some suggested modifications of day's theory of harmony

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Some SuggestedModifications of Day'sTheory of HarmonyEbenezer Prout B.A.Published online: 28 Jan 2009.

To cite this article: Ebenezer Prout B.A. (1887) Some SuggestedModifications of Day's Theory of Harmony, Proceedings of the MusicalAssociation, 14:1, 89-117, DOI: 10.1093/jrma/14.1.89

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MARCH 5, 1888.

DR. BRIDGE

IN THE CHAIR.

SOME SUGGESTED MODIFICATIONS OF DAY'STHEORY OF HARMONY.

BY EBENEZER PROUT, B.A.

WHEN, at the beginning of the present session, I promisedour Secretary to read a paper before this Association onDr. Day's System of Harmony, I ventured to indulge thehope that the chair would be taken on that occasion by theeminent musician to whose efforts we are indebted for themost complete exposition of that system. By the sudden andlamented death of Sir George Macfarren, England has losther greatest theoretical musician ; and his name is so inti-mately connected with the subject on which I am to have thehonour of addressing you this afternoon, that I cannot refrainfrom paying this tribute to his marvellous gifts, and ofacknowledging my own indebtedness to his writings for amore thorough insight into what I believe to be the truetheory of music, than I had been able to obtain from othersources.

In commencing my remarks, let me in the first place mostearnestly disclaim any attempt to dogmatise. I make noprofession to set myself up in any degree as an authority inmatters of theory. My object this afternoon is quitedifferent. In the course of my experience as a teacher, Ihave several times met with difficulties in connection withDay's system, and have given considerable thought to theway in which such difficulties could best be overcome. It isat least not improbable that others who have taught on thissystem may have experienced similar troubles to my own ;and my purpose is to show in what way the objections towhich Day's system is fairly open may, in my humble opinion,be best met.

I trust I shall not be deemed unduly egotistical if I heredigress for a few minutes to give my own practical experienceof the system for teaching purposes. Up to a comparativelyrecent period I was entirely unacquainted both with

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go Suggested Modifications of Day's Theory of Harmony.

Dr. Day's work and with Professor Macfarren's book foundedupon it. Some seven or eight years ago, one of my pupils atthe Royal Academy told me that he was reading Macfarren's" Harmony," but could not understand it without help ; andhe asked me if he might bring it to me to read over with him.I consented, and we went through the book together, fromthe beginning to the end. Before this time I had alwaysused Richter's work as my text-book with pupils, and I foundso much that was quite new to me in Macfarren's views, thatat first I was inclined t6 be incredulous. As we read on, thecompleteness, reasonableness, and consistency of the systemrevealed itself to me more and more; and by the time wehad reached the end of the book I had arrived at a con-clusion, which I have never since seen reason to alter, thathere, and here alone, so far as my experience goes, is a theorysufficiently complete to explain alike the simple progressionsof the old composers of the seventeenth century and themodern harmonies of Schumann, Brahms, or Wagner, whichis throughout perfectly consistent with itself, and which isprobably the only system which has not nearly or quite asmany exceptions as rules.

Let me say at once that I am not going to ignore the ob-jections that may be made to Day's system as he left it. Inthe very first year of the existence of this Association, Mr. C.E. Stephens read a very able paper (with which, he willprobably be surprised to learn, I in the main thoroughlyagree), pointing out Day's inconsistencies. With this matterI shall deal later ; but, for the sake of those of my hearerswho may not have read either Dr. Day's or Professor Mac-farren's books, it will be well first to set forth as briefly aspossible the salient points of Day's system. I cannot do thisbetter than by quoting the summary given by Macfarren inhis " Six Lectures on Harmony " :—

" The ancient style is distinguished from the modern, andthe rules for each are shown to be clear, consistent, and com-prehensive. The confusion that prevails in many minds asto what are the limits of the elder, and the extent of the laterstyle, may be dispelled by an observance of this distinction.

" The ancient, contrapuntal, strict, or diatonic style re-gards the interval of the fourth from the bass as invariablydissonant; allows of no unprepared discords, save onlypassing-notes; and applies its uniform rules of progressionand combination equally and alike to all the notes of the key,admitting not the inflection of any of these, save in thespecial cases stated in our second lecture (i.e., the alterationof the sixth and seventh degrees of the arbitrary minorscale).

" The modern, fundamental, free, or chromatic style admitsinflected notes that change not the key; acknowledges

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Suggested Modifications of Day's Theory of Harmony. 91

exceptional treatment of certain notes in the key as distinctfrom others, which is exemplified by the second inversion—including the consonant fourth—of three exceptional chords,and by the selection of three exceptional notes as roots offundamental harmonics, and accepts the natural generationof discords in place of their artificial preparation.

" The rules for concords and for the three classes of dis-cords—passing-notes, suspensions, and essential discords—inthe strict style, are general, uniform, and exceptionless.

" The rules for concords and discords in the free style arespecial to particular notes, having always a reference to therelationship of each to the tonic. The exceptional treatmentof the concords of the key-note, the subdominant, and thedominant, refers to this relationship; the treatment of eachof the chromatic concords refers to and depends upon thisrelationship; the natural resolutions of each and all of thechromatic discords refer to this relationship."—(Macfarren:" Six Lectures on Harmony," pp. 215-217.)

Here we have very succinctly the outline of Day's wholesystem. With the first part, dealing with the differencebetween the strict and the free styles, we need not troubleourselves this afternoon, because the matter is one of merelyhistorical interest. The chief objections urged to the theoryare owing to Day's deriving his chords in the key from theharmonics of their roots. Mr. Stephens, in the paper towhich I have referred, showed clearly the inconsistencies towhich Day stands committed. Let me take perhaps themost striking example of these inconsistencies. In derivinghis fundamental discords in a key, Day takes first the tonicas a root, then the dominant, as the first harmonic generatedby the tonic, next the supertonic, the first harmonic gene-rated by the dominant. Here he stops, saying: " Thereason why the harmonics of the next fifth are not used, isbecause that note itself is not a note of the diatonic scale,being a little too sharp." As a matter of fact, the note haseighty-one vibrations, where the true minor third below thetonic would only have eighty. But though Day is so par-ticular on this point, he allows the harmonic flat seventh,which is ^ too flat, and the harmonic eleventh which is fatoo sharp ; the difference being in both cases much greaterthan that which he disallows in the instance first-named.Furthermore, he declines to admit the chord on the mediantin the fundamental position, because the seventh of the scaleis not a perfect fifth above the mediant, being too flat, in theratio of eighty to eighty-one, though, as I have just said, heallows the harmonic seventh, which is distinctly flatter.Again, the minor ninth from the root (the seventeenth har-monic) will obviously be a shade flatter than the true dia-tonic semitone, the ratio of which is fifteen to sixteen.

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92 Suggested Modifications of Day's Theory of Harmony.

From what has been said, it will, I think, be evident, thatno system of harmony can work in actual practice in whichabsolute purity of intonation is insisted upon. As a matterof fact, in the tempered scale, with twelve semitones to theoctave, no interval except the octave itself ever is, or can be,in perfect tune. All perfect fifths must be shghtly flat, andall major thirds rather sharp. But here the " adjusting powerof the ear," of which Macfarren has spoken, comes into play.The ear accepts a sound which varies very slightly from trueintonation as the equivalent of the real sound which shouldbe produced. Nay more, the same sound produces anentirely different effect according to its surroundings. Listento these two progressions of chords—

(«) (*)

Ex I.

frr 6 t>7

The third chord at (a) followed by the chord of G majorproduces distinctly the effect of an augmented sixth in thekey of C. The very same notes on the piano approached inprecisely the same way at (b) followed by the chord of D flatproduce no less distinctly the effect of a dominant seventh inD flat. Examples of this adjusting power of the ear may beheard at all orchestral concerts. On the violins the openstrings are tuned in perfect fifths; but in the wind instruments,with a fixed tempered scale of twelve notes to the octave,such as the flute or oboe, the fifths must necessarily be slightlyflat. Will anyone maintain that if the violins play two oftheir open strings, say A, E, giving the fifths perfect, and theoboes play A, E in unison with them that the sounds are notfor all practical purposes identical ? With the natural hornsand trumpets, as they were written for by the older mastersbefore the introduction of the slide or valve, the intonationwas pure; but this does not prevent their combination eitherin unison or in harmony with other instruments in which onlythe tempered scale was possible. But the argument may becarried even further. Handel has actually used the seventhand eleventh harmonics of the trumpet (the natural trumpet,be it remembered) as the equivalents of the notes of thetempered scale. In the third part of "Judas Maccabaeus "is a song, now always omitted in performance," With honourlet desert be crowned." The song is in A minor, and hasa trumpet obbligato, the trumpet being in D. In this song werepeatedly find both C natural (the seventh harmonic) andG natural (the eleventh) used not only for rapid passing notes,

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Suggested Modifications of Day's Theory 0/ Harmony. 93

where a little impurity of intonation mght pass unnoticed,but as holding notes of the harmony in solo passages. Theeleventh harmonic is frequently met with in other places inhis works; the seventh I do not remember to have comeacross elsewhere; but it seems evident that Handel consideredthe harmonic seventh as the practical equivalent of thetempered seventh. Even supposing that the slide were inuse in Handel's time (of which I have been unable to findany proof), the fact still remains to be mentioned that in thissame song he has in one long and important passage doubledhis trumpet by the oboes in unison, the former certainlyplaying with pure, and the latter with tempered intonation.

Again, many of us are familiar with the common experi-ment of making a low string on the piano sound its harmonicsby " sympathetic resonance "—by raising the damper fromthe low string, and then striking the notes which are• harmonics of that string. After the finger is removed fromthe upper note, the harmonic sound will be distinctlyheard produced by sympathy on the lower string. I havemyself on a Broadwood grand piano often obtained theseventh and eleventh harmonics on the low strings in thisway. The ear could detect the difference of pitch ; but thefact that the perfect fourth was able to set even the perceptiblysharper eleventh harmonic in vibration certainly appears tome to furnish an additional argument in support of my viewthat the two notes are practically, though of course notmathematically, identical.

Some of you will probably by this time have anticipatedthe proposal that I am about to make with the view ofremoving the difficulties arising from these questions ofintonation. As all our music is founded upon the temperedscale, I would take that scale as the basis of harmonicinvestigations. No other appears to me to be possible. Mr.Stephens, in the paper I have already referred to, suggeststhe founding of a system of harmony merely upon thecommon chords of the roots, rejecting the sevenths and thehigher harmonics, such as the eleventh and thirteenth, asbeing out of tune. But, as a matter of fact, in the temperedscale actually in use neither the thirds nor the fifths areperfectly in tune. Let us take the facts as we find them. Iam quite aware that I may be told that the notes of which Iam speaking are not the real and actual sounds. I admitthe fact, and I add that I do not care whether they are ornot; they are practically quite near enough to be acceptedas tl e equivalents of the theoretically correct notes. Forexample, in the key of C, I find three different C's, varying inpitch according to their harmonic derivation and surround-ings. Beginning with the flattest, I have C the seventh of thefundamental chord on the supertonic, next comes C the root

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of the tonic chord, and highest of all, C the harmonic eleventhof the dominant. The ratios of these three C's to one anothercan easily be proved to be respectively 63, 64, and 66 ; andmy contention is that for practical purposes these three notesare identical.

I therefore start with the tempered scale as my ground-work, and hope to be able to show you that with this modifi-cation Day's system is singularly complete and thoroughlyworkable. Let me try, as clearly as I can, to set before youa few of the chief facts which present themselves when viewedunder this modified aspect. The existence of the harmonicseries is of course a matter not of theory but of fact. I takethe notes of the tempered scale as the equivalent of theharmonics for precisely the same reason as I take thetempered thirds and fifths as the equivalent of the trueintervals.

First let us clearly understand what constitutes a Key. Ibelieve that the following definition will meet with generalacceptance.—" A key is a collection of twelve notes withinthe compass of an octave, of which the first is called thetonic, or key-note, to which note the other eleven bear a fixedand definite relationship." The chords in a key are formedby various combinations of these twelve notes. When it issaid that twelve notes constitute a key, it is meant that notmore than twelve can be employed as harmony notes in onekey. In other words, two notes which are enharmonics ofone another̂ —e.g-., D sharp and E flat—cannot both be used inchords of the same key, though it would be quite possiblein the key of C, for example, where E flat is a harmony note,to use D sharp as a chromatic passing-note. If, however, achord be introduced of which D sharp is really one of thenotes, there is a modulation, and we are no longer in the keyofC.

The tonic or key-note is the root from which the wholekey is more or less immediately derived. (I am speakingnow, let me remark in passing) of a major key; of the minorI shall have something to say later.) From the tonic aregenerated naturally the series of sounds known as harmonics.As these are very numerous, we should, if all were used, getso many more than twelve notes derived from the tonic asto obscure all feeling of a key, as that term has been defined:we therefore make a selection, taking first the lowest notes ofthe harmonic series, as springing more immediately from theroot, and therefore having the first claim to a position ofrelationship to that root. Even those of you who maydisagree most entirely with the views I am endeavouring toset forth this afternoon will hardly controvert my statementthat the perfect fifth and major third are derived from anyroot as its third and fifth harmonics. Let me remind you of

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the hannonic series as far as the seventeenth note, taking C asa root:—

1 3 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17Of these the seventh (and of course its octave the four-

teenth) also the thirteenth are a little too flat, and theeleventh somewhat too sharp; the other notes are perfectlyin tune. The even numbers are all octaves of some lowernote, and we need not concern ourselves with them. Forour key we first form a chord on the tonic by the successivesuperposition of thirds. In this way we first obtain themajor common chord, then the chord of the fundamentalseventh, always to be known by its containing a major third,perfect fifth, and minor seventh from the root. The reasonwe take B flat and not B natural for the seventh of thechord is that it is produced earlier in the series, and thereforehas a prior claim, and further that while B flat is a primaryharmonic of the root, B natural is only a secondary harmonic,being really the fifth harmonic from G, the third note inthe harmonic series of C (5x3=15). B flat being thusestablished as a note of the fundamental harmony of C, itfollows that B natural is necessarily excluded, because of theimpossible false relation of the two notes in the same chord.Thus far a fundamental chord always contains the sameintervals; but beyond this point alternative notes presentthemselves. The next third above the seventh is of coursethe ninth ; this may be either major (the ninth harmonic) orminor (the seventeenth harmonic). The eleventh occurs inonly one form—the eleventh harmonic ; but for the thirteenthwe can either take the major (the thirteenth harmonic) orthe minor thirteenth—a secondary harmonic, being the minorninth of the dominant. It will be obvious that our chord-building can be carried no further, because the next thirdabove the thirteenth is the duplication of the root, afterwhich the series recommences.

Let us now look at the notes which we have already in thekey derived from the tonic root, and forming part of thetonic chord:—

Ex.3.

(The (b) show the alternative minor ninth and thirteenth.)We nave now all the notes that can be obtained directly

from the tonic, and even two secondary harmonics—(»'.«.,

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harmonics of a harmonic)—D natural and A flat—which areprimary harmonics of G ; but our key is not yet complete.I have spoken of the tonic as the source whence the wholekey is derived; let me use a familiar illustration, and call itthe head of the family. Then the different harmonics youhave seen springing from the root may be looked upon asC's children, G (the first new note) being the eldest son. Inprocess of time G grows up, marries, and has a family—C's grandchildren—what we call the secondary harmonics ofC. The laws of nature being invariable, the notes of theharmonic series from G will bear the same relation to theirroot or generator as we saw in the case of C. I need notwrite down the series ; but you will see at once that we canmake a fundamental chord on the dominant G, up to themajor and minor thirteenth, exactly as we can from thetonic. Here is the full chord:—

Ex. 4. TjK—(gy

We have here two new notes for our key, the leading note(fifth harmonic of the dominant) and the minor third of thekey—a secondary harmonic of the dominant, as the minorthirteenth from C is the secondary harmonic of the tonic.

We have still one note wanting to complete the key; wehave nothing yet between F natural and G natural. Whereare we to go to get it ? Not to the harmonic series of E,the second note derived from C. This note, as we shall seedirectly, is unavailable as a root in the key. Just as, afterexhausting the resources of the tonic we take its fifth—thedominant—so we now go to the supertonic, as the firstharmonic generated by the dominant. Building up, asbefore, a fundamental chord on this root, we obtain:—

Ex.5.

We have now the material for the key of C quite complete,and we go no further. Why, it may be asked, not go on toA, the fifth of D, and build a series of fundamental chordsupon it ? Not in the least, I conceive, for the reason thatDay gives, because it is too sharp ; for if it is not too sharpto be used in the chords of the key I fail to see why it shouldbe too sharp to use as a root. No, there is a very muchstronger reason than this. If A be taken as a root in the keyof C it will generate C sharp as its fifth harmonic. But wehave already D flat in the key, derived as a primary harmonic

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from the tonic, and therefore having a prior claim; and twonotes having the enharmonic diesis cannot co-exist in thesame key, or there will be more than twelve sounds in thekey. You will now see why after G we took D, and not E,as the next root; because E as its major third would giveG sharp, which is incompatible with the A flat already in thekey as the minor ninth of the dominant. For a similarreason we cannot take the subdominant F as a root, becauseits minor ninth, G flat, makes the enharmonic diesis withF sharp, the major third of the supertonic. No other rootsthan the three already shown can be taken in any one keywithout giving more than twelve notes in that key.

From the three roots now given—the tonic, dominant, andsupertonic we obtain all the chords, whether diatonic 01chromatic, of the key. For instance, the chord of the super-tonic is a portion of the dominant harmony—the fifth, minorseventh, and major ninth; the mediant chord is anothersection of the same harmony—the major thirteenth, root, andthird. And here I may say that, as I ignore, in accordancewith the requirements of the tempered scale, the slightdissonance of the fifth of this interval, I decline to acceptDay's prohibition of the use of this chord in its fundamentalposition. The occasions for its judicious employment are, Iadmit, rare ; but I see no reason why it should be absolutelyforbidden, and precedents for its use may undoubtedly befound in the works of the greatest masters. To continue—the subdominant chord is composed of the seventh, majorninth, and eleventh of the dominant chord, and the frequentprogression from subdominant to tonic harmony is on thisview merely the ordinary passage of dominant harmony totonic, from which, as we have seen, the dominant harmonyis derived, and which thus obeys a very general law as to theprogression of the harmonies of fundamental roots to chordsderived from another fundamental root. Similarly, thesubmediant chord is the fifth, seventh, and major ninth ofthe supertonic; while the imperfect triad on the leading noteis only the third, fifth, and minor seventh of the dominant,with the root omitted. In the same way may be shown thederivation of the chromatic chords in the key—one willsuffice ; and I purposely choose an extreme one. The chordon the minor second of the key, the first inversion of whichis the so-called " Neapolitan sixth," consists of the minorninth, eleventh, and minor thirteenth of the tonic. All otherchords can be explained on this system with equal ease, andthe discords no less readily than the concords.

You will now see, by examining the three fundamentalchords already shown, the true names of the semitones in thechromatic scale. That between C and D is D flat, the seven-teenth harmonic of C, not C sharp, which does not exist in

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the key. Similarly we find E flat and F sharp, both derivedfrom D, the supertonic, A flat derived from G, and B flatderived from C. As a matter of expediency, and when usedas passing-notes, these semitones are often written with adifferent notation, but their true derivation and their har-monic relationship show that their real names must be thosethat I have just given.

One of the most useful points in Day's system is his treat-ment of the question of chromatic chords in the key. Nosingle chord can possibly define a key; every chord thereforemust be either in the key of the chord which precedes it orof that which follows it. Whenever these chords are bothin the same key, the intermediate chord must also be in thatkey; and if one or more of its notes be inflected by an acci-dental, it will be a chromatic chord in the key. As a simpleillustration of my meaning, take the two following progres-sions of chords:—Ex. 6. (a)

At (a) the fourth chord produces a modulation to the keyof E flat, and is therefore considered as belonging to thatkey. At (6) the same chord is followed, as well as preceded,by chords in the key of C, and is, therefore, in this case, chro-matic in the key of C, and not diatonic in the key of E flat.This simple and reasonable explanation does away with thenecessity of explaining these chords as transient modulationsinto other more or less remote keys. This theory of chro-matic chords also furnishes us, as I shall endeavour to showpresently, with some of the most effective means of modu-lation.

It is now time to say something about the minor key, asubject on which I have not yet touched. I need not dwellupon the point that the major and minor keys most closelyconnected are not the so-calJed " relative " major and minor{e.g., C major and A minor), but those which have the sametonic—C major and C minor. On this I think there will beno difference of opinion. The real difficulty in the scientificexplanation of the minor key seems to me to arise from thefact that, whereas every root generates a major third as oneof its harmonics, no root can possibly generate a true minorthird. This is capable of a very simple mathematical demon-stration. The ratio of the minor third, as you will see from

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the harmonic series given above, is 5:6. The minor thirdwill therefore be f above its root. But as the root is always(after the prime) a power of two, an even number in theseries (2, 4, 8, 16, &c), none of which can ever be divisible byfive without a remainder, and consequently cannot be multi-plied by six without fractions in the result, it is clear that noharmonic of a root will ever stand to that root in the ratio of5 :6. The only tonic minor chord to be found among theharmonics is that composed of the seventh, minor ninth, andeleventh of the supertonic; and this explanation is inad-missible, because the tonic being, so to speak, the head andfoundation of the key, the derivation of its .chord from a rootwhich, itself one of the secondary harmonics of the tonic, isas much an inversion of the natural order of things as to saythat King George III. was descended from Queen Victoria.Besides this, the minor third, even supposing we admit it asthe nineteenth harmonic from the root, is excluded from thechords of that root for the same reason for which we excludethe major seventh (the fifteenth harmonic) because it wouldmake a false relation with the major third which is alreadypresent, and which, being generated earlier in the series(as" the fifth harmonic), has the prior claim to a place in thechord. The major common chord is a natural, the minor anartificial product, formed by an arbitrary lowering of theinterval of the third, for which no scientific explanation canbe given. Admit this chord as a postulate, and the wholebecomes clear; for every other chord in the minor key isderived from one or other of the fundamental roots, which,it may be remarked, in passing, is the reason that all thechords of the minor key, with the single exception of thetonic chord itself, may be used as chromatic in the majorkey.

Day's theory of fundamental harmony furnishes us withvery clear guidance as to what discords may be taken withoutpreparation. All the fundamental discords, whether in theircomplete form or, as frequently met with, with certain notesomitted, may be thus treated, because the notes forming themare already generated by nature when the roots are sounded;and these discords are therefore not artificial combinations,but reinforcements of sounds which already exist. Did timeallow, it would be very interesting to go into this subject, andto show you how all these discords resolve either upon theirown roots, or on the roots of the other fundamental discordsin the same key; but this would lead us too far. I mustrefer you to the treatises of Day and Macfarren for the eluci-dation of this important subject.

The system I am discussing also gives us valuable assis-tance in understanding the means of modulation and therelationship of keys. Let me trespass on your patience while

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ioo Suggested Modifications of Day's Theory of Harmony.

I say a few words on each of these important points. Oneof the most frequent, I might almost say the most frequentmethod of modulating is by taking a chord as belonging toone key, and leaving it as belonging to another. I am re-ferring now not to enharmonic modulation, as in the case ofthe chord of the diminished seventh, but to much simplerchanges of key. Take, for instance, an ordinary modulationfrom C to A minor:—

Ex. 7.

Here the fourth chord is taken as the subdominant of C major,and quitted as the submediant of A minor. By admittingchromatic chords in the key the resources of this method ofmodulation are immensely enlarged ; so much so indeed thatit is possible in this way to modulate from one key to anyother. The chord may be diatonic in the first key andchromatic in the second, or chromatic in the first and diatonicin the second, or, lastly, chromatic in both. Let me giveyou one example of each :—

Ex.8.

In this really very simple modulation from C major to Bmajor, which I have purposely chosen as being to one of themost remote keys, the third chord is taken as the first inver-sion of the tonic chord of C and left as the " Neapolitansixth" (the first inversion of the chromatic chord on theminor second of the key) in B major. The modulation Ihave already given you from C to E flat (Ex. 6, a) will serveas an illustration of the second method. Here the fourthchord is taken as the chromatic minor chord on the sub-dominant of C, and quitted as the supertonic chord in E flat.I will purposely choose a concord, and not a discord, for mythird illustration of a modulation by means of a chordchromatic in both keys. I am going to modulate from C toG, not in a manner that I should adopt or recommend for

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ordinary use, but merely to show you what is possible bymeans of these chromatic chords:—

Ex. 9.

Here the fourth chord is taken as the chromatic chord on theminor sixth in the key of C and quitted as the chromaticchord on the minor second of G. It would be easy to spendan hour in illustrating this point, especially if discords areused as well as concords ; but I trust I have said enough toshow you how the whole subject of modulation becomessimplified by this system.

I further believe that this theory gives us a clear insightinto the question of key-relationship. Everyone admits thatthe keys of F major and G major are the most nearly relatedto C. Why ? Not, I think, for the reason frequently given,because their scales contain most notes in common with thatof C; because, if that were the reason, the next nearest keyswould be on the one side B flat, and on the other D; andthese keys, as I shall endeavour to show you directly, arenot the next nearest. I believe the real reason to be thatF and G are the two notes which stand in the simplest har-monic ratio (2 : 3) to C, the former as a generating, the latteras a generated note. I would carry this view further, andsay that the next nearest key to C is E major on the gene-rated, and A flat major on the generating side, because thetonics are in the next simplest ratio, 4 : 5 ; 3 : 4 being, ofcourse, merely the inversion of 2 : 3 and giving no new note.In support of this view, let me adduce the practice ofBeethoven. In the first movements of his sonatas he mostlyfollows the example of Haydn and Mozart, and in a majorkey puts his second subject into the key of the dominant;but when he makes an innovation, what key does he choose ?Never in any one instance the key of the supertonic, or ofthe flat seventh, but invariably a key the tonic of which isdistant either a major or minor third from the tonic key ofthe movement (I need hardly remind you that the nextsimplest ratio to the major third, 4 : 5, will be the minorthird, 5:6). In the " Waldstein " Sonata, Op. 53, we see thefirst subject in C major, the second in E major; similarly, inthe Sonata in G, Op. 31, No. 1, the second subject is in Bmajor, in both cases a major third above the tonic. Again,

8 Vol. 14

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in the Quintet in C, Op. 29, the second subject is in A major;and in the great Trio in B flat, Op. 97, and the Sonata, Op. 106,in the same key, the second subjects are in G major, a minorthird below the tonic. (In each case I am speaking of thefirst movements.) It is worthy of notice that in everyinstance the second subject has more sharps or fewer flatsthan the first; in other words, it is on the generated, and noton the generating side of the tonic. A modulation to a keywhich generates the tonic necessarily produces a more dis-turbing effect on the feeling of tonality than one to a gene-rated key, because in the former case the tonic sinks intoa subordinate position, while in the latter it retains itssupremacy.

Let us look for a moment longer at this subject, which isof great practical importance. We have seen that the keyof E is the next nearest, after F and G, to C ; and that Aruns it very close. A is of course one of the nearest relatedkeys to E ; now let us take the other nearest related—viz.,B major. We are no further from E than we were, but weare now in quite a remote key from C. Why ? Obviouslybecause while E has to C the ratio of 5 : 4 and A has to Cthe ratio of 5 :6, B has to C the much less simple ratio of15 : 16. To take a few more remote keys—C has to C sharpthe ratio 24 : 25, while to F sharp C has the still more com-plex ratio of 32 : 45. All of you who understand theharmonic series can easily verify these figures for yourselves.From what I have put before you I deduce the followingimportant general rule as to key-relationship:—The degreeof relationship of two keys depends upon the ratios of theirtonics to one another. If that ratio be simple (i.e., if thenumbers are small), the keys will be nearly related; if theratio be complex, the keys will be more remote.

I fear I have already trespassed unduly upon your time andpatience, and must hasten to a conclusion. You will, I think,see that, while firmly believing in the general truth of Day'ssystem, I am not prepared to swallow it whole. In somethings I consider him incorrect, and in others dogmatical.From his prohibition of the chord on the mediant in thefundamental position, I have already expressed my dissent.I object also to his statement that there can be no chordof the eleventh except on the dominant. Let me give you abeautiful example of a tonic eleventh from one of the mostcharming passages in the first movement of Schumann's Con-certo in A minor—I quote merely the harmonies, without thefiguration for the piano :—

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The fifth chord of the passage is incapable of any otherexplanation, in its context, than that of the minor seventh,minor ninth, and eleventh of the tonic. I believe I havealso in modern music occasionally met with a supertoniceleventh; but I cannot at this moment recall an example.But there is certainly no reason why the eleventh cannot beused on the tonic and supertonic as well as on the dominant.Again Day, without giving any reason, forbids the supertonicchord to be followed by the chord of the tonic, unless bothare in their first inversion, or the tonic chord is in its secondinversion. This dogmatic rule is contrary to the practice ofthe great masters. Here are two examples from Handel's" Israel in Egypt," in the first of which you will hear thefundamental position of the supertonic followed by the firstinversion of the tonic, and in the second the first inversion ofthe supertonic followed by the fundamental position of thetonic:—

HANDEL—" He spake the word " (Israel in Egypt).

Ex. 11.

HANDEL—" He gave them hailstones " (Israel in Egypt).

Ex. 12.

In the " Benedictus " of Gounod's Cecilian Mass is a beautifulexample of the supertonic followed by the chord of the tonic,both having their roots in the bass. The treatment of theprogression, like many others, requires care ; but I can seeno reason at all for its absolute prohibition. Day andMacfarren further declare that second inversions are onlyallowed with three chords, those of the tonic, dominant, andsubdominant. I grant that they are extremely rare onother degrees of the scale, but that they are not impossibleis proved by such passages as one that I shall give you fromSchumann's " Paradise and the Peri," where the second

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104 Suggested- Modifications 0/ Day's Theory of Harmony*

inversion of the chord on the supertonic is introduced withcharming effect:—Ex. 13.

Other examples may occasionally be met with, especiallyin the works of the older composers.

I have now arrived at the end of a to me deeply interesting,but by no means easy task. I am not so Quixotic as toexpect to induce all of you to agree with the views I haveset forth. On the contrary, I know that there are so manyexcellent musicians who are strongly opposed to Day's theorythat I feel that, in selecting such a subject for my paper, Ishow something of the rashness of a man who puts his handinto a hornet's nest. Pray understand that I intend nodiscourtesy to my audience in the comparison; all I mean isthat I expect to be fiercely assailed on all sides. I haveendeavoured to speak to you to-day not as a scientist—tothat I lay no claim—but as a practical teacher. I have pointedout, with a view to the assistance of such of you as mayteach from this system, the modifications which experience hastaught me it is advisable to adopt; and whatever theoreticalobjections (and they are doubtless many) may be urged againstmy method, I can only say that in practice it works admirably.By its help I have analysed with little difficulty many of themost chromatic progressions of modern music, and have thusfulfilled the prophecy uttered by Sir George Macfarren morethan twenty years ago at the Royal Institution. I cannot dobetter than conclude by quoting his words, which I heartilyendorse:—

" I firmly believe that everything which may be hereafterwritten with good effect, like everything which has up to thepresent time been produced, will be, must be, clearly account-able and fully explicable by the theory it has been myprivilege to lay before you."—("Six Lectures on Harmony,"p. 220.)

It only remains to thank you for the patience with whichyou have listened to a long, and necessarily very technicalpaper.

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DISCUSSION.THE CHAIRMAN..—Ladies and gentlemen, I am sure you

have all been most interested in Mr. Prout's enthusiastic andadmirable lecture. It is well known to all of us that this Daysystem was very eloquently and pertinaciously advocatedand supported by the distinguished man who we all regret isnot with us now. I cannot occupy your time this evening ashe undoubtedly would have done with remarks on thesystem which he did so much to foster. It certainly hasstrong supporters, and very strong opponents. We haveone on my right who is ready for the fray. I have lookedforward to hearing an admirable discussion between thosetwo champions, as I am sure you all do. Probably thataccounts for the very good audience we have this afternoon.I may say for myself that I have derived an immense deal ofhelp from this system. I have of course read Day's book—the original book—and it is wonderful to me that forty yearsago he should have seen so far ahead as to have enunciateda theory which after all is much borne out by modernpractice. I think Sir George Macfarren showed wonderfulacumen in so strongly supporting it at that time, which hedid at a great deal of personal inconvenience and loss.There are many points in the theory to which, as a man whoteaches a great deal, I am not quite able to reconcile myself,and Mr. Prout has in most cases put his finger upon the veryblots which I suppose any teacher of experience has comeacross. But I am bound to say that I do not know anyother book, or any other system, which has, after all, helpedme over ugly corners with my pupils so readily as thissystem of Dr. Day's. I will not detain you longer, but willat once call on Mr. Stephens.

Mr. CHARLES E. STEPHENS.—Mr. Chairman, I am boundto accept your challenge. As an uncompromising opponentof Dr. Day's system, it would have been strange if I had notbeen here to-day, and it would be affectation in me to saythat I did not expect to be called upon to address to you a fewwords on the subject. I should feel inclined to postpone myremarks to an occasion when I should have more time, forto-day I fear I must necessarily weary the patience of someof my listeners. I can but implore you to bear with me inthe task of disputing opinions so eloquently and admirablyexpressed as those of Mr. Prout. I wish you to take noticethat Dr. Day adopts three roots in the key—viz., the tonic,dominant, and the supertonic, and he does not carry thisprocess any further. Thus, in speaking of the key of C hedoes not go on to take A as a root, because, as he says, thisA is " a little too sharp." I do not dwell on this point now,

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io6 Suggested Modifications of Day's Theory of Harmony.

I merely ask you kindly to bear it in mind, as I shall haveoccasion to return to it presently, since this very note, whichhe here says is too sharp, and, therefore, cannot be taken asa root, is afterwards especially assigned as such by himself.Now I maintain that in taking the supertonic as a root thereal and true principle of tonality is entirely lost sight of.Let me define a key in my words, which will differ veryessentially from those of my friend Mr. Prout. I maintainthat a key is a diatonic series of notes, and not the twelvechromatic series of sounds. I point out to you that asingle chord taken—the chord of C, for instance—does notnecessarily denote the key of C, it may belong to variousother keys. If you take a chord of C in juxtaposition withthe dominant of that key, that is, a chord of G, still you donot define your key. What is wanting to define the key ?The subdominant harmony, which, I think, every musicianmust feel is a most important element in the constitution ofa key. Take the triad on C, followed by that on G—it maybe a plagal cadence, in the key of G, or a dominant cadencein the key of C. Take a triad on G followed by a triad onC—it may be a perfect cadence in the key of C, or asubdominant cadence in the key of G. It is the note Fwhich is absolutely essential to decide the key. You thenhave the dominant root on one side of the tonic, and thesubdominant on the other (which I conceive to be the trueprinciple of tonality) and the key is determined and complete.Those three roots furnish you with the entire diatonic scale,with perfect intonation—the C, E, and G from the tonic, F,A, C from the subdominant, and G, B, D from the dominantroot. Having thus endeavoured to show that the sub-dominant triad, or even the subdominant note alone is agoverning power in the key, it must surely be admitted thatthis subdominant must be of truthful intonation. Will thisbe admitted by adherents of the Day system ? May I askMr. Prout whether he admits that the subdominant shouldbe the true fifth below the tonic ?

Mr. PROUT.—Not necessarily.Mr. STEPHENS.—Then you compel me to quote from the

High Priest of this creed.Mr. PROUT.—I do not care what Macfarren says about it.Mr. STEPHENS.—However, I will read it: " The fifth of a

keynote and its dominant, or of a keynote and its subdominant,are, in the scale of nature, perfectly true in intonation ascompared with each other." That is what Macfarren says.Now comes the astounding fact that the Day theorynever supplies us with a truthful subdominant at all. AnF of some kind is, in the Day theory, supplied by tworoots; that from the tonic C gives the F considerablytoo sharp for the true F in the scale, and that from

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the dominant G is too flat. Therefore, the true governingnote in the key is wholly unprovided for in the Daysystem. To pursue my argument. The so-called sub-dominant triad in the Day system is actually derived, asMr. Prout has shown us, from the dominant root, as itsseventh, ninth, and eleventh. Now the seventh is, as alreadysaid, too flat for the note in the scale, the A is somewhat toosharp, and the C is so much too sharp as not to be claimableas C natural at all.

Mr. PROUT.—The whole lot are out of tune—I grant it.Mr. STEPHENS.—And yet this is one of the governing

powers in the key! Let me now go through the whole ofthe notes in the scale, and point out to you that to the key-note itself Dr. Day's system assigns no less than three differentpitches. Can that be a proper system, when two of thosepitches contradict the key-note itself, which is supposed tobe the root and origin of all ? Thus he gives three pitchesfor the key-note, of which two are at variance with thatnote; three for the mediant, two of which are wrong ; twofor the subdominant, both of which are wrong; two for thedominant, one of which is wrong ; two for the submediant,both of which are wrong; and two for the leading note, oneof which is wrong. And this is the system we are asked toaccept as the truth! In other words, what I have saidamounts to this, that in Day's system the tonic gives itsown triad and the supertonic note correctly; the dominantnote gives its own triad correctly, and everything elsewrong; and the supertonic gives every note but itself atvariance with the scale. Now temperament does not helpDr. Day out of this difficulty. Temperament is an expedientto alter slightly that which is right in order to meet certainthings, and thereby falsify to a certain extent for the sakeof compromise; but, on the contrary, the Day theory asksus to take notes all incorrect in pitch, and temper themto make them correct, which is quite the reverse of thetrue principle of temperament. With much left untouchedon this part of the subject, I pass on, and as Mr. Prouthimself acknowledges that this method is full of incon-sistencies, I am sure I shall have his support in whatI am now about to say. Dr. Day amongst his dogma-tisms says you cannot modulate upon a pedal. Let meask you when you go home to turn to Beethoven's sym-phonies, and refer to the second part of the slow move-ment of No. 1, and there you will find modulations upon adominant pedal which no amount of sophistry can possiblyreconcile with Day's dogma that there can be no modulationupon a pedal. Then Dr. Day himself contradicts his ownlaw, since on a dominant pedal, say G, he allows an A to bemade use of as a root in order to modulate the key of the

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108 Suggested Modifications 0/ Day's Theory 0/ Harmony.

minor supertonic. Of course he found that particularmodulation so very common under such circumstances thatit was utterly impossible to ignore it. But in allowing thishe selects the very note as a root which he himself haspreviously told you cannot be used as such, because it is alittle too sharp.

Mr. PROUT.—Quite right; he does that many times.Mr. STEPHENS.—He utterly contradicts himself. Then I

hold that Day treats his subject in an illogical spirit; he for-bids the use of a triad on the mediant, on which Mr. Proutagrees with me, but nevertheless Dr. Day allows that chordto be freely used in its first inversion. What can be thecommonsense or the logic of considering a chord in the key,in its first inversion, but out of the key in its original position !I need hardly say that in the Day theory the second inver-sion of the chord is not recognised at all. Yet I ask you,when you return home to turn to some of the charming littlepieces of Heller, in which you will find it continuallyoccurring, and also in Mendelssohn's March from " Athalie,"where, on the re-appearance of the initial motivo, he har-monises it with a descending chromatic bass. There, on thethird note of the melody, the A, he has the very chord—thesecond inversion of this particular triad. Here I ask you tobear in mind that Dr. Day, when he speaks of the secondinversion of chords, as Mr. Prout has brought before yournotice, allows them on three notes in the key—that is the tonic,dominant, and subdominant. Now there Dr. Day uncon-sciously supplies me with an argument in favour of the sub-dominant, as the true governing principle in the keys. Ifurther charge Dr. Day with being incomplete. We mustnot visit him too severely on the question of the minor mode.As Mr. Prout has very ably said, there are great difficultiesattending the consideration of that question; but this devoteeof the Day system must acknowledge that Dr. Day, no lessthan all his predecessors, has failed to give anything like asatisfactory explanation of the minor mode. He also fails togive an explanation of some of the most ordinary things,some of which I have put on a diagram, to make them moreobvious to you—viz.:—

g r 1 r r — 1 —Omit the second C in the tenor part, and we then have therethe triad on the mediant. With the said C we have the first

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inversion of the diatonic chord of the seventh on the tonic.Now Dr. Day does not admit the triad on the mediant,because he cannot find a root for it, and that seems to be. theonly reason why he excludes it.

Mr. PROUT.—Day's reason for excluding the triad on themediant is not that he cannot find a root, but -that the Enatural and B are both reckoned from G, and do not give aperfect fifth, but 27 to 40.

Mr. STEPHENS.—My analysis of it is that it is a perfectfifth. At all events, among his remarks, Dr. Day says thischord " cannot have the tonic for its root, the major seventhof the tonic forming part of i t" ; which is of course tanta-mount to saying there can be no chord of the major seventhon the tonic. Now, the progression on the diagram is verysimple indeed. I have purposely written the discord with-out preparation, as I do not want to be told that it is a sus-pension. It is a very commonplace progression, but it is notonly not explained, but is absolutely forbidden by the Daysystem of harmony, according to which the B in the secondchord cannot be B natural, but must be B flat, derived fromthe tonic root. I am now going to a rather more recherckiinstance. If you turn to the introductory movement," Chaos,"in Haydn's Oratorio the " Creation," you will find there inthe sixth bar this chord—

which the Day theory does not account for in any way what-ever. Its adherents say that it explains everything that canbe explained, but I say the prophetic words of Macfarrencannot be realised when this, which was written by thegreat Haydn, is not explained by the Day theory and cannotbe explained by it.

Mr. PROUT.—Macfarren does not explain it, but it is abso-lutely incorrect to say it cannot be explained! Why, it isas simple as possible on the Day theory. The root of thischord unquestionably as it stands is the G, B is a majorthird, the D is a perfect fifth, the A flat is a minor ninth, andthe F sharp a fifteenth, a secondary harmonic—it is anunusual form of the chord of the augmented sixth. Excusemy interrupting Mr. Stephens, but when he says it cannot beexplained on Day's theory I must set him right on that. Itis as clear as daylight.

Mr. STEPHENS.—Anyhow, I have not found it in Day'stheory.

Mr. PROUT.—It can be explained all the same.Mr. STEPHENS.—Then among the natural inferences of this

theory is this—that as the roots change, we ought, strictly

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no Suggested Modifications of Day's Theory of Harmony.

speaking, to change the pitch of certain notes. Now, althoughwe admit the tempered scale to reconcile difficulties, andso on, I think we ought to claim that chords should be atleast tolerable in their truthful intonation. Here I havewritten a plain progression:—

E.—80C—64

16

8163

Ii

786672

I

18=

My remarks chiefly refer to the E and C. As derived fromthe tonic they are in truthful intonation ; but when derivedfrom the supertonic the E is too sharp and the C too flat,and when derived from the dominant, the E is too flat andthe C too sharp. The ratio of these notes as derived fromthetbnic is five to four, but I have used higher numbers—viz.,eighty to sixty-four, to avoid fractions in the other cases,which correspondingly give eighty-one to sixty-three, andseventy-eight to sixty-six, that is, a difference of about three-quarters of a tone between these last two derivations of whatis still claimed as a major third. Can you support such asystem as that ?

Mr. PROUT.—Certainly ; there is not the least difficulty inthe world. All I say is that those three C's and three E'sare practically identical. I do not say they are theoretically.

Mr. STEPHENS.—Let me call your attention to the fact thatthe difference of the numbers is sixteen, eighteen, and twelve ;and assuming these numbers as representing proportionatevibrations, with the assertion that eighteen and twelve arepractically identical I cannot agree! I now wish to give youvery briefly an auricular test of the results of the Day system.I have written here, intending the key of C, a passage—viz.,C, C sharp, D, D sharp, and E. That is wrong according toDay's system ; the notation should be C, D flat, D, E flat, andE. Now, some may ask, what is the consequence which waywe write those notes ? I will tell you. My friend Mr. Proutmay say there is no great shakes in my argument; but I amnevertheless going to make a series of shakes on those notes.I shall of course shake on C, and also on C sharp, with D ;but if D flat be written instead of C sharp, I must shake withan E of some kind. I do not want to take any unfairadvantage, so I ask what E should I use for the shake uponthat D flat ? I am going to accompany it with the secondinversion of the minor ninth on the tonic, which comprises

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Suggested Modifications of Day's Theory of Harmony, m

an £ natural. Shall I shake with E flat or £ natural ? Youmay say which I like. I like neither; the E flat is very ugly,and the £ natural will sound ridiculous. Then when I cometo my D sharp I shall shake with E ; but if E flat be writteninstead of D sharp we must shake with F.

Mr. PROUT.—I think the ugliest thing is to want to makea shake on notes in that position at all, and I do not thinkMr. Stephens would think of doing so.

Mr. STEPHENS.—I am only showing you what is the prac-tical result of this system. I certainly should not write ashake on such notes, according to Dr. Day's notation. [Mr.Stephens here played the shakes as they would be in thereceived notation, and in Dr. Day's.]

Mr. PROUT.—That is not Day's system.Mr. STEPHENS.—What shall I shake upon—anything you

like ?Mr. PROUT.—I am really afraid, unless I answer some of

these extraordinary matters as they come up (for Mr. Stephenshas given me such a lot of material) I shall forget half of it,and you will think there is no answer. If Mr. Stephens willexcuse me interrupting, this is not a question of harmony;the shake is not a harmony note. It is simply a question ofnotation of passing notes, and nothing else.

Mr. STEPHENS.—I beg your pardon.Mr. PROUT.—The note with which Mr. Stephens makes

the shake is not a note of the chord at all; it is a passingnote. As a matter of expediency, if it were necessary towrite those notes, I should very likely write that note asC sharp, but the shake note is not a harmony note, and youmay have a note for a shake which is not in the key. Iexpressly mentioned that as a passing note you may haveC sharp in the key or D sharp in the key. The shake doesnot affect the question.

Mr. STEPHENS.—I never claimed that the note.with whichthe shake is made is a harmony note. I merely say I amgoing to make shakes on D flat and E flat: what notes am Ito make them with ?

Mr. PROUT.—I object to your torturing this into a questionof harmony.

Mr. STEPHENS.—I beg your pardon, the D flat and theE flat are notes of harmony. We are recommended thissystem of harmony, and I say, see what it drives us to. Ithink nothing can be more intolerable than the effect of thoseshakes according to the Day notation, whereas with theaccepted notation they are perfectly good, and it would seemthat the partisans of Day would require the composer torefrain from writing shakes on particular notes, simplybecause, according to the notation of Dr. Day, they producea bad effect. It appears to me that is begging the question

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H2 Suggested Modifications of Day's Theory of Harmony.

entirely. I have already trespassed very much on yourpatience, but I go on to maintain there is no necessity forany such cumbrous explanations as are generally given in theDay theory of chromatic harmonies. I look upon them aseither modulations or transient glances at other keys, andsometimes as modifications of notes already existing in thekey, and they all so admit of explanation. I will take oneinstance, that of the chord Mr. Prout mentioned, the chordon the minor second of the scale, the first inversion of which ispopularly called the Neapolitan sixth, of the derivation ofwhich Dr. Day gives no explanation whatever, but merelysays it may be taken. According to my theory, that ofborrowed harmonies from other keys, this chord, assuming Cas our initial key, is borrowed from that of F minor, and if,after the tonic triad of C, we simply interpolate the tonictriad of F minor, and then use what is known as theNeapolitan sixth in the key of C, the whole thing is madeobvious. I understand it to be a rule of the Daytheory that his chromatic chords must be immediatelyfollowed by something especially characteristic of the key, inorder that the tonality may not be obscured. That is reallythe strongest corroboration which can exist of my theory thatthese chromatic chords are glances at other keys, broughtround to the original key by something especially charac-teristic of that key. I will now give you an instance, fromBeethoven, of the chord in question in its first position, andthen supply one chord which will again make my viewsobvious:—

My ear cannot admit this chord of £ flat as being in the keyof D minor at all. I will now supply the modulating chord,and you will see at once that Dr. Day's cumbrous theory istotally unnecessary for things which are quite simple:—

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Mr. PROUT.—What is the next one ?—one chord is in onekey, the chord of E flat in G minor, and the next chord—isthat also in G minor ?

Mr. STEPHENS.—In D minor.Mr. PROUT.—If so, one chord is in one key, and another

in another, without any connection.Mr. STEPHENS.—Let me set you right there. There are

modes of changing key, without resorting to actual modula-tion, by pure transition. For instance, I am in the key of C[Illustrating]. Now I am in the key of A flat [Illustrating],

Mr. PROUT.—Not necessarily.Mr. STEPHENS.—I can so go through the whole cycle of

keys.Mr. PROUT.—You can go, but there is no connection,

because you are in no key at all when you have simply thosethree chords one after the other.

Mr. STEPHENS.— In the instance I have given fromBeethoven, I cannot feel myself to be in D minor allthrough, and supplying the dominant of the key of G minorshows the whole thing. I have already trespassed toomuch upon your patience, but you will admit that withsuch an opponent I have a very formidable task. I haveendeavoured to meet him, and we have discussed in thegreatest good humour, and I hope we shall continue to do so.All he has said to-day does not in the least shake my con-viction that the endeavour of Dr. Day—a most earnestendeavour—is nevertheless the outcome of a purely empiricalsystem; and I do maintain that all these chromatic harmonieswhich he asserts to belong to the key are not in the key,which I have defined to you as consisting of the diatonicharmonics on the tonic, dominant, and subdominant, the oldwriters on theory being thoroughly justified in giving thosenames to those three roots, in accordance with their trueharmonic position. They also open up the whole systemof cadence, and much besides that it is essential for a composerto understand. I admit that other theorists have failed toexplain many things, but I hold that the grand desideratumhas not been met by Dr. Day, and Mr. Prout, so far as I canunderstand, agrees with me that he has not supplied all thatis necessary. Let us hope some theorist will arise who willremove all difficulty and reconcile all things that appearcontradictory, but I maintain it is impossible to altogetherdiscard the mathematical view from the question, becauseyou must remember that the major third, and the fifth, andthe octave, and other intervals, are not the invention of man.They are not arbitrarily assumed things, but they are thingsthat are given us by nature itself, and every monochord,every bit of string you stretch, every pipe, contains those veryelements which are really the foundation of our musical system.

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i i4 Suggested Modifications of Day's Theory of Harmony.

Therefore, I maintain we cannot discard the mathematical view.The tempered scale is one which is expedient, under numerouscircumstances which I cannot enter into now. A long arrayof brilliant names may occur to you as of those who havewritten on the subject of theory—viz., Rameau, Derode,Sabbatini, Sorge, Schnyder, Jelensperger, Albrechtsberger,Abbe Vogler, Gottfried Weber (chief among the unbelieversin physical demonstration), and Baron Blein, who wouldadmit nothing that could'not so be proved; but I think that thetrue theory of harmony has yet to be written. Possiblyamong those who are here to-night we may find the futuregreat theorist, but I do emphatically deny that position toDr. Day.

Mr. SOUTHGATE.—May I ask one question ? Dr. Day'ssystem, so far as I can understand it, has one patent defect, thatis the want of providing the means of getting the true sharpenedtonic, if I may so call the note. Mr. Prout said that Dr. Dayobtained what is practically the sharp tonic by means of aflat instead of a sharp notation; I would ask him,carrying out the analogy which he has given us of thederivation of the different chords, and their children or grand-children, why does he not go one step further, and take the Aas the root, which is almost perfect to the C, and then fromthat A why he does not derive the major third, the C sharp ? Iwould also remark Mr. Prout would say it is a questionsimply of altering the notation, but in writing I am quite certainhe would not dare to write a modulation into the key of Dwith the A root, and practically with D flat as the leadingnote.

Mr. PROUT.—I should not think so. I should think themodulation took place on the chord of A; it is a simple thingto answer Mr. Southgate's question with regard to the Csharp. I mentioned it incidentally in the paper. We do nothave C sharp in the key from the root of A, because we havealready D flat from the root of C, C being the tonic, D flatbeing the highest note I wrote down of the harmonic series.Being there already by virtue of its derivation from the tonicit ousts any possible enharmonic—any possible C sharp. Itis there " fixed in its everlasting seat," as Handel's chorussays, and consequently no C sharp can dispute its place. Ifthe D flat is going to resolve as a chromatic semitone as amatter of expediency, I should write it as C sharp, althoughI knew it was really D flat.

The CHAIRMAN.—If no other lady or gentleman wishes tospeak, I will ask Mr. Prout to reply.

Mr. PROUT.—I have kept you already so long that I willnot say more than just a word or two I really find verylittle to answer in Mr. Stephens's remarks, because he hasbeen " bowling wides " all the time. He has been attacking

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Day, and I have been attacking Day in many respects, andin a great deal of what Mr. Stephens says as to this accuracyand inaccuracy of intonation I go with him entirely: he isperfectly right. But what I say is this. You cannot takeany of these absolutely correctly intoned upper notes; it willnot work in practice. Mr. Stephens objects to the temperedscale as being a compromise. It may be, but it is a com-promise on which Bach wrote his forty-eight Preludes andFugues, and on which all the great composers have written.It was good enough for them, and it is good enough for me.I should rather make theory conform to the practice of theold masters than try to stretch the practice of the old mastersby out-of-the-way things to suit a mathematically-correcttheory.

Mr* STEPHENS.—I entirely go with you with regard to thetempered scale. I merely point out that we should be ableto bear all chords with their just intonation. But there is nomusic without the tempered scale—none at all.

Mr. PROUT.—I say let us take the tempered scale, which isthe only thing we can go upon. Mr. Stephens comes hereand tells me one thing, I tell you something entirely different.If all the gentlemen who teach harmony in this room were tocome together, I doubt if we should find any two of themagree on any one thing, and probably there would be five orsix systems before the discussion was over.

Mr. STEPHENS.—Allow me once more to interrupt. Youhave not been adhering to Dr. Day, you have been merelydefending the tempered scale. You start from the temperedscale.

Mr. PROUT.—Excuse me, I say let us take the temperedscale as the equivalent—that is what I want to make clear—the equivalent for all practical purposes of the true notes.Mr. Stephens himself says he believes in the tempered scale,without that we can have no music, but talks about perfectthirds and fifths which we do not get on the piano or in theorchestra. The tempered thirds and fifths are practicallyequivalent to the perfect intervals. AH I do is to carry thesame principle of temperament a little further and say I dealwith the tempered seventh and the tempered ninth, andtherefore I take the harmonics and carry them up. If youtake that system, and instead of adhering absolutely to astrict mathematical basis, you adopt the tempered scale inpractice, whatever theoretical objection you may make to it,it works splendidly. Mr. Stephens said Dr. Day could notmake out that chord from the " Chaos" overture, but it is assimple as possible.

Mr. STEPHENS.—I cannot make it out from Day.Mr. PROUT.—I can see around me half-a-dozen of my

Royal Academy pupils, any one of whom I would back at

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u6 Suggested Modifications of Daft Theory of Harmony.

six to one to tell that chord as quickly as I did, if I hadasked them.

Mr. STEPHENS.—How about the subdominant as a rulingpower of the key I

Mr. PROUT.—I do not admit it is a ruling power of the key,because it would reduce the tonic to a subordinate positionas a harmonic of the subdominant, instead of its being thehead and foundation of the key. I should like in return toask Mr. Stephens, if we had time, when he adds his D tothe subdominant, and makes it what is commonly called thechord of the added sixth, whether he still regards it as thesubdominant chord which will have just the same effect inmaking the cadence. That chord of the added sixth is onestrong argument in favour of the generation of the sub-dominant not being from the tonic. It is the third inversionof the dominant eleventh.

Mr. STEPHENS.—Do you admit the term added sixth ?Mr. PROUT.—It is convenient.Mr. STEPHENS.—It is altogether wrong.Mr. PROUT.—It does not accurately express it, but that

is a matter into which one could go at any length. It justcomes to this, that Mr. Stephens and I differ very widely onmatters of theory, but we should both agree to a great extenton most of these matters if it came to a question of practice.We are simply looking at different sides of the same shield.One thinks one way is the best for getting over difficulties,and the other thinks another is the best. I have given youthis afternoon what I consider the best way of explainingthem, and the simplest way in the long run, although it wasa little bit intricate to myself at first; but when you do getover the difficulty, I believe you get such a grip of thesubject as you can get in no other way. I believe that isthe experience of most people who have studied this systemthoroughly, especially teachers who have taught from it. Ihave spoken to many who tell me the longer they teach fromit the more conveniently the thing works. Mr. Stephenswill teach on quite a different system, and perhaps someother afternoon you will have an opportunity of hearing himexplain the principle on which he would teach. I do notknow whether there is much else to say ; I can only say thatMr. Stephens has not convinced me in one single instance,or converted me from the error of my ways, if error it is.

The CHAIRMAN.—There is one point on which I mustsay I am bound to go with Mr. Prout and Dr. Day's theory,and that is more particularly about the definition of the key.I think it is such an advantage to define a key as includingall those twelve notes, and not merely that very meagrearrangement of notes that Mr. Stephens wants. It is anadvantage also to recognise that you are able to use all the

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chromatic chords in the major key, without having toexplain that you have just dashed for a moment out of yourhouse into the next, and back again. You can take anumber of chords apparently foreign and you canuse them without introducing modulation. That definitionof a key as including twelve notes I think is extremelyvaluable and useful. If I continued I might be involved ina sort of triangular duel between Mr. Stephens, Mr. Prout,and myself, in which I fear I should come to grief. Itherefore move a very cordial vote of thanks to our dis-tinguished lecturer. It is a great thing to get a man in Mr.Prout's position, who, as you know, is one of the mostprominent musical critics we have, to give his valuabletime, not merely in coming here to read a paper, but toprepare a paper of this sort. It has given us all a greatdeal of pleasure. I have never seen an audience so in-terested or antagonists so good-tempered. I will thereforeask you to join in giving Mr. Prout our most cordial thanks.

Mr. STEPHENS.—Allow me to have the pleasure of second-ing that. Mr. Prout has come forward and stated his con-victions. Our object here is not hero worship, or anythingof that kind, but to try and elicit truth. Mr. Prout hasendeavoured to do it according to his view, and I haveendeavoured to dispute what he said according to my own.I have the sincerest pleasure in seconding the resolution.(The vote of thanks was carried unanimously.)

Mr. PROUT.—I have already occupied your time so longthat I shall only thank you for the vote of thanks as wellas for the very patient attention you have given to a some-what intricate discussion.

Mr. SOUTHGATE moved a vote of thanks to the Chairman,which concluded the proceedings.

9 Vol.14

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some suggested modifications of day's theory of harmony

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