A Treatise on Harmony Part 3_Joseph Humfrey

A

TREATISE

ON

HARMONY

WITH

EXERCISES

JrHUMFREY ANGER

Professor of_Harmony,etc.,attheTorontoConservatoryofMusic

Hon. Mus._{Doc., Trinity University,}Toronto;

Mus.Bac., Oxon.; F. R.C.O.

InThree Parts

PART

III

BOSTON, MASS.

U

THE

BOSTON

MUSIC

COMPANY

Copyright, 1912,by G. SCHIRMER, Boston

Forallcountries

Depositadoconformealaleyde RepublicaMexicanaen

PREFACE

THE

theoryof thechromatic elementin_{music, as}enunciatedin_{the present}

volume, is a departure from that which usually obtains in text-books. It

has beenin constant use,however, since thecommencementof the twentieth century,andis,therefore, offered to theworldofmusic onlyafterdue

delibera-tion. _{This, the}EnharmonicTheory,as it is_called, is intermediatebetween

that which restricts chord construction to the limits of an octave, and that which extends fundamental discords to the eleventh and thirteenth. The

theory wasfirst_explained,and published,ina_pamphletentitledThe Modern

Enharmonic Scale,and theauthoristhus enabled, hereand_{now, to reply to}

certain _questions which have been raised in connection _{therewith; questions} which, notimprobably,

may

ariseagain from time to time(in the minds of thosewho are patientenough toexamine a newmethodof _presentinganold subject),andto ignorewhich mightpossiblyinferaninabilityto reply tothem.

i. It hasbeen said that the enharmonic theory iscalculated to

circum-scribethecomposer.

A

text-bookonHarmonysimplybears thesamerelation tothe art ofmusic thatatext-bookongrammarbears toliterature;and, just as thelatterisnot "calculated to circumscribe" the poet, so theformerisnot"calculated to circum-scribe"thecomposer. Thereal _object of a _{work on harmony} should be the theoretical education of the student of music, and the present treatise is no

exception to the rule. There are, it

_may

be said, two classesof composers, namely,thetone-authorand the tone-poet,andthe_{student who,}after_having

pursued a course of _harmony,evinces a taste for_composition, willnaturally entertheranksoftheformer_class,andwilldoubtless writeinaccordance with the generallyacceptedprecepts. Shouldhe,however,possess thatvital_spark

of genius,whichentitleshimtoa_placeinthelatter class,hewill_then,indeed not be circumscribedby any workwhateveron harmony. To the tone-poe'.

academic lawsarehis"veryhumble andobedientservants."

ii. It has beensaidthat the extensionoftheseriesofchordsofthe dimh.-ishedseventhis_unnecessaryandinexpedient.

Oneoftwothingsis,without any doubt, both unnecessaryandinexpedient, namely, thisextensionof thechordsof thediminished _{seventh, or the}

conten-FOOT-NOTE. "This_{pamphlet, published}in_{the year1907,having}fulfilleditspurpose,will

notbe republished, the essenceoftheworkbeing incorporatedinthe presenttreatise."

iv PREFACE

tion_{that the greatcomposershabituallyemployed}falsenotation. The

enhar-monicscale,as the basisofchromatics,comprisesthenotation_adoptedin the greatest and bestmusical compositions; andtheharmonicorigin of this scale

istobefoundintheextendedseriesofdiminishedtetrads(seeEx.15,page385).

Itisnot onlyexasperating,butitalso_{savors of egotism, to}beconstantlyunder the necessity oftellingstudents thatsuchandsuch a chord can only beexplained

bytheexcuseof falsenotation, thatisto_say, thechordis_{incorrectly written,} and,thatisto say, the composeris_{incorrect, for,of course, the teacher could} not be incorrect (sic). The enharmonic theory (with its extended series of

diminished tetrads) at leastenables the theorist to give alogical explanation of_{every chordemployed}intheworksofthe_{acknowledged masters}of classical

music,withthenotationadoptedbythecomposerhimself.

Inthis_connection,onecritic, afterreading (?)the above mentioned

pam-phlet,refers_{to the following progressions}inthefirstmovementofBeethoven's Sonata in

E

flat,Op. 31, No. 3, measures 241-244, and statesthat since the chordsat(a) and(6)are identically thesame uponthe piano, while thepassage is_undeniablyinthe keyof

E

flat,therefore, one ofthechordsmustbefalsely

notated.

P

Accordingtothe limitationsofthe"

_Day

"theory, thecritic,ofcourse,is

correct, for, thenote

F

sharpis _regardedas being foreign to the tonality of

E

flat. _Accordingto theenharmonic theory, thenote

F

sharpisnotforeign to_{the tonality}of

E

flat,andbothofthesechordsare availableforusein this key; that at(a) isthe (superior)diminished tetradon the inter-dominant, and that at (b) the (inferior) diminished tetrad on the hyper-supertonic. These chords,itwillbeseen,occurasNos.3and6, intheabove-mentioned series of

pentads(Ex. 15,page385). Now,the "

Day" theorist forbids the use of the chordat (b),but Beethoven employsit,andtheenharmonic theoryjustifiesits

use; surely, there is _{something unnecessary}and inexpedientin _regard to this

matter. This_{particular passage,} which was chosen by a critic (not by the author)withthespecialpurposeof_{refuting the}enharmonictheory,seemstobe a_{peculiarlyhappy} illustration ofits_{true significance}andof itsexact

applica-PREFACE

tion tomusicalcomposition; further reference toit,therefore,

may

beexcusable. Threepoints areworthyofnotice,namely,(i) that thechordat(ft)is_employed at_all,which indicatesthat the_composer recognizedboth chords; (2) that it isemployedafterthechordat_(a),which infers that,under certain conditions, the notationofthechordat(b) ispreferable to that at(a)and(3)that thechord

at(6)isimmediatelyrepeated,which emphasizes thecomposer'spredilectionfor

thechordasthuswritten. Furthermore,

may

therenotbeanaestheticreason

for the change in notation? The portamento phrase mark over the chords at(a)wouldimplythatthisdiminishedtetradis_regardedasanimportantchord,

andit isthus treated as_{a fundamental} discord; whereas, the legatomarkover the chord at(b), withthechordnowoccurring upon aweak beat,mightwell implythatthisdiminishedtetradis regardedasa_{fortuitous chord, the}

F

sharp

and

A

natural beingchromaticauxiliary notes,andthe

C

a_{diatonic auxiliary} note.

G

flat isnotanauxiliarynoteto

G

natural. Wherefore, thediminished tetrad at (6),mayeven be regardedas not being an independent chordat all, but asa chordof

E

flat, inits second inversion, and with its thirdand fifth

temporarilydisplacedbyornamentations. Theanalysis of thechordfromeither standpointisjustified by the enharmonic theory, see chapters

XX,

240

and 241, and XXII, 267.

iii. Ithasbeensaid_{that the author,} who vigorouslydenounces Dr.Day,

isvirtually aDaytheorist.

This statement, which possesses a certain element of truth, is, for that veryreason,allthemoreinsidious. The author isa_Daytheorist_onlyin

re-spect to thefactthat he _{accepts the} tonic, the dominant and the supertonic as generators of fundamental discords; he rejects absolutely, the

Day

theory ofchordsof theeleventhand_thirteenth,and he rejects alsotherestrictionof generators (at least in themajor mode) to the above-mentioned three notes;

two features in the

_Day

theory without which it would cease to exist. The

following quotation fromThe Modern Enharmonic Scaleshould be sufficient

to _prove that theauthor canlay very little claim to_{being a disciple of}Dr.

Day.

"

Asthe strength ofa chain dependsuponitsweakestlink,soone_example

fromaclassicalworkwill besufficienttodemonstratethe inconsistency of the

Day

theory. Thefollowingpassage fromSchubert'sSonatain

A

minor,Op.42, Moderate

VI PREFACE

first _movement, measures 33-35, is unquestionably, in the _key of

C

major.

Thechordmarked* is_{unquestionably a chromatic chord}inthe_{keyof C,for,} it is_{immediately preceded}andfollowedbychordswhich aredefinitely in this

key. The name of this chord _{is, unquestionably,} the common chord of

B

major, and it

_may

be termed the leading-note major triad. As a borrowed chord,itoccurs as thedominanttriadinthe_keyof

E

_{minor, one}ofthe borrow-ingkeys (page 387). This_chord,itwillbeseen,occurs as theconsonantbasis ofthe_{primary pentad}onB,inthekeyof_C,Ex.15,page385. While,its_use,

byoneofthe greatclassicalmasters,justifiestheformationof_{chromatic major}

triadsondiatonicnotes otherthanthe supertonic(which the

_Day

theory

for-bids), justifiesthesystemof_{borrowing from}therelativeminor keys (which the

Day

theoryalso forbids), andindirectlyjustifiesthe completeseriesof_primary

pentads,whichconstitutes theharmonicbasisoftheenharmonic_theory. "_From

thestandpointofthe

_Day

theory,thischordwouldbeexplained

asfollows: sincetheJiarmonic formof thechromaticscale does_{not comprise}

thenote

D

sharp,thisnote can only be regarded asaninstance offalse

nota-tion, andthecomposer shouldhavewritten

E

flat, incidentallygiving riseto

falserelationwiththe

E

naturalinthetreble ofthepreceding chord,butthis

point

may

be _overlooked; withthischange, the three notes of the chordwill

be foundinthefundamental chordofthe thirteenthonthe supertonic, as the

third, theminorninth, andthemajorthirteenth,and thename ofthe chord,

therefore, will be the sixthinversion of the _supertonic minor ninth and _major

thirteenth,withtheminorninthfalsely notated,andtheleading-note doubled. Poor

Schubert!"

According to the virtually-a-Day-theorist itis, simply,thecommonchord of

B

major, see ChapterXIX. Inthisconnectionit

_may

beadded that,

al-though noneofthe threetriads,

A

major,

E

majorand

B

major,is_recognized

inthe

_Day

theoryasa chromatic chordinthe _{keyof C, yet}theiruse assuch cannotbedenied;andwiththeir_{acceptance, the}wholefamilyof_{attendant keys} becomesevenmoreclosely unitedwiththeoriginaltonicthanheretofore. One

Day

theorist, inacknowledging theirclaims as chromatic_chords,venturesto

callthemthe"falsetriads;"butis itnotthetheoryratherthanthetriads,to

which the term "false" should be applied? The fact that these triads are acceptedatall_{by anydisciple of this}schoolisatacit_{acknowledgment}ofthe limitations of the

_Day

_theory.

Anothercriticopines thatDr.

Day

isdead, thathewasaworthy man,etc.,

butwhydisturbhisashes. Wouldthat the "worthy

man

"wereas "dead"

asthecriticseemsto imagine! Thedateof hisearthlydemise(1845),however,

practically coincides, phcenix-like, with the birthof hisharmonictheory,and

itis_{the theory,}nottheman,whichisunderconsideration. Theauthorwould

beoneofthelasttodisturbhisashes,butthese asheshave beenmetamorphosed

into seeds which have been sown broadcast in all _{English-speaking,} not to mentionother, lands. Everyteacherwhopinshis faithtochordsoftheeleventh

andthirteenth, orwho,in_pedanticmannertalksof falsenotation,is_{a standing} witness to thefactthatDr.

_Day

is stillaliving force.

May

the presenttreatise becomea toolwith whichreposcat inpacewillbe carved_uponhis final

PREFACE

Vii

Theworksofthe bestcomposers, both ancient and _modern, teem with chromatic_{chords, the analysis of}whichfromthestandpointofthe

_Day

_theory, isjust ascomplexandillogicalasthat of themajortriadintheSchubertsonata, but which, fromthestandpointoftheenharmonictheory,is_simpleandnatural.

Thatthechromatic elementinmusic has nothithertobeen reducedtoa compre-hensive system,isnoproof that the greatcomposersdidnot_{employ anysystem} at all in _writing chromatic _{progressions.} Practice comes before theory in matters_{appertaining toharmony;}thatis_{to say, the}lawsoftheoryarederived

anddeducedfromthe practice of the bestcomposers.

The enharmonic theory is consistent and comprehensive; it _justifies the harmonicprogressionsintheworksofthe_composersof_{the past,}anditwillbe applicable to those of the future, so long as the art ofmusicisbaseduponthe principles ofequaltemperament. Astoitsintrinsic_value,time the_supreme

critic inmatters musical alonewilldeliveranultimate_judgment.

ToMr.

Edmund

Hardy, Mus. Bac., for valuedsuggestionsin reviewing the manuscripts;and toMissN. _{R. Hearn,} for indefatigable workin

copy-ing the examples and exercises, the acknowledgments and thanks of the author _{are due,}and areherewith cordially tendered.

The author, furthermore, desires to expresshis _appreciationof thework

accomplishedby Mr. H. Clough-Leighter, of theBostonMusic Co.'s Edito-rial staff, theeminent criticandcomposer, whose erudite musicianship has been invaluable in_preparing this treatise for_publication.

J.

HUMFREY ANGER

Toronto,Canada, June3, IQII.

GLOSSARY

Thefollowing terms,someofwhicharenotin_ordinaryuseinconnection with_{Harmony, some}ofwhichare variouslyemployedbywritersonthe_theory of music, while othershave beenoriginatedinorder tomeettherequirements of_{a systematic treatment}ofthissubject, areemployedin_{the present} workas explained below.

ActiveNotes,ornotesofmotion,the supertonic,subdominant, submediant

andleading-note.

Affinitive _{Scales, those} whichhave thesame tonic, beinganalogous with

relativescales, thosewhich have thesame signature. Affinitivescalesare in chromatic relationship, while relative scalesare indiatonic relationship, with one another.

ArtificialTetrad,a chordof_{the seventh, the}real_{root or generator of}which

is_{a major}thirdbelowthenominalorapparentroot,suchasthechordsofthe minoranddiminished sevenths, whichare regardedas derivatives of pentads. Associate _Scales, the tonic, dominant and subdominant, together with

boththeirrelativeandaffmitive_scales; thesescales,nineinnumber, comprise

thecomplete(diatonicandchromatic)materialofa givenkey.

Chromatic Modification, a chordin which one of the notesis _absolutely displacedbyits_upperorlowerchromatic_{auxiliary note.}

When

two ormore

notes are so displaced, thechordis_regardedasa_compoundmodification.

CompoundSuspension,thecombinationoftwoormoresinglesuspensions, or the_compoundingofoneormoresuspensionswith anotherdiscord.

Deceptive Cadence, the dominant chord followed by the submediant. This _progression is sometimes called the "interrupted

"

cadence; the real

purposeof_{the cadence,however,}is_deceptionratherthaninterruption. Diatonic Modification,achordin which oneofthenotesis_absolutely

dis-placedbyits_upperor lower_{auxiliary note.}

When

two or morenotes are so displaced thechordisregardedasa_compoundmodification.

Displacement, (i)absolute, asinthe caseofamodification, (2)temporary, asinthe caseofasuspension.

Dorian Cadence, aname given to the progression whicharises when the Phrygian cadence(ofthemajormode)istranscribed totheminor mode.

EnharmonicScale,The Modern,achromaticscale inwhich certainnotes are employed under two different names. This scale

_may

be regaided asa convenientmethodof_{exhibiting thecomplete(diatonic}andchromatic)material

GLOSSARY IX

Extreme Chromatic Scale, thenameemployedfor theinstrumental form ofthechromatic_{scale,which comprises seventeen}notes to the octave.

Fortuitous Chord, one arising from the use of auxiliary notes. Such a chordis_{not regarded from}thefundamentalstandpoint.

Half-Note, Quarter-Note, Etc.,termsindicatingminim, crotchet,etc.

Heptad, a fundamental discordcomprising seven differentnotes, a chord ofthe thirteenth.

Heptadic Theory,the _systemof_{harmonywhich comprises chords} of the eleventhandthirteenth.

Hyper, aprefixindicating"chromatically raised," suchas hyper-tonic. Hypo, aprefixindicating

"

chromaticallylowered;"it isemployedonlyin

connectionwiththeextreme chromatic hypo-dominant.

Imperfect Chromatics, the minorsupertonic, the minor mediant and the minor submediant, with their enharmonic equivalents the hyper-tonic, the hyper-supertonicandthehyper-dominant.

InterruptedCadence,thedominantchordfollowedbyarest.

Interdominant, the chromatically raisedsubdominant; this note, being a perfect chromatic, a _{distinguishing} term is employed in _{preference to}

hyper-subdominant.

Leading-Note Tetrad, a chord of the seventh on the leading-note, the leading-note being the actual root, as distinguished from the leading-tetrad, inwhichthe leading-noteis_onlythenominalroot.

MajorChromaticScale, themelodicformofthechromaticscale.

Melodic Triads, triads formedfrom thenotesof themelodic minor scale (sometimes called the additional triads), as distinguished from the harmonic

triads,thoseformed fromthenotes of theharmonic minorscale.

MinorChromaticScale, theharmonic formofthechromaticscale.

Minor Tetrad, an artificial tetrad in which the seventhis _minor, as

dis-tinguished from a diminishedtetrad, in whichtheseventh isdiminished. In bothcasesthe thirdisminorandthefifthdiminished.

Modifications,modified_{chords, diatonic}andchromatic,q. v.

Natural Tetrad,a chordof theseventhwhichnaturally occursina scale,

as distinguishedfromanartificialtetrad,aderivative ofa pentad.

Note, (i)acharacteremployedto representa musical sound, suchas half note, quarter-note,etc.,and (2)atermemployedto indicatea musical sound, such as leading-note,highnote,wrongnote. Theterm"tone"isnot_employed to indicatea musical sound,except indefinitely, as, forexample, the toneof a piano. Thisterm, togetherwithsemitone,is_employedchiefly inthe

measure-mentof intervals,hencetheterm"diatonic."

Passive Notes,ornotes ofrest,thetonic,mediantanddominant.

en-X GLOSSARY

harmonicallychangeable,exceptforuse aschromatic_auxiliaries,orin connec-tionwiththe rarelyemployedextreme chromaticchords.

Pentad, a fundamentaldiscordcomprising fivedifferentnotes,a chordof the ninth.

Pentadic Theory, the systemof_{harmony which comprises chords}of the seventhandninth,butrejectschordsoftheeleventhandthirteenth.

Phrygian Cadences,themediant majortriadprecededby, usually,a chord ofthe sixthonthesubdominant.

Precadential Chord, a chord employed before _{the perfect cadence;} such chords do notcontain the leading-note.

Retardation,a notetiedoverfrom onechord, towhichit_{belongs, into the} next, towhichitdoes not_{belong;but which,} unlikea_suspension,moves

dis-junctly.

SubordinateChromatic Chord, one not borrowed from anassociatescale,

as distinguished from an ordinate chromatic chord, which

may

be regarded as_{having been borrowed,}inthefirstinstance,from anearly relatedscale.

Sub-suspension, a suspension formed by the use of the lower auxiliary note,andwhich,therefore,rises inresolution.

Subtonic, thenoteatonebelowthetonic.

Superior Tetrads and Pentads, those derived from the tonic, dominant and_{supertonic, as generators,}inboth modes;inferiortetradsandpentads occur

in the_major mode only, andare derived fromthe submediant, the mediant

andleading-note as generators.

Symbol,a stenographicmethodof_namingchords.

Tetrad,afundamentaldiscord comprisingfourdifferentnotes,achordof the seventh.

TetradicTheory, the_systemof harmony which limits chordconstruction tothe confines ofanoctave.

Tonal Intensity,a termapplied to the relative _{degrees of acuteness} and

gravityinconnection with bothdiatonicandchromaticnotes.

Tonal Vision, the faculty of appreciating theeffect of_{music through}the

mediumofthe eye,sometimescalled"hearingwiththe eye."

Transition, passingfrom onekeyintoanotherwithoutthe useofanychord derivedfromthedominantofthenewkey.

Turning Note,anauxiliarynote formingpart ofaturn; thepassingnote passesonwards,the turningnotereturns.

CONTENTS

PART

III

Introduction -

...

355

XIX

Chromatic Concords - -391

XX

ChromaticDiscords

-417

XXI

Diatonic Modifications - ₄₄₀

XXII

Chromatic Modifications

-457

XXIII

Enharmonic

Modulation

-481

XXIV

_Contrapuntal

_Harmony

- -505

XXV

Basses

and

Melodies -

-524

XXVI

Five-part

Harmony

- 551

XXVII

Instrumental

_Harmony

-

-597

Additional Melodies

-633 Appendixiii

Harmonics

- i

IndextoPartIII

-659

General Index - ₆₆₂

The titles of the above chapters refer to _{the principal subjects} under consideration. In Chapter

XX,

fundamentaldiscordsonlyare explained. In Chapter

XXVI,

referenceismadeto_harmonyin_two,three,six,sevenandeight

parts. In Chapter XXVII, the pianoforte, the organand the stringquartet

alone are considered,andeven these instruments_{are necessarily treatedupon} verygenerallines.

A

TREATISE

ON

HARMONY

PART

III

INTRODUCTION

THE

chromatic element is said to have been first introduced

into musical_composition

by

Orlandus Lassus (Roland de Lattre),

1520-1594; but, though

employed more

or less freely

by

all

suc-ceeding composers,

no

attempt, apparently,

was

made

to treat

this _{important subjectsystematically until}aboutthe middleof the

nineteenth century,

when

a notable

work on

_Harmony

_by

Dr.

Alfred

_Day

was

published.

It is _customary in works on

_Harmony

to treat the diatonic element in a _{systematic manner, but,} unless the author is

an

avowed

disciple ofDr.

Day,

notsothe chromatic element; the

more

frequently

employed

chromatic chords are _merely enumerated

and

briefly explained, on the lines adopted hi _Chapter

XVIII

(Part II) of the present treatise. If it be _{possible, however,} the

chromatic element, as well as the _diatonic, should be treated,

not _only in _{a systematic,} but also in a _logical

and

_a

comprehen-sive manner.

The

_Day

_{theory, as}

was

said above, is _systematic,

and

it is_logicalasfarasit_goes, butit is

_by

no

means

comprehen-sive. This _theory, moreover, has given rise to

abundant

contro-versy, being accepted

by

many

theorists _{as the only} correct basis of thechromaticelement,

and

being rejected

by

others as

unsound

and

erroneous, from boththe diatonic

and

chromatic standpoints.

Reference has _already been

made

(in the Introduction to

Part II) to the fact that there are, from the diatonic standpoint,

threedistincttheories ofchordconstructionin_vogueat_{the present}

day,

and

the

modern

student, in _{whatever path} his _steps are

directed, should at least _possess

some

knowledge of their

charac-teristic features. These theories, in _regard to fundamental dis-cords, are asfollows:

366 INTRODUCTION

1.

That

which_comprises chordsof _{the yth only,} which

_may

be termedthe_{Tetradic theory;}

2.

That which

_comprises both chords ofthe _7th

and

_gth,

which

_may

betermedthe _{Pentadic theory}

;

and

3.

That which

comprises chordsof the _{yth, gth,}

nth

and

1_3th,

which

_may

be termedthe

_Heptadic

_theory.

In the tetradic

and

_pentadic theories, the chromatic element

is _{regarded chiefly} from the melodic standpoint; in _{the heptadic} (Day) theory it is_{regarded chiefly} from theharmonic _standpoint.

The

present treatise, in relation to the diatonic element, is based

on

_{the pentadic theory,} but in relation to the chromaticelement,

a _departure is

made

from the _{original theory,}

and

this element is

regardedchieflyfrom theharmonicstandpoint.

THE

TETRADIC

THEORY

According to the tetradic _theory, chord construction is

con-fined to the limits of an _{octave; pentads, therefore,} are

unavail-able,

and

consequently, unrecognized. Chords

known

as _pentads

in theothertheories,are

known

underother

names

in this theory,

the _{ninth being regarded as} an unessential dissonance a

sus-pension or an auxiliary note. Chromatic chords are formed,

almost _exclusively,

_upon

the _principle of _{chromatically altering}

the notes of diatonic chords;

when

otherwise,they are_regarded

asborrowedchords.

On

first _thought it_{might appear} reasonable to_arguethat the

tetradic _theory is above criticism, for, since the tonal material of

music is _comprised within the limits of an octave,

and

since one

octave, except in thematterof _pitchalone, is_identically the

same

as another, so _{every chord} in music

_may

_{presumably be obtained}

withinthis _{restricted sphere.}

On

second thought, however, it will be found that there are

some

_{valid objections} to this _theory in connection with both the

diatonic

and

thechromatic elements.

The

restriction of music to the limitsof

an

octave isnot

INTRODUCTION 357

is

_{by no means}

limited to

an

octave;

and

the harmonic chord of

nature, the scientificbasis of chord _{construction, extends} over an

unlimited

number

of octaves. _While, in _{art, the suspension 9} to

8, one of the

most

frequently

employed

discords, is,of course,

im-practicable, without exceeding such narrow confines.

Further-more, the various octaves in themselves are not of equal value

from the harmonic _standpoint, for, a chord which is _acceptable

in oneoctave

may

beintolerable in another;compare,for_example,

the lower octaves on _{the pianoforte} with those in the middle of

theinstrument.

Wherefore,

no

apparent purpose is _served,

and no

material

advantage obtained,

by

thislimitation inchord construction.

The

_theory of octave limitation _precludes, of _course, the

formation of _pentads; but the non-recognition of these chords is

in the_{light of} a _{retrogression in} the_development of

_harmony,

_for, from an historical _{standpoint, pentads were accepted before they} were _rejected.

That

the ninth is _{frequently treated as a}

suspen-sion or an _{auxiliary note,} is not a sufficient reason _{for rejecting}

the chord,for, thechord

may

be,

and

frequentlyis,

_employed

asan

independent discord, as a chord of motion, each of its constituent

notes _proceeding to a note of the tonic chord the chord of

rest.

In regard tothe derivatives of_{pentads, the} tetradic_theory of

harmony

is _very defective.

The

chord of the _{leading-seventh,}

for_example, is not_{regarded as a}

dominant

_discord,

notwithstand-ing that it _naturally resolves

_upon

the tonic chord; while the

leading-note is _{regarded as the absolute root} of the chord.

That

the leadingseventhisa

dominant

discord,however,

may

be_readily

proved. For, the firstinversion of the triad on the _{leading-note,}

wjien it resolves

_upon

the tonic chord, is a

dominant

discord,

and

is

_employed

as such at the cadence in_{counterpoint,}

whenever

the

Canto

Fermo

isin the _{bass; this}chordis,therefore,aderivativejof

the

dominant

seventh. In like _manner, if a seventh be

added

to

this _triad,

and

the chord be resolved

_upon

the tonic, it will still

358 INTRODUCTION

the

dominant

ninth.

The

addition of the seventh does not

abso-lutelychange thewhole character of a chord, itis _{simply an}

har-monic

_enrichment; the root is _{not changed thereby,}

and

_the

gen-eral _purposeofthechord isnotaffected.

Wherefore,

no

apparent purpose is _served,

and no

material

advantage obtained,

by

this rejection ofchordsof theninth.

The

tetradic _{theory, though} restricted in _regard to the dia-tonic element, is the _reverse, in _regard to the chromatic.

_Upon

the _{principle of} chromatic alteration, even the simplest concords

may

be converted into unreasonable discords.

Nor

isthere letor hindrance

_by

law or precept,in _{this respect,}otherthan _{the general}

rule that _{the part} which

_{moves by}

a chromatic semitone

must

proceed

by

a diatonic semitone. _Composers, it is _well-known,

write chromatic_{progressions largely}

_upon

these_{very lines,and, as}

a matter of _fact,there is little if

_any

_{objection to} the _principle,

the objection liesin the fact that thereis _apparently

no

limitation

to the use of chromatic _notes; in this _{respect, good taste,} a

variable _quality

on

the part of the _{average student,} alone

determines where the lines

must

be drawn.

The

_following

ex-ample

illustrates the _{principle of} chromatic alteration as applied

to thetriad of

C

major.

Ex.1

(a) (*) (c) (d} (e) (/) (g) (k) (/) (/)

The

triad at (a) is converted into-a minor triad at (b), into

an

_augmented

triad at (c),

and

into a diminished triad at (d).

Of

the

G

flatin the triad at (d), it

_may

besaid that thisnoteisof

veryrare occurrence in the _key of _C,

and

students

would

be w0ll

advised not to

_employ

it at all;

and

the

same

remark

may

be

applied to the anomalous triad at (e).

The

diminished triad on

the chromaticallyraised root, at (/), is _practicable,

and

thistriad

togetherwith thoseat (b), (c), (d)

and

(e), will all beconsidered in

em-INTRODUCTION 369

ployed as chromatic fortuitous chords,for, perfect intervals

must

not be

_employed

under other

names

in chord construction; the

triadat _(g) containsan

_augmented

third,

and

that at (h) a doubly

diminished fifth, both of these intervals being the enharmonic

equivalents of aperfect fourth.

The

triads at (i)

and

(j) are, in

a word,unmusical; it isoneof the first laws inmusic that

no

note

ofa diatonicscale

_may

_{be enharmonically changed under}

_any

con-ditions, unlessitbe forthe _purposeof _modulation,inwhich case, of course,it is

no

_longer a note of the scale under consideration.

The

_{chromatic changesexemplified} in _{these particular} triads _(i)

and

(j) simply lead to unbridled license on the part of the

student; they are not to be found in the worksof the great

clas-sical _composers,

and

_they

do

not tend toelevate theart.

Wherefore,

no

apparent purpose is _served,

and no

material

advantage obtained,

by

this _extravagance in chromatic altera-tion. _Wherefore, also, the tetradic theory, as a theory, fails to

fulfil the_requirementsof

_{modern harmony.}

THE

PENTADIC

THEORY

The

_{original pentadic theory} has been _explained in Parts I

and

II of_{the present} treatise; itneed_only be

added

here that the

chromaticelement in this _theoryis _practicallythe

same

as thatin

the tetradic_theory,

and

itis, therefore,opento similar objections.

The

chromatic elementin themodern_{pentadic theory, as adopted}

in _{the present volume,} will _{be duly explained} after

some

reference

hasbeen

made

to the

work

_accomplished

_by

Dr.

_Day

in thecause

of

_Harmony.

THE

HEPTADIC

THEORY

"Alfred_{Day, M.D.,}theauthorofanimportanttheoryof_harmony,

wasborn in _London, inJanuary, 1810. In accordance with the wishes

of his father,hestudiedin London andParisfor themedicalprofession, and, after _taking a degree at _{Heidelberg, practised} in London as a

homceopathist. Hisfather's want of _sympathy for his musical inclina-tions in his earlier _{years having prevented} him from attaining a

suffi-360 INTRODUCTION

cient _degreeof_practicalskill inthe art, he turnedhisattention to the

study of its_principles, and formed the ideaof_makingaconsistent and

complete theory of _Harmony,to_{replace the}chaosof isolated rules and

exceptions, founded chiefly on irregularobservation of _{the practice} of great composers, which, till _{comparatively}lately,was all that _{in reality} supplied the place of system.

He

took some years in maturing his theory, and published it _finally hi 1845, three years only before his death,February

n,

1849."*

Dr. _{Day's theory,} even before his

work

on

_Harmony

was

published,

was

endorsed

by

Sir _{George Macfarren (1813-1887),}

who, in _{1860, published} his

own

_work, entitled,

"The

Rudiments

of

_Harmony,"

in _{which he not onlyadvocated,} butalso

somewhat

further developed, the principles originated

by

Dr. Day.

Mac-farren succeeded Sir Sterndale Bennet as Professor of

Music

at

Cambridge

in 1875,

m

the _{following year}

became

_{principal of} the

Royal

Academy

of _{Music, and,}in _recognition of his services in

the cause of_music,

was

_knightedin 1883.

With

sucha_champion,

the

_Day

theories were _{rapidly promulgated,}

and

before the

end

of thenineteenth _century

_many

eminent theorists in both

_Europe

and

America

had

_adopted the_same,

had

_taught the

same

to their pupils,

and had

written further treatises

upon

the same. Yet,

notwithstanding its

numerous

adherents, since, theorists equally

eminent,

have

refused to accept the

Day

theory, it

_may

be

as-sumed, that,

up

to a certain _{point, this theory}

must

be _logical,

and

the _{arguments conclusive,} but that

_beyond

_{this point,} it

and

theygiveriseto _{diversity of opinion.}

Dr.

_Day

divides _{the subject} of

_Harmony

into

two

distinct

parts, namely, (i) the Diatonic or strict,

and

(2) the Chromatic

or free; each part is treated _{independently} of the _{other, but they}

are united, as it _were,

_by

a _chapter on

what

he terms, Diatonic

free

_harmony.

Since the _principles enunciated in Part I have

not been _{adopted, so} far as

_any

original treatment of the subject

is _concerned,

_by

_any

_{of his successors,}

and

since _{they have}

practi-cally

no

bearing on the subject under consideration, namely, the

INTRODUCTION 361

chromatic element in music,it willnotbe_necessary todevote

_any

attentiontothem. Itisinhistreatmentof "_{Chromatic harmony,}

or

_harmony

in the _{free style,"} as hecalls it, that Dr. _Day's repu-tation

was

established.

"His _explanationof the_{chromatic system}was quite new, and his prefatoryremarksso well explainhisprinciples that they

may

be _fitly

quoted. After _pointing out that the laws of diatonic

_harmony

had

been so stretched to _apply them to modern _styles that they seemed

'utterly opposed to _practice,' he _{proceeds: 'Diatonic discords require}

preparation because they are unnatural; chromatic do not because they

may

be said to be _{already preparedby} nature' sincethe

har-monicsof arootnotegive the noteswhichform withitthecombinations

hecallsfundamentaldiscords. 'The harmonicsfromanygiven note are

a major third, perfect fifth, minor seventh,minoror _{major ninth,} elev-enth,andminoror _major thirteenth.'

And

this series gives the com-plete category of the fundamental chords of _{Day's chromatic system.}

Moreover, withtheviewofsimplifying the tonaldevelopmentof_music,

and giving a larger scope to the basisofa _{singlekey and, thereby,}

avoiding the consideration of innumerable short transitions hegives

a number of chromatic chords as _{belonging essentially 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 supertonic and tonic as well asonthedominant. In respect of thishe

says: 'The reason

why

the tonic,dominant,and supertonic arechosen

for roots is, because theharmonics in nature rise in the same manner;

first theharmonics of _{any given note,} then those of its fifth or

dom-inant, then those of the fifth of that dominant, being the second or supertonic of the original note.

The

reason

_why

the harmonics of the next fifth are not used is, because that note itself is not anote of the diatonic scale,being a little too sharp (as the fifth of _the super-tonic), and can only be used as a partof a chromatic chord.'

The

advantages of this system of _{taking a} number of chromatic chords

under thehead ofonekey, willbe obviousto_any one

who

wishes fora

complete theory to _{analyse the progressions} of keys in modern music

as well as their harmonic structure. For instance, even in the early

'Sonata Pathetique'of Beethoven, under aless _{comprehensive system,}

it would be held _{that, in} thefirst_bar, there was a transition from the original keyof

C

minor toG; whereas underthis_system thefirst

modu-S62 INTRODUCTION

lationwould beheld totake place in thefourth bar, to

E

flat,which is

farmorelogicalandsystematic."*

From

this extended _quotation it will be seen that Dr.

_Day

advocates theformation of a chord

_upon

the

dominant

_consisting

of a series of _{super-imposed} thirds,

and

comprising every note in both the _major

and

minor

_mode;

this _chord,

known

as the domi-nant _thirteenth,

_may

be

_employed

in _{the following four} different forms:

Ex

2

INTRODUCTION 303

idea of adding another third above a chord of_{the ninth, so forming a}

chordof_{the eleventh,}and thenanother_{third, soforming a chord}ofthe thirteenth,was,presumably, originalonthe partofDr._Day.

The

most

important feature in the

_Day

_{theory, however,} is the _systematic methodin which the chromatic element in music is

treated.

Fundamental

(or primary) sevenths on the tonic

and

supertonic are, for the first timein _{the history}of _{music, regarded}

as chromatic _chords;

and

the_{arguments brought forward} to_prove

that these chords

may

be

_employed

in _{a key without inducing} a

modulation, should be sufficiently convincing to _every

modern

theorist. Chords of the _major

and

minor ninth

on

the tonic

and

supertonic are next considered.

The

chord of the eleventh

on

these notesisforbidden

by

Dr.

_Day:

"The

chordof_{the eleventh,"} he _says, "unlike the chords of the seventh

and

_ninth, can _onlybe taken on the dominant."

The

reason given for _excluding this

chord, is "because its _resolution,if taken on either tonic_or super-tonic,

would

be out of the key." Finally, the chordsof the

thir-teenthon thetonic

and

_{supertonic, the ninth}

and

_{thirteenth being}

either _{majoror minor, as} in the case of the

dominant

_thirteenth,

are considered.

Although Dr.

_Day

excluded the eleventh

from

the tonic

and

supertonic thirteenths,

and

notwithstanding that Macfarren

en-dorses this _particular feature of the _{theory, yet}

more modern

authors have _{not only included the eleventh} in _{these chords,} but

have alsoadvocated the use of the chords of the tonic

and

super-toniceleventh.

The

three chordsofthe thirteenthintheir_{complete forms}

_may

beexhibited thus:

364 INTRODUCTION

These

_{chords, having once been established,}

became

the

har-monic

basis of the chromatic scale which Dr.

_Day

_adopted,

and

which has since

become

known

as the harmonic formof the

chro-matic scale. _But, it

_may

be said, so faras this scaleitselfis

con-cerned, that it is obtainable from the chords of the minor ninth

on

the _tonic,

dominant

and

_supertonic, without

_any

reference to

theelevenths orthirteenths.

Mention

_may

be

made

of _{Dr. Day's} valuable _arguments on the difference between the diatonic and chromatic semitone.

He

conclu-sively proves that, theoretically, the diatonic semitone is the larger of the two,inother words, that the interval

C

to Z)flatis _{slightlygreater,}

from the standpointof acoustics, than the interval

C

to

C

sharp. This point, however, is of no practical importance in connection with

Har-mony,for,according tothelawsof_{equaltemperament,}these semitones

are identically the same in _point of size; they differ _only in notation anduse.

Reference should also be

made

toone _particular chord which

is consideredunderthe_headingofthe

dominant

eleventh, namely,

the chord _generally

known

as the "

Added

sixth." This chord is

claimed to be the third inversion of the

dominant

eleventh, the root

and

third ofwhichare, ofcourse, omitted.

After the treatment of the fundamental discords has been

exhausted, Dr.

Day

next considers the chords of the

_Augmented

sixth. For_{these chords, probably}for thefirsttimein _{the history}

of music, a double root is claimed.

The

bass-note of the

follow-ing chords, for example, is _regarded as a _{primary harmonic} of the

generator G, the dominant, while the uppernotes are _regarded as

INTRODUCTION 365

primary harmonics of the generator Z?, and, therefore, as second-ary harmonics of G.

By

secondary harmonic is understood a

note generated from one of the harmonics of a _{given generator.}

Dr.

Day

states that the interval of the

_augmented

sixth "should not be inverted, because the upper note, being a secondary

har-monic

and _{capable of belonging only} to the _{secondary root,} should not be beneath the _lower, which can _{only belong} to the _primary root."

The

three forms of this chord

_may

_{occur, not only on} the

minor submediant,as exemplified in Ex.4, but,also,

on

the

minor

supertonic;in thelattercasethe tonic

and dominant

are theroots.

The

chromatic element in music is _finally considered in con-nection _{with "Passing} notes in the free _{style," when,} in certain cases, the necessity for the use of the melodic form of the chro-matic scale, which has hitherto been excluded, is _recognized.

After which a _chapter

on

enharmonic modulation _brings this

uniquetreatise on

_Harmony

toa conclusion.

On

first _thought it _{might appear}reasonableto_argue that the

heptadic theory is above criticism; in the first _place because it is

founded

_upon

a system,

and

surely

some

system is _{preferable to}

none atall; and, in the second place, because it is a convenient

theory; convenient, because the

dominant

thirteenth _comprises

all thenotes of the diatonic_scale, so that

_{any and}

_every diatonic

chord

_may

_presumably be explained, and, convenient, because

the three fundamental discords _comprise the notes of a

com-plete chromatic scale, so that

any and

every chromatic chord

may,

presumably, beexplained.

On

second thought, however, it will befound that there are

some

valid _objections to _{the heptadic theory,} in connection with

boththe diatonic

and

thechromatic elements.

With

_regard tothe chordof _{the eleventh,} it

_may

besaid that

evenDr.

_Day

himself_{apparently regards}this chordin the_{light of}

a _{stepping stone to the} chordof the_{thirteenth, for,} in the case of

the tonic

and

supertonic fundamental discords, the use of the

eleventh is _{forbidden, notwithstanding} that the thirteenth is

366 INTRODUCTION

eleventh, according to Dr.

Day,

seems to be in connection with

the above-mentioned

much

abused chord of the

"Added

_sixth,"

which, as

was

said above, he regards as the

dominant

eleventh

with the

dominant and

the leading-note omitted; reminding one

ofthe_playof_Hamlet,with

Hamlet and

_{Ophelia omitted} ! _And, if

this kind of _analysis can be _defended, then

_why

should not the

tonic chord be called "the fifth inversion of the

dominant

thir-teenth?" _According tothe tetradic_{theory, the root} of this chord

is _{the supertonic, but Helmholtz, the great}

German

_philosopher,

and Rameau,

anoted French _author,both claim thesubdominant

as the root of this chord.

The

_{mystery surrounding} the chord is

probably attributable to the fact that there are two chords con-sisting of identically the

same

notes, one of which naturally re-solves

_upon

the tonic chord, while the other naturally resolves

upon

thedominant. (PartII, Chap. XII.)

Ex.5

INTRODUCTION 367

the plagal cadence. This chord is _really a

subdominant

triad,

to which asixth is _{added, hence} the

_name;

the sixth is, therefore,

the dissonant note,

and

the chord

_may

_{be regarded} as a diatonic

modification.

The

sixth is _frequently introduced as a _passing

note, as at (b),

when

it is_very evident that the root of the chord

is the subdominant. According to the

_Day

_theory this chord is

a

dominant

eleventh,

G

being the root; but as the root

and

third

are omitted the seventh

and

ninth

become

consonant intervals, while, of course,

D

thesixth isalsoconsonant.

The

chord

marked

* in Ex. ₅ (c), is the first inversion of a secondary tetrad on the supertonic; thefifth, therefore, is

now

the dissonant _{note, and, as such,} it is _prepared

and

resolved in

the usual manner.

The

whole_passage is _{a sequence} of _secondary

sevenths, a

dominant

sequence, that is _{to say,} the roots rise a

fourth or falla fifth, consequently the root of this chord is

_D;

_F,

in this case, could not be theroot. _According tothe

_Day

_theory

G

is the rootnot_onlyof thischord butalso ofalltheother tetrads

in this _passage, with the result that consonances

become

disso-nances,

and

dissonances, consonances;

and

the theory, whetherit

be _regarded as convenient or _anything else,

becomes

contrary,

illogical,

and

inconsistent.

With

regard to the chord of the _{thirteenth, though,} at first

sight,it

_may

_appearconvenient tobeable tocall

_any

chord

_by

this name, which cannot be satisfactorily explained in

_any

other _way,

yet, since there areso

many

possible combinations of the notes of

this _{chord, the}

name

dominant

thirteenth

becomes

_indefinite,

and

especially is this the case,

when

it is

remembered

that the

dominant

minorthirteenth (with minor_ninth) is available for use

in the_{major mode.} Dr.

_Day

furnishes

no

lessthan _seventy

differ-ent _examples of the useof this _chord,

some

in _four-part

and

some

in _{five-part harmony,}

_many

of the latter being of _{an extremely}

harsh character.

Each

of these _examples is _{accompanied with}

one or

more

rules, to which there are sometimes exceptions, so

that the whole_chapter results in a

mass

of _{complications leading}

368 INTRODUCTION

Two

forms of this chord

_may

be _{briefly considered,} that at

(a), Ex. 6 which comprises the

dominant

with its third

and

thirteenth,

and

that at (b) which is the _{same, chord with} the

seventhincluded.

Ex.6

.

()

(*)

6 7

6

The

chordat _(a)is_{figured 6,}

and

the

name

"dominant

sixth" is, therefore, justas applicable

and

quite asjustifiable as the

name

"dominant

thirteenth." This chord is _figured

^

_by

_Macfarren,

but it is _{not customary} in _figured basses to

_{employ any}

figure

higher than the

number

9, that is to say,

any

double figures at

all.

The

dominant

sixth,

must

not be confused with the first

inversionof themediant triad; thelatter chordis _rarely

_employed

except in _{sequences, while the}former isof

common

occurrence.

The

chordat (b), Ex. 6 is figured^,

and

thisis the chord,

the one chord, to whichreference is almost _invariably

made when

the term

"dominant

thirteenth" is_employed; the

name

"domin-ant sixth

and

seventh,"however, ismore _definite, forit_exactly

de-scribes this _particular chord, whereas

"dominant

thirteenth" is

employed

fora great

number

_{(sixty-nine?) of}other chords.

Upon

the treatment of thischord Ex. 6, (b), depends

en-tirely the question as towhetherornotitshould havea distinctive name.

_When,

as in Ex. 7, the treble note is either preceded as

at (a), orfollowed asat (b),

by

thenote below, that is _{to say,}

_by

the fifth of the root, the chord is _simply a

dominant

seventh,

withthe sixth introduced _ornamentally as an _auxiliary note; at

(a) the sixthis

_employed

asa free _{turning note;} at (b) the fifthis

temporarily displaced, the sixth being either a suspension or an

INTRODUCTION 369

that is _{to say,}

when

the fifthis _{absolutely displaced (as in} Ex. 6),

then a distinctive

name

is desirable,

and

a definite

name

is

preferable to an indefinite

name

in this

and

in all cases.

Ex. 7

(a)

()

370 INTRODUCTION

Ex.9

Dr.

_Day,

thesechords are not _{only identically} the

same

from

an

instrumental standpoint, but they are also _identically the

same

in

name

and

in _effect;

and

this

_may

besaidto bethecrucial _{point in} the

_Day

_theory.

The

_names, "

dominant

eleventh

and

thirteenth"

can at least be _tolerated, but

when

Dr.

_Day

claims that an

aug-mented

fifth,a dissonantinterval,is_identicallythe

same

asa

minor

sixth, a consonant interval, it can _readily be seen

_why

so

_many

eminent musicians have_rejected his _theory. Dr.

_Day,

_moreover,

does not stop at this _point; he states

and

claims that

whenever

the chord at (a) Ex. 9 is _employed, the _composer

whoso-ever he

_may

be is _{guilty of false notation, in other words,} has

written

an

incorrect note. It is _truly a _{convenient theory,} forit

is _certainly a matter of

much

convenience,

when

a progression

occurswhichisnotin accordancewith the_{principles of the theory,}

to be able to

_qualm

the conscience

_by

saying that the composer

is _wrong; but,ifmusicalworksare to_{be judged}

and

_analysed

on

theselines, it will be found that all _{the great composers}are

sys-tematically

wrong

in theiruseof certainchromatic notes, namely,

the hypertonic, the hyper-supertonic (the note

employed

in the

present instance)

and

the hyper-dominant.

The

prefix

"hyper"

hereindicates _{"chromatically"}raised. _(See

_Appendix

I, PartI.)

It

was

stated above that Dr.

_Day

drew

a line of distinction

between the diatonic

and

the chromatic semitone, claiming that the latter

was

theoretically smaller than the former; admitting

INTRODUCTION 371

semitone being in the ratio _{of 15:16,}

and

_{the chromatic, 24} : _25,

he might at least have credited _{the great composers with}

recog-nizing the

same

difference,

and

might have granted

them

the

privilege ofemployingthatonewhich,forthetime_{being,appealed} to their emotions. This privilege, however, is _emphatically de-nied,

and

the theory, therefore, contradicts itself; it _accepts the difference butrejects thedistinction.

Reverting

now

to thechordsin Ex.9, it

may

besaid that the

augmented

fifth of the

dominant

in the _{major mode,}

and

the

minor sixth of the

dominant

in the _{minor mode,} are often intro-duced ornamentally in connection with the _perfect cadence, as

shown

in _{the followingexample:}

Ex.10

.

()

(*)

372 INTRODUCTION

E

flat in each case. Musical notation

_may

be defined as the art

of _transcribing to _paper the effect of musical sounds;

and

this

effectisas _readilyappreciatedwith the_eyeas withthe ear

_by

the

true musician. If, therefore, the above progressions differ from

one another

when

_{played onthe piano, then they}shouldbe notated differently

when

transcribedtopaper. Beethoven,itiswell

_known,

composed

his _greatest works whilst _suffering from total _deafness; he, through the eye alone,

was

able to _{appreciate the majestic} harmonies of the "Choral

_Symphony."

_Wherefore, it is a

reflec-tion

_upon

the genius of _{the great composers, to} state that _they

wrote one thing but intendedanother, or to claim that they

heed-lessly or needlesslysacrificed correct notationforconvenient nota-tion.

Now,

the notes

D

_sharp

and

E

flat, instead of being

intro-duced _afterthe

dominant

chord,

may

be

_employed

in the

dominant

chord, in the treble, in place of the fifth of the root, which thus

becomes

_{absolutely displaced,} as

shown

in Ex. 9.

The

chord at

(a) thus

becomes

a chromatic modification,

and

the chord at

(b), a diatonic modification, but the resolution is the _same,

and

the resultant effect _practically the _same, as

when

the notes

D

sharp

and

E

flat are _{introduced ornamentally,} as in Ex. 10.

They

are, therefore,

two

entirely different chords, they naturally

belong to

two

different_{keys (C major}

and

C

minor), and,

more-over, they

must

have

two

different names. _{Their names,} respec-tively, are "the

dominant

augmented

fifth

and

seventh,"

and

"the

dominant

minor sixth

and

seventh," names, which

may

not

be as convenient as the

name

"dominant

thirteenth," butwhich

atleast_{possess the merit of definitely determining} the character

of _{the chord,} a feature far

more

desirable than that of indefinite

convenience.

With

regard to Dr. Day's statement that the interval of the

augmented

sixth cannotbeinverted,it need_onlybesaidthat this statement is not in accordance with facts, for the third inversion of the

German

sixth,

when

theinterval is,of course, inverted, has

INTRODUCTION 373

Dr. Day's statement with regard to the fifth of _the super-tonic, which he claims is not a note of the diatonic scale, being

"a

little _{too sharp,"}

_may

be

and

is_{perfectly true} from the

stand-point of acoustics, but, in his endeavors to establish a _principle,

heoverlooks _{one very importantpoint, namely,} that acoustics and

equaltemperamentare _byno

means

one

and

the same_thing.

Acous-tics is thebasis of the science of _music, but _equal

_temperament

is the basis of the art.

B

_flat, for _example, as the minor seventh of C, is alsoout of tune (again a little_{too sharp),} with

B

flat the

seventh harmonic of _{the generator C, nevertheless} it is the

out-of-tune note which is

_employed

in _{equal temperament; while,} even

the_majorthird of art is not_{perfectly in} tune with the _major third ofnature.

Anotherillustration of this_pointwill beseen in the_{major and}

minor tones.

The

interval between the first

and

second degrees

of the _{major scale,} is a _{major tone,} with the vibration ratio of 8

to _9, while the interval between the second

and

third degrees is a

minor _tone, with the ratio of ₉ to 10.

_Now,

the first

and

second

degreesinthekey of _D,for _{example, are exactly thesame,}

accord-ingtothelaws_{ofequaltemperament,} asthesecond

and

third _degrees

in the_key of C;wherefore, if the note

D

has, say, 288 vibrations,

thenthe_{note E,}inthe_keyof _D,will _{have 324vibrations,} whereas

in the_keyof

C

thisselfsame_note, ,will haveonly 320vibrations.

Theoretically,thisisafact,practically,it isabsurd.

The

difference between _theory

and

_practice in this

and

in _{similar instances,}

_may

be explained on the grounds, that the

demands

_{of equality} in art

have necessitated the _{sacrifice of inequality} in science.

Even

sup-posing that the violinist

and

thevocalist _{can exemplify}the

differ-ence between the _major

and

the minor _tone, the effect

would

not be in accordance with the _principles

_upon

which the

modern

major diatonic scale is _constructed,for thefundamental _principle of this construction is_{equal temperament.}

Equal

temperament

may

be defined as the division of the

octave into twelve exactly equal semitones. These semitones

374 INTRODUCTION

and

five chromatic; but, whetherdiatonic or chromatic, theyare

all _{mathematically equal.} It follows, therefore, that allthe _perfect

fifths willalso beequal,

and

equal in _{every respect; for,} it is

_upon

this _{very principle} of _{equality that the}

modern

system of scale

construction is based, every

new

scale _{being the exact}

counter-part of the last, exact in every particular except that of _pitch alone. _{Wherefore, Dr.} Day's statement, thatone of these_perfect

fifthsisoutof _tune, iserroneous.

In the old _meantone, or _unequal

_temperament

_{system, the}

semitoneswere not all _{equal, and, consequently, the perfect}fifths

were not all _equal, with the result that certain _keys, called "wolves," could notbe

employed

at all.

_Johann

Sebastian _Bach,

however, struck the death blow to this _system in the

"Wohl-temperirtes Klavier," in whichthere are

two

preludes

and

fuguesin every key; and,after his death _(1750), the old _{system gradually}

passed away,

and

the

modern

or equal

temperament

became,

and

has since _{remained, the}definitelyestablished system.

In _{the equal}

_temperament

system the perfect fifths are

tem-pered (hence the name), that is _{to say,} tuned_slightly flat; butso slight is this flatness that the

human

ear can _{scarcely appreciate}

the differencebetweenthe _tempered

and

the natural fifth.

To

be exact, the tempered fifth is

_^

of a

comma

flatter than the

nat-ural fifth;

and

a

comma

(the

name

for the difference between a

major

and

a minor tone) is about ₅ of a diatonicsemitone. This

tempering, which is the basis of the art of _{piano-forte tuning,} is

indispensable for the purposes of

_{harmony, and}

_equal

tempera-ment

alone enables the_composertowritein

_{any and}

in_everykey,

and

to modulate to or from

_any

_key

_{by means}

of enharmonic

changes.

There are, therefore,

now

no

"wolves,"

and no

fifths

"a

little _{too sharp,"}

and no

difference between the diatonic

and

the

chromatic semitone; wherefore, the primary minor ninth on the

submediant, the mediant

and

the leading-note

may

be taken for

the _purpose of _{extending the principles of} the chromatic element