thus — THE PRINCIPLES OF HARMONY. 53 fl J^^ / d ^^ ^ r^o* (f3 ^ t/ rm\' <"^ l^- v_^ c^ though of course this is regarded as a licence. Another relaxation of the rule is permitted sometimes when another note on the same bass intervenes between a dissonant interval arid its resolution; thus -t^- J- S ^ -C2- z:^: j^^ s f^- - -o- r- iq: -- — Gh- isz 1^21 -eS*- -(^ irz: Z2: :z2: -K^ 15. In thorough-bass figuring it is usual, when a note is resolved on the same bass, to figure the intei^al to which it passes, even if it be an octave, a fifth, or a third ; thus fctd: :q: or #£ -Gh- ^=d or m^ -Gt- or @5 icz: -f^ r^ -D / p* ((^ V y r^ ' 1 r»xiL ' 1^- ft 1 >~^ 1 1 And when the other parts do not move, the fact is usually indicated by horizontal lines drawn from them ; thus — 54 THE PRINCIPLES OF HARMONY. ii^ S -o- ■^ m r -- i^z: and the same lines are drawn when the bass moves, while the other parts stand still ; thus i w- -&- ::?:5: 'C> m^. 0| 7 ^ ^ -- which last example might also have been figured thus m^ m 3 -Gh- We will now give a figured bass and treble, and the student is to fill in two inner parts, according to the figiu-es, carefully avoiding conse- cutive fifths and octaves, and scrupulously resolving every dissonance according to the preceding rules : ^ ^=ii r J ^ ' 'ZJl. '&- -&- @^ la: 22: ± -o^ -^ — ^- 22: -&- -o- Z2: ■is>- zz: i THE PRINCIPLES OF HARMONY. 55 *i -s>- :izL ^2: -^± Z2 -- ^&- :^- 9J I — f lT- 2,n^ 1 l> -« r-H e3 fl f3 o n ^ (M o o o Oi 10 o o CO CO 11 CO ^ 6X) 12 > CO 13 CO 14 CO '^ tH CO o 15 CO a a 16 17 CI 05 18 > > cS n -t? -;j o C) o o ,£3 ,d -4-3 "^ •^ d d 05 19 > d CO o d 20 -(J rd -4-3 -^ d This paradigm of harmonics has been placed at the head of this Chapter, because frequent reference will have to be made to it. The THE PRINCIPLES OF HARMONY. 57 student is advised to study and copy it, as upon it all our superstructure will be built. 2. '" In the first place, it will be remarked that in this series of har- monics the distances or intervals between the adjacent notes become progressively less as we proceed upwards. Thus : the interval between the generator and the first harmonic is an octave — the next interval is a perfect fifth above that — the next is a j)erfect fourth — the next is a major third— the next a minor thkd — and so on. Now, rejectmg those sounds which are out of the key, B flat and F sharp, let us take the three notes which come between them, C, D, and E, and we find that they form a real diatonic progression (i. e. the fii'st three notes of the diatonic scale of C major). Let us then take these three notes as a melody to be harmonized. For this purpose, the fii'st thing will be to discover the fundamental basses. Now, seemg that C and E are harmonics of the root C (as above given), and that D is not only so, but is also the second harmonic sound produced by the root G, as we have seen in Chapter II ; let us take these sounds, C, G, C, as oiu- basses, and we shall at once effect oiu* object. 8 5 3 ]/ ^ — 1 rv A r^ rrn "W J ^' Ov^ 1 ^'^-^ V^ r^ 1 n . * This section is entirely derived from Logier's admirable " System of the Science of Music ;" London, 1827, pp. 48, et seqq. 58 THE PRINCIPLES OF HARMONY. The fundamental basses being thus discovered, we can now proceed to add harmonies, according to the rules laid down in this book. Our result is as follows : — r-f ^ ^ -1 > rj s^^ rv ff\ ^*^ ^ J "r^ "^ ^^ ' t/ /S^- ^^ 1 (€>■ v_^ rj r^ Here, then, we get a harmonized scale of three notes. How are we to get beyond these ? Let us turn to the general paradigm at the head of this Chapter, and we shall find that the harmonic corresponding to no. 7 (or ^ of that string or tube whose whole length would give the generator C) is B flat. This note is not a part of the scale of C, but it is its fundamental minor seventh ; and when it is added to the triad of C, the immediate result is that C ceases to be a tonic root and becomes a dominant. Then the ear is immediately seized with an irresistible craving after a tonic resolution. To satisfy this, the seventh must descend and the third must rise, as we have already explained in Chapter II, and thus we find om'selves landed in the key of F. This process is called a moclidatio7i, and will be further explained in a future Chapter. We have by this means added one more note to our original scale of three ; and that note is F natural. Taking F, then, for our new tonic generator, and treating it exactly as we treated C, it mil supply three notes, F, G, and A, corresponding THE PRINCIPLES OF HARMONY. to the C, T>, and E which we took for our original scale of three notes. Combining these two series, and connecting them harmonically by means of the dominant seventh, B flat, we shall produce a scale of six notes, or a hexachord, as follows : — :c2: :S2: &S=; -O- S: S: :g-- Z2_ hi Z2: :z2: 1^21 122: "O- -^5' Our scale of melody, taken alone, belongs to the key of C, but our harmonies belong partly to the key of C and partly to that of F. If we wish to add three more notes according to this system, and by a similar process, we shall modulate into the key of B flat, of which F is the dominant. And in the same way, by pursuing a similar course, we should next modulate into the key of E flat, of which B flat is the dominant. To this process there is no limit. z^ — - ^ 2Z 'ZJ- S: -(S*- &c. SES ^Q- -^- -Gf- ^Q L£^ ^ : ^&- 221 -Gh ^&- -- "C5- b7 hi bl 1)7 &c. itM h ^ CDW 2 Generator 1 I 2 60 THE PRINCIPLES OF HARMONY. From these premises Logier * draws the conclusion that " no scale can naturally consist of more than three sounds, for which there are only two fundamental basses required, viz. the tonic and dominant." He also shews that the " subdominant" (or the fifth below the tonic, or the fourth of the scale ascending) is in reality no true root of any part of a natural scale, but only the generator, or tonic, of a new one. From all wliich he sums up the following general rule : — " Whenever we use a [fundamental] seventh, and thus proceed to a new key, we modulate into that key." 3. From the discoveries in the preceding section, it has been seen how to form a compound scale which shall modulate continually and progressively into new keys : but it has not yet been shewn how to derive from nature a true diatonic scale which shall begin and end in the same key. As far as the sixth note of the scale, we have to a certain extent succeeded in our endeavour to form and harmonize the scale ; but there we broke away altogether, and hopelessly. Having wandered into the key of the subdominant, our only way of returning must be by a modulation into the original tonic again. But, by the above rule of Logier's, this is to be effected by the inter- position of the chord of the dominant seventh. * "System of the Science of Music," pp. 50, 51. THE PRINCIPLES OF HARMONY. 61 It will be observed here, that at the sixth note of this scale the E flat (or dominant seventh leading to the key of B flat) has not been introduced. The sixth chord may then be considered as a tonic chord, and at rest. But our object now is not to rest there, but to complete the scale of C in our melody, and at the same time to modulate back from the key of F into that of C by means of the dominant harmony on G. The minor seventh of G, being F, here supplies a combining note for the harmony, (indicated in the above example by a bind,) otherwise these two chords would be totally disjoined. It will also be remarked that the introduction of the leading note, B natural, in the seventh place of the scale, causes the semitones to faU between the 3rd and 4th, and 7th and 8th, according to the rule of all diatonic major scales. By this forcible introduction of the dominant harmony of G the ascending major scale of C has been harmonized. But still there remains an awkward harmonic disconnection between the 6th and 7th of the scale, which can never be entirely got over. The fact seems to be, that the leading note has so strong a tendency upwards to the octave of the tonic that it cannot bear reference to any lower interval. Its appropriate place would rather appear to be helow the key-note from which the scale is started. ^ -^ -^ — < " ^ as then every chord by which the scale is harmonized would be naturally and strongly connected with the next. If such be a true view, then the leading note would be simply a preparatory note, introducing a scale of the compass of a hexachord — ascending. This view of the natural formation of a harmonized ascending scale will be seen to be of some value when we come to consider the minor scale. But that must be i-eserved for the present. 62 THE PRINCIPLES OF HARMONY. 4. By the use of the inversions of the common chord, and of the chord of the dominant seventh, great variety can be introduced into the harmonic accompaniment of the ascending major scale. It will be a very useful exercise for the student to take the scale of and harmonize it according to the following varied basses and figures. And after that he can still further improve himself by trans- posing his work into other keys. No. 1. 3 :c2: -B -.r± i^: 22: iq: Z2: -& -^^ IQ -^- 'r^ r ^ e :^ C2: :^ 1^- :^2: :^ 4. /mr \ ^^ f-^ ^^ II l^- h-' fZi rj h^ v»^ _rJ _ _ . L^_ \ 1 ^ rj II ' L_ L_ '. \ L l6 b5 5. (S>- ^- ^2: ^ t:2i f^- T2-- -iSh- '.ZJl 5. The descending diatonic scale now comes for consideration. Here, of course, the leading note will not help us, as it rises but THE PRINCIPLES OF HARMONY. 63 does not fall. By treating the seventh of the scale as simply part of the common chord of the tonic G, we can get over this difficulty ; thus ^ IQ -Gi -^- 22: :z2 The next stage in our proceedings will be to consider ourselves in the key of G, and to regard the three notes B, A, and G as a natural scale in that key, just as we did with respect to the scale of three notes ascending. Our result will be as follows : — -jr^- -^^ :5^ -^- -«s>- -s>- - -& zc? 22: -s^ B: S -s>- G :^2: -^ -(S*- -@- Generators I C (The lines drawn from note to note indicate the resolutions of discords.) (The dotted lines indicate the crossing of the parts to avoid consecutive octaves.) THE PRINCIPLES OF HARMONY. 65 This last method answers best when inversions are used ; thus — :q; :?2: '7:j- :q: -ZJlL Sf Z2: '^U" -Ci: 1^:2: -Gt- -e^ :r2: -G>- -- V2: s ^- There are many other ways of harmonizing both the ascending and descending diatonic scale, but they involve certain rules of harmony which have not yet been reached in this Treatise, and are therefore for the present postponed. K CHAPTEE Y. 1. We must now consider the origin of what is called the "minor mode." {Scarcely any subject in the whole science of music has so much perplexed theorists. The explanation here offered is therefore put forward with some diffidence, although the author is himself convinced of its coiTcctness. The necessity for such an explanation arises from the conviction that " nothing which is agreeable to the ear can be contrary to nature : " but " the minor mode is agreeable to the ear ;" therefore " the minor mode is not contrary to nature" — L e. it is derived from natural phaenomena, and, consequently, can be explained by deductions from those phsenomena. In order to render such explanation intelligible, it is necessary here to make a digi'ession, for which, it is hoped, no further apology is needed. 2. Referring to the paradigm of harmonics at the head of Chap. IV, and having regard to those which cori'espond to prime numbers, i.e. 1, 2, 3, 5, 7, 11, 13, 17, 19, it will appear that their degree of ferfection gradually decreases as they ascend in pitch, at least so far as No. 13. To explain this by examples. The octave, or the ratio 1:2, is absolutely perfect. The fifth above this, which is at the distance of a twelfth from the root, or the ratio 1:3, is also generally called perfect. And so it is in a certain sense. But it has one imperfection, viz. that if we continually ascend by intervals of a fifth, we shall never exactly reach an upper octave of the root. For the cycle of twelve fifths, C, G, D, A, E, B, FJ, C#, G#, DJ, A#, EJJ, and B#, is not the same in its results as seven octaves of C : in other words, after i-educing the two results to the compass of one octave, it is found that B# is sharper than the octave of the root, giving (instead of the ratio 1 : 2) an interval which is represented by 2^® : 3^^ THE PRINCIPLES OF HARMONY. 67 Now, the true octave is iu the ratio 1 . 2, while this interval is 2^* : 3^^, or 2G2144 : 531441. The difference, then, between them will be found by comparing these ratios : which is done by multiplying the 1 : 2 by 262144 ; i. e. 1 : 2 = 262144 : 524288 : whence it appears that the interval arrived at by ascending in fifths exceeds the true octave by the fraction 5^4088 ' ^ residuary interval, which is called the " Pythagorean comma." For this reason all the fifths, in tuning keyed instruments, are tuned a little flatter than perfection, by one twelfth of the Pythagorean comma, an interval so minute that the ear cannot detect it. The next interval corresponding to a prime number is the major third, which occurs in the second octave, and thus bears the ratio 1 : 5 to the root. This interval is much more imperfect than the fifth, and, in tuning a keyed instniment, will require much gi-eater alteration, or tempering (as it is called). For the cycle of three major thirds, C, E, Gtf, and BJJ, falls short of the true octave ; giving the ratio 4^ : 5^, instead of the ratio of the octave, 1 : 2, which exceeds it by the comma yff , or " enhar- monic diesis." This imperfection is obviously much more appreciable than that of the fifth, especially when it is remembered that in " tempering" it has to be equally distributed among no more than three major thirds. The next prime number in the series (no. 7) gives the interval of the fundamental minor seventh. But, as has been already observed, it gives it very much too flat. So much so, that most theorists have demurred to accepting it as the origin of dominant discords and of modulation. Rameau *, who, I believe, first noticed it among the acute harmonics of a vibrating sti'ing, called it " le son perdu," the lost sound, and passed it over as insignificant. But, even in the last century, there were some writers who to a certain extent recognized its place in the harmonic system f. It is admitted by Chladui | to be midway between consonances and dissonances, although he deprecates its use. He, however, uses these remarkable words : " On pent cependant presumer que la cause, pour laquelle I'accord le la septiSme, (ut, mi, sol, si ^), et celui de la sixte supei-flue, (ut mi> sol, laj), ne sont pas aussi desagreables ^ I'oreille, que Ton pourrait le croire, d'api-es leurs * " G^n^ration Hannonique;" Paris, 1737, (vide Chap, i.) + Vide Pizzati, " La Scienza de' Suoni e dell' Armonia;" Venice, 1782, (Part. IV, Chap, ix.) Also Tartini, "Trattato di Musica j" Padua, 1754, (p. 126.) X "Traits d' Acoustique ;" Paris, 1809, (p. 28.) K 2 68 THE PRINCIPLES OF HARMONY. nombres compliqu6s, tient k ce que I'oreille substitue h. ces nombres les rapports 4:5:6:7, dans lesquels Fintervalle ^ differe de la septieme -- du comma g|, et de la sixte supei-flue du comma encore plus petit jff •" But surely, as we admit of the temperament of fifths aud thirds, without denying the genuineness of those found in nature, so we may regard the ordinaiy minor seventh as a tempered modification of the fimdamental seventh found among the harmonic sounds of nature. Similarly, we may regard the harmonics corresponding to the prime numbers 11 and 13 in our paradigm as repi'esenting the augmented fourth and the minor sixth in the third octave : — although the former is much too flat, being nearly midway between the perfect and augmented fourth ; and the latter is much too sharp, being also intermediate between the major aud minor sixth. The next prime number, 1 7, however, gives us the interval of a minor ninth, very nearly in tune, being only a very little too flat, and almost identical with that intei-val on an equally tempered instrument*. This number, then, we may at once admit among our fundamental discords, and use as freely as the major ninth or the minor seventh. The next prime number is one of the utmost importance (No. 19), as it gives us the minor third of nature, and may be regarded as the source of the whole system of the minor mode. It is almost in tune. Indeed it is more satisfactory to the ear than the minor third as usually represented by the ratio 5 .; 6. Reducing it to the first octave, it is represented by the ratio 16 : 19, and only falls short of the usual ratio 5 : 6 by the very small comma gV. Regarding the major triad as 16 : 20 : 24, and the minor triad, as here derived, as 16 : 19 : 24, nothing can be more simple and natural than their relations one to the other. Even Chladni admits this t, where he says " Peutetre quand on se sert quelquefois de I'accord parfait mineur au lieu du majeur, ou du majeur an lieu du mineur, I'oreille est raoins hlessee{'.) parce qu'elle substitue 'k la tierce mineur | I'intervalle yf, en entendant une variety de rapports, comme 16 : 19 : 24, et 16 : 20 : 24." Let us, then, assume the fundamental minor third of nature to be |-|, or that produced by taking |g- of the length of a string or tube. It will only remain to shew why the usual figures, 5 : 6, cannot be taken for this pur- * This derivation of the minor ninth is duly recognized by Catel, in his well-known " Traits d' Har- monie;" Paris, 1802, (p. 6.) t Chladni, " Traits d' Acoustique ;" (p. 29.) THE PRINCIPLES OF HARMONY. 69 pose — in other words, why the minor triad is not correctly represented by the ratios 10 : 12 : 15. 3. Referring once more to our general paradigm at the head of Chap. IV, and seeing which are the notes represented therein by the ratios 10, 12, and 15, we find them to be E, G, and B, or the minor triad of E. But the root of all the notes in this series is C, not E. And C cannot be the root of the minor triad of E. Therefore the numbers 10, 12, and 15 do not correctly produce a genuine minor triad. Therefore the ratio 10 : 12 (or, which is the same, 5 : 6), does not represent a real fundamental minor third. But it does really give the interval between the thii-d and fifth of the major triad, as the paradigm will shew. There are, then, in nature two different minor thirds, only one of which is fundamental ; or, as it may be expressed, one of them, ^, is the root with its minor third, the other, |, is the third and fifth with the root omitted. 4. It may, then, from these arguments, be fau-ly assumed that the mmor triad has its origin in nature, analogously to the major, and that it is part of the tonic series. The dominant must always have its third or leading note major ; otherwise it would be too far from the note to which it leads, and from which it is necessarily a semitone distant. 5. On referring to Chapter I, it will be seen that every major key has a minor key connected with it, called its " relative minor." This connection can hardly be said to be of natural origin, inasmuch as the harmonics of the root of the major key do not give us the common chord of its relative minor. But as several of the intervals of their scales are common to both, and as their " signature" (or the flats or shai'ps in the stave) is the same, and as, moreover, it is very easy and very usual to go either by modulation or by harmonic progression, (which term shall be explained hereafter,) from one to the other ; it will be most 70 THE PRINCIPLES OF HARMONY convenient on the whole to ilhistrate the derivation and harmonization of the minor mode by starting from the key-note A, and taking as our model the process adopted for the scale of C major in the preceding Chapter. 6. For this purpose we will shew the scale of three notes in the major and minor modes, side by side — Major. Minor. a 122: -- -- -o- -o- 22 f^- s "^z: 221 Z2: whereas the harmony of the second note of the minor scale, as written here, is not, and cannot be, dominant, inasmuch as it contains no leading note, the third being minor. It is simply the minor triad of E, and has no connection with the chords with which it is here associated. THE PRINCIPLES OF HARMONY. 71 To correct this, it is merely necessary to put a sharp before the G, when it immediately becomes a leading note : — iq: ^- ^^ hS»- xz: 1^2: icz: to which we may add the seventh if we please — .- :g^ ^& ^ 1^21 'JZZ. :_ tizi^z s In forming the harmonized major scale, the fii'st three notes were followed by three others, similar, and similarly harmonized, in the key of the subdominant. Pursuing the same course, as far as possible, with the minor scale, the result will be as follows — 22: --g-- s s "^IT" -/S>- ?^s m -jr:2i :z2: 221 hS- -^- 221 i i This will do very well. But it differs from the scale in the major, inasmuch as there is no modulation into the key of D betw^een the third and fourth note. This is inevitable, because the tliird note of the scale 72 THE PRINCIPLES OF HARMONY. is necessarily a minor third to the root A, and therefore cannot be used as part of a dommant chord. Still, there is sufficient connection to satisfy the ear, and a real modulation does take place at the next note, where the leading note, G% is introduced. As we have thus got out of the key of A into that of D minor, we are obhged to modulate back again to our original key : and tliis obliges us to introduce the leading note, Gj$, into the melody as the seventh note of the scale. The whole will then stand as follows — # -«s>- -o- 22: -s>- -Gh- -- 1^2: Z2: } It will not escape notice, that between the sixth and seventh notes of the minor scale, as here shewn, there is the interval of an augmented second. This constitutes the chief characteristic of a regular minor scale. 7. In Chapter IV, towards the end of section 3, it is shewn how awkwardly the sixth and seventh notes of the diatonic scale hang together. This is even more apparent in the minor scale, in consequence of the introduction of the very dissonant interval of the aug- mented second between those notes. If the leading note be, as it were, prefixed to the minor hexachord (or scale of six notes), all this awkwardness vanishes at once ; thus — :i^=^:^=^: P: :£ ^2: :e Every chord connects itself with those next it, without difficulty, and without harshness. There is therefore no small i-eason to surmise that this is the most natural form of the scale. THE PKINCIPLES OF HARMONY. 73 8. Before proceeding to harmonize the descending minor scale, it will be necessary to introduce a new dominant interval, or fundamental dis- cord — the minor ninth. This is one of the most important elements of modulation, and affords a greater variety of resources than any other known combination of notes to the experienced harmonist. It will be well, therefore, to devote a Chapter to it alone. CHAPTER VI. 1. It will be seen on reference to the paradigm prefixed to Chapter lY, that the 17th harmonic gives us the minor ninth in the fourth octave from the root. It is therefore a natural harmonic. But it is clearly no part of the tonic harmony, for it is out of the key. Nor is it in the hey of the dominant ; for the minor ninth, for example, of the dominant G is A flat, which neither belongs to the scale of G nor to that of C major. But it does belong to the scale of C minor, although it is a harmonic of the dominant G. It therefore is a very important addition to the dominant harmony, because it only belongs to it as such, and by no means belongs to the same root considered as a tonic, being necessarily and essentially domi- nant, and leading downwards to the fifth of the tonic quite as irresistibly as the leading note leads upwards to the octave of the tonic, or the dominant seventh downwards to its third. When the tonic is of the minor mode, the force of the dominant seventh is weakened, because, instead of falhng a semitone to its reso- lution, it has to fall a whole tone, since the tonic third is minor. Compare the two — THE PRINCIPLES OF HARMONY. -9 1^— 75 sa 'Gh- SEE semitone r whole tone :^2 But the minor ninth at once makes up for this defect, by its own strong dominant tendency to the fifth of the tonic. i -f^- -is- ^i ^ isi: The two together form as strong a dominant chord as can be required. And the additional indication of the minor mode which this mterval gives, renders it of especial value when we wish to introduce and fix that mode. The major ninth is not suitable for the minor mode, on account of its incongruity with the sixth note of the minor scale, which is mmor, as we have seen, and with that same note considered as the third of the subdominant triad, which, as we have also seen, is minor also. Therefore the minor ninth must be substituted for the major in the minor mode. Its resolutions, inversions, and figuring are analogous to those of the chord of the added major ninth, only that the accidental flats and sharps involved will of course be different ; and in tliis difference there is also this advantage, i. e. that there is no danger of mistaking the inversions L 2 76 THE PRINCIPLES OF HARMONY. or figuring of the chord of the minor nmth for those of the dominant seventh ; for in the former the accidentals are always combined of a con- trary Tcind'^f which is never the case in the latter. A few examples wiU shew all these points at one view. m ^ a -&- 22: 221 The chord of the minor ninth uninverted, properly- resolved and figured. J The same, with the ninth resolved to the octave on the same bass, before the resolution of the '^ ' other parts. The next two examples give the ninth in an inner part, and below the leading note. This is always allowable with the minor ninth, and gives it a great advantage over the major, — where such a position is not recommended. -(^3- :^ s r- - ' -- -SH 'jr^L. I * i. e. shai'ps against flats ; thus ^- is easily to be distinguished fi-om THE PRINCIPLES OF HARMONY. 77 S= ^ i -- 1221 When the ninth is resolved in this way, the hidden consecutives between the extreme parts may be tolerated (as in bar 2), but it is very much better as in bar 1. 78 THE PRINCIPLES OF HARMONY. ^ s '^ Q '^ o I This method of resokition takes away all difficulty about the descent of the bass. It may equally apply to the first inversion if required. -- 32: & -o- The third inversion, regularly resolved. -s>- CJ 1^ s S: -^ -O- i^z: The same, with the ninth resolved before the rest. 6-6 f" 2 iJ 1^2^21 -«S- -- q: ^ I Two ways are here shewn of improving the resolution of this ugly chord. The former is not very satisfactory, but the latter is good. In tliis inversion, the ninth being below the fifth, no consecutive fifths can occur. 2. Hitherto we have been speaking of the chord of the minor ninth and its inversions, without omitting any interval. Like the chord of the added ninth, however, (see Chap. III. 13,) this chord is all the better for such curtailment ; and the general rules for the omission of its various intervals in each inversion are similar to those given in the case of the chord of the added ninth. But when the octave of the root is omitted from the chord of the minor ninth, a very remarkable and important series of chords is dis- covered, which demand special consideration. 80 THE PRINCIPLES OF HARMONY. 3. The first inversion of the chord of the minor ninth, omitting the octave of the root, is known as the " chord of the diminished seventh," and is so named after its characteristic interval — It is susceptible of three inversions, all of them being of the greatest use m harmonizing — =^ ■o- m. T 3 -^ -^ ^l 6 f Now on examining the chord of the diminished seventh, it will be seen that it consists entirely of minor thirds superposed — GltfB, BD, and DF. And on examming the inversions, they will be found to consist in each case of two minor thirds and one augmented second ; thus — BD, DF, and FGJI; DF, FGjf, and GflB; and FGJf, GJfB, and BD. On all keyed instruments, the same key stands for Gil and Ab, And although they are not the same note really^ yet they are so nearly the same that one may be always substituted for another with impunity. Indeed, the construction of our instruments, the method of tuning now universally prevalent, and the requirements of free modulation, render such interchanges imperatively necessary. Note. The interval which actually exists between two such notes is named the " Enhar- monic Diesis," and is represented by the fraction i|4. Sec short Examples at the end of this work, Nos. 6, 7, and 12. THE PRINCIPLES OF HARMONY. Any note, then, may be changed into that which is nearest it in pitch, and which is represented by the same key on all keyed instru- ments, e.g. Gjl into Ab, Aj$ into Bb, Bfl into CQ, CJ|: into Db, DJ into Eb, M into FQ, F^ into Gb ; and conversely, Gb into FJf, Fb into EQ, Eb into m, Bb into C^, Cb into Bt3, Bb into Aj?, Ab into Gjf. This sort of interchange is called an "Enharmonic Change!' The ear is, as it were, deceived by it, just as it is by the tempering applied to the fifths, thirds, and sevenths, as described in Chapter V. Whence it follows that the chord of the diminished seventh and all its inversions may be alike regarded as in fact composed of three minor thirds ; thus — 1st minor third < Gft F or Ab F B or Ab Cb or B Ab or GJf D or Ebb B or Cb 2nd minor third \ ^ B F D F D Ab F Ab or GJj: F or E# B or cb Gj$ or Ab 3rd minor third 4 B D B F D F or EJf D or Cx GJf or Ab Eif or F The minor thirds may also be reckoned the other way (i) F to Gft or Ab; (2) G$f or Ab to B or cb ; (3) B or cb to D or Ebb; &c. For every one of the intervals composing this chord may be enharmoni- cally changed, as is here shewn. M 82 THE PRINCIPLES OF HARMONY. Hence it may be seen how very useful this chord is in modulating from key to key. For example — d= B F3- ^. -^^ t m -jtX -- it -^ ~C7 -"^^ 4p: (S»- :^^ =^==g^ -^ -x± Roots A E A A D A Keys D t X. fa^ ^2: :<:2: -is>- 4S?- -^ -s*- 51 -^ -^' KS'- Bb Eb #= J2 p ^ ^^nzqp^Jgz::: ^21 ipzizzp^ :|^ ^ Roots Eb BQ EQ B KeysEb Ep E E A B E E A ; E In this example — Bar i. contains the minor triad of A, and the first inversion of the dominant triad of E, with the seventh omitted. Bar ii. contains the minor triad of A ; and the third inversion of the chord of the minor ninth of A without the octave of the root : being therefore the second inversion of the chord of the dimi- nished, seventh, leading to the next chord. THE PRINCIPLES OF HARMONY. 83 Bar iii. contains the first inversion of the minor triad of D ; and the first inversion of the chord of the minor ninth of A, without the octave of the root — being the chord of the diminished seventh. Bar iv. contains an enharmonic alteration of the last chord, whereby it is converted into the fom'th inversion of the minor ninth of C, without the octave of the root — being therefore the third inversion of the chord of the diminished seventh. In the last half of the bar, the bass resolves the minor ninth by descenduig to the root, and the rest of the chord remains as a chord of the dominant seventh on C. The student ^vill now be able to carry on this process of analysis, by the aid of the rules already given. He is begged, however, to take especial notice of the enharmonic change in bar vi., which he vd]\ have to examine very carefully. He will do well, also, to observe the roots and keys, as indicated in capital letters below the bass stave. 4. In bar x. of the preceding example, the common chord of E major is marked thus f . This has been done to draw attention to the fact that here the chord of the minor ninth has been followed by a major triad. This licence is always allowable; although the character of tliis chord is more essentially minor than major, as has been sufficiently shewn. By means of this licence, an even more extended usefidness is gained, and the composer is enabled by it to modulate stiU more freely. 5. It is then possible from one chord of the diminished seventh to modulate directly into a great variety of keys : for instance, taking M 2 84 THE PRINCIPLES OF HARMONY. our start from A minor, the following resolutions of the diminished F seventh ^ will at once appear : — ^^ 3 ~i:jl s 22 ^=1 - -^ «E :tt?2: &=J|^z±f^=^: © A major C minor C major -A == 1 I J V \ \ - m r~'^r=^ o- «p= # -^<- Fjl minor FJf major D jf major i^i=fe w5^^r$^S^g=#=^ fl^ J-^S*- f - fezzz^ - itcz ^ "g^- 5!2: D Jj! minor Eb mi mmor Eb major ^ ^^^=|i^WE^i^ 3 12^21 :^g: ^^ z:?" izs: B^: — b ^ [221 Gb major Gb minor THE PRINCIPLES OF HARMONY. 85 At f , bars viii. and ix., the equivalent notes have been omitted, and the enharmonic change taken for granted, This is the usual way of writing such modulations, for the sake of simplicity. We may here quote a very good example from Catel (Traite d'Harmonie), which clearly exhibits the enharmonical resources of the chord of the diminished seventh — >Q_ 11. ^^ 111. 4f^ 4tQ_ 4fQ. ife: n:2L ^m- \&- -G>- -Gh -G»- m^ Z2: '-^- =S^ g; ivv - -^- -«s>- -- -f^^- : 3^2: b5 Roots >Q_ b5 6 t b7 5 «5 >E=i 22: :(=z :^^; *=: \' 1 1 C9 85 In this example the diminished seventh >h and all its inversions are successively presented in bars ii. — vi., and all the various roots given from which they are severally derived. Yet, if played on a keyed instrument, the chords in these five bars remain unchanged, as will be at once perceived on playing them. 6. Before proceeding any further, it will be well to give a rule by which to discover the root of any fundamental harmony. 86 THE PRINCIPLES OF HARMONY. For this purpose it will be necessary to refer to the general table of the keys with their signatures, given in Chapter I, sect. 5. Now, it will be found a convenient plan, to class the key-notes according to their signatiu-es, calhng that key the sharpest key which has the greatest number of sharps ; and, generally, in comparing two or more keys together, calling that the sharpest which has the most sharps or the fewest flats in its signature. And we may apply the same mode of speaking to the ludiviclual notes themselves : for instance, E Q may be called a sharper note than Dp, because the signature of that key of which it is the tonic has two more sharps than belong to the key of D. Similarly, F may be called a sharper note than B b ; and, generally, we may call the notes in the subjoined Hst sharper or less sharp accord- ing to the order in which they stand; those to the left hand being sharper than those to the right. Thus, in the list of notes — Ctt, FjJ, B, E, A, D, G, C, F, Bb, Eb, Ab, Db, Gb, Cb ; CJt is the sharpest of all notes; then comes F^, and so on, decreasing in sharpness till we reach Cb, which is the flattest of all. This is simply a convenient mode of naming the various notes by way of comparison, the special advantage of wliich we shall now proceed to shew. 7. " In every fundamental chord, the leading note is the sharpest note to be found." To this rule there is no exception, and by means of it the root of every fundamental chord can easily be found. THE PRINCIPLES OF HARMONY. 87 For example, let us try to discover the root of the following chord — M :q On reference to the Hst or table in the last section, it will be seen that of the notes composing this chord the sharpest is A ; therefore A is the leading note ; therefore F is the root ; and it is evidently the first inver- sion of the chord of the minor ninth without the octave of the root, otherwise called the chord of the diminished seventh. And from this it follows, in the regular course, that the chord belongs to the key of Bt^ minor. This rule is so compendious, so simple, and of such general apphca- tion, that it ought to be thoroughly mastered, and continually applied, by every student of harmony. To it may also be added the following subsidiary rule — " If there is a minor ninth in any chord, that ninth will be the flattest note." Thus, in the above example the flattest note is Gb, which is thereby known to be the miaor ninth of F, the root. If, however, there be no minor ninth in a dominant chord, then the flattest note will be the dominant seventh. 88 THE PRINCIPLES OF HARMONY. Thus, in the annexed chord, the flattest note is D, which is the dominant seventh of E, and the whole chord is a chord of the added ninth on that root : =i^ .^2_ Z2: 99 i 8. Let us now apply these rules to the enharmonic variations of the chord of the diminished seventh and its inversions — ^^- '^^ &- ^ ,«s»- -m- "oio^- "gg: In No. i. the sharpest note is CJf, therefore the root is AQ, and the L chord is a diminished seventh ^J. In No. ii. the sharpest note is EQ, therefore the root is CQ, and the .6 chord is in its last inversion 04 . % In No. iii. the sharpest note is Ajf, therefore the root is Y%, and the 6 chord is the first inversion of a diminished seventh Q4 . 3 In No. iv. the sharpest note is Gp, therefore the root is Eb, and the chord is the second inversion of a diminished seventh Q5 . In the first case the chord leads to D minor (or major) ; in the second THE PRINCIPLES OF HARMONY. 89 to F minor (or major) ; in the third to B minor (or major) ; and in the fourth to Ab minor (or major). The student will now analyse the following exercise, detecting the roots, and describing every chord — S^ 2i i --^^=^ ® :§fl=4 -S»- Gh- .J. -&- g^^ £i 1 -^ ~jzi: ^ ±^ i^ :Jf^ :^=3P -Gh "1 6 f t —^^J^ ■^Gh i ist :^ «s- -«si — &i- ^fe^ l^it^^^^g: i^ &^^~b3 E ■b a 1^21 :rj: -^s*- i^ b5 -«s b7 b5 t> b7 b d: /^Ts i 22 aS [^ — g zzLz^ -^ ^^= "c^ ^^ -^- b^ -(S>- S is>- ^^=B^= 2:± :^ :^z 3i :zii: ^ Itc^' -iS^ "^>" ^O' 9 8 I b7 Note. If the root and leading note be both omitted from the chord of the minor ninth, it is reduced to an imperfect or diminished triad — i «y N 00 THE PRINCIPLES OF HARMONY In this form it is impossible to detect its root, excei)t by a consideration of the succeeding cliord. If that chord be on the tonic A, the preceding example will of course be an imperfect chord of the minor ninth on the dominant root E. -S- E A But if the succeeding chord be on the tonic C, then the doubtful chord must be regarded as an imperfect chord of the dominant seventh on G. f^ W=^^ G C It is therefore the most equivocal and unsatisfactory of all the simple chords in music. CHAPTER VII. 1. Having introduced and explained the chord of the minor ninth and its derivatives in the last Chapter, we may now proceed to apply it to the harmonization of the descending minor scale. In Chapter V, section 6, the ascending minor scale was worked out : and it was shewn in the following section, that between the sixth and seventh degrees of this scale the chromatic interval of an augmented second occurs. In forming and harmonizing the descending minor scale, care will be required in the treatment of this awkward interval. ^^^^EEP^^^^EEg :^± ^AEEEEAEA On setting out the melody, it will be observed that all the notes of it belong to the dominant chord of E, except the first, sixth, and eighth. We need not, then, unless we choose, have any dealings with the subdomlnant D, or with any harmony but that of the tonic and dominant. N 2 92 THE PRINCIPLES OF HARMONY. :a: g— ^-eg: 22: "C5- 1^21 -s>- 1221 -(S*- :e2: 12:2: 22: :z2: Roots A A E E E 6 E E A Keys A E A But here there appears a somewhat irregular treatment of the domi- nant seventh between the third and fourth chords ; thus — fS>- :2:2: 5E^= -f^ :g =p= F And although this progression is tolerated by licence, on account of the contrary motion between the extreme parts, yet it is by no means elegant. It will therefore be better to look on the note E in this scale as part of the tonic harmony, and to harmonize thus — E -IS»- q=ip2: g=^ -- :q: :q: -- :??^E^ 1^2: 94 THE PRINCIPLES OF HARMONY. Such an alteration necessitates a total change in the harmonization. The following appears to be the best method : — I iS -- :q: -^^-IQ- S s i^SzIg: JUL -O- - rj rj 7^ —7-J- ^ Z^^Z "r^ s :z^ ry --g: e> — <^- -<&- -o- _c*t- -(S^ -e>- -s*- # 122: 22: -- 'IZL. 4J^2Zz^zttQ: Z2: Z2: 221 6 Q5 I 3. It will be useful now to give examples of the harmonizing of scales minor and major when the scale is given to the bass part. As an exercise to the student, only the bass and figures will be given, and it will be for him to supply the three upper parts, and to assign the proper roots of each chord : #a -^- -f^^- -s>- 4Q-ip jQ_ J^lJ2^ 1^=22 ■tS)- :^2z 6 B5 6 6 ' f m -&- I^Z -o- Z2: ^^=-^-- :^=^: =^=p= -t^- - f^—^—^ _ D6 ■^—&- 221 6 6 -^' -Gi- rj rjr_ :i^:±=S 1^21 6 4 6 ^=3= r^ CHAPTER VIII. 1. All the discords hitherto described have belonged to one class, viz. " fundamental discords." That is, they have all been parts of the dominant harmony, as derived from nature. But besides the funda- mental discords, there are others of various kinds, some of which it is time to explain. 2. It may be well, in this place, before going fiurther, to define a few terms more accurately than has yet been done in this Treatise. i. A Concord is a combination of root, third, and fifth, such as the major and minor triads — (and perhaps their inversions also, though they are imperfect concords). ii. The sounds of which a concord consists are called Consonances. iii. When any one of the sounds composing a concord is removed, and some other sound substituted in its place, the perfection and satisfactoriness of the concord is destroyed, and a different and contrasting effect produced. The intruded new sound which pro- duces this result is called a Dissonance. iv. The chord in which the dissonance is heard is called a Discord. 96 THE PRINCIPLES OF HARMONY. 3. The discords and dissonances of fundamental dominant harmony liavc been discussed. It is necessary now to speak of Dissonances by Suspension ■*'. Let the following melody be played, accompanied only by its funda- mental bass. It is satisfactory to the ear, though tame and bald : I _Q_ 'G>- Z2: -G>- S i^z: -s»- 1221 :z2: But let the sound G in the melody be continued through part of the succeeding bar, without altering the bass (as in the following example), and the ear will immediately begin to long for the delayed note F. And if each note in this melody be similarly treated, we shall have 22: «s>- 221 2IZ izz: Wlien an interval of a melody (or of an inner part) is thus kept back in descending, it is called a susjoension. Thus, in the preceding example, G suspends F, F suspends E, E suspends D. * This is principally derived from Logier's "System," ut sup. pp. 62-G5. THE PRINCIPLES OF HARMONY. Or, if viewed in relation to their bass notes, The ninth suspends the eighth, The fourth suspends the third, The sixth suspends the fifth. These include all the dissonances by suspension. The following are examples of each : — 97 -- ''Z±k .Q_ ^?^:^^B m i^z: :q: 22: icz: 221 :q: X2_ i^z t z \ 3 Here the dissonance of the fourth is produced by suspending the third. -s*- -- ;i Q_ ^± J^ (S>- Z2: Si ,Si -«s>- S— n -- -- -- -o- -Gf- 22: izz: :22: Z2: -- -- 9 8 9 8 Here the dissonance of tlie ninth is produced by suspending the eighth. 'TJ — C 22: :g: r rJ -(S>- ^zi ^^f^ -C^ - ^^^ -!S>- T^ 22: 22: 22: 22: 221 22: 22: Z2: 6 5 6 5 6 5 Here the dissonance of the sixth is produced by suspending the fifth. 98 THE PRINCIPLES OF HARMONY. 4. The suspension, then, being produced by a lagging note — a note behind the rest in the progression from one chord to another, — it follows that these dissonant notes must exist as consonances in the previous chord, before they became dissonances by being, as it were, left behind. The rule, then, may be thus stated : — "All dissonances must be introduced by consonances," or, in other words, " The sound which constitutes the dissonance must first be heard in the preceding chord as a consonance." And tliis is called preparing a dissonance. And it is also evident from this, that in whatever part (treble, bass, or inner) the dissonance occurs, in that same part it must also be prepared. It will also be noted, that the dissonance always descends one degree upon the following consonance. And this is called resolving a dissonance. A few examples will make this clearer : — 221 -@- -O- -- m- 1221 -O- Z2: If we wanted to introduce the dissonance of the fourth into this har- mony, we might do so, perhaps, as follows : — :c2E ,- Z2: :^2: -<^ THE PRINCIPLES OF HARMONY. 99 But on inspecting the progression of these parts, it will soon be seen that the fourth is not i^repared. To remedy this defect, it must be introduced into the same part in the preceding bar ; thus — F Z2T2 (S>- E -G>- rU -S^ m 22: -- Z2: And the same process would enable us to introduce the dissonance of the ninth into the last bar ; thus — :22ZC2 ^ ^2 -- o- ICZ ■^:2: m Z2: -s>- :q: And the dissonance of the sixth should be prepared and resolved in the same manner. The following rules will be useful : — I. The dissonance of the fourth may be introduced whenever the fundamental bass (or root) descends a fourth or ascends a fifth. II. The dissonance of the ninth may be introduced whenever the fun- damental bass ascends a fourth or descends a fifth. o 2 100 THE PRINCIPLES OF HARMONY. The preceding example will serve to illustrate both these rules. 5. The preceding remarks refer only to dissonances by suspension occurring in uniiiverted chords. By employing inverted basses, a great variety of effects by suspension may be obtained. In every case care must be taken " never to let the suspending note be heard together with the note it suspends.^' Thus ^ :^ :^ :^: -cr :^2= would be wrong ; as the suspended B is heard in the tenor part. In a free style, however, such a combination is occasionally met with. Still, it is necessary for the student to keep to the rule ; excepting only in the case of the ninth by suspension, wbich may be used with the eighth, 2')'^^ovided the parts proceed by coiitrary motio7i, and only then. Indeed, even in this allowed case, the licence should be sparingly used. Thus, for instance, would bo allowed by licence, while the following — THE PRINCIPLES OF HARMONY. 101 is*- ^- i ;^E^^ -G>~ ^ ITj- -^ would be altoo'ether inadmissible. For it is evident that hidden coiise- cutive octaves would be produced — -- I^ y«^- 1221 Here the leading note, B, is allowed by licence to fall to G, to avoid the transgression of the above nile for the resolution of dissonances. In this inversion the fLmdamental dissonance of the sixth becomes a fourth to the inverted bass ; as thus — 102 THE PRINCIPLES OF HARMONY. Roots -y- — 1 1 — 1 1 1 1 n ^^ — ^ — ^H J — ^ feh-fe — _cJ eL- \—^— _c-^ (S'— ■ tJ . ... f -J f^^— f^— r^ j-^ rj v^ 4 3 I 4 3 /m^' -^ vfj- — -^ V-^ ^-^ rj 1 — & — 1 6 5 6 6 In the second case here given, the dominant seventh, F, has been allowed to fall to C instead of E, in order to avoid breaking the same rule. In this inversion the fundamental dissonance of the fourth necessi- tates the suspension of the inverted bass itself ; thus — Roots ^ -C2_ -O- g f^-D -^ ^ 22: :q: -Gf- i^- cn Z2: for the inverted bass is always to be looked upon as an upper melody transposed into the lowest place, and therefore to be treated in all respects as such. THE PRINCIPLES OF HARMONY. 103 In the second inversion of the common chord, the fundamental dissonance of the fourth becomes a seventh to the inverted bass, thus — Roots g JCZ. :^: r iq: &- 122: In this inversion the fundamental dissonance of the ninth becomes a fifth to the inverted bass, thus — Roots #^^^^ A- -r^ -& ^ -- In this inversion the fundamental dissonance of the sixth in- 104 THE PKiNCIPLES OF HARMONY. volves the suspension of the inverted bass itself, producing the fol- lowing : — Roots -i -r± Z2: -o- - ^— r^ -.■s*- 1^21 In the inversions of the chord of the dominant seventh the various dissonances will appear as follows. The student will understand them without further explanation. ^^ -P -Gh ^ ~rjr:z^ -Gf- :gr"-sep»-^ Z5: 122 ^2=jz^=:t ^ d=d: I^ gd] -ri- b^=Hzg2 1^21 --^-g- ici 22: 1^2 22: :q; -«s>- JZ^ Z2 Z2 Z2: "-—(S*- 7 - t 8 :^S> ' 6 - 5 - 4 3 ^ z:2 221 :zq: :22: ziz: :z2 22i:r^z ::- :^S>- -- -- Roots 4 3 7 t 3 THE PRINCIPLES OF HARMONY. 105 m^ -^ :q: -o- -«s»- iq: Z2: :?^=;^ 1^21 Z2: 7 6 6 - 6 5 4 7 6 i = :r±: Z2 z:2: ^i?3: zz: Z2 i^z: Z2: :^- B/oots :- -- :«S>- i I 3 3 - 4 3 In none of these cases has the dissonance of the ninth been admitted, because it becomes identical, practically, with the chord of the added ninth, already described ; or, if minor, with the chord of the minor ninth. The student will now be able to add dissonances by suspension to the chords of the added and minor nmths and their inversions, by carefully adliering to the rules given above. We will therefore give a bass with figures, and with the roots indicated, for him to harmonize. To render this easier, however, the following rules will be useful. The first and second have already been given in other words. I. When the fundamental bass (or root) descends a foui'th, or ascends a fifth, the dissonance of the fourth may be introduced, prepared by the octave. i =F 12:2: 'Z^ ^m 2^ -G>' ?- lOG THE PRINCIPLES OF HARMONY. II. When the root ascends a fourth, or descends a fifth, the dis- sonance of the ninth may be introduced, prepared by the fifth. :C2: -G>- =F W^ S>z 1^21 -«S>- III. When the root descends a fourth, or ascends a fifth, the dis- sonance of the sixth may be introduced, prepared by the third. ly. Wlien the root ascends a second, the dissonance of the ninth, prepared by the third, and that of the fourth, prepared by the fifth, may be introduced. i -^ s ^: -(S>- -«s>- -1^- ~r^ 5 — 'JIJZ -G^ 1^21 -- V. Wlien the root ascends a third, the dissonance of the sixth, pre- pared by the third, may be introduced. THE PRINCIPLES OF HARMONY 107 m 'j:^ 2:^: VI. When the root ascends a sixth, or descends a third, there can be no dissonance. VII. When the root descends a second, or ascends a seventh, the dissonance of the sixth, prepared by the fifth, or that of the fourth, prepared by the third, may be introduced. - "^ . .^1 J iq: -iS>- -G^- -^■ A- -JZll -fE^- -^ "C?" -^2_- *^2=H -iSi- jf5 [When inverted basses are used, and dominant harmonies added, certain additional rules may be added.] VIII. When the root of preparation bears a dominant seventh, and ascends a fourth, or descends a fifth, the dissonance of the fourth, prepared by the seventh, may be mtroduced. -St r^ T^ -G>- A r -IS'T 22^21 P 2 108 THE PRINCIPLES OF HARMONY. IX. When the root of preparation bears an added ninth or a minor ninth, and ascends a fourth, or descends a fifth, the dissonance of tlie sixth, prepared by the ninth, may be introduced. fc^ =^ :^2: hS^ -^- 9 ^— 6 I ^ 0= ^iSh 221 :P= 1^21 -(^^ 1221 Exercise on Dissonances. )h-t -G^ '-9- -Gt^ 1^2: -- G -& ~«^ -Gt- 9 8 G -s>- G D D 7 6 G D F=P= 1^- :r=t -- hS»- -(S*- -- z:i -- 3 6 -v — G ^v_ D 6 7 6 ) c 8 7 % - D 6 - 5 5 4 4 V G D -H'&H- t 3 G 7. When two dissonances can be regularly introduced on the same root, according to the foregoing rules, they may be introduced simul- THE PRINCIPLES OF HARMONY. 109 taneously. They are then called double dissonances, and the chord into wliich they are introduced is called a double discord. Thus we may have the double dissonance of the fourth and ninth. For example — i XT 3 -^Eif- ^ -^S- -f^- iq: We may likewise have the double dissonance of the sixth and fourth. For example — J. -^± 1^2: It is important to observe carefully the identity of the notes composing this last double discord with those forming the second in- version of the common chord. In nine cases out of ten, a 9 must be 4 treated as a dissonance, and not as an inverted triad; i.e. the dissonant notes must be prepared and resolved according to the rules given in this Chapter. Whenever a J is followed by a § on the same bass, we may be sure it is a dissonance, and must act accordingly. The root will then, of 110 THE PRINCIPLES OF HARMONY. course, be different from what it would be, were the chord the inversion of a triad. Thus the root of the first note in this bar, "^7 ^ is not G, but D ; and bears the double dissonance of the sixth and fourth. Therefore both the B and the G must be sounded, in the same parts, in the previous chord ; thus — -«SI -!&'- "^ z-tS ^. -^ T^~ Ib7 5 ">% i. The dissonance of the ninth may be resolved on the sixth, if the bass ascends a third — 1221 (S»- .^ '^^- ~jrjr. ii. The dissonance of the sixth may be resolved on the third, if the bass ascends a third — i --g-- id: IC^I THE PRINCIPLES OF HARMONY. ] 1 1 iii. The dissonance of the sixth, may be resolved on the sixth, if the bass descends one degree — m C21 -B -TTJ- iv. The dissonance of the fourth may be resolved on the sixth, if the bass descends a fourth, or ascends a fifth — -«s- g icz: Z2: -c;^ -<©- -&- T^ :^ 4-21 i^ :z2: 1 9. It Tvdll also be useful to remember that while fundamental dis- cords are resolved on another bass, discords by suspension are resolved on the sa^ne hass. Also it is well to bear m mind that while funda- mental discords require no |9repa?Tt^zo?i, but only resolution, discords by suspension require both ^^j^e^^ara^io?^ and resolution. CHAPTER IX. 1. In" Chapter VIII, section 3, it was said that " When an interval of a melody (or of an inner part) is kept back in descending, it is called a suspension. It remains now to treat of the other case : " When an interval of a melody (or of an inner part) is kept back in ascending, it is called a retardation!' The principle of the two is the same, only that the name of suspension is not, strictly speaking, suitable to the case of a note kept from asce7iding. 2. When the leading note is retarded, the dissonance of the major seventh hy retardation is produced. i XT 221 -fSh- -<^- -G^ 1^2: m^ ^ 22: A 1^21 This is prepared by the tliird, and can be introduced whenever the root falls a fifth, or rises a fourth. THE PRINCIPLES OF HARMONY. 113 When the fifth is retarded, the dissonance of the second hy retardation is produced. I :g: =F z:± is: -C2. -G>- 1^2: ■^ id: :^2" -Gh- i^z: This is prepared by the fifth, and can be introduced whenever the root falls a fifth, or rises a fourth ; if the fifth is so placed as to be able to rise a degree. These are the only fundamental retardations. The rules for their preparation and resolution are much the same as those which refer to suspensions. 3. When the two retardations are used together, a douhle discord hy retardation is produced ; thus — -A -Gh- T^ 1^2: The student will be able to work out the application of these dis- sonances to chords with inverted basses, by reference to the method employed in the case of suspensions. Q 114 THE PRINCIPLES OF HARMONY. 4. Eetardations and suspensions may likewise be variously combined of which we will now give some examples.* i^tett^ ^Eg gj , — &^ ^m 'JOH f^- :^: :JJ^ -et- 'JUl- -^- TIT" 9 § ^ P5 -^ -G>- t^-=^ 3 -&r ;*i -Gh- .■q: -- -<^- -JZT- % S! -^ -fS' "O" -- ^^ C^ 1^ m ^-=^ 2d=^ :c2: -TD rcz: 53 -- -- ^— ^ rD— \—r:3 P=^ (S>- :^ :?2: T^~ iS>- -(S*- 7 6 2 6 * This chord will be explained in a future Chapter. -<^y(S^S(S/t)^y^>-^ Q 2 CHAPTER X. 1. In Chapter IV, section 2, Logier's rule was quoted, that " Wlien- ever we use a [fundamental] seventh, and thus proceed to a new key, we modulate into that key." And in the same place a regularly pro- gressive modulation of this kind was given. We will now give it in a more concise form. g ^^^ -Gh 7^ a a^g ^aiS^: 22__L^ fe^=^i -^^ s^ K>g3 l^'Sj^- lu^ r:^ a 122^ ^51 e: ip2=S m (S>- (S>- :z2 ^ isi 6 b5 b7 be b5 be b7 b .be b5 b ^ -t9- :^2: 't^. 3§^ -<5<- Z2" '=is: -m^m^ :^2z:bs^: s-^ ^ «: t^ «S— " i^ 3^ ^3^ ^e^«i ^!S>- xz 1^ t^ 122: tz=± :^± :^ e 4 i D7 57 THE PRINCIPLES OF HARMONY. 117 This may be still further curtailed by the following method, whereby the octave changes to a dominant seventh in each successive chord. ^^ r r :^*£=^W ^ fer '^^ v^- VSh- m£ 'IZL. --^ tec ^ i*: b5 be b.5 be /rs 1^^- ^m- S*— jt? tl*- l3: It-i g _a t:;^ -(S«- -^^ ^ I^X z^: I ^^ — '1 I I -^ be -^— r-^i- iS: 1f^ 4j- 2 41- 2 (i C7 \:^ But the progression can even further be shortened by omitting all the tonic triads and inversions, and regarding each successive chord of the dommant seventh, or its inversion, as embracing the proper resolution of the preceding one. In this form the result will be as follows : — :^^=^; ^E fc ^&- -&- &g=5g %gHg='S=fi^ Jld -d- A ^S I^S ^21 2?:^ *!^- ^«S- b5 D6 b5 where we may observe, 1st, that the leading notes all appear as though 118 THE PRINCIPLES OF HARMONY. they were resolved downwards, iii defiance of nile. But the ear supplies the omitted intervening tonic chord in each case, and thus is satisfied. Moreover, the progression of the roots, each becoming a dominant to the next, quite overi'ides the effect of the UTegular resolutions of the leading notes, and carries the ear with it headlong in its downward course. There- fore this mode of resolving leading notes is allowed, as a licence. We observe, 2ndly, that the bass and treble both proceed chromatically by similar motion, and at the interval of a diminished fifth. And we observe, 3rdly, that the first and third inversions are used alternately. Another form of this series is the fundamental position, thus — 'J-l^^- (S»- E^i^S ^^ ^m ^^^ 1C5: :il^ m Z2L G^ — t'^ 'ZH. ^^ i^ i^: -cr D7 D b7 b7 b7 b7 D7 E7 &c., &c., &c. Such a series is called a " sequence of dominant sevenths." 2. Chords of the added ninth, as well as of the minor ninth, are susceptible of similar treatment. The latter particularly so, in the form of diminished sevenths, thus — sd^ i BADE where, by the enharmonic substitution of Gtt for A^ in the fourth chord THE PRINCIPLES OF HARMONY. 119 we change the bass from G (which it would have been) to E, and thus confine our progression of roots to the three, E, A, and D. By similar enharmonic alterations of the intervals of any of the chords, we can of course introduce any roots we please, as was shewn in Chapter IV, section 5. We may call this series a " sequence of minor ninths," or, if we please, a " sequence of diminished sevenths." 3. If we wish to modulate in the contrary direction, i.e. from tonic to dominant continually, our process of curtailment will not be quite so complete as that just described. It will be well to begin our work thus — ^- ^: -H — e* — 1± — 1^1^. *^p: -& — «i^. :2: e>- Z2^ sa os^ i gU-4^— 3 i T^ B- _^_ IQ- cz: (S>~ '^- \?±. 1^ -^- ^£S=ii=^a=^s ^^S?i M-=jf q: S3i #^ i^: @^p^=^ jzi -:^=^$^- ' 22: and then lower the bass note, as before- * Logier in his " System of the Science of Music " calls this chord simply the " chord of the sharp sixth." But as the sixth is not sharp only, but augmented, it is better to call it, as Day does, the " chord of the augmented sixth." THE PRINCIPLES OF HARMONY. 127 i t^ ^ e/ 15^21 we produce a discord,"' which has been called by the same authorities the " German sixth." These names are very unmeaning and unsystematic; but as they are frequently employed, it is well for tlie student to make himself familiar with them, and with the chords they are used to denote. 5. Now it is clear that in each of these chords every note, except the loioest tvhich has been altered, is derived from the root D, and belongs to that key of which D is the dominant, viz. G. But if the chord be played several times in succession, alternately with the chord of G, thus — S^ -&■ T T T T T T r f " r T^ if r f r IS r r -G>- the ear will be anything but satisfied, and will desii'e to hear the har- mony of C ; as thus — * Logier Ciills this tlie •' compuuiid sharp sixtli." 128 THE PRINCIPLES OF HARMONY. -et- ■x^ r -^Ijr :^: 1^2: ii^ -& :^: 12^ -s»- This is enough to excite a suspicion that G is here rather a dominant than a tonic root. And this suspicion is strengthened by the fact that, after the chord of the augmented sixth, the ear will not tolerate the minor common chord of G ; as thus — :Jf^ \y^- -o- -&- In this case, the ear imperatively calls for the majo?'; thus proving that the leading note is required, which is the only third in the dominant harmony. We may therefore fairly conclude that the chord of the augmented sixth must be followed by a dominant chord. Therefore its root (so far as it has been discovered hitherto by us) must be the dominant of a dominant, i.e. the supertonic. Therefore it belongs to the key of which it is the supertonic, and of which the chord into which it resolves is the dominant. THE PRINCIPLES OF HARMONY. 120 6. But now the question arises : What is the root of tliat altered bass note which could not be derived from the supertonic root 1 Clearly it is the minor ninth of the dominant, and being itself consequently an essentially dominant interval, it is the cause of that tendency to the real tonic wliich was pointed out in the last section. 7. It will be seen that the interval of the augmented sixth is formed between the minor ninth of the dominant root, and the leading note of the supertonic — and that the only interval derived from the dominant root is its minor ninth. All the other intervals of the chord, whether it be in the forms commonly called the French, the Italian, or the German, are obviously derived from the supertonic root. Examples — 3 ^ ^:^ -G>- -G^ s -- fS^ % i^z: - — -- -^^- m- -fer- '-^- ^-^ A e 1221 iq: :cz 8 D7 8 a? 8-pD7 E/OOts. 7 ^: -- ^^sJ-^ 1^21 ^S*- -(S>- :^ :q: ]2to: zz: r=r jis- Roots. ''9 3 b5 D6 4 :^ b9 -& 1^21 -cz: -^ b9 \ S 2 b9 -O- 22: 132 THE PRINCIPLES OF HARMONY. No. 7. No. 8. No. 9. e (S>- T^ 1^21 Roots. 7 -- ^21 'TTJ' ^ t HSt s Z2: ^ g — ^ B 6 07 B4 4 6 -^ 3 ^21 ^g-- 122: zz: :^2: -e? b9 o - 4 3 b9 6 4 -^■ b9 6 d7 4 5 No. 10. No. 11. No. 12. i Z3=^ 3 i^ 2:3: 3 1^21 :22Z 1^2: «r^ :js -(Si- "C7" :S^ rr ^^ :?=: •TTT r=fe ^2: :^: ^ ^21 -©- -s^ Z2: t 3 Roots. 7 1) b9 22: :^ 22: b9 b9 22: THE PRINCIPLES OF HARMONY. 133 No. 13. No. 14. No. 15. ^ — <^ — s 22: -^x -^^ -rj ^ k^- r ■^- =»s^ -(S*- T:^ Z2Z ^ rr- : y-R ^ ?2: ^0-^ I — I I — ^ irz. 'i::iL. - 8 07 Roots. 7 ^- n4 - t 3 ? - G4 - S iq: :^ :^z: icz: •59 8 C: 59 4 3 (59 6 5 4 3 No. 16. No. 17. No. 18. ^a *=^=R= ^r-q»- *»-5 ^ zz -iS'- e 221 Z2: P -TTT 8 D7 Roots.''? = « -. ^ Z2: 8 07 b5 - ^? = 59 1221 5 — 4 3 bo b9 - -« #- 99 fl6 4 '% i Z2: 134 THE PRINCIPLES OF HARMONY. No. 19. No. 20. No. 21. No. 22. No. 23. No. 24. ^ ~r2L 1ZL ^='^: :z2: -Gh- icz: ^__&jJ_J_^_g_tii:if^— 5^ ~-^y-^-^-= :b^i=^ Z2: i^z: i^ 221 E2 6 B7 Roots. 7 ^ 22: G7 Q7 7 h9 57 ^2: 1^21 6 B7 -- 1^21 ^S- 221 B6 4)- Roots. 7 % b9 C7 be 4 221 D6 7 % b9 'I 221 07 7 ^- b9 "b7~ be 4 2z: 136 THE PRINCIPLES OF HARMONY. No. 31. No. 32. No. 33. -JTJlZ ^-g-- -^x -«s- :^i$^^2zte: _cz; ^(S> 221 ^<^- iq: ^- ^t^^it;^ -& b7 ,6 5 B5 6 ^6 ^6 6 4 B^ B5 4 I ,b6 b4 b5 b£ 4 5 B5 4 &c., &c., &c. This formula may be found useful when a sudden modulation to a remote key is required. 10. The chord of the augmented sixth can be inverted- FiTst Inversion. ^ < g^ % Roots. 1% D G -- U- ^ -&- j^. -TTT bfi 4 G -t b C i -^^ =i^ b5 D G :i ^3=^ tted: :z2: G -^ iq: B7 THE PRINCIPLES OF HARMONY. 139 U ^ s; :z2: ■^- '^ -^- ^gzz=t:g: ^- ^ '^ ^ :z2: €r-^p - ie ^ b7 Roots. 1 DT G i% D G G Second Inversion. i _^ r^z ^E^ ^^ -^ -C^ ^ P P- :?2: -^^ — ^ :f^ -C^ i?^ -- ^ v^ ^^ i^i: ^ Z2: -fe- Roots. D G G ii n G G «'6 D G 6 G i ^§=t r^ <^ (S I 'T^ L^' Roots. D G G Third Inversion. -^ Z2: or, ^^=^- Roots. -s»- D G C7 G T 2 140 THE PRINCIPLES OF HARMONY. Note. — Dr. Day denies that the interval of the augmented sixth can be inverted, on the ground that the harmonics derived from the secondary root must not be placed below those derived from the primary. But in the first place we may imagine the roots to be distant as many octaves below as we please, so that we can always suppose the two roots to be at their proper interval apart, viz. a twelfth. And, in the next place, the chord has been used in an inverted position by many of the most esteemed composers with excellent effect. The earliest instance being in Weldon's Anthem " Hear my crying," near the end of the concluding movement, (Vide Boyce's Collection of Cathedral Music, vol. ii. p. 218, of the editions of 1768 and 1788.) Weldon was a pupil of Henry Purcell, and died in 1736.* 11. As this seems the most fitting place for explaining the meaning of what is called " false relation," or " cross relation f it may be stated that by these terms are signified certain harmonic incongruities between two different parts or voices, which are exceedingly offensive, and generally forbidden. The general rule is, that " when a note of the same name occurring in two successive chords is altered by an accidental, it must be sounded by the same part or voice, otherwise forbidden false relations will ensue." Thus — -23=1 IS wrong, but fm — 3 . ^~" i^ right. -^- -^r The reason of which is, that the false relation gives the impression of two different keys simultaneously used; for, in the former and incorrect example here given, the treble — C2: • See Short Examples at the end of this work, Nos. 1 and 2. THE PRINCIPLES OF HARMONY. is part of a melody iii C minor, while the other parts — 141 2!> -& clearly belong to C major. If, however, the note to be altered is doubled, it need only progress accordmg to the above rule in one of the parts, otherwise consecutive octaves would ensue — —9 f^- S E -iS*- P- i By licence false relations are allowed in purely dominant modu- lations, when occuiTing between different symmetrical phrases or groups; thus — •tS»- ^ r^rr -is>^ 5? i -tr r r T r where the two phrases are enclosed in brackets, and the false relation of the G and Gitt is not harsh or unpleasant. Where the roots proceed regularly -from dominant to tonic in a descending cycle of fifths, or where such regular cycle is symmetrically reversed, false relations may sometimes be tolerated: for example — :^: =z± ^ or z:^ iSh- -^1 G D ' — 3j^ D G 142 THE PRINCIPLES OF HARMONY. because "the reo-iilar resolution of the dissonant notes in each case con- soles the ear, and mitigates the harshness of the false relation. Great composers have occasionally used considerable licence in the matter of false relations ; but we do not recommend the young com- poser to follow their example. We have given some curious and exceptional instances at the end of this work.* It may also be observed that in many cases the interposition of an intervening chord will not save the false relation : for instance — ^ or even /m;— ^rL:*r^ In each of these cases the harsh effect of the false relation remains, because the intervening dominant chords belong equally to the major and minor mode, and therefore do not interfere with the impression produced by the two incongruous tonic chords between which they are placed. In strict counterpoint the rules against false relations are even more exactly enforced. The above observations, however, will suffice for the student's ordinary guidance. * See Short Examples, Nos. 16, 17. CHAPTER XII. 1. In dealing Avitli the chord of the augmented sixth, it was only- found necessary to make use of one interv^al belonging to the dominant root, i.e. the minor ninth; all the rest bemg derived from the secondary or supertonic root. But whenever we find a chord with a minor third and a minor seventh, we may be sure it comes from two roots. For the minor seventh demands a majoi' third, or leading note, to enable it to resolve regularly. If we substitute a minor third, our root is necessarily altered. Thus — Js ,G>^-=^ « :g: %- -&- 7 is a regular resolution of the chord of the dominant seventh of D to the tonic common chord of G. Here, of course, the leading note FJf rises to the octave of the tonic, G. 144 THE PRINCIPLES OF HARMONY. But let us substitute F natural, and a totally different effect will be produced on the ear — -- I^X ziz: For here, evidently, we must regard the F natural as the minor seventh of the dominant root G, while the A and the C are as evidently the fifth and minor seventh of the supertonic or secondary root D. This chord then has clearly a double root — -Gf- S: -^- ■- 'CJ' s -&- -- "Z2; Z2: 7 D G 7 G and the supertonic discord is resolved first, and then the dominant dis- cord, just as was found to be the case in the chord of the augmented sixth. 2. This is a most important chord in several ways ; for, in the first place, its first inversion has been known to almost all theorists and didactic writers on harmony as the " chord of the added sixth," or " great sixth " — THE PRINCIPLES OF HARMONY. 145 ^q: :^2 -- -»s- -Gh- ' 7^- Of course it may be said that this argument also applies to the explanation given in this section. But it should be borne m mind that we hioio the downward tendency of the fundamental seventh, and connect it in our minds with the great flatness of that interval as it a^Dpears in nature ; whereas we know that a real perfect fourth or eleventh does not exist anywhere among the harmonics, and that the interval which occupies the eleventh degree of the series, and is supposed to represent the fundamental eleventh, is so very sharp as to be much more like an augmented fourth or eleventh than a perfect one. THE PRINCIPLES OF HARMONY. 147 Therefore it is against the indications of nature to treat (jf the fundamental eleventh as thougli its natural tendencies could form the basis of any argument. The next objection is the unnecessary intrusion of the eleventh as a fundamental harmony. For in all cases in which this interval is supposed to exist, it may equally well be considered to be either the minor seventh of a secondary root, or a dissonance by suspension. Wliy then intrude it here ? 3. Without going further into this matter, let it then be assumed that the " chord of the added sixth," as it is generally and erroneously called, is nothing more than the first inversion of the chord of the minor seventh with a minor third ; and that it is derived from two roots, the dominant and the supertonic. This last chord has three inversions — ^ -- 7 B iP ^^_ _c:z: 7 E -o- C2z: -^ Z35: _Q T^ B IE/ 1*^ t ?:^ _^ :?2: A =^^ B VE T^- 7 E l^^ -(S>- i^: ^ A 6 4 2 B 6 5 E -- 4. Sometimes, especially in modem music, the regular resolution of the first inversion of this chord is curtailed by the omission of the dominant chord, which usually intervenes between the double -root chord and its final tonic resolution. This curtailment may take place either in its major or minor variety. Thus — S' B ^ IS" 221 or -s>- 1^21 -&: ^ icz: 1^21 -^ m- -Gh- 221 5 F D 5 D 6 5 B E Sometimes the minor discord is followed by a major tonic triad, after the manner of what is called a " Tierce de Picardie " (see Chapter XIII, 3), when the follomng result is produced — * * See Short Examples at the end of the work, No. 18. THE PRINCIPLES OF HARMONY. 149 'Gh ^ S -^ ^ 122: And sometimes this occurs when the whole passage is in the major key, and when the preceding chord is also itself major, thus — ^^ -G^ 221 In all these cases the chord has been preceded by that of the sub- dominant; but this is far from being always the case. Moreover, the uninverted chord of the minor seventh and third, as well as its second and third inversions, may occasionally be treated after a similar manner; as will be seen by the following examples — n tf 1 1 1 I ^ J y ft u. ^ r^r-i ^^ <-^ 1 '^-J,^-) ^^ >^ f^^i^^ s< ■ -f-j /T it H f^^ c^ r^ hr^^^ •^^ ^^ if\ ^ ^