anger - the modern enharmonic scale as the basis of the chromatic element in music

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(A NEW PHASE IN THE SCIENCE OF HARMONY)

A CertitrrDelnrered at the Toronto Conservatory of Mumc

(mider ihe awpicc* o( the TeroM* Ck( CkibXDecember Ift. 1906

11I, I

if

BY

J. Ifttttifrrg Attgrr

TOBONTO

:

WM. TYRRELL h CO.

ft53 ' I

Entered according to Act of the Parliament of Canada in the yearone thousand nine hundred and wven, by J. HuuraBT Amgbk,of Toronto, at the Department of Agriculture.

THE MODERN ENHARMONIC SCALE.

fe,

aLTHOUGH the subject of this paper is nominallythat of a scale, yet the object, and real ptirpose inview, is to offer to the world of music a new theory

of Harmony ; a theory having reference to the origin,basis and use of the chromatic element in Music; and atheory to supersede that originated by Dr. Day, whichfails to explain satisfactorily the progressions employedby.modem composers.mCsir Hubert Parry, the nmsical director of the RoyalCollege of Music, in the article on " Day," in Grove'sDictionary of Music and Musicians, says : " This is notthe place to point out in what respects Dr. Day's hypothesisis vulnerable." Is not this an avowal on the part of one ofthe most eminent of living conqx>sers and theorists, thatthe Day theory is vulnerable ? and vulnerable means opento attack and defeat. Again quoting from this article :" No other theory yet proposed," says the author, " canrival it in consistency and comprehensiveness." Thisdistinguished musician, therefore, is presumably of theopinion that a theory may yet be proposed which will beboth nK>re consistent and more comprehensive.

A modem author of a valuable text-book on Harmony,**Dr. Charies Vincent, author of " Hannony, Diatonic and

Chromatic." The theories on th

4 THE MODERN ENHARMONIC SCALE,

yi that he bases his system " on the Diatonic, Chromatic,and Enharmonic scales, and ventures to prophesy thatthe text-book of the future must and will follow somesuch plan, in order consistently to explain modem combin-ations, which, under the theoretical systems at presentin vogue, require a wrong application of the laws of acousticsand a complex multiplication of roots,"

Another work on Harmony, one of the most recentworks on this subject,* in the chapter on Chromatic Triuds,contains the followinti significant statement: " The worksof modern comjioscrs, such as Dvorak, Grieg, and Wagner,show that it is jmsiblc to employ a major triad on everychromatic degree of the scale. For the present, however,the student should confine himself to the chords heregiven." Again, in the chapter on Other Chords of theSeventh, a similar statement is made, namely, that " theworks of modern composers, e.g. Dvorak. Grieg^ and Wagner- furnish examples of many other chromatic sevenths usedin a key without modulation. It is well for the studentat present to use only those already given." Here we seethe genuinely honest intentions of the authors completelybaffled by the limitations of their theory. The work isadmitted to be based on the system advocated by Mac-farren, the immediate successor of Day; it is therefore oneof the many existing treatises on the Day theory, and con-sequently the tivrks of the best modern composers are noturitten in accordance with this theory. With regard to theexpression, "for 7 he present," it may be said that the" present " lasts throughout the work, so that the student(the hapless student) is left to discover for himself whatchromatic triads and chords of the seventh are at hiscommand when he enters the realm of composition.L- r

",A '-'o"'^ of Harmony," by Sir Frederick Bridge and Dr.frank J. Sawyer, a work written at the request of the Council ofthe Royal College of Music.

in

THB MODERN ENHARMONIC SCALR.

With regard to the theory that is about to Iw proposed,in one sense it may not be new; it is, possibly, the theoryupon which the older classical masters, such as Bach andBeethoven, wrote their immortal compositions, and it isprobably the theory adopted by modem composers, suchas Dvorak, Grieg, and Wagner. This theory is simplyreduced to a practical system, so that the works of the greatmasters may be analysed without the necessity of havingto offer any apology, or what is worse, without the neccs-ity of having to state that they the great masters-employed incorrect notation, which is one of the mostunfortunate features and defects of the theory advocatedby Dr. Day.

And with regard to the Modem Enharmonic Scale, it Usimply a chromatic scale, in which qertain notesto beexact, three of the chromatic notesappear each undertwo different names. This scale is intended for studentsof harmotiy only, and particularly for those who are takingup composition, so that they may see at a glance whatchromatic notes are available for use, and that they maybe able to understand the principles upon which the greatcomposers employed the chromatic element in music.

It must be apparent to every lover and connoisseurof music Ihat a great change has been effected in the artof music during the past 30 years or so. This change maybe said to have been inaugurated by Richard Wagner.The school, of which Wagner is the prototype, was, thirtyyears ago, ironically alluded to as " the music of the future,"but many a true word is spoken in jest, and the music of thefuture it proved to be, notwithstanding the critics of the lastgeneration. Wagner now holds the honored position ofbeing regarded as one of the greatest of musicians. Thenames of Dvorak and Grieg have already been mentioned

;

to these may be added such names as Brahms, Coleridge-


Taylor, Elgar, Leoncavallo, MacDowell, Mascagni, Parry,Puccini, Saint-Saens, Strauss, Tschaikowsky, Widor, andmany others whose compositions are not only fresh andoriginal, but are also illustrative of the marvellous andinexhaustible resources of the art.

A brief review of the past will bring to mind the factthat a change in the style of musical composition is not con-fined to one generation, it would seem to occur in a greateror less degree with every generation. Thirty years priorto Wagner's triumphs at Bayreuth, we find Schumann atthe height of his fame, and honored with the degree ofD. Phil. (1840) ; while it may also be mentioned that thefirst Conservatory of Music was founded by Mendelssohnin this same decade (1843) at Leipzig. Thirty yearsearlier, 1810-1820, Beethoven, the generally acknowledgedgreatest genius of all, was in "his freest and most joyouscreative period." Thirty years earlier, again, and we findMozart at the zenith of his career (1780-1790); while hisgreat contemporary, Haydn, about this time founded themodem orchestra, and established the form of the sonataand symphony. And, one generation earlier brings us tothe days of Bach and Handel, in whom is to be found thefountain-head of the modem art of music. To Bach, be itsaid, we are indebted for the establishment of equal tem-perament, that is to say, the division of the octave intotwelve exactly equal semitones, as now employed in music.But with all the advance that has been made in compo-sition since the middle of the nineteenth century, whatadvance has been made in that science which is the funda-mental basis of compositionHarmony ? Is not thesystem employed by Beethoven, and the other great com-posers, sufficient for all time, so long as music is basedupon the laws of equal temperament ? Can the progres-sions employed in modem compositions be satisfactorily


explained by the theories at present in vogue ? Suchquestions as these will now be considered, but, it may herebe said that little or no advance has been made in thescience of Harmony since the year 1850; that the theorywhich arose about that date fails to meet modem require-ments; and that the system employed by Beethoven isthe only true system, but that it has never, hitherto, beeneither consistently or comprehensively expounded.

Just as the diatonic scales constitute the basis ofdiatonic chords, so the origin of chromatic chords is to befound in a chromatic scale.

The introduction of the chromatic element into musicalcomposition is attributed to Orlandus Lassus (Roland diLattre), 15 20- 1594, and though employed more or lessfreely by all succeeding composers, yet from neither amelodic nor harmonic standpoint was any attempt appar-ently made to systematise the subject until about themiddle of the nineteenth century, when the notable workon Harmony by the above-mentioned Dr. Day appeared.

The text-books of Germany, the home of the greatestmasters of music

; of France, a country noted for itsscientific researches

; of Italy, the birth-place of modemmusic

;and of England,* second to none in her love for

the art; all simply refer to a few of the most frequentlyemployed chromatic chords, explaining their treatmentand giving examples of their use; but, while the diatonicelement is treated systematically, in no case is the chromaticelement so treated, in no case is the origin, from whichchromatic notes are derived, explained, and in no case isthe theory expounded in such a manner that the student

* Whenever reference is made in this paper to the " Enelish "theory, it is understood to be that which was in existence prior tou J *"Ji.^ ^'^^ *? theory. The English theory comprisescnoros ot the ninth and their derivatives, a feature not jienerally

accepted by the German authors.

8 THB MODERN ENHARMONIC SCALE.

can grasp the principles upon which the great composersobtained their chromatic effects, or leam what material isavailable for use in his own compositions.*

The theory enunciated by Dr. Day has given rise tomuch controversy in the world of music, though rejected bymany of the most eminent theorists as being illogical, it has,nevertheless, been accepted by other very eminent theoristsas the correct basis of the chromatic element ; consequently,the modem student, whatever his views may be, shouldat least be familiar with this theory.

The following brief account of Dr. Day is taken fromthe above-mentioned article in Grove's Dictionary.

" Alfred Day, M.D., the author of an important theoryof harmony, was bom in London, in January, 1810. Inaccordance with the wishes of his father, he studied inLondon and Paris for the medical profession, and, aftertaking a degree at Heidelberg, practised in London as ahomoeopathist. His father's want of sympathy for hismusical inclinations in his earlier years having preventedhim from attaining a sufficient degree of practical skill inthe art, he turned his attention to the study of its principlesand formed the idea of making a consistent and com-plete theory of harmony, to replace the chaos of isolatedrules and exceptions, founded chiefly on irregular observa-tion of the practice of great composers, which till com-paratively lately was all that in reality supplied the placeof system. He took some years in maturing his theory, andpublished it finally in 1845, three years only before hisdeath, February 11, 1849."

Dr. Day's theory, even before his work on harmony waspublirhed, was endorsed by Sir George Macfarren (18 13-

*This assertion is made in good faith ; the author is notaware that any attempt has been made to systematize the chromaticelement in music in any work on Harmonv other than that byDr. Day, or in a work founded on the Day theory.

THE MODERN ENHARMONIC SCALE. 9

1887), who in i860 published his own work, entitled, " TheRudiments of Harmony," in which he not only advocated,but also somewhat further developed, the principles origi-nated by Dr. Day. Macfarren succeeded Sir StemdaleBennett as Professor of Music at Cambridge in 1875, in thefollowing year became principal of the Royal Academy ofMusic, and in recognition of his services in the cause ofmusic, was knighted in 1883. With such a champion, theDay theories were rapidly promulgated, and before theend of the nineteenth century many eminent theorists inboth Europe and America had adopted th^ same, hadtaught the same to their pupils, and had written furthertreatises upon the same. Yet, notwithstanding its numerousadherents, since, theorists, equally eminent, have refusedto accept the Day theory, would it not appear, and may itnot be assumed, that up to a certain point this theory mustbe logical, and the arguments conclusive, but that beyondthis point it and they give rise to diversity of opinion ?

Dr. Day divides the subject of Harmony into twodistinct parts, namely, (i) the Diatonic or strict, and (9)the Chromatic or free ; each part is treated independentlyof the other, but they are united, as it were, by a chapteron what he terms. Diatonic free harmony. Since the prin-ciples enunciated in Part I. have not been adopted, so faras any original treatment of the subject is concerned, by anyof his successors, and since they have practically no bearingon the subject under consideration, namely, the chromaticelement in music, it will not be necessary to devote anyattention to them on this occasion. It is in his treatmentof " Chromatic harmony, or harmony in the free style," ashe calls it, that Dr. Day's reputation was established.

" His explanation of the chromatic system," says SirHubert Parry, in the able article on Day, above mentioned," was quite new, and his prefatory rem- :s so well explain

xo THE MODERN ENHARMONIC SCALE.

his principles that they may be fitly quoted. After pointingout that the laws of diatonic harmony had been so stretchedto apply them to modem styles that they seemed ' utterlyopposed to practice,' he proceeds ; ' Diatonic discordrequire preparation because they are unnatural ; chromaticdo not because they may be said to be already prepared bynaturesince the harmonics of a root note give the noteswhich form with it the combinations he calls fundamentaldiscords. The harmonics from any given note are a majorthird, perfect fifth, minor seventh, minor or major ninth,eleventh, and minor or major thirteenth.' And this seriesgives f- complete category of the fundamental chords ofDay's "h jmatic system. Moreover, with the view ofsimplifying the tonal development of music, and giving alarger scope to the basis of a single keyand therebyavoiding the consideration of innumerable short t/ansitions

^he gives a number of chromatic chords as belongingessentially to every key, though their signatures may not besufficient to supply them, and with the same object buildshis fundamental discords on the basis of the supertonicand tonic as well as on the dominant. In respect of thishe says : ' The reason why the tonic, dominant, and super-tonic are chosen for roots is, because the harmonics innature rise in the same manner ; first the harmonics ofany given note, then those of its fifth or dominant, thenthose of the fifth of that dominant, being the second orsupertonic of the original note. The reason why the har-monics of the next fifth are not used is, because that noteitself is not a note of the diatonic scale, being a little toosharp (as the fifth of the supertonic), and can only be usedas a part of a chromatic chord.' The advantages of thissystem of taking a number of chromatic chords under thehead of one key will be obvious to any one who wishes fo^a complete theory to analyse the progressions of 4ioytfm

cAj\d4


modem music as well as their harmonic structure. Forinstance, even in the early ' Sonata Pathetique ' of Beet-hoven, under a less comprehensive system, it would be heldthat in the first bar there was a transition from the originalkey of C minor to G; whereas under this system the firstmodulation would be held to take place in the fourth bar,to E flat, which is far more logical and systematic."

From this extended quotation it will be seen that Dr.Day advocates the formation of a chord upon the dominantconsisting of a series of super-imposed thirds, and com-prising every note in both the major and minor mode

;

this chord, known as the dominant thirteenth, could beemployed in the following fc.r different forms.

,i

Fig. I.

(a) With the major ninth and major thirteenth, (b)with the minor ninth and minor thirteenth, (c) with theninth major and thirteenth minor, and (d) with the ninthminor and the thirteenth major.

It is not to be supposed that all the notes of thesegigantic chords were intended to be sounded simultaneously,although such a discord might under certain conditions bepossible

;the special use of these chords was apparently

in connection with the analysis of chords. The convenienceof the system is most patent, for, since the chord at a, forexample, contains all the notes of the major diatonic scale,so every diatonic chord might be regarded as a more or lessincomplete form of this chord, and therefore could bereadily explained.

la THE MODERN ENHARMONIC SCALE.

Chords of the ninth, eleventh, and thirteenth, it mayhere be said, are mentioned in the work on Harmony b;*Albrechtsberger, 1736-1809, but the chords in this caseare formed by adding a third, a fifth and a seventh, respec-tively, below a chord of the seventh. The idea of addinganother third above a chord of the ninth, so forming achord of the eleventh, and then another third, so forminga chord of the thirteenth, was presumably original on thepart of Dr. Day.

The most important feature in the Day theory, however,is the systematic method in which the chromatic element inmusic is treated. Fundamental (or primary) sevenths onthe tonic and supertonic are, for the first time in the historyof music, regarded as chromatic chords ; and the argumentsbrought forward to prove that these chords may be em-ployed in a key without inducing a modulation should besufficiently convincing to every modem theorist. Chordsof the major and minor ninth on the tonic and supertonicare next considered. The chord of the eleventh on thesenotes is forbidden by Dr. Day : " The chord of theeleventh," he says, " unlike the chords of the seventh andninth, can only be taken on the dominant." The reasongiven for excluding this chord, is "because its resolutionif taken on either tonic or supertonic, would be out of thekey." Finally, the chords of the thirteenth on the tonicand supertonic, the ninth and thirteenth being eithermajor or minor, as in the case of the dominant thirteenth,are considered.

Although Dr. Day excluded the eleventh from thetonic and supertonic thirteenths, and notwithstandingthat Macfarren endorses this particular feature of thetheory, yet more modern authors have not only included theeleventh in these chords, but have also advocated the use ofthe chords of the tonic and supertonic eleventh.


The three chords of the thirteenth in their completeforms may be exhibited thus :

Fig. 2.

These chords, having once been established, became theharmonic basis of the chromatic scale which Dr. Day adopt-ed, and which has since become known as the harmonic formof the chromatic scale. But, it may be said, so far as thisscale itself is concerned, that it is obtainable from the chordsof the minor ninth on the tonic, dominant and supertonic,without any reference to the elevenths or the major or minorthirteenths.

Mention must be made of Dr. Day's valuable argumentson the difference between the diatonic and chromatic semi-tone. He conclusively proves that theoretically thediatonic semitone is the larger of the two, in other words,that the interval C to D flat is slightly greater, from thestandpoint of acoustics, than the interval C to C sharp.This point, however, is of no practical importance in con-nection with harmony, for, according to the laws of equaltemperament, these semitones are identically the same inpoint of size

; they differ only in notation and use.Reference must also be made to one particular chord

which is considered under the heading of the dominanteleventh, namely, the chord generally known as the " Addedsixth." This chord is claimed to be the third inversion ofthe dominant eleventh, the root and third of which are, ofcourse, omitted.

M THE MODERN- ENHARMONIC SCALE.After the treatment of the fundamental discords has

been thoroughly exhausted, Dr. Day next considers thechords of the Augmented sixth. For these chords, probablyfor the first time in the history of music, a double root isclaimed. The bass-note of the following chords, forexample, is regarded as a primary harmonic of the generator

Fig. 3.

(a) (d) (e)Italian French. Oerman.

G, the dominant, while the upper notes are regarded asprimary harmonics of the generator D, and, therefore, assecondary harmonics of G. By secondary harmonic isunderstood a note generated from one of the harmonicsof a given genr

-ator. Dr. Day states that the interval of theaugmented

-ixth " should not be inverted, because the uppernote, being a secondary harmonic and capable of belongingonly to the secondary root, should not be beneath the lower,which can only belong to the primary root. ' ' The three formsof this chord may occur, not only on the minor submediant,as exemplified in Fig. 3, but also on the minor supertonic

;

in the latter case the tonic and dominant are the roots.The chromatic element in music is finally considered in

connection with " Passing notes in the free style," when,in certain cases, the necessity for the use of the melodicform of the chromatic scale, which has hitherto been ex-cluded, is recognized. After which a chapter on enhar-monic modulation brings this notable treatise on harmonyto a conclusion.

THE MODERN ENHARMONIC SCALE. >5

Up to the chord of the dominant seventh, the Germanthe English, and the Day theories of harmony are practi-cally in perfect accord, from this point onward, howeverthey commence to diverge. Chords of the seventh, ortetrads, as they may more conveniently be called, on otherdegrees of the scale, arc acknowledged by the German andthe English, and sometimes by the Day theory; at othertimes such chords are regarded as being derived from thedominant, with the dominant omitted, and not as inde-pendent chords.

Chords of the ninth, or pentads, are acknowledged byboth the English and the Day theories, but according to theGerman theory, the ninth is regarded in the light of a sus-P'.nsion or an auxiliary note; and though undoubtedlythe ninth is treated as one of these discords in the greatmajority of cases, yet there are exceptions, as in the fol-lowing transcription from a string quartet bv such a com-poser of the strictly German school, as Havdn

:

Ji2ii J- i

Pig. 4.

According to the English theory the chord at * is thedominant major ninth. It would be interesting to knowwhat name would be given to this chord bv a theorist ofthe German school

; possibly the ninth would be regardedas a " Changing note," a term which seems to be appliedby some writers tc any kind of auxiliary note when em-ployed not in accordance with the generally accepted laws.

i6 THB MODERN BNHAKMONIC 8CALB.

The chord of the seventh on the leading-note of themajor mode, according to the Day theory, is known as"the first inversion of the dominant major ninth "; but,since this chord does not contain the dominant, and con-sequently not the distinguishing interval of the ninthabove the dominant, the name is little short of a misnomer.According to the English theory this chord is regarded asa derivative of the dominant ninth, and is known as " theleading seventh "; but even this name, though satisfactorywhen applied to a derivative of the dominant ninth, is notapplicable to a derivative of any other chord of the ninth,such as the tonic or supertonic ninth.

The chord of the seventh on the leading-note of theminor mode, according to the Day theory, is known as" the first inversion of the dominant minor ninth," alsofor the same reasons as given above, really a misnomer.According to the English theory this chord is also regardedas a derivative of the dominant ninth, and is known as thechord of " the diminished seventh " ; Dr. Day, be it noted,also refers to this name for the chord, but Macfarren totallyignores the name. It is also called the chord of thediminished seventh by German theorists, but they seemto have no name for the leading seventh, other than " thechord of the seventh on (or of) the seventh degree."

Since the derivative of a fundamental or primary minorninth is called a chord of the diminished seventh, after thename of the interval between the third and ninth of thegenerator, so the derivative of a fundamental or primarymajor ninth may be called a chord of the minor seventh,after the name of the interval between the third and ninthof its generator. Although the interval of the minorseventh, unlike that of the diminished seventh, occurs inother chords, yet the use of the term " minor seventh "will only be necessary when the chord comprises, like that

THBMODBRN BNHARMO MC SCALE. 7

of the " diminished seventh," a diminished fifth and a minorthird. The interval of a minor seventh occurs not only inthe chord under consideration, but also in the dominantseventh and in all primary sevenths, and in the secondaryseventh on the supertonic and on other degrees of the scale,but no confusion will arise in this respect, for, all chords ofthe seventh other than the minor and the diminishedseventh are named after the root upon which they areformed.

The convenience of this term, " chord of the minorseventh," or "minor tetrad," wilt be seen from the followingexample. The chord at a is the diminished tetrad on themediant

;that at b, the minor tetrad on the mediant ; in each

case the tonic, C, is the generator. The name " leadingseventh " for the chord at b would be incorrect, for, E (theroot) is not the leading-note

; while the name " firstinversion of the tonic major ninth " is only applicable toa degree, for, there is no tonic present, nor according to thefiguring would the presence of the tonic be justified, whilethis name is distinctly applicable to the chord at c. whichthough similar to, is not the same as, that at b.

Fig. 6.

In other respects, that is to say, setting aside the dif-ferent views with regard to chords of the ninth and theirderivatives, the English and the German theories may besaid to be practically the same ; the Day theory, however,

It THE MOOBRN BNNARMUNIC SCALE.

not only proceeds to the chords of the eleventh and thir-teenth, thereby differing from the other theories in thematter of diatonic chords, but it alio treats of the chro-matic element in harmony from an entirely different stand-point. In the Day theory, as was stated kjx)ve. chromaticharmony is reduced to a system, but this is not the casewith the older theories. That the great classical compo-sers based their chromatic progressions upon sotne systemthere can be little or no doubt, and it is often said that thiasystem was based upon the happy-go-lucky rule thatchromatically raised notes must be resolved by ascendingand chromatically lowered notes by descending; but itshould rather be said that this rule arose from the system,and not the system from the rule. In any case this freeand easy rule is not applicable to the Day theory of chro-matic harn.jny. wherefore, the great composers did notwrite in accordance with this theory.

The adherence given to the Day theory by so manyemment musicians is probably traceable to two considera-tions, firstly, a desire to have some system, rather than noneat all. in connection with chromatic chords, and secondly,the fact that it is a comvuietit theory

; for. since the domi-nant thirteenth comprises all the notes of the diatonicscale, so any and every diatonic chord may presumably beexplained, and since the three fundamental discords of thethirteenth comprise the notes of a complete chromaticscale, so any and every chromatic chord may presumably beexplained.

With regard to the chord of the eleventh, it may be saidthat even Dr. Day himself apparently regards this chord inthe light of a stepping stone to the chord of the thirteenth,for. in the case of the tonic and supertonic fundamentaldiscords, the use of the eleventh is forbidden, notwithstand-ing that the thirteenth is regarded as an essential note.

THI MODRRN ENIIAKMOXIc vALK. 19The principal use of the dominant eleventh. acronlinK toDr. Day. soenu to be in connection v-ith the above-nwn-tioned much abused chord of the " Added ixth." whichZ^uV'i *'^'''' ''" "'^'''^' " '^' 'J""**"""' ^>venthwith the dommant and the Ieadini;-note omitted ThisjJmo,t remind, one of the play of Hamlet, with Hamlet andOpheha (the leading lady) omitted. According to theSll^r,. T '^" '^' ^' '^''' "^'"'^ ' ^he .supertonio. butnotlTp i I ^!'' ^'"*^" philosopher, and Ramcau. a^ted French author both claim the subdominant as theroot of this chord. The mystery s,:rrounding the chord isprobably attributable to the fact that there are tuv chordsconsisting of identically the same notes, one of whichnaturaly resolves upon the tonic chord, while the othernaturally resolves upon the dominant.

Fig. 9

The chord marked * in the example at a is the " Addedsixth, and Its resolution ui)on the tonic chord constitutes

ao THE MODERN ENHARMONIC SCALE.

a variation of the plagal cadence. This chord is really asubdominant triad, to which a sixth is added, hence thename; the sixth is, therefore, the dissonant note, and thechord may be regarded as a diatonic modification. Thesixth is frequently introduced as a passing note, as at b,when it is very evident that the root of the chord is thesubdominant. According to the Day theory this chord isa dominant eleventh, G being the root ; but as the root andthird are omitted the seventh and ninth become consonantintervals, while of course Dthe sixthis also consonant.

The chord marked * in the example at c is the firstinversion of a secondary tetrad on the supertonic; thefifth, therefore, is now the dissonant note, and as such it isprepared and resolved in the usual manner. The wholepassage is a sequence of secondary sevenths, a dominantsequence, that is to say the roots rise a fourth or fall a fifth,consequently the root of this chord is D; F, in this case,could not be the root. According to the Day theory G isthe root not only of this chord but also of all the othertetrads in this passage, with the result that consonancesbecome dissonances, and dissonances, consonances ; and thetheory, whether it be regarded as convenient or anythingelse, becomes contrary, illogical, and inconsistent.

With regard to the chord of the thirteenth, though atfirst sight it may appear convenient to be able to call anychord by this name, which cannot be satisfactorilyexplained in any other way, yet since there are so manypossible combinations of the notes of this chord, the namedominant thirteenthis most indefinite, and especiallyis this the case, when it is remembered that the dominantminor thirteenth (with minor ninth) is available for use inthe major mode. Dr. Day furnishes no less than seventydifferent examples of the use of this chord, some in four-part and some in five-part harmony, many of the latter

THE MODERN ENHARMONIC SCALE. ai

being of an extremely harsh chpracter. Each of theseexamples is accompanied with one or more rules, to whichthere are sometimes exceptions, so that the whole chapterresults m a mass of complications leading to " confusionworse confounded."

Two forms of this chord must be briefly consideredthat at a which comprises the dominant with its third andthirteenth, and that at b which is the same chord with theseventh included.

Pig. 7.

,5,tl^'- ^r f ^'' ^^'^"^ ^' ^^ *^^ "^"^ " dominantsixth IS therefore just as applicable and quite as justifiablej^thenanie dominant thirteenth." This chord is figured3 by Macfarren. but it is not customary in figured bassesto employ any figure higher than the number 9. that is tosay any double figures at all. Now place C in the tenorinstead of B. and the chord is converted into a 6-4 on theK"^^l'u^"]

according to the Day theory it should stillbe called the dommant thirteenth; and now. place C in thebass, retaining C in the tenor and not changing the treble

and alto notes, and the chord is converted into the commonchord of C. but why should this chord not be termed thefifth inversion of the dominant thirteenth ? If. becauseIt would be incorrect to double the eleventh, then place Gm the tenor instead of C (doubling the dominant as in thechord at a),

.t will still be the common chord of C. and theridiculous extent to which thl. theory may be carriedbecomes abundantly evident.

33 THE MODERN ENHARMONIC SCALE.

This chord, the dominant sixth, must not be confusedwith the first inversion of the mediant triad; the latterchord is rarely employed except in sequences, while theformer is of common occurrence.

This chord at 6, is figured J, and this is the chord, theone chord, to which reference is almost invariably madewhen the term ' dominant thirteenth ' is f-mployed ; '

name " dominant sixth and seventh," however, is m..definite, for it exactly describes this particular chord,whereas " dominant thirteenth " is occasionally employedfor a great number (sixty-nine ?) of other chords.

Upon the treatment of this chord (Fig. 7, 6) dependsentirely the question as to whether or not it should have adistinctive name. When, as in Fig. 8, the treble note iseither preceded as at a, or followed as at b, by the notebelow, that is to say by the fifth of the root, the chord issimply a dominant seventh, with the sixth introducedornamentally as an auxiliary note; at a the sixth is em-ployed as a free turning note ; at b the fifth is temporarilydisplaced, the sixth being either a suspension or an accentedauxiliary note. When the fifth is not introduced at all,that is to say, when the fifth is absolutely displaced (as inFig. 7), then a distinctive name is desirable, and a definitename is preferable to an indefinite name in this and in allcases.

Fig. 8.I

j=i(a)

T

mi7 -5 6

I

^w

TI7 -S

t

THE MODERV EXHARMONMC SCALE. 3

These chords (Fig. 7, a and b) are equally available foruse in the minor mode, when the sixth, which is almostmvariably placed in the treble, becomes a minor interval.The general treatment of these chords is the same, whetherthe mode is major or minor. The natural resolution ofthe dominant sixth and seventh is shown in Fig. 9 ; a, in themajor mode. ,.nd b, in the minor.

Fig. 9.

rff-^(a)

ilS. B =q(6)

s^f^^^ rllz^d^=M ! ly =#=

^

The following chords, which, it will be seen, are identi-cally the same on the piano, must now be considered.

Fig. 10.

According to Dr. Day, these chords are not only identicallythe same from an instrumental standpoint, but they arealso identically the same in name and in effect ; and thismay be said to be the crucial point in the Day theory. Thenames, " dominant eleventh and thirteenth " can at leastbe tolerated, but when Dr. Day claims that an augmentedfifth, a dissonant interval, is identically the same as a

a4 THE MODERN ENHARMONIC SCALE.

I

minor sixth, a consonant interval, it can readily be seenwhy so many eminent musicians have rejected his theory.Dr. Day, moreover, does not stop at this point ; he statesand claims that whenever the chord at a is employed, thecomposer^whosoever he may beis guilty of false notation,in other words, has written an incorrect note. It is trulya convenient theory, for it is certainly a matter of muchconvenience, when a progression occurs which is not inaccordance with the principles of the theory, to be able toqualm the conscience by saying that the composer is wrong

;

but, if musical works are to be judged and analysed onthese lines, it will be found that all the great composersare systematically wrong in their use of certain chromaticnotes, namely, the hyper-tonic, the hyper-supertonic (thenote employed in the present instance) and the hyper-dominant. The prefix " hyper " here indicates " chro-matically " raised.

It was stated above that Dr. Day drew a line of dis-tinction between the diatonic and the chromatic semitone,claiming that the latter was theoretically smaller than theformer ; admitting this to be the case from the standpointof acoustics, the diatonic semitone being in the ratio of 15 :16, and the chromatic, 24 : 35, he might at least havecredited the great composers with recognizing the samedifference, and might have granted them the privilege ofemploying that one which, for the time being, appealed totheir emotions. This privilege, however, is emphaticallydenied, and the theory, therefore, contradicts itsel* ;it accepts the difference but rejects the distinction.

A passing reference may here be made to a modemGerman text-book, A Manual of Harmony, by Dr. S. Jadas-sohn, in which it is claimed that the chromatic semitone isthe larger of the two, because of its tendency to rise, whilethe tendency of the diatonic semitone is to fall. This view

THE MODERN ENHARMONIC SCALE. as

of the difference between the two semitones is not generallyheld, in fact, it is erroneous, and the following irregularst^ccession of sounds given as illustrating " the true pitchof the tones " is contrary to the first principles of scaleconstruction.

Fie. 11

The above is called " the enharmonic-chromatic scale,"and it is here quoted in order that no confusion may arisebetween this and " the modem enharmonic scale," whichis about to be considered.

Reverting now to the chords in Fig. lo, it may be saidthat the augmented fifth of the dominant in the major modeand the minor sixth of the dominant in the minor mode,are often introduced ornamentally in connection with theperfect cadence, as shown in the following example.

Fig.ia.

At a the fifth of the dominant (in the treble) is followedby a chromatic semitone ; at b, it is followed by a diatonicsemitone. In each case the natural resolution of thechord is given

; without the resolution, the emotional effectof the semitone would not be obtained. Let each of these


progressions be played slowly, two or three times, on thepiano, and notwithstanding the fact that the D sharp andthe E flat are identically the same in pitch, it will be foundthat a difference in effect is clearly discernible

; an sestheticdifference, the augmented fifth of the rootthe hyper-supertonicis tinged with joy, while the minor sixth of theroot^the minor mediant is tinged with sorrow. Accord-ing to the Day theory, the D sharp is false notation, thenote should be E flat in each case. Musical notation maybe defined as the art of transcribing to paper the effect ofmusical sounds; and this effect is as readily appreciatedwith the eye as with the ear by the true musician. If,therefore, the above progressions differ from one anotherwhen played on the piano, then they should be ttotateddifferently when transcribed to paper. Beethoven, it iswell known, composed his greatest works whilst sufferingfrom total deafness

; he, with the eye alone, could appreciatethe majestic harmonies of the " Choral Symphony."Wherefore, it is a reflection upon the genius of the greatcomposers, to state that they wrote one thing but intendedanother, or to claim that they heedlessly or needlesslysacrificed correct notation for convenient notation.

Now, the notes D sharp and E flat, instead of beingintroduced after the dominant chord, may be employed inthe dominant chord, in the treble, in place of the fifth ofthe root, which thus becomes absolutely displaced, asshown in Fig. lo. The chord at a thus becomes a chro-matic modification, and the chord at b, a diatonic modifi-cation, but the resolution is the same, and the resultanteffect practically the same, as when the notes D sharp andE flat are introduced ornamentally, as in Fig. 12. Theyare, therefore, two entirely different chords, they naturallybelong to two different keys (C major and C minor), and,moreover, they must have two different names. Their

THE MODERN EXHAKMONIC SCALE. a;

names, respectively, are " the dominant augmented fifthand seventh," and "the dominant mmor sixth and seventh,"names, which may not be as convenient as the name" dominant thirteenth," but which at least possess themerit of definitely determining the character of the chord,a feature far more desirable than that of mere convenience.

With regard to Dr. Day's statement that the intervalof the augmented sixth cannot be inverted, it need onlybe said that this statement is not in accordance with facts,for the third inversion of the German sixth, when theinterval is, of course, inverted, has been employed by thebest composers.

It is unnecessary to consider other points in this notabletheory, sufficient has been said to show that it does notfurnish the correct chromatic material at the command ofthe composer, and consequently it fails to furnish a satis-factory explanation of certain chromatic chords. As atheory, so far as chords of the ninth are concerned, andthese in relation to the tonic, the dominant and the super-tonic, as roots or generators, it holds good for the minoror artificial mode only

; when applied to the major ornatural mode it signally fails. The whole question maybe summed up thus : if the Day theory of the chromaticelement in music is correct, then the principles upon whichchromatic progressions have been written by the greatestexponents of the art of music, are incorrect, and, exnecessitate ret, the converse must hold good.

But, it is one thing to oppose a theory, and it is anotherand a very different thing to propose a theory. The theoryupon which the chromatic element in music is based, asnow about to be explained, it may be said, is not the resultof a momentary deliberation

; it has stood the test of con-stant application for several years, and though presumablyit has never hitherto been expounded, yet there seems to be

a8 THE MODERN ENHARMONIC SCALE.

no doubt whatever that it fonns the basis on which thegreatest composers of the past have written their chromaticprogressions, and in which the composers of the future mayfind new and unending combinations of tone color, like theever varying changes of the kaleidoscope.

A chromatic chord may be defined as a chord whichcontains one or more notes foreign to the diatonic scaleof the key in which it is introduced, but which does noteffect a modulation. A chromatic scale, though usuallydefined as a scale ascending or descending regularly bysemitones, should be regarded, not from the melodic stand-point alone, as a succession of semitones, but also rrom theharmonic standpoint, as the basis of chromatic chords. Achromatic scale may be said to be the tonal extension of adiatonic scale; it, therefore, comprises seven diatonic andfive chromatic notes; but the chromatic, as well as thediatonic, all belong equally to the key.

There are three forms of the chromatic scale in use atthe present day ; these in relation to the note C as a tonic,are shown in the following example.

Fig. 13.

I\

i

i

THB MODERN ENHARMONIC SCALE. 89

The scales at a and b are respectively known as theharmonic and melodic forms of the chromatic scale; thescale at c is employed in instrumental music only, and thenotation is the result of the simple plan, adopted frequentlyby even the best composers, of forming the chromatic notesby accidentally raising diatonic notes in ascending, and byaccidentally lowering them in descending, in order to . duceas much as possible, the number of absolutely necessaryaccidentals. This, the instrumental form of the chromaticscale, will not be considered further, as it is strictly amelodic scale, for, the notes A sharp (ascending) and G flat(descending) possess no harmonic relationship whateverwith the key of C ; they can only be employed in this key aschromatic auxiliary notes.

The notation of the sca!e at a, it will be seen, is thesame in descending as in ascending. In the notation ofthe scale at b, while three of the chromatic notes in ascend-ing differ from those in the scale at a, yet in descending,these two scales are precisely the same. Similar scales maybe formed from any other key-note, but it is necessary toemploy identically the same intervals in relation to thekey-note as are employed in the above scales. The chro-matic notes in these scales are regarded as borrowed notes

;

the source from which they are borrowed will be dulyconsidered.

The scale at a was adopted by Dr. Day as the basis ofhis theory of the chromatic element in music ; the scale atb was totally ignored by him as the basis of chromaticchords; it was employed solely in connection with chro-matic auxiliary notes.

The scale at a, as will be duly shown, constitutes thebasis of the chromatic chords of the minor mode ; that atb, the basis of the chromatic chords of the major mode. TheDay theory, therefore, is applicable to the minor mode alone.

JO THK MODERN KNHARMnNIC SCALK.

The scale at

TIIK MOtlKKN KMMkMOXU SCALK. 31In the fourth, fifth and sixth harmonks of a lmvci,

generator ts to \h: found the i-ommon chord of nature andthe major and minor thirds which comfxise this chordconstitute the basis of all chords. Having obtained onecommon chord, or triad,

-.s it is also called, two attendantcommon chords are formed, one of which has for its rootthe fifth of the given chord, while, the other has for itshfth the root of the given chord; in these three chordsknown as the primary triads of the kev. are to be found thJnotes of the major diatonic scale. The notes thus obtainedfrom nature are tempered, or tuned, for the purposes of artan order that all the semitones may be exactly equal so'that any note may be taken as a key-note. The majorcale is divisible into two similarly constructed tetrachordsand by means of these tetrachords the attendant majorscale, arc formed, and then their attendants, and so onunti I major scale is formed on every possible note.

I alike the major, the minor common chord is notderived from t he harmonics of a given generator ; the majortriad IS natural but the minor is artificial. The minor triads obtained from the major triad. It mav be formed bvtwo entirely diflferent methods, the one. diatonic, the othe^chromatic. The diatonic or relative minor triad is obtainedby placmg a minor third helou' the given major third thusreyersmg the order of nature; the chromatic or affinitiveminor triad is obtained by accidentally lowering the thirdof a major triad, thereby making it minor, and at the sametime making the upper third major.

The minor triads thus obtained become the tonic chordsof related minor scales, respectively known as the relativeand the affinitive scales, the former being in diatonic and thelatter m chromatic relationship with the given major scaleThe minor triad, like the major, has two attendant minortnads. and in these three triads is to be found the signature

i I

3nt form of the minor scale ; the seventh degree ofthis scale is now chromatically raised again for the pur{)oseof art, it then becomes a projwr leading-note. The minorscale, therefore, like the minor chord, is also artificial. Thescale, as thus obtained, is known as the harmonic form ofthe minor scale; the distinguishing feature of this form ofthe scale is the interval of an augmented second whichoccurs ttetween the sixth and seventh degrees; in order toavoid this unmelodious interval, another form of the minorscale, known as the melodic form, is frequently employed,but the harmonic is the true form of the modem minorscale. Other forms of the minor scale also exist, but asthey are only in occasional us, and as they do not in anyway affect the chromatic element in music, it will not benecessary to consider them in this connection.

Besides the major and minor triads, there are also thediminished and the augmented triads, the former beingnaturally diatonic and the latter invariably chromatic;but all four triads play a very important part in the forma-tion of chromatic chords.

The fourth, fifth, sixth and seventh harmonics of a givengenerator constitute a chord of the seventh, or tetrad, as itis called. When a tetrad comprises a major third, aperfect fifth and a minor seventh, as in this case, it iscalled a primary tetrad, to distinguish it from other chordsof the seventh, known as secondary tetrads.

The fourth, fifth, sixth, seventh and ninth harmonicsconstitute a chord of the ninth, or pentad, as it may becalled ; in this case, as the ninth is major, the chord will bea primary major pentad. A primary minor pentad isobtained from the eighth, tenth, twelfth, fourteenth andseventeenth harmonics.

THB MODIRN KNHARMONIC RCAtR. If

It is unnecessary to continue this series of fundamentaldiscords l)cyond chords of the ninth.

By omitting the root of a pentad, a tetrad is obtained,known as a derivative of the p(>ntad; but these derivativesarc obtained from primary pentads only. The derivativeof a major pentad is called u minor tetrad, and that of aminor pentad, a diminished tetrad. These are the onlycases in which a tetrad is called after the name of its com-pommt seventh; in all other cases the tetrad is namedaccording to its root.

A triad, a tetrad, and a pentad, may be formed uponany note of the major and minor (harmonic) scales. Thepentad on the leading-note, however, on account of itsextreme harshness, is of very rare occurrence.

The tetrads on the dominants of both modes are natur-ally primary discords; when formed upon other notes theyare called secondary tetrads.

The pentads on thr dominants of both modes are nlsonaturally primary discords; when formed upon other notesthey are called secondary pentads. The ninth in a secon-dary pentad is generally treated as a suspending note, oras an auxiliary note, in which case the chord may be re-garded as a modified tetrad.

The chromatic element in music is obtained from aseries of six primary minor pentads. The tonic, as thegenerator of all that appertains to the key, is taken as theroot of the first pentad; then the fifth of the tonicthedominantis taken as a root; and, thirdly, the fifth ofthe dominant^the supertonicis taken as a root. So farthis is in accordance with the theory of roots as advocatedby Dr. Day. The reason why these notes are chosen, hestates, is " because the harmonics in nature rise in the samemanner." At this point Dr. Day stops, in fact, he gives adefinite reason why the harmonics of the next fifth are not


used, namely, " because that note itself is not a note of thediatonic scale, being a little too sharp (as the fifth of thesupertonic), and can only be used as part of a chromaticchord. It is this very note, the submediant, howeverwhich must be taken, in spite of Dr. Dav. and upon whichanother primary minor pentad is formed; followed byanother upon its fifth, the mediant; and lastlv. by another

*

upon Its fifth, the leading-note.A series of fundamental discords is thus obtained the

natural evolution of which, from the original tonic as agenerator, will be seen in the following example

Fig. 14.

The whole-notes in the above example are the roots ofthe successive chords

;the quarter-notes, in each case, form

a diminished tetrad (or. chord of the seventh) Thesechords, it will be seen, extend over a compass of exactlyfour octaves, commencing and ending on the tonic of thekey.

'- A scale may now be formed from the notes compri.sedby the above chords, a scale which will meet all the require-ments of the modem composer, a scale containing not onlythe diatonic and the chromatic, but also the enharmonicelement, and which, therefore, will be termed

The Modern Enharmonic wScale op C.Fie. 15.

THE MODERN ENH VU.\..>Ni.' SCALE. 3S

The value of the notes i . the abow s ;Je in no wayrefers to the question of tin . in perform .ncc; the scaleshould be played as a regular successio;; of semitones. Thehalf-notes are the diatonic notes of the scale of C major;the two quarter-notes represent what may be called theperfect chromatics, which are not enharmonically change-able; while the eighth-notes, which may be called theimperfect chromatics, may be employed in either form.

The six generators occur each three l imes ; t he subdomin-ant (the only note of the scale which is not employedas a generator) and the perfect chromatics occur twice;while the imperfect chromatics occur once only in eachform. The subdominant may be regarded as the "patri-arch" of the key. for it is among the upper harmonics ofthis note, namely, numbers 24, 27. 30, 32, 36, 40. 45 and 48,that the natural scale of the tonic is found.

The term "modem," as applied to this scale, may, ifdesired, be omitted; it is employed in the first place to dis-tinguish the scale from the ancient enharmonic scale ofthe Greeks, and, in the second place, to distinguish it fromthe weird unmusical scale, containing intervals smaller thana semitone, which may be performed on the violin or by thevoice. The term " enharmonic " is employed to designatea scale comprising certain notes, the pitch of which may berepresented by two different names.

The notation of this scale is applicable to the key of Cmajor only

;by transposition, however, it may be employed

as the harmonic basis of any other major key.The origin of the chromatic scales is to be found in the

above series of primary minor pentads. The pentads onthe tonic, the dominant, and the supertonic furnish thenotes of the harmonic form of the chromatic scale, whilethe notes of the melodic form are obtained from the seriesas a whole. The modem enharmonic scale, it will therefore


be seen, is simply the melodic form of the chromatic scalewritten in another form, the enharmonic form, and sowritten in order to avoid the necessity of writing this scaleboth ascending and descending.

The melodic form of the chromatic scale is the basis ofthe chromatic element of the major mode; the harmonicform, it will be duly shown, is the basis of the chromaticelement of the minor mode.

Chromatic chords are divided into two classes, namely,the ordinate and the subordinate. Ordinate chromaticchords are regarded as borrowed chords, the source fromwhich they are bonowed being the scales most closelyrelated to the given scale, namely, the relative and affinitivescales, together with the dominant and sub-dominant, andtheir relative and affinitive scales. These scales' areknown as the " associated scales." Subordinate chromaticchords are such as may be formed from the modem enhar-monic scale, but which do not belong to any of theassociated scales.

The associated scales of C major are shown in thefollowing table :

Major Scales

Relative minors(Diatonic relationship)

Affinitive minors(Chromatic relationship)

Subdominant

F major

D minor

F minor

Tonic

C major

A minor

C minor

Dominant

G major

E minor

G minor

The above scales comprise the notes of the modem enhar-monic scale of C, no others, and none omitted. For thepurpose of comparison these scales are shown in the fol-lowing example.


Pig. 16.

It will readily be seen that the number of chromaticchords at the command of the modem composer is con-siderably in excess of those available for use under therestrictions of the Day theory.

The associated scales of C minor are shown in the fol-lowing table :

Minor Scales

Subdominant Tonic1 Dominant

F minor C minor G minorRelative majors

(Diatonic relationship^ A flat major E flat majorB flat majoiAffinitive majors

(Chromatic relationship) F major C major G major

The above scales comprise the notes of the harmonicform of the chromatic scale

; no notes occur under twonames; there is, therefore, no necessity for an enharmonicscale in connection with the minor mode. For the purposeof comparison, these scales are shown in the following,example :


Fig. IT

i po:^ -lE^nS^^^^^^ .jo'i l^uoPc pocit has been shown that the minor triad is artificial , that

the minor scale is artificial, and now it will be seen that thechromatic chords of the minor mode differ essentially fromthose of the major. Furthermore, whereas in the majormode, in addition to the chords borrowed from the asso-ciated scales, there are certain extra, subordinate, chro-matic chords, in the minor mode the very opposite obtains,for not only are there no subordinate chromatic chords, buteven some of the major and minor triads of the associatedscales are not available for use as borrowed chords.

Having now considered the origin of the chromaticelement in music, it is necessary to revert to Dr. Day'sargument with regard to the fifth of the supertonic, whichhe states is not a note of the diatonic scale, being " a littletoo sharp." This statement is perfectly true from thestandpoint of acoustics, but Dr. Day, in his endeavors toestablish a principle, overlooks one very important point,namely, that acoustics and equal temperament are by nomeans one and the same thing. Acoustics is the basis of thescience of music, but equal temperament is the basis of theari. B flat, for example, as the minor seventh of C, is alsoout of tune (again a little too sharp), with B flat the seventh

THE MOIJERX ENHARMONIC SCAI.K. 39

harmonic of the generator C, nevertheless it is the out-of-tune note which is employed in equal temperament

; andagain, even the major third of art is not perfectly in tune withthe major third of nature.

Another, and an interesting illustration of this pointwill be seen in the major and minor tones. The intervalbetween the first and second degrees of the major scale, isa major tone, with the vibration ratio of 8 to 9, while theinterval between the second and third degrees is a minortone, with the ratio of 9 to 10. Now ihe first and seconddegrees in the key of D, for example, are exactly the same,according to the laws of equal temperament, as the second andthird degrees in the key of C ; wherefore, if the note D has,say, 288 vibrations, then the note E, in the key of D, willhave 324 vibrations, whereas in the key of C this selfsamenote, E, will have only 320 vibrations. Theoretically thisis a fact, practically it is absurd. The difference betweentheory and practice in this and in similar instances, may beexplained on the grounds, that the demands of equality in arthave necessitated the sacrifice of inequality in science. Evensupposing that the violinist and the vocalist can exemplifythe diflerence between the major and the minor tone, theeffect would not be in accordance with the principles uponwhich the modem major diatonic scale is constructedfor the fundamental principle of this construction is equaltemperament.

Equal temperament may be defined as the division of theoctave into twelve exactly equal semitones. These semi-tones constitute a chromatic scale, seven of the semitonesbeing diatonic and five chromatic; but whether diatonic orchromatic they are all mathematically equal. It follows,therefore, that all the perfect fifths will also be equal, andequal in every respect; for, it is upon this very principle ofequality that the modem system of scale construction is


based, every new scale being the exact counterpart of thelast, exact in every particular except that of pitch alone.Wherefore, Dr. Day's statement, that one of these perfectfifths is out of tune, is erroneous.

In the old meantonc, or unequal temperament system,the semitones were not all equal, and consequently theperfect fifths were not ail equal, with the result that certainkeys, called " wolves," could not be employed at all. Themighty genius of the great Johann Sebastian Bach, however,struck the death blow to this system in the immortal" Wohltemperirtes Klavier," in which there are two pre-ludes and fugues in every key; and after his death (1750)the old system gradually passed away, and the modem orequal temperament became, and has since remained, thedefinitely established system.

In the equal temperament system the perfect fifths aretempered (hence the name) , that is to say, tuned slightly flat

;

but so slight is this flatness that the human ear can scarcelyappreciate the difference between the tempered and thenatural fifth. To be exact, the tempered fifth is rr of acomma flatter than the natural fifth; and a comma (thename for the difference between a major and a minor tone)is about J of a diatonic semitone. This tempering, whichis the basis of the art of piano-forte tuning, is indispensable

for the purposes of harmony, and equal temperament aloneenables the composer to write in any and in everykey, and to modulate to or from any key by means of enhar-monic changes.

There are, therefore, now no " wolves," and no fifths" a little too sharp," and no difference between the diatonicand the chromatic semitone ; wherefore, the primary ninorninth on the submediant, the mediant and the leading-note-may be taken for the purpose of extending the principles


of the chromatic element beyond the inconvenient undillogical limitations of the Day theory.

The reason for not extending this system beyond theleading-note is twofold

; in the first place the perfect fifthabove this note, not being a diatonic note of the scale couldnot be taken as a generator; and in the second place, thesystem is complete without it.

With the enharmonic scale as the basis of all the possiblechords in the key of C major, three important new triads,namely, A major, E major, and B major, besides others,are added to the already accepted chromatic concords.These, it will be seen, are the dominant triads of the threeattendant minor keys, and by their acceptance the wholefamily of attendant keys becomes even more closely unitedwith the original tonic key than heretofore. By the addi-tion of these triads, together with their sevenths and ninths,a theory of harmony is obtained by which every chord inthe works of .he great composers can be justified. Further-more, the modem composer will find at his command amajor triad on every semitonal degree of the key, exceptalone on F-sharp (or G-flat), which, having no note commonto the diatonic scale, is naturally altogether foreign o thekey. At the same time, seemingly boundless possibilities inthe realm of tone color are presented to the composer, possi-bilities which indeed may ftever be exhausted so long as theart of music is based upon these very principles of equaltemperament.

The chief defects of the Day theory may be briefly sum-marized as follows :

(i) It was altogether unknown to the great classicalmasters of the Bach to Beethoven period (and even later)

;

indeed, it frequently happens that chords employed by thesecomposers cannot be explained by the Day theory, in whichcase the chords are said to be written in False notation.


(a) The compound intervals of the eleventh and thir-teenth being represented in figured basses by their simpleformsthe fourth and ixth. the names " dominant fourthand dominant sixth " are just as applicable and quite ajustifiable as the names " dominant eleventh and thir-teenth."

(3) These names, furthermore, are very indefinite. Dr.Day furnishes thirty-two different examples of the use of thedominant eleventh, and as many as seventy diflFerent ex-amples of the use of the dominant thirteenth.

(4) The theory completely overthrows the generallyaccepted theory of roots, and, consequently, root progres-sions, for any diatonic triad or chord of the seventh may beregarded as an incomplete form of tho dominant thirteenth.

(5) The liarmonic form of tlie chromatic scale being thebasis of this theory, and this scale being derivable from thechords of the ninth on the tonic, dominant and supertonic,nothing is gained by adding the eleventh and thirteenth tothese generators.

(6) It is founded (as has been shown above) upon afalse estimate of the true significance of equal temperament,in which, for the purpose of enharmonic changes, all thesemitoneswhether diatonic or chromaticmust be re-garded as being absolutely equal.

(7) Finally, when applied to the works of the greatmodem composers

Wagner, and his contemporaries andsuccessors, this theory signally fails, for. in the chromaticextensions of the modes now in vogue, a primary seventh(to mention one chord alone) may be employed not onlyupon the tonic, the dominant and the supcrtonicas advo-cated by Dr. Day^but also upon all the degrees of themajor scale.

The chromatic major triads in the key of C major, accord-ing to the enharmonic system, arc shown in the following

THE M(DbKN EXIIARMOMC SCALE. 43

example;the triads at a, b, c, e, and

a; art- admitted in theDay theory, but the triads at d f. and It are not recognizedin this theory.

Flg.18.(a) m

From tlie above example it will be seen that, with theexception alone of the major triad on F-sharj) (or G-flat)all the major triads in music are a\ ailable for use in the keyof C major. To each of these triads except two (those at aand e). a minor seventh may be addiil. when the chordbecomes a primary tetrad. The chromatic primarv tetradsin the key of C, according to the enharmonic system, areshown in the following example :

Of the above chords, those it a and h are alone recog-nized as "fundamental sevenths" in the Day theory:those at e and

a,arc also recognized, but as incomplete forms,

respectively, of the "dominant and tonic minor thir-teenths." The notes of the chord at d occur in the har-monic form of the chromatic scale, but this chord does notbelong to any one of the three great chords of the thirteenthof the Day theory^ with all their chromatic modifications.

The chords at^* and ^, not being dominant sevenths inany cf the associated keys, are subordinate chromaticchords, but they are available for use equally with theborrowed chords.

Although a minor seventh cannot be added to the majortriads on the minor supertonic and minor submediant. yet

44 THE MODBRN ENHARMONIC SCALE.

its enharmonic equivalent, an augmented sixth, may beadded to these triads, when two chords of the German 6thare obtained. These chords, but no other German 6ths,are recognized in the Day theory, while in the enharmonicsystem there are three others ; the former two are shown ata and b, and the latter three at c, d and e, in the followingexample :

Fig.ao.

The normal bass-note of the German sixth is regardedas the root of the chord, and this form of the chord of theaugmented sixth, together with the Italian and Frenchforms, and certain other chroiiuuic chords, are known as"chromatic modifications."

It has already been seen from Fig. 14, that accordingto the Day theory there are three chords of the diminishedseventh in the key of C, and that according to the enhar-monic sy;jtem there are three other chords of the diminishedseventh; the latter three are but enharmonic equivalentsof the former three, of course, for, from an instrumentalstandpoint there can be but three chords of the diminishedseventh in any case, but their place in the present chro-matic theory is justified by their use in the compositionsof the great masters.

When to the chords which have already been mentionedare added the chromatic minor triads, secondary tetrads,primary major pentads, and their derivatives, togetherwith the various chords included under the name of chro-matic modifications, it will be seen that the present theoryof the chromatic element in music, is in reality, at leastfrom the standpoint of the major mode, more comprehensivethan that advocated by Dr. Day and his successors.

THB MODERN ENHARMONIC SCALE. 45

Chromatic chords should be studied systematically afterthe mamier of diatonic chords; they may be divided intothree classes, namely :

I. Chromatic Triads.Concords, major and minor.Discords, augmented and diminished.

II. Chromatic Tetrads and Pentads.Primary tetrads.Primary major pentads and minor tetrads.Primary minor pentads and diminished tetrads.

III. Chromatic Modifications.Auxiliary notes.Chords of the augmented fifth.Chords of the augmented sixth.Secondary sevenths, etc.

The works of the best composers, both ancient andmodem, teem with chromatic chords, the analysis of whichfrom the standpoint of the Day theory is complex andaiogical. but from the standpoint of the enharmonic systemis simple and natural

; that the great masters employed thelatter system as the basis of their chromatic progressionsthere can be and there is not the least shadow of doubt.That the chromatic element in music has not hitherto beenreduced to a system is no argument in favor of the state-ment that the great composers did not employ any systemat all in writing their chromatic progressions. Practicecomes before theory in this case; that is to say, the lawsof theory are but the result of the practice of the bestcomposers.

As the strength of a chain depends upon its weakestlink, so one example from a classical work will be sufficientto demonstrate the inconsistency of the Day theory. Thefollowing passage is taken from the Sonata in A minor.Op. 42, by Schubert (1797-1828).

46 THB MODERN ENHARMONIC 8CALB.

F%.fll

This excerpt is unquestionably in the key of C major,it occurs in the first movement of the sonata, bars 194-6.The chord marked is unquestionably a chromatic chordin the key of C, for, it is immediately preceded and followedby chords which are definitely in this key. The name ofthis chord is unquestionably the common chord of B major,or the leading major triad. As a borrowed chord it occursas the dominant triad in the key of minor, the relativeminor of the dominant of C, one of the afore-mentionedlx)rrowing keys; this chord, it will be seen, occurs in No. 6,of the primary minor ninths in the key of C, Fig. 14 ; while iuuse, by one of the great classical masters, justifies the forma-Mon of major triads on diatonic notes other than the super-tonic, justifies the system of borrowing from relative minorkeys, and indirectly justifies the complete series of primaryninths, which constitute the basis of the enharmonic scaleof C.

If this chord be examined from the standpoint of theDay theory, the analysis will be as follows : since, theharmonic form of the chromatic scale of C does not containthe note D sharp, this note is therefore an in?'anceof falsenotation, and the composer should have written E-flat(incidentally giving rise to false relation with the E-naturalin the treble of the preceding chord, but this may be over-looked)

; with this change, the three notes of this chord willbe found in the fundamental chord of the thirteenth on the


Bupcrtonic, us the third, the minor ninth and the majorthirteenth, and the name of the chord, as employed bySchubert, will therefore be " the sixth inversion of theupertonic minor ninth and major thirteenth, with theminor ninth falsely notated. and the thirteenth doubled "Poor Schubert I

Further quotations would be superfluous.When, in order to justify the presence of simply a major

common chord, it becomes necessary to resort to suchabsurd extravagance, it will be seen that the present theoryof the chromatic element in music, is in reality, at leastfrom the standpoint of the major mode, more consistentthan that advocated by Dr. Day and his successors.

Such is the general substance of the theory which isherewith offered to the musical world at large, as a sub-stitute for that originated by Dr. Day. As to its intrinsicvalue, timethe supreme judge in musical mattersalonewill determine; but, whatever the judgment ultimatelymay be, the author will feel that he has not laboredm vain, if it be admitted that the enharmonic theory,as a theory, not only rivals but also supersedes the Daytheory in both "consistency and comprehensiveness."

Conservatory of Music,Toronto, Canada,

December ist, 1906.

A ir^atiae on famongwtt4 Ewrriara

BY

3. fntttfrro Awjpr.

Canadian EditionWtt. Tyrrell & Co., Toronto.English EditionThe Vincent Music Co., London.United States EditionThe Boston Music Co., Boston.

IT has been the pnvileRe of the author to teach the theory ofmusic, and notktn^ but the theory, for the past ten years. Duringthis extended period he has naturally had exceptional oppor-tunities for discoverins the difficulties which confront the

average student;and the special purpose of this present workthe outcome of inany requests from both past and present pupils

as well as froni fellow teachers, is to assist the student to meet andmaster these difficulties, so that Harmony, which is one of themost important and at the same time, one of the most charmingstudies in the art of music, may become a source of pleasut* andenjoyment, instead of being, as so frequently happens, a source ofworry and annoyance. j ft , "ij wi

The author has endeavoured, in his treatment of the subiectto iDe as simple, as concise, and at the same time as thorough aspossible;

simplicity being a necessity for the beginner in everysubject

;conciseness, a nece^ity at the present day, when harmony

Is but one of the many studies which claim the attention of thestudent

;while a thorough explanation of the subject is not onlv

absolutely necessary but this, indeed, must also be on somewhatoriginal lines m order to justify the publication of yet anotherrreadr^ntiJiw'"" ->'*- --V --"-t treatises have

*i, J^ '^ nu *^^ supposed, however, that any very originaltheories wi 1 be advance

47

ISpl'''*i^-n''5*"''the dishonesty of plagiarism should ever be

rJHn. TTu' ^1^^^". "''""St endless possibilities in the modifi-cations of the chords in general use, either by the employmentofaAUiary notes, or by the introduction of the chrcmat cSen?!found ?t,,^V*^'"*'**y *,*'"* T^y ^ " ' P-"t "'k *ai befo mpU* K ^ '* " ^ '***' î'^P*^'"- ^'''' J'e author, in orderto meet the requirements of modem compcsers. advocates theof,^)ÎI ^V"^' "?'" ^'."^ ^"^ ^- PK "). the principalfelturâLHv

the ccmbmat.cn of tfie enharmonic element with thlalready accepted chromatic extension of the major scale.

worWo"!!*"""^'*attention has been devoted, throughout thiswork, to the exercises at the end of each chapter. Thi-se it win

t^ fi!^;.Ty.^^^numerous and of a varied character. In additTon

exe&ll,^^!^ '';'""*' ""^y ^ TcgB^rded as the staple form offnlrod^fc;* nn7,f? A^^ exercises on the analysis of chords, on the^n .^ V

(includmgthe preparation and resolution) of discords.WnWhtT""''Vu"t.^ melodfes and unfigured bas.es' on clothtagi.^ li^^ . * '^'*}u

harmcny. and en composing original passagesetc. In all cases the exercises have been carefully graded, and hivebeen arranged with the special view of assisting candidates ,êpar-ing for exammations. the student, however is not necessarilyexpected to work all the exercises at the end of each chTp^r^Hrnôf1.,*r ^.f'"^K ^ *

suggestive character, may. at the Ke-'ticn of the teacher, be omitted, or on the other hand others of ahimself.^^

'"^^ ^^ ^'^^"^ ^''^" ^y *>* ''^^'- r by the student

t,n?H T"û ^",J^ ? ^*^ question as to whether the student^,"nnina 't"i,"'*^ Vu' ^^- *?".'?* *." ha.mcnize melodies from thebeginnmg. The author is mclined to think not. but evervthinereally depends on the student

; for though one student will gm"?the principles readily enough, and another will fail to do so. bothoi them may in the end prove to be equally successful in theirgeneral work. This matter must also be left with the teacher, andthP oL*#!!;."-^fi? \"I^P'V^^ melodies, etc.. ait; omitted when^re'viet'S at'a^Ltt" day:

*'^^ "'" P'^''>' ^ '^""^ "-^"' -

No illustrations from the compositions of the great mastenrhave been included, for though of considerable interelTfo th^ruThl~fif?; V^ ^ question whether such examples are really of anyW !^Ll **"' ^*I*''*S*- Speaking generally, 'it may l,e siid. thatI^llwJ^TP'!,'"^'*ûf P^^ " "> another example couldSwKi.r^f h""**' f^'^'K^ ''*' "^""^^^ ^ the same composer!which might he quoted to disprove the rule ; indeed, to furnishon hT"""*,!,'",!!''

****' exceptions and explanations, rSina t^ ?

would become an extremely bulky volume, far exceed-tag both the intentions of the author and the limits of the present

Hiv!H!.H?J!fm^'''i"".u'*'''^'y dimensions, this treatise has beenas foUows ^ ^ ~*>''

f ^hi* may be summarized

^*"T,Ji" ^1"' ^^'"' "?* r^J?*" *"

PART I.

CONTENTS.

ST

Chapter.

Introduction.

I. The Major Diatonic Scale.

II. Intervals.

III. The Minor Dtatonic Scale.

IV. The Common Chord.

V. Harmonic Progressions.

VI. Cadences, Sequences, etc.

VII. The Chord of the Sixth.

VIII. The Chord op the Six-Four.

IX. The Dominan-' Seventh.

X. The Inversions of V7.

XI. Natural Modulation

Appendix i.Symbols.

Index

sa

PART II.

CONTENTS.

Chapter.

Introduction.

XII. Secondary Sevenths.

XIII. Chords of the Ninth

XIV. The Derivatives op V9.

XV. Simple Suspensions.

XVI. Compound Suspensions.

XVII. Auxiliary Notes.

XVIII. Extraneous Modulation.

Appendix ii.

Figures.

Index.

In Chapter XVIII., Arpeggios. Pedals and Ground Bassesare explained, and a brief reference is made to some:of the morefrequently employed chromatic chords.

PART III.

53

CONTBNTS.

Chapter.

Introduction.

XIX. Chromatic Concords,XX. Chromatic Discords.

XXI. Diatonic Modifications.XXII. Chromatic Modifications.

XXIII. Enharmonic Modulation.XXIV. Three-Part Harmony.XXV. Fivb

.>ART Harmony.XXVI. Contrapuntal Part-Writing.

XXVII. Instrumental Harmony.

Co.NCLUSION.

Appendix iii.Acoustics.Appendix iv.History.Index.

General Index.

Part III. and a Key to the Exercises in P.r* tpared for the press.'zeroises m Part I. are being pre-

APPRECIATIONS BY AUTHORITIES.

" A Treatise on Harmony, by J. Humfrey Anger, is preferableto almost any of the existing worlcs on the subject."

George Coleman Gow, Vassar College.

" I have looked it over with much interest, and I certainlyfind it more intelligible than any similar work I have ever lookedat." Kate S. Chittenden, AmerkuH Institute of Applied Music.

" I have carefully and critically examined A Treatise onHarmony, by J. Humfrey Anger, and must say that I never had abetter book dealing with the subject, in my hands."J Joseph Baunach, St. Joseph's College.

".4 Treatise on Harmony, by J. Humfrey Anger is concise,

charmingly and clearly expressive in language, and covers everyconceivable point, and many which no other book even mentions.The work in its entirety is a monument to the author's genius as ateacher."

Frederic H. Pease, State .Wormal College, Michigan.

" I have been very ipuch impressed with the excellent workon harmony which has recently appeared from your pen. The verypractical and logical manner in which you treat this importantsubject, and the admirably conceived original examples given ineach section of the work render the treatise a welcome and mostvaluable addition to the best existing literature on Harmony."

A. S. WOGT, Conductor of the Mendelssohn Choir,Toronto, Canada.

" Mr. Anger's Treatise on Harmony is certainly of great strength,carefully thought out, and founded upon the most sensible plan ofteaching the student ' what t-j do ' instead of ' what not to do,'which so often fails to accomplish any result other than to com-pletely befog the pupil.

" The author has accomplished just what is so sadly needed,a book that is simple, concise and thorough ; and with the highstandard maintained throughout, he makes it deserving of a heartywelcome from teacher and pupil."

Charles PEMBERTON,l/tv

APPRECIATIONS BY AUTHORITIES. 55

" I believe that few present-day Eneltsh theorists are morecapable and better equipped than Dr. J. Humfrey Anger, for thework of evolving and extending out of the old a new and practicabletheory that will cover all the requirements of the modem composer.

" The advent of his Treatise on Harmony is significant in itstimeliness and applicability to the present-day needs of the studentof harmony who, if he be of an inquiring and sensitive mind, willmore than likely feel the influence of the subtle spirit of ultra-modernity as expressed by 'he younger and freer school of musicwriters. That way lies perversion and madness for him unless heis well balanced and guided aright.

" Not that Dr. Anger has radically diverged from the straightand somewhat narrow path of traditional tenets and doctrines ofmusic invention, but that out of the old dispensation and rulinghe has augmented, developed and widened tne hitherto crampedlatitude in which, by its very restrictions, originality of thoughtand expression was well nigh exhausted. To this much-neededend his new scale" The Modem Enharmonic Scale of C "seemsto me the outcome of a natural evolution in scale formation; andIts theoretical value is self-evident by at once providing a funda-mental basis for the consistent p.iid logical accounting of the manyseeming irregularities in chord structure present in the works f)f theromantic and modem composers. Apart from this innovation Iwould eamestly commend the work for its lucidity and conciseyet thorough treatment of an inexhaustible study ; and to mymind it should appeal most strongly to the conscientious teacherand earnest student."

H. CLOUGH-LEIGHTER, Composer and Critic. Boston, Mass.

anger - the modern enharmonic scale as the basis of the chromatic element in music

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