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FACULTY

Presentedtothe

./MUSIC

Facuk

>-

f

^c

1^

ElizabethTempleton

UNIVERSITY

(3)

117

Carmen

Ave.,

WINNIPEG,

-

MAN.

(4)
(5)

A

TREATISE

ON

HARMONY

WITH

EXERCISES

By J.

HUMFREY

ANGER

NEWLY

EDITED

AND

REVISED

BY

H.

CLOUGH-LEITER

from posthumous annotations of the author

In Three Parts

PART

I

$2.00

BOSTON, MASS.

(6)

Copyright, 1006, byG. SCHIRMER,JR.

Copyright,1919,byTHE BOSTONMusicCa

forallCountries.

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PREFACE

Ithasbeentheprivilege oftheauthortoteach the theoryofmusic, andnothingbutthetheory

',forthe past tenyears. During

thisextended period he has naturally had exceptional opportunities for discovering the difficulties which confront the average student; and the special purposeof thispresent work, theoutcomeofmanyrequestsfrom both pastandpresent pupils,aswellasfromfellow teachers,isto assistthe

student to meet and masterthesedifficulties, sothat Harmony, which

is one of themost important and, at thesame time, oneofthe most

charmingstudies intheart ofmusic,maybecome asourceofpleasure and enjoyment,instead of being,as so frequently happens,asourceof worry and annoyance.

Theauthor hasendeavoredin histreatmentofthesubject,tobeas

simple, asconcise, andatthe same time as thorough aspossible; sim-plicity being a necessity for the beginner in every subject; concise-ness, anecessity atthe present day, when harmony isbut oneofthe manystudieswhichclaim the attentionofthe student; whilea thorough

explanation of the subject is not only absolutely necessary, but this,

indeed, mustalsobe onsomewhat original lines inorder tojustifythe

publicationofyet anotherworkona subjectuponwhichsomany excel-lent treatiseshavealreadybeenwritten.

Itisnottobesupposed, however, that any veryoriginal theories

willbeadvancedinthe early chaptersofthe presentwork; onthe

con-trary,the authorisconvincedthatthe first objectofthe student should betoacquireaperfectknowledge, asfaraspossible, ofthat purityof harmonicstructureuponwhichtheimmortalcompositionsofaMozart

werebased. The modern composer, to

whom

a new effect is always

anobject tobeattained forthefeeblep'atitudeandthe dishonestyof plagiarism should everbeavoided willdiscover almost endless

possi-bilitiesin the modificationsofthe chordsin general use, eitherbythe

employmentof auxiliary notes,orbythe introductionofthechromatic element. Whateveroriginality there

may

beinthe presentwork will befound,therefore, in alaterchapter, whereinthe author, inorder to meetthe requirementsofmoderncomposers, advocates the adoptionof anew scale (seeEx. 5, page xi), the principal featureofwhichis the combination of the enharmonic element with the alreadv accepted chromaticextension ofthemajorscale.

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IV PREFACE

Considerable attention hasbeendevoted,throughoutthiswork, to

theexercises at the end of each chapter. These, it will be seen, are both numerous andofavaried character. In addition tofigured basses, which may be regarded as the staple form of exercise, there arealso exercises on theanalysis ofchords, onthe introduction (includingthe preparation and resolution) of discords, ontheharmonizationof

melo-diesandunfiguredbasses,onclothingblank rhythmswithharmony, and on composing original passages, etc. In all cases the exercises have beencarefullygraded,and have been arranged with thespecialviewof assistingcandidates preparingforexaminations. Thestudent,however, isnotnecessarily expectedtoworkalltheexercises attheendofeach chapter; many of them being of a suggestive character, may, at the

discretion oftheteacher, beomitted, oronthe other hand, othersofa similar type may be added either by the teacher or by the student

himself.

Itseems tobe amootquestion as to whetherthe student should, or should not, be taught to harmonize melodies from the beginning.

Theauthorisinclinedtothinknot,buteverythingreallydepends onthe student; forthough onestudentwillgrasp theprinciplesreadilyenough,

and anotherwill fail to doso, bothof them may inthe end proveto beequally successfulintheirgeneral work. Thismattermustalsobe

leftwith theteacher,andiftheexerciseson harmonizingmelodies,etc., are omitted when the chapter is first studied, they will probably be founduseful inreviewworkatalaterday.

Another feature, and one to which the author attaches much importance,isa systembywhich bothtimeandspacemaybe savedin

writing the names of chords. By the use of the signs, + for major, - for

minor, o for diminished, and x for augmented, combined with certainRomanandArabic numerals, eachchordmaybedesignatedby meansofa symbol (somewhatresembling thesymbols usedin

Chemis-try),placedbelowthebass-note. Thesavingintimeandspacewillbe readilyseenfrom oneortwoexamples, thus:

SYMBOL SIGNIFICATION

Vv1 Thefirstinversion

ofthe dominant-seventh.

I-a Thesecondinversion ofthetonicminorcommonchord.

x!V?o The diminished seventh on the chromatically raised subdominant.

Suspensions, andin facteverychord employedinmusic,

may

also be represented. In no caseit may be said, is the generally accepted systemof figures (employedforthefigured bass),tamperedwith.

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PREFACE

V

Inthissymbol system,whichisexplainedinAppendix1, thefirst

sixnotesofthescaleare representedby thefirst six

Roman

numerals, theLeadingnote,however,isrepresentedbytheletter L, inorder that the attentionofthestudentmaymorereadilybedirected tothe

char-acteristics of this note,whichnot only has, as arule, afixed

progres-sion, butwhich, unlike the other notesofthemajorscale, isnever the root of a diatonic common chord, and never the tonic of an attend-ant key. Theminorseventh inthe minorscale willbe knownas the

subtonic, inpreference to thesomewhatawkwardexpression, "flattened leading-note," and is represented by the

Roman

numeral VII. This system,it ishoped,willsupplyalong-feltwant,forinsteadofregarding theanalysis ofchords asasomewhatirksometask,the earnest student,

it has been found, takes a delight in trying to discover the correct symbolsforthe variouschordsastheyoccurinanexercise.

Another feature of this work, to which the author would draw

attention,isthe theoryofthebesttreblenote. Incertain chords,such asthesixthonthesubdominant, the cadential

4,thesecond inversion

of the dominant-seventh, etc., onenotein particularis often prefer-ableinthetreble,andthisnotehasbeenpointed outinevery instance Innocase, however,isthetreblenoteforthefirstchordofanexercise

suggested bythe use of a figure below the bass note; a plan which, while frequently adopted in text books, savors, in the opinion of the author, of mistaken kindness. Before commencing to worka figured

bass thestudentwilldowell to jot inhereandthere,whereverpossible,

the best treble notes, especially at the cadences; these will not only

becomesteppingstones, asitwere,andsoprevent that aimless groping inthe dark, so commonwith beginners, butwill also,asarule, ensure

an interesting melody, a feature often lacking in an otherwise correct exercise.

The leading and diminished seventh chordsare treated as being

derived, respectively,fromthedominant major and minorninthchords,

bytheomissionofthedominant,butin neithercaseisthechordcalled

thefirstinversionofthedominantninth; thestudent being taughtthat

inthe last named chord thedominant isalways present, whilein the chordsoftheseventhonthe leading-note thedominantisnever present.

A

distinction isthusmadewhere adifference, if perhaps only aslight one,certainlyseemstoexist.

In thetreatmentofsuspensions theauthorteaches thattheconcord isthesuspendednote,the discord being the suspending note; forit isthe concordoverwhichthereissuspense while the diacordisbeing heard.

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vi PREFACE

Thus,inthe suspension9to8,theninthisthe suspendingnote,andthe octave thesuspendednote,the ninthandtheoctavetogether, of course, constituting the suspension. Suspensions are also taught asbeing of

twokinds, risingandfalling,sothat theterm'retardation

'

is unneces-saryhi thisconnection; asuspensionisalwaysregardedas falling unless

thewordrisingisused, inthesamewaythatascaleisalwaysregarded

as beingmajorunlessthewordminorisused. Theterm'retardation'

isemployed fortheso-called'

drivingnotes,'and isconsidered in con-junction with the 'anticipation,' of whichit maybe regarded as the converse.

Secondarysevenths are treatedundertheheadingoffundamental

discords', a fundamental discord being regarded as a discord derived

from a fundamentalnote orroot. Thus,thechord Ex. 1 at (a), Ex. 1, in the key of C, isthe supertonic

seventh, II?; inthekeyof fi-flat this same chord

would be the mediant seventh, III?. The chro-matic chord at(6),inthe key of C,istheprimary

seventhon thesupertonic, II+7. Ineachcase the root is the same, viz.

D. The seventh on the dominant, though regarded as a primary seventh, isalwayscalledadominant-seventh.

Much

attention has alsobeen devotedtothe subjectofharmonic progressions. The best progressions both to and from the various chords, astheyseverally occur,areconsideredineveryinstance,sothat thestudentmaylearn towrite shortpassages,introducinggiven chords;

abilitytodo whichisnow.necessaryin theoreticalexaminations.

Theline ofcontinuationisusedto indicatethe retentionofoneor

morenotesfromthe preceding chord; this, theoriginalmeaningofthe term, avoids all confusion, and avoids also the necessity of adding a

mark (ofanykindwhatever)tothe bass-noteofacommon chord.

Theillustrationsthroughoutthisworkhave beengiveninthe keys of

C

major andeither

A

orCminor,butthey shouldinevery casebe transposedintootherkeys,andthemoreimportant examplesshouldbe committedtomemory.

No

illustrationsfrom the compositions ofthe great masters have beenincluded, forthoughofconsiderableinteresttothe true musician,

it isaquestionwhethersuchexamples arereally ofanybenefit to the student. Speaking generally, itmay be said, that forevery example chosen to prove a rule, another example could probably be found, possibly intheworksofthesamecomposer,which might be quotedto disprove the rule; indeed, to furnish adequate illustrations, with

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PREFACE Vll

exceptions, and explanations, a work on harmony would become an extremelybulky volume,farexceedingboththe intentionsoftheauthor andthelimitsofthe present work.

In order to avoid unwieldly dimensions, this treatise has been dividedintothreeparts, the contentsofwhichmaybesummarizedas follows:

PARTI. The major and minor diatonic scales, intervals, the

com-mon chord and its inversions, cadences, sequences, the dominant-seventh,andnaturalmodulation.

PARTII. Secondary sevenths, the dominant-ninth with its

deriva-tives the leading and diminished seventlis,

suspen-sions, auxiliarynotes,andextraneous modulation.

PARTIII. Chromatic concordsanddiscords, enharmonic modulation,

certain modified chords, harmony in other than four

parts, contrapuntal partwriting, andthestringquartet.

No

mention,itwillbeseen,ismadeintheabovesynopsis, ofchords

of theeleventh and thirteenth. To thistheory of chord construction theauthorisaltogetheropposed,thoughatthesametimefully appreci-ating the greatworkwhichAlfredDay,M.D. (1810-1849), the origina-tor ofthe theory,accomplishedfortheadvancementoftheart,andfully

recognizingalso the strong supportwhich ithas receivedat thehands ofmanyofthemost eminenttheorists inEngland andAmerica.

Bach,to

whom

weareindebtedfortheestablishmentofequal tem-perament,thevery basis ofallmodernmusic, diedpracticallyahundred

years before the Day theory was promulgated. Moreover, to Beet-hoven, the acknowledged greatest musical genius the world has ever seen, thistheorywasabsolutelyunknown. Thestudent,therefore,who would examine and study the works of the great classical masters,

willfindtheDaytheory, tosay theleast,unnecessary. Furthermore,

itcannotsomuchasbeassumedthat the great composerswrote,even unconsciously, in accordance with this theory, for the notation of certain chords which occur in their works, is

entirely at variance with this theory. The

chord at (a), Ex.2,for instance, is disallowed,

andwhen thuswritten,the Z)-sharp is said to be 'incorrect notation employed for conve-nience/and should,it isclaimed, beE-flat, as at (b) the chord being, according to the Day theory, the dominant minor thirteenth, and

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Vlll PREFACE

advocatedbythe author, the chords arenotthesame; that at(a)isthe

^7^i.e. thedominant-seventhwith thefifth augmented, in thekeyof

C major; while the chord at (6) is the Vj_i.e.the dominant minor sixthandseventh,inthekeyof C minor. This chord,itmaybe said, willbe explained induecourse, and the name, whichatfirstmay ap-pearsomewhatstrange,will,it is felt,bejustifiedbythe explanation.

Herethen aretwo conflicting theories. In the onecase, thetwo

chords are regarded as being the same, andthe composer, whoeverhe

be,thatwritesD-sharp,iswrong; intheother,the chords aredifferent,

withdifferentsymbols and names, in different keys,and even possess, aswillbeshownbelow,differenteffects,whilethecomposermaywrite whicheverhisfancydictates. Thefactthat thesechords areidentically

thesameonthe pianoisnoproof whateverin favoroftheDaytheory, for this mayalso besaid oftheGerman formof theaugmentedsixth, which is identically the same on the piano as the dominant-seventh; andthe same may alsobe said ofthe various enharmonic changesof

thediminished seventh, butnoincorrectnotationis claimedforthese chords.

Thedifference inthementaleffect ofthe chordsat(a) and (6),to

whichreferencewasmadeabove,isdueinparttothedifferencebetween

G

to D-sharp, whichisanaugmented fifth, and

G

to#-flat, whichisa minorsixth; inthelattercasetheintervalisconsonant, inthe former,it isdissonant. Thisdifference in effectwillbereadilyappreciatedifthe natural resolutions of these chords, shown at (a) and (6), Ex. 3, are playedonthe piano. Theprogres- Ex. 3

sion at(a)is tinged withjoy,that (a) (6) at (6) with sorrow.

A

veritable

illustration of musical antithesis.

Now

effects such as these are as readilyappreciated with theeye as withtheear by the true musician. Beethoven, itiswell known,

com-posed his greatest works whilst suffering from total deafness. He, with the eye alone, could appreciate the majestic harmonies of the 'Choral Symphony.' Itistherefore hardlyright to supposethat the great masterswrote onethingand intended another; andit is hardly

justtoclaim that they heedlessly or needlesslysacrificedcorrectnotation

for convenient notation. Examples of incorrect notation

may

perhaps occuroccasionally,butthey are veryrareindeedintheworksofthe best composers; eventhewell-known pre-cadential chord intheAndanteof

Beethoven's Sonata appassionata (measure

G), in which the

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Trir-PREFACE IX

livefifthsarehidden fromtheeye,onlybythe notationemployed, can beexplained,and thecomposerabsolvedfromthesin of incorrect

no-tation.

Itmay be claimed that certain chords can only be explained as

chordsof the eleventh orthirteenth; to this the author would laya counter-claim, that such chordsdo notexist in theworksof Bach, of

Beethoven, or indeedof anyofthe old masters; forsurely thesegreat composerswroteinaccordance withsometheory,andithasbeenshown

thatthiswasnottheDaytheory. Furthermore,thistheoryisevenmore

inadequate when applied to modern compositions; for C-sharp is

employedas freely as D-flat, Z)-sharp as l-flat, and(7-sharpas

A

-flat,

inthekey of Cmajoratthe present day. The old cryof 'incorrect

notation'

isnolongerplausible; it islittleshortofaninsulttothegood composer; andthedayhasnowcomewhenthe theoristmustbeprepared toaccount forallsuch chromatic progressions.

A

modernauthorl

of avaluablework on harmony, says that he baseshissystem "onthe Diatonic, Chromatic andEnharmonicscales,

andventures to prophesy that the Text-book of the futuremust and

will follow some such plan, in order consistently to explain modern combinations, which,underthetheoreticalsystemsatpresentinvogue, require a wrong application of the laws of acoustics, and a complex multiplication of roots." With these sentiments the author of the presentworkisin perfectaccord,andthe theorywhich he advancesto meetthesemodernrequirementswillbe propoundedin PartIII of this

work. Here it will be shown thatthe complete key of C major, the

material, soto say, atthecommandofthe composer,

may

beobtained

by theformation of a series ofprimary ninths, commencing, like the Daytheory, with thetonic astheoriginalgeneratorofthe key; thisis

followedbya primaryninthonthedominant, asthefifthofthetonic; andthis,byanotheronthesupertonic, asthefifthofthedominant. So far thisisin keeping withthe Daytheory. "Thereason," toquoteDr. Day'sownwords, "whythetonic,dominant, andsupertonicarechosen

for roots, isbecause theharmonicsin naturerisein thesamemanner;

firsttheharmonicsofanygivennote,then thoseofitsfifthordominant, then thoseofthefifthofthatdominant, being the second or supertonic

oftheoriginal note. Thereason," andtothistheauthorwould draw

particular attention, "whytheharmonicsofthenextfiftharenotused

is, because that note itself isnota noteofthe diatonic scale, beinga

'Dr.Charles Vincent,authorof"Harmony,DiatonicandChromatic." The

theo-riesonthe origin of theChromatic elementinmodernmusic,advocatedinthe present

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X PREFACE

littletoosharp (asthefifthofthesupertonic),andcanonlybe usedas apartofachromatic chord." Thelatterpartof thisquotation

may

be, andis, perfectly truefrom the standpoint of acoustics; but acoustics andequaltemperamentarebynomeansoneandthesame thing. J3-flat,

forexample,astheminorseventhof C,isalsooutoftune(againalittle toosharp),with -flattheseventhharmonicofthe generatorC, never-thelessit istheout-of-tunenotewhichisemployedinequaltemperament; andagain, eventhemajorthird of artisnotperfectly intune with the majorthird ofnature.

Another,and aninteresting illustration of thispointwillbeseenin

themajorandminortones. Theintervalbetweenthefirstandsecond degreesofthemajorscale, isa majortone, with the vibrationratio of 8 to 9, while the interval between the second and third degrees is a minortonewith theratio of9to 10.

Now

thefirstand seconddegrees

inthekeyof D,forexample, are exactly the same, accordingtothelaws

ofequal temperament, as thesecondandthirddegreesinthe keyofC; wherefore,ifthe note

D

hassay,288vibrations,then the note E,inthe keyofD,willhave 324vibrations,whereasinthekeyof

C

thisselfsame

note, E, will have only 320 vibrations. Theoretically this is a fact,

practicallyit isabsurd. Thedifferencebetweentheoryandpractice in thisandinsimilar instances,

may

beexplainedonthe grounds, that the

demandsof equality in arthavenecessitatedthe sacrificeof inequality in science. Even supposing that the violinist and the vocalist can exemplify the difference between the major and the minor tone, the

effectwould notbein accordancewith the principlesupon which the

modern major diatonic scale is constructed; for the fundamental principle of thisconstruction isequality,the octave being dividedinto

twelvemathematicallyequal semitones.

In the old meantone, orunequal temperamentsystem, the semi-toneswerenotallequal,with theresultthatcertainkeys,called"wolves," could not be employed at all; when, therefore, the Wohltemperirtes

Klavier, in which there are preludes and fugues in every key, made itsappearance, theold system gradually passedaway, andequal tem-perament became, and has since remained, the definitely established

system.

Returningnowtothe pointin question, since allthe semitones are exactlyequal,itfollowsthatallthe perfectfifths willalso be equal,and

equalineveryrespect, forit isuponthisveryprinciplethat themodern system of scaleconstructionisbased, everynewscalebeing the exact counterpartofthelast, exact inevery particularexcept that of pitch

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PREFACE

Dr.

Day

states is"alittletoo sharp," cannot bethesame note asthat whichisemployedforthesubmediantinthemodernmajorscale, all the fifthsinwhichare equal.

Itwasthenthisveryquestionof perfect equalitywhich prompted

the author to continue theseries ofprimaryninths from the pointat whichDr.Dayends. For,sincethefifthsareallequal,thesubmediant

willbeinthesamerelation tothe supertonicasthe supertonicistothe dominant, and as the dominant is to the tonic. Having, therefore, formedchordsoftheprimary minorninthuponthetonic, thedominant andthesupertonic,thesubmediantistakenforthenext chord, alterwhich themediantistaken,andfinally the leading-note; finally,because the

perfect fifth above the leading-note can never form part of a diatonic scale.

A

series of chords is thus obtained, the natural evolution of

which,fromtheoriginal tonicasa generator,willbeseen inthe

follow-ing example:

Ex.4

I

Thewholenotesintheabove exampleare the rootsofthe succes-sivechords; the quarter-notes,in eachcase,form a chord ofthe dim-inishedseventh. Thesechords,itwillbeseen, extend overa compass ofexactly fouroctaves,commencing andendingonthetonic ofthe key.

A

scale may now be formed from the notes comprised by these

chords,ascale whichtheauthorbelieveswillmeetallthe requirements

ofthemoderncomposer, ascale containing notonly the diatonic and the chromatic, butalsotheenharmonicelement,and whichmay there-forebetermed

Ex.5

The ModernEnharmonicScaleofC

Thehalf-notesintheabove exampleare the diatonic notesofthe

scale; the quarter-notesarechromatic, andare called theperfect chro-matics,being diatonictotheattendant keys they are never

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enharmoni-Xii PREFACE

cally changed; the eighth-notes are called the imperfect chromatics, and maybe employed in eitherform.

Ifthis scalethenistakenasthebasis forallthepossiblechordsin

thekeyofCmajor, threeimportantnewtriads,viz.,

A

major,

E

major and

B

major, besides others, are added tothe already accepted chro-matic concords. These, itwillbeseen, arethedominanttriads ofthe

threeattendantminorkeys,andbytheiracceptance thewholefamilyof

attendant keys becomes even more closely united with the original tonic key than heretofore. By the addition of these triads, together with their sevenths and ninths, a theory of harmony is obtained by

whicheverychordinthe worksof the greatcomposers can bejustified. Furthermore, themoderncomposer willfind at hiscommand a major triad on every semitonal degree of the key, except alone on F-sharp

(or Cr-flat),which, having no note common to the diatonic scale, is

naturally altogether foreign to the key. At the same time, seem-ingly boundlesspossibilities in the realm of tone-color are presented to the composer,possibilities which, indeed, maynever beexhausted so long asthe art ofmusic is based upon these very principles of equal temperament.

Thechiefdefects oftheDaytheory

may

bebrieflysummarizedas follows:

(1) Itwasaltogetherunknown tothe greatclassicalmastersofthe

Bach to Beethoven period (and even later); indeed, it frequently

happensthatchordsemployedbythesecomposers cannot beexplained

bytheDay theory,in whichcasethechords are said to "bewrittenin

Falsenotation.

(2) The compoundintervals ofthe eleventh and thirteenth being represented in figured basses by theirsimple forms the fourth and

sixth,the names 'dominant fourth'

and '

dominantsixth'

are just as

applicable and quite as justifiable as the names '

dominant eleventh'

and 'dominantthirteenth.'

(3) These names, lurthermore, are very indefinite. Dr. Day

furnishes thirty-two different examples of the use of the dominant eleventh, and as many as seventy differentexamples of the useof the dominantthirteenth.

(4) The theory completely overthrows the generally accepted theoryof roots, and, consequently, root progressions, for anydiatonic

triadorchordof theseventh may be regarded as an incompleteform of thedominantthirteenth.

(17)

PREFACE Xiil

thistheory,and this scalebeing derivablefromthe chordsofthe ninth onthetonic, dominant andsupertonic,nothingisgainedbyaddingthe eleventhandthirteenthtothesegenerators.

(6) Itisfounded (ashas beenshownabove) uponafalseestimate

ofthe truesignificance ofequaltemperament,inwhich,forthe purpose of enharmonic changes, allthe semitones whetherdiatonic or chro-maticmustberegardedasbeingabsolutely equal.

(7) Finally,when appliedtotheworksofthe great modern com-posers Wagner, and his contemporaries andsuccessors, this theory

signallyfails, for,inthechromatic extensionofthemodesnowinvogue, a primaryseventh, (tomention one chordalone) maybe employednot onlyuponthetonic, thedominant andthesupertonic, asadvocatedby

Dr. Day, but alsouponallthedegrees ofthemajorscale.

The various theories of harmony, however, after all is said and done, maywellbelikened to thedifferentpathsup mountParnassus.

Theoriginator ofatheory issimplythe discovererofanewpath, the teacheristheguide,andthestudentisthepilgrim. Toreach the sum-mitisthe objectofall. Arewenotbuttoo frequentlypronetobelieve

that thepathweourselvestrodisthe onlypath?

May

not the pilgrim

occasionally take another path? Should not the guidebe acquainted with everypath? Isitnot the partofthe discoverertofind,ifpossible,

thesmoothestpath,toremoveobstructionsandtograde theroadwhere necessary,andthustoprepareawaywhich fromhishearthefeelswill bewelcomedbyall?

A

pathofjoy to the guide,apathofcomfortto thepilgrim.

Toallhis pupils,pastas well as present, theauthordedicatesthis work; and if itbe the means of elevating theirmusical taste, or the meansofarousingakeenerdesirefor,and abetter appreciationof,the higher typesofmusic, or themeansof leadingtheminto therealmof

composition, when perhaps they may add something, if only a small

contribution, tothatwealthofmusicofwhichwearealready thehappy

possessors,then,indeed,will he feelrepaid,and amply repaid, for the

timeandlabor spentuponthesame.

Theauthor, furthermore,desires toexpress hisappreciationof the work accomplished by Mr. H. Clough-Leighter, of the Boston Music Co.'s Editorial staff,the eminent critic and composer, whoseerudite musicianship hasbeeninvaluable in preparing this treatise for publi-cation.

J.

HUMFREY ANGER

TORONTO, CANADA, Junes,1911

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CONTENTS

PART

I

CHAPTER PAGE

Introduction 1

I.

The Major

Diatonic Scale .. 6

II. Intervals 17

III.

The Minor

Diatonic Scale 31

IV.

The

Common

Chord

41

V.

Harmonic

Progressions 56

VI. Cadences, Sequences, etc 78

VII.

The Chord

ofthe Sixth 102

VIII.

The Chord

of theSix-four 114

IX.

The

Dominant-Seventh 134

X.

The

Inversions of V? 148

XI. Natural Modulation 166

Appendix

Symbols

i

Index.. ix

The titles of the above chapters refer to the principal subjects undertreatmentinthe various chapters; other cognatesubjects, how-ever, are often introduced, as in Chapter III,where a brief reference

is made tothe chromatic scale; andagain in Chapter VI, where the

principle of modulation is explained, and reference is made to the 'Additionaltriads'oftheminor mode.

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(21)

A

TREATISE

ON

HARMONY

PART

I

INTRODUCTION

THE

importantperiod atfactorwhichin

Harmony

music, appearsfirst beganto haveto bebeenan about the end of the sixteenth century, being

contempo-raneous with the establishment of the

modern

diatonic scales, with the gradual development of

Form

in musical composition,

and

with the rise of the opera

and

oratorio, all of which are the direct results of the Renaissance, the

period, therefore, in which

modern

music

had

its birth. Priorto this period, all music

was composed

ona basis ofCounterpoint. Counterpointwas,

and

is,theart of

com-bining

two

or

more

melodies.

The

first school of contra-puntalcomposersaroseinthe fourteenth century,

and

before this era,

Harmony,

as

we

understandit, did not exist.

In

Harmony,

musicisregardedfromthe perpendicular

point ofview; in Counterpoint, fromthehorizontal.

The

highest types of

modern

classical music

may

be said to be

built

upon

a union of

Harmony

and

Counterpoint.

From

the

commencement

of the 17th tothe middle of

the 18th century considerable advance

was

made

in theart of music generally. Scarcely a chord exists at the present

day

whichisnotto befoundintheworks of

Johann

Sebas-tian

Bach

(1685-1750),

and

George Frideric

Handel

(1685-1759).

To

Bach

is due the

consummation

of the

(22)

2 INTRODUCTION

About

theyear 1750,

and

inagreatmeasure duetothe

influence of Bach, equal

temperament

gradually

came

into use, taking the place of theold mean-tone, orunequal

sys-tem

of tuning.

The

pianoforte (the

most

important, per-haps, of all musical instruments)

now

began to

become

popular; the

modern

orchestra, also,

was

established about this time, while the Sonata

and

Symphony

(the highest

types of musical composition) were brought to perfection atthe hands of Josef

Haydn

(1732-1809), Wolfgang

Ama-deus

Mozart

(1756-1791),

and

Ludwig van

Beethoven

(1770-1827).

Since thedeathofBeethoven,

who

isgenerallyregarded asthegreatestmusical genius theworld haseverknown, no

materialadvance has been

made

inthescience of

Harmony.

Modern

composersareindebted for

any

originalityin their

harmonies almost entirely to the

employment

of discords

formed either

by

the use of auxiliary notes, or

by

the

intro-duction of the chromatic element. In the matter of

har-monic

progressions,it isa questionwhether any advancehas

been

made

upon

the masterpiecesofthe great composers,or, indeed, whether

any

advance can be made, as long as the

division ofthe octaveis based

upon

equal temperament.

The

theoryofmusic,it

may

herebesaid, comprises the followingsubjects; the

Rudiments

ofmusic,

Harmony

inall fys branches,

Form

in composition, Counterpoint (simple

and

double),

Canon and

Fugue, Orchestration, Acousticsas

appliedtomusic,acriticalknowledgeofthegreatestmusical

works,

and

the general History of theart of music. This course of study will occupy the attention of the average

student fora period of at least three years, afterwhich he

will begin to learn

how

little he really knows; for having

climbed to the

summit

of one hill, he will find, rising inall

directions, higher

and

higher still, the

summits

of other

(23)

INTRODUCTION 3

inthe valebelow.

Thus

it

was

withBeethoven,

who

atthe

zenith ofhis career,exclaimed,"Ihavenot studiedenough."

The

subject of

Harmony,

forthe purposesof study,

may

be dividedintothe following branches:

I. Introductory

The

major

and

minordiatonicscales,

and

intervals. II.

The

Common

Chord

Harmonic

progressions, cadences

and

sequences.

III.

Fundamental

Discords

The

dominant-seventh, modulation, secondary

sevenths, major

and

minor ninths

and

their derivatives.

IV. Suspensions

Single, double

and

triple, bothrising

and

falling.

V. AuxiliaryNotes

Modifiedchords, pedals

and

arpeggios.

VI. Chromatic Chords

Triads, primary sevenths

and

ninths,

and

chro-maticallychanged notes.

In additiontothe above, whereinthe subjectisusually

treated from the four-part vocal standpoint, the following

features, also, will be consideredin the present work:

Harmony

for

more

orless thanfourvoices,

Harmony

in itsrelation to counterpoint,

and

Harmony

forinstruments,especiallythestringquartet.

It is very important for the student to possessa good generalknowledgeofallthat appertainstotherudimentsof

music before

commencing

the study of harmony.

A

thorough knowledge ofthe diatonic scales

and

of intervals

beingespecially important, thesesubjectswill betreated of inthefirstthreechapters.

The

scales,indeed, arethe verys

(24)

INTRODUCTION

put together to form words, words to form sentences, and

sentences to form a poem, so, in music, notes are put

to-getherto form chords, chords to form periods,

and

periods to formthe '

song withoutwords.' In the scale, onenote at atime isconsidered, each note havingits

own

technical

name;

in the interval,

two

notes at a time are considered,

theeffectbeingeitherconsonantor dissonant; while, inthe

chord, three or

more

notes at a time are considered, the

combined effect beingeithera concord or a discord.

Suchquestions as the valueofnotes

and

rests,the

mean-ing of the Great Staff, the various clefs

and

their use, the

value of the measure (including time

and

accent),

and

the significance of the musicalterms

and

signs in ordinary use,

belong to the subject of rudiments,

and

it would be out of place to treat of

them

inthe presentwork. Letthe student

not fail to have a perfect mastery over all these matters

before

commencing

tostudychapter IV; for,soonerorlater,

he willlearn that theharmonic structurewhich heisabout

to erect requires, like architectural structure, a firm and solid foundation.

Furthermore, it is

most

desirable that the student should possess

some

knowledge of piano playing

and

sight-singing; if, indeed,it onlyissufficientto singthemelodies,

and

to play theprogressions, containedin his exercises; for

an intelligentappreciation of everything he writesis indis-pensable.

He

should also learn that music

may

be

appre-ciated through the

medium

of the eye, as well as through that of the ear. This faculty of reading music with the understanding, as one

would

reada book, isnota property

appertainingtogeniusalone; itisrather a property

apper-taining to talent,

and

talent is the reward of perseverance

and

anaptitudeforwork. In ordertoacquirethis faculty,

let thestudent, in the first place, try to imagine the effect ofeachprogression,

and

here thesigHt-singingwill be found

(25)

INTRODUCTION

invaluable,

and

inthesecondplace,let

him

testtheaccuracy

of his conclusions,

and

this

may

be done

by means

of the

piano. If this plan be conscientiously adopted, hewill, in

due course, be able to see the effect of a passage without

singingover theparts,

and

without

making

use ofthe piano atall; for, inthe imagination, every chord

may

be played,

every

melody

sung,

and

thefull intent of every progression

perceived

and

appreciated.

The

student, moreover, isstrongly advised, in working

exercises, togive to each chordas itoccursitsproper tech-nical name,

and

to write under the bass-note the symbol

employed to represent the chord. It is not sufficient to

calla chord

by

itsalphabetical

name

alone.

The

common

chordof C, forexample,inthekeyof C, differsaltogetherin effectfromthe

common

chordof

C

inthekeyof

F

(majoror

minor),

and

fromthe

same

chord inthe keyof G, or in the

key of

E

minor; but the alphabetical

name

is the

same

in allthese cases.

The

characterof a chord depends entirely

upon

thekeyinwhichitoccurs.

Now,

by

the

employment

oftechnicalnames, for which symbols

may,

ifpreferred, be

substituted, the apparently complicated functionsofchords

are reduced to a remarkable degree of simplicity, as there is but oneset of technical

names and

these, with certain modifications, serve the purpose for every key, major or

minor.

In conclusion, let it be said,

and

it cannot be too strongly emphasized, that the faculty of '

tonal vision/ if such a term

may

be employed, or,

'

hearing with the eye/

asit is frequently called,

must

in

any

case

and

atall costs

be acquired before the student can entertain the hope of

(26)

CHAPTER

I

THE MAJOR

DIATONIC SCALE

1.

A

musicalsoundistheresult of regular vibrationsin

the air; irregular vibrations result in

what

is

commonly

called noise.

Every

musical sound possesses three char-acteristics:

(1) Pitch:"the acuteness orgravity ofthe sound;

(2) Force: itsdegree of loudness;

and

(3) Quality, which depends

upon

the source of the sound.

The

sound

known

as 'middle-C/ is regarded as the standardof pitch in music.

Itiscalled middle-Con accountof the central position

whichthenote,

employed

torepresent thissound, occupies inthe GreatStaff:

i

Middle-C,

may

be said to be the starting point in the

studyof

Harmony.

The

difference in pitch between

two

musical sounds is called an interval. Intervals are measured

by

semitones;

a semitone, i.e., a half-tone, being the smallest difference

between

any two

soundsinmusic.

A

semitoneistherefore

known

as theunit ofmeasurement.

2.

A

succession of sounds ascending (or descending)

regularly is calleda scale (Latin: scala, a ladder). There

are

two

kinds of scales used in music, the Diatonic (Greek: dia,through,

and

tonos}a toneorsound),

and

theChromatic

(27)

THE MAJOR DIATONIC SCALE 7 Theconstruction of thechromatic scale, which consists of semi-tonesonly,willbeconsideredinalaterchapter.

Ofthediatonicscale,inwhich bothtones

and

semitones occur, there are

two

kinds the

Major and

theMinor.

The minorscale willbe consideredin chapterIII,a knowledgeof intervalsbeing necessary inorder tounderstanditsconstruction.

Anotherkindof scale

known

as the

Enharmonic

should

also be mentioned; it can be played on instruments ofthe

violincharacter,

and

canalsobe sung

by

thevoice. Inthis scale, the soundsare closer togetherthan a semitone; it is,

therefore, not used in

Harmony.

The

term enharmonic, at

the present day, isexclusivelyusedfora changeinthe

name

ofanote,such as C-sharpto Z)-flat, there beingno changeof pitch whatever.

The

term

key

is

employed

to represent the notes of a

diatonic scaletaken collectively. This term, however, has really a wider significance, for it comprises also certain

chromatic notes.

3.

A

diatonic'scaleis

named

afterthe noteon which it

commences,

and

this note is called the key-note.

Any

sound in music

may

be taken as a key-note. There are

sevendifferent notes,each with adistinctivelettername, in a scale; to these the octave of the key-note is invariably

added to complete the effect.

Each

note or step is also called a degree.

In amajor scale the semitones occur between thethird

andfourth,

and

between the seventh and eighth degrees; all

the other spaces beingtones.

Middle-C is taken as the key-note for the first scale, whichis therefore calledthescale of

C

major, orsimply, the scale of C, the

word

major being understood. It is also called the Natural,

and

sometimes, the

Normal

scale.

(28)

THE MAJOR DIATONIC SCALE

The

semitones occurbetween

E

and

F,

and

between

B

and C. These are

known

as diatonic semitones, to dis-tinguish

them

from the chromatic semitone, which is

ex-pressed

by

the use of oneletter only, as, for instance,

C

to

C-sharp.

Musicalnotation

somewhat

fails, itwill be seen, to in-dicate the difference between the tones

and

semitones; all

the notes in the above scale being apparently equidistant.

The

construction of a scale

may

be exactly represented

in the form of a ladder. In

the

accompanying

diagram the

roundsof the ladder are of

two

kinds,the broad linesrepresent

the notes of the major scale, showing the position

and

the relative size of each tone and

semitone; while thethin,dotted lines indicate the notes which exist between the degrees that

area tone apart; if the scale of C,isplayed on the piano, these

noteswill bethe black keys.

4.

A

major scale is divisi-bleinto

two

equal parts, of fou.

%

notes each, called tetrachords (Greek: tetra, four,

and

chorde, a string or note). These tetrachords are of exactly similar construction, the semitones occurring between the

third

and

fourth degrees in each case, the other spaces beingtones.

Upper

Lower

Tetrachord Tetrachord

(29)

THE MAJOR DIATONIC SCALE 9

In the foregoingexample the tetrachordsare saidtobe

disjunct.

When

the last note of one tetrachord becomes

thefirst ofthenext,the tetrachords are saidtobeconjunct.

If a scale is extended for

two

or

more

octaves, the succes-sive tetrachordswill occurdisjunctly

and

conjunctly, alter-nately, throughout.

Although

any

alphabetical group offournotes

may

be

calledatetrachord,yetthistermin

modern

musicinvariably

signifiesa tetrachord in whichthe notesoccurin theabove order, namely, tone, tone

and

semitone.

5.

The

principal use ofthe tetrachordisin connection with the formation of

new

scales.

A

new

scale

may

be formed

by

takingeither ofthe tetrachordsinthescale of C,

and

addinga

new

tetrachordtoit..

The

new

tetrachordwill,

in every ease, necessitate the use of anaccidental in order that thesemitone

may

occur in the right place; otherwise

thescale

would

stillbeinthe keyof C.

The

uppertetrachord inthescale of

C may

betakento

form the lower tetrachord in the scale ofG;

and

in order thatthe

new

uppertetrachordshallbe properly constructed,

itwillbenecessary forthe

F

to be

made

sharp.

1

F-sharp,

now

beinganintegralpartofthescale,

is placed immediately after the clef-signature,

and

becomes

known

as the key-signature, thereby in-dicating that every

F

in the keyisto be sharpened.

The

upper tetrachord in the scale of

G

may

next be

takentoformthelower tetrachordinthescale of D,

and

in

addingthe

new

uppertetrachord the

C

will

now

havetobe

made

sharp.

The

key-signature in this case will therefore

(30)

10 THE MAJOR DIATONIC SCALE

Inlikemanner,

by

taking theuppertetrachordofeach

successive

new

scale, thescales of A, withthreesharps; E,

withfour; B, with five; F-sharp, with six;

and

finally, C-sharp, with seven sharps,

may

severally be formed.

The

above scales, except

C

(thenaturalscale),are

known

asthe

*

sharp

'

scales.

6.

The

lowertetrachordinthescale of

C may

betaken

toformtheuppertetrachordinthescaleofF;

and

in order thatthe

new

lower tetrachordshallbe properly constructed,it

will be necessary forthe

B

to be

made

flat.

B-flat,

now

beinganintegral partof the scale,

isplacedin the key-signature.

The

lower tetrachord in the scale of

F

may

next be taken to form the upper tetrachord in the scale of B-flat,

and

inaddingthe

new

lower tetrachord the

E

will

now

have

to be

made

flat.

The

key-signature will therefore contain

two

flats.

In likemanner,

by

taking the lower tetrachordof each

successive

new

scale, the scales of J-flat, with three flats;

A

-flat, with four; D-flat, with five; Or-flat, with six;

and

finally, C-flat, with seven flats,

may

severally be formed.

These are

known

as the *

flat'

scales.

7.

The

following tables

show

the signatures of all the sharp

and

flat keys:

(31)

THE MAOR DIATONIC SCALE 11

S

fc

B-flat ^1-flat D-flat 6?-flat C-flat

It willbe seen that the relative positions ofthe sharps

and

flatsareinvariably the

same

inboththetrebleandbass

staffs. This accounts for the upper

G

having been chosen

as the third sharpin the treble for the key of

A;

also for

thelower

F

as thelast flat in thekey of C-flat.

The

signatures ofC-sharp

and

C-flat, especially, should becommitted to

memory,

forthey respectively contain the

signatures ofallthe other sharp

and

flatscales.

The

signa-ture ofthekeyofE,forexample,willbeseenifthelastthree

sharps aretaken

away

from the signature of C-sharp; and

the signatureof #-flat will beseen if the last four flatsare

taken

away

fromthe signature of C-flat.

These signatures, C-sharp

and

C-flat,

when

employed

forthe alto-

and

tenor-clefs, are written as follows:

Alto-clef

Tenor-clef

Thekey of C-sharp is rarelyemployed in music, its placebeing takenbythekeyofD-flat (withfive flats),whichisidenticallythesame

key. The same

may

alsobesaidofthekey ofC-flat,whichis identi-callythe same as thekey of

B

(with five sharps). When two scales

are thesamein pitch,but different inname, they aresaid to be 'en-harmonicequivalents.'

8.

The

te.trachordal system of scale construction

may

(if desired) be continued beyond the scales of C-sharp and

(32)

12 THE MAJOR DIATONIC SCALE

with eight sharps, (one double

and

six single sharps); after which

would

come

D-sharp, with nine sharps;

and

then

A*

sharp, with ten;

and

finally #-sharp, with eleven; for the next scale, B-sharp, isidentically the sameas the scale of C.

In like

manner

the flat scales mightbe continued as far as

A

-double-flat; the next scale, D-double-flat, being also identically the

same

asthe scale ofC.

The

following diagram illustrates the cyclical system

upon

whichthe scalesare constructed.

G

Intheabovediagram, the sharpscalesareonthe outerside,and theflatscalesonthe innerside,ofthecircle.

By

means

ofthe tetrachords a closedegree of

relation-ship

may

be said to existbetween

any

given scale and the

scaleswhich immediatelyprecede

and

followit; thesescales, therefore, are said to be related tothe given scale,

and

are called '

attendant '

(33)

THE MAJOR DIATONIC SCALE 13

The

scale of

C

iscalledthefirstscale, simply becauseit isthefirstto be considered; for in the theoryof scale

con-struction there is no first scale,

any more

than there is a

beginningto a circle.

9.

Each

degree of the scale has a technical

name,

by

the use of which reference

may

be

made

to the functionsof

any

given note, irrespective of its alphabetical name,

and

regardless of the pitch of the key. For convenience in writing, these technical

names

may

be represented

by

symbols.

The

technical

names

forthe degreesof the typi-cal scale of C, together with their symbols, are given in

the followingtable, which shouldbe read upwards.

(34)

14 THE MAJOR DIATONIC SCALE

of the upper tetrachord, and, therefore, the key-note of the succeedingscale.

Submediant signifies thethird belowthe tonic, just asthe mediantis

the third above; or it may be regarded as the note midway

betweenthetonicandthesubdominant.

TheLeading-note(Anglo-Saxon: laedan, tolead orgo)issotermedfrom the strongtendencyof thisnotetorisetothetonic. In French

it isappropriatelyknownasLanotesensible, thesensitivenote.

ThesymbolLisemployedfor this note,insteadofVII,asmight have beenexpected because,inthefirstplace,it iseasiertowrite;

and, in the second place, because a distinctive symbol more

readily directsattentiontothecharacteristics of this note,which,

(itmayherebesaid),not onlyhas,asarule,afixed progression,

butwhich, unlike the other notesofthemajorscale,isnever the root of a diatonic common chord, and never the tonic of an attendant key.

SUMMARY

1.

The

characteristics of a musical sound.

Middle-Cthestandardof pitch. Definitionofsemitone.

2. Scales in general.

Diatonic (major andminor), Chromatic and Enharmonic.

3.

The

scale of

C

major.

Degreesofthescale representedintheformofaladder.

4.

The

tetrachord.

Orderofthe notes tone, toneandsemitone.

5.

The

uppertetrachord.

Thescale ofG,andallthe sharpscales.

6.

The

lower tetrachord.

Thescaleof F,andalltheflatscales.

7. Key-signatures.

Thesignaturesof C-sharpandC-flatcontain the signatures ofallkeys.

8.

The

cycle of keys.

Illustrativediagram. Attendantscales. 9. Technical names.

(35)

THE MAJOR DIATONIC SCALE 15

EXERCISES

1. State the characteristics ofamusical sound.

2. Define

and

compare

the terms, Diatonic,

Chro-matic,

and

Enharmonic, as employed in connection with the scales.

3. Explain the useofthetetrachord,

and show

that

any

major scale

may

be formed

by

tetrachordal construction

fromitsattendant scales.

4. Writethemajor scales,the key-signatures of which are:

5. Writeonthetreble staff, thescale of #-major;

and

onthe bassstaffthescale of

A

-flatmajor;

and

mark,with a slur, theposition of thesemitones in each.

6. Give the

meaning

of the terms, Tonic, Mediant,

Dominant and

Leading-note;

and

state

why

the

Submedi-ant is so called.

7. Give the technical

name

of the note A,

when

it

occursinthe keysof G,J5-flat,E,D-flat, /''-sharp

and

C-flat.

8.

Name

the tonic of the scale of which the domi-nantis:

then writethis scale,

and

write also theenharmonic

equiva-lent of this scale;

and

in each case mark, with a slur, the tetrachords.

9.

By

the

employment

of accidentals,wherenecessary,

convert the following passages into major scales, the first

(36)

16 THE MAJOR DIATONIC SCALE

10. Correct the following key-signatures:

() (*) ..() (<*) 00

11. Writethe key-signaturesoftnescales ofwhichthe

keynotesare:

hK

b

and

12.

Name

the scales in which the following intervals occur.

(<*)

W

00 (4) () (/)

13.

Name

(1) the subdominant

and

(2) the

leading-noteinthekeysofD, B-flat, E, A-flat, F-sharp

and

(?-flat.

14. Writeovereach of the notesinthe following

pas-sage the

symbol

indicatingitstechnicalname.

15.

Name

the key in which the following passage is written,

and

re-writeit,omitting theflats

and

employingthe

correct key-signature; also insertthe symbols.

1

i*

16.

Add

the necessary clefs

and

accidentals (but

no

key-signature) to convert the following

melody

into (1) the

keyof B,

and

(2) the keyof D-flat.

(37)

CHAPTER

II

INTERVALS

10.

The

terms major (greater)

and

minor (less), as

appliedtothediatonicscales, require

some

explanation, for

it

must

not be supposed that

by

their use it is intended to

implythat onescaleis greaterthananother. These terms referto theintervalwhichexistsbetween the key-note

and

thethird degree, or, to use the technicalnames, the Tonic

and

the

Mediant

ofthescale. Informertimes thescale of

A

-minor, for example,

was

known

as '

the scale of

A

with

thelesser third,' a designation

more

definite, perhaps, but

certainly less convenient than that which obtains at the present day.

Some

knowledge of intervals is therefore

necessary beforethe construction ofthe minor scalecan be thoroughly understood.

By

an interval is understood the difference in pitch

between

two

musical sounds.

11. Intervals, likethelines ofthe staffand, aswill be seen later, the notes of a chord, are always reckoned

up-wards, unless the contraryis expressly stated.

An

interval is measured

by

the number of semitones

whichitcontains.

Every

intervalhasbothaparticular

and

ageneral

name

;

thelatterbeinganordinalnumber,signifyingthe

number

of degrees, orletters,concerned; theformerbeingaqualifying

name, dependent

upon

the

number

ofsemitones concerned.

The

unison (Latin: unus sonus, one sound), orprime,

though not actually an interval, isconsidered as such for

convenience in classification.

The

intervalswhich existbetweenthe tonic ofa major scale,asthe lower sound,

and any

degreeinthatscale,as the

(38)

18 INTERVALS

upper, are termed Normal.

The

normal intervals in the

scale of C, withtheir respective names,

and

the

number

of

semitones contained in each, areasfollows:

(39)

INTERVALS 19

oflettersconcerned,

and

theneither

by

counting the

number

ofsemitonesit contains, which isgenerally best forsmaller

intervals, or applying,

what

may

be termed, the

'

normal

test,' which iscertainly best for the larger intervals. For

example, the interval

E

to

G

is a minor third; for since there are three letters concerned, viz., JS,

F

and

G, it is a

third;

and

since the intervalcontains three semitones, itis

minor. Or, to applythenormal test, if

E

isregarded as a

tonic, the key will be

E

with four sharps,

and

since the

normal or major third is(r-sharp,

E

to

G

natural, being a semitone less, will be minor.

The

interval

F

to

D

is a

majorsixth, for there aresix letters concerned,

and

F

to

D

normal in the key of F.

The

interval

B

to

A

is a minor seventh, for, in the key of

B

(withfive sharps) thenormal

(ormajor) seventhis

A

-sharp.

All the fourths in the above example are normal in their respective keys,

and

therefore perfectinthekey of C, except alone

F

to B, which, being a semitone greater than perfect, issaid to be

augmented;

likewise allthe fifths are perfect, except

B

to F, which, being a semitone kss than

perfect, is said to bediminished.

The augmentedfourth,whichoccurs onlyuponthesubdominantof

the key, wasformerlyknown as, andis stilloftencalled, the Tritone,

(Latin: Tritonus, three-toned). It has alsobeen called the Extreme, Sharp, Superfluous, and Pluperfect fourth. The diminished fifth has

alsobeencalledtheFalse, Flat,andImperfectfifth. Theabove names, however, aeebyno means in ordinary use; infact, they are rapidly becomingaltogetherobsolete.

13.

By

employingthesigns,

+

for major,- for minor,

X

foraugmented,

and

o fordiminished, the namesof inter-vals

may,

for convenience in writing, be represented

by

symbols. Thus, forexample:

2+

indicatesa major second;

3- " a minorthird;

4x

" an

augmented

fourth;

(40)

20 INTERVALS

The

sign

+

may

be regarded as representing normal

intervals generally, so that

when

employed in conjunction

witheither4or5, it

would

indicateaperfect interval. This sign

may

often be omitted altogether, being taken for

granted,especially

when

referring toperfect intervals; just

as thescale ofC-majorisoften calledthescale of C,theterm major being understood.

14.

The

following table comprises all the simple dia-tonic intervals; showingalsothesymbol employedforeach interval,the

number

ofsemitones containedineachinterval,

and

the frequency of occurrence of each interval in

any

major

scale.

NAME

(41)

INTERVALS 21

The

simplediatonic intervals inthekeyof C,the octave (and unison) excepted, together with their symbols, are as

follows: Seconds

^

2+ 2+ 2- 2+ 2+ 2+ 2-Thirds

^i

-&-3+ 3- 3- 3+ 3+ 3-Fourths

44

44+444

Fifths h?r

(42)

22 INTERVALS

themajor third at (a)

may

be reduced in sizea chromatic

semitone, either as at (6) or as at (c). Care

must

betaken,

insuchcases, never to change theletter

name

ofeithernote

ofthe interval, for although theintervalsat (d)

and

at (e), are respectively the

same

on the piano asthose at (6)

and

at (c), yet not only do theydifferin

name

(being seconds),

but, aswill be dulyexplained, theydiffer also ineffect,

(a) (6) (c) (d) (e)

The

diatonic intervals above the given note

C

are as follows:

2- 21- 3- 3+ 4 4X

50 5 6- 6+ 7- 7+ 8

16. Theoretically speaking, all the major

and

perfect intervals

may

beaugmented,

by

being increasedachromatic semitone;

and

all the minor

and

perfect intervals

may

be diminished,

by

being decreased a chromatic semitone.

Practicallyspeaking, however,

some

of these intervals, not-ably the

augmented

third

and

the diminishedsixth, contain exactly the

same

number

ofsemitonesas aperfect interval,

and

for this reason are not used in harmony.1 Intervals so

derived are almost invariably chromatic; the

augmented

fourth

and

thediminished fifth, however,areambiguous, for

they are sometimes chromatic, and, as already has been

seen, sometimes diatonic.

1

Although some of these very intervals have occasionally been

introducett-intomoderncompositions bytheemploymentofchromatic

auxiliary notes, yet, nevertheless,theydonotexistbetweenthe notesof anyofthe accepted chordsinharmony.V

(43)

INTERVALS 23

The

chromatic intervalsabovethe given note

C

are as follows:

IX 2X

-&-30 40 5X

6x 7o tk>

The

following example comprises all the simple inter-vals, bothdiatonic

and

chromatic, asemployedin

harmony

abovethe given note C. CHROMATIC by Diminution DIATONIC CHROMATIC byAugmentation Unisons Seconds Thirds Fourtlis Fifths Sixtlis jfl

(44)

24 INTERVALS

17.

Compound

intervals have the

same

qualifying

terms as their corresponding simple forms. Thus, the in-terval at (a) isamajorninth, ora

compound

majorsecond;

at (6), it is a minor tenth; at (c) it is a perfect eleventh.

With

the exceptionoftheninth, however,

compound

inter-vals are almost invariably regarded as though they were

simple intervals; the interval at (6), being called a minor third,

and

that at (c), a perfect fourth.

(a) (&) (c)

9-1- 10- 11

18.

An

interval is said to be inverted

when

thelower

soundis placed above the upper, or vice versa. Perfect in-tervals

when

inverted remain perfect; major intervals

changeto minor,

and

minorto major;

augmented

intervals

change to diminished,

and

diminished to augmented.

The

numerical

name

of an inverted interval

may

be found

by

subtracting the numerical

name

ofthe (uninverted)interval

fromthe

number

nine. Thus, the inversions ofthenormal

intervals in 11 willbe as follows:

-a

1- 6- 5 4 3- 2- 1

An

interval is said to be melodic

when

the

two

soundsareheardin succession,

and

harmonic

when

theyare

heard incombination.

19-3

19.

Harmonic

intervals are divided into

two

namely, consonant,

when

the effect is pleasant

and

dis-sonant,

when

otherwise.

The

consonant intervals are sub-divided into

two

classes, namely, perfect

and

imperfect; the perfect

con-sonant intervals comprising the fourth, the fifth

and

the

octave, includingalsothe unison; theimperfect, comprising

References

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