FACULTY
Presentedtothe./MUSIC
Facuk>-
f^c
1^
ElizabethTempleton
UNIVERSITY
117
Carmen
Ave.,WINNIPEG,
-MAN.
A
TREATISE
ON
HARMONY
WITH
EXERCISES
By J.HUMFREY
ANGER
NEWLY
EDITED
AND
REVISEDBY
H.
CLOUGH-LEITER
from posthumous annotations of the author
In Three Parts
PART
I$2.00
BOSTON, MASS.
Copyright, 1006, byG. SCHIRMER,JR.
Copyright,1919,byTHE BOSTONMusicCa
forallCountries.
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 alwaysanobject 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.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.PREFACE
V
Inthissymbol system,whichisexplainedinAppendix1, thefirst
sixnotesofthescaleare representedby thefirst six
Roman
numerals, theLeadingnote,however,isrepresentedbytheletter L, inorder that the attentionofthestudentmaymorereadilybedirected tothechar-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.
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 andeitherA
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
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, ofchordsof 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, diedpracticallyahundredyears 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
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, andG
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. 3sion at(a)is tinged withjoy,that (a) (6) at (6) with sorrow.
A
veritableillustration 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 intheAndanteofBeethoven's Sonata appassionata (measure
G), in which the
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
modernauthorlof 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
beobtainedby 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
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 seconddegreesinthekeyof 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,whereasinthekeyofC
thisselfsamenote, E, will have only 320 vibrations. Theoretically this is a fact,
practicallyit isabsurd. Thedifferencebetweentheoryandpractice in thisandinsimilar instances,
may
beexplainedonthe grounds, that thedemandsof 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
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 ofwhich,fromtheoriginal tonicasa generator,willbeseen inthe
follow-ing example:
Ex.4
IThewholenotesintheabove 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 thesechords,ascale whichtheauthorbelieveswillmeetallthe requirements
ofthemoderncomposer, ascale containing notonly the diatonic and the chromatic, butalsotheenharmonicelement,and whichmay there-forebetermed
Ex.5
The ModernEnharmonicScaleofCThehalf-notesintheabove exampleare the diatonic notesofthe
scale; the quarter-notesarechromatic, andare called theperfect chro-matics,being diatonictotheattendant keys they are never
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 andB
major, besides others, are added tothe already accepted chro-matic concords. These, itwillbeseen, arethedominanttriads ofthethreeattendantminorkeys,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.
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 pilgrimoccasionally 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
CONTENTS
PART
ICHAPTER PAGE
Introduction 1
I.
The Major
Diatonic Scale .. 6II. Intervals 17
III.
The Minor
Diatonic Scale 31IV.
The
Common
Chord
41V.
Harmonic
Progressions 56VI. Cadences, Sequences, etc 78
VII.
The Chord
ofthe Sixth 102VIII.
The Chord
of theSix-four 114IX.
The
Dominant-Seventh 134X.
The
Inversions of V? 148XI. Natural Modulation 166
Appendix
Symbols
iIndex.. 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.
A
TREATISE
ON
HARMONY
PART
IINTRODUCTION
THE
importantperiod atfactorwhichinHarmony
music, appearsfirst beganto haveto bebeenan about the end of the sixteenth century, beingcontempo-raneous with the establishment of the
modern
diatonic scales, with the gradual development ofForm
in musical composition,and
with the rise of the operaand
oratorio, all of which are the direct results of the Renaissance, theperiod, therefore, in which
modern
musichad
its birth. Priorto this period, all musicwas composed
ona basis ofCounterpoint. Counterpointwas,and
is,theart of com-biningtwo
ormore
melodies.The
first school of contra-puntalcomposersaroseinthe fourteenth century,and
before this era,Harmony,
aswe
understandit, did not exist.In
Harmony,
musicisregardedfromthe perpendicularpoint ofview; in Counterpoint, fromthehorizontal.
The
highest types of
modern
classical musicmay
be said to bebuilt
upon
a union ofHarmony
and
Counterpoint.From
thecommencement
of the 17th tothe middle ofthe 18th century considerable advance
was
made
in theart of music generally. Scarcely a chord exists at the presentday
whichisnotto befoundintheworks ofJohann
Sebas-tian
Bach
(1685-1750),and
George FridericHandel
(1685-1759).
To
Bach
is due theconsummation
of the2 INTRODUCTION
About
theyear 1750,and
inagreatmeasure duetotheinfluence of Bach, equal
temperament
graduallycame
into use, taking the place of theold mean-tone, orunequalsys-tem
of tuning.The
pianoforte (themost
important, per-haps, of all musical instruments)now
began tobecome
popular; the
modern
orchestra, also,was
established about this time, while the Sonataand
Symphony
(the highesttypes of musical composition) were brought to perfection atthe hands of Josef
Haydn
(1732-1809), WolfgangAma-deus
Mozart
(1756-1791),and
Ludwig van
Beethoven(1770-1827).
Since thedeathofBeethoven,
who
isgenerallyregarded asthegreatestmusical genius theworld haseverknown, nomaterialadvance has been
made
inthescience ofHarmony.
Modern
composersareindebted forany
originalityin theirharmonies almost entirely to the
employment
of discordsformed either
by
the use of auxiliary notes, orby
theintro-duction of the chromatic element. In the matter of
har-monic
progressions,it isa questionwhether any advancehasbeen
made
upon
the masterpiecesofthe great composers,or, indeed, whetherany
advance can be made, as long as thedivision ofthe octaveis based
upon
equal temperament.The
theoryofmusic,itmay
herebesaid, comprises the followingsubjects; theRudiments
ofmusic,Harmony
inall fys branches,Form
in composition, Counterpoint (simpleand
double),Canon and
Fugue, Orchestration, Acousticsasappliedtomusic,acriticalknowledgeofthegreatestmusical
works,
and
the general History of theart of music. This course of study will occupy the attention of the averagestudent fora period of at least three years, afterwhich he
will begin to learn
how
little he really knows; for havingclimbed to the
summit
of one hill, he will find, rising inalldirections, higher
and
higher still, thesummits
of otherINTRODUCTION 3
inthe valebelow.
Thus
itwas
withBeethoven,who
atthezenith ofhis career,exclaimed,"Ihavenot studiedenough."
The
subject ofHarmony,
forthe purposesof study,may
be dividedintothe following branches:
I. Introductory
The
majorand
minordiatonicscales,and
intervals. II.The
Common
Chord
Harmonic
progressions, cadencesand
sequences.III.
Fundamental
DiscordsThe
dominant-seventh, modulation, secondarysevenths, major
and
minor ninthsand
their derivatives.IV. Suspensions
Single, double
and
triple, bothrisingand
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
formore
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 scalesand
of intervalsbeingespecially important, thesesubjectswill betreated of inthefirstthreechapters.
The
scales,indeed, arethe verysINTRODUCTION
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
technicalname;
in the interval,two
notes at a time are considered,theeffectbeingeitherconsonantor dissonant; while, inthe
chord, three or
more
notes at a time are considered, thecombined effect beingeithera concord or a discord.
Suchquestions as the valueofnotes
and
rests,the mean-ing of the Great Staff, the various clefsand
their use, thevalue of the measure (including time
and
accent),and
the significance of the musicaltermsand
signs in ordinary use,belong to the subject of rudiments,
and
it would be out of place to treat ofthem
inthe presentwork. Letthe studentnot 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 possesssome
knowledge of piano playingand
sight-singing; if, indeed,it onlyissufficientto singthemelodies,
and
to play theprogressions, containedin his exercises; foran intelligentappreciation of everything he writesis indis-pensable.
He
should also learn that musicmay
beappre-ciated through the
medium
of the eye, as well as through that of the ear. This faculty of reading music with the understanding, as onewould
reada book, isnota propertyappertainingtogeniusalone; itisrather a property
apper-taining to talent,
and
talent is the reward of perseveranceand
anaptitudeforwork. In ordertoacquirethis faculty,let thestudent, in the first place, try to imagine the effect ofeachprogression,
and
here thesigHt-singingwill be foundINTRODUCTION
invaluable,
and
inthesecondplace,lethim
testtheaccuracyof his conclusions,
and
thismay
be doneby means
of thepiano. If this plan be conscientiously adopted, hewill, in
due course, be able to see the effect of a passage without
singingover theparts,
and
withoutmaking
use ofthe piano atall; for, inthe imagination, every chordmay
be played,every
melody
sung,and
thefull intent of every progressionperceived
and
appreciated.The
student, moreover, isstrongly advised, in workingexercises, togive to each chordas itoccursitsproper tech-nical name,
and
to write under the bass-note the symbolemployed to represent the chord. It is not sufficient to
calla chord
by
itsalphabeticalname
alone.The
common
chordof C, forexample,inthekeyof C, differsaltogetherin effectfromthe
common
chordofC
inthekeyofF
(majororminor),
and
fromthesame
chord inthe keyof G, or in thekey of
E
minor; but the alphabeticalname
is thesame
in allthese cases.The
characterof a chord depends entirelyupon
thekeyinwhichitoccurs.Now,
by
theemployment
oftechnicalnames, for which symbols
may,
ifpreferred, besubstituted, 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 orminor.
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
inany
caseand
atall costsbe acquired before the student can entertain the hope of
CHAPTER
ITHE MAJOR
DIATONIC SCALE1.
A
musicalsoundistheresult of regular vibrationsinthe air; irregular vibrations result in
what
iscommonly
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
soundknown
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 thestudyof
Harmony.
The
difference in pitch betweentwo
musical sounds is called an interval. Intervals are measuredby
semitones;a semitone, i.e., a half-tone, being the smallest difference
between
any two
soundsinmusic.A
semitoneisthereforeknown
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
theChromaticTHE MAJOR DIATONIC SCALE 7 Theconstruction of thechromatic scale, which consists of semi-tonesonly,willbeconsideredinalaterchapter.
Ofthediatonicscale,inwhich bothtones
and
semitones occur, there aretwo
kinds theMajor and
theMinor.The minorscale willbe consideredin chapterIII,a knowledgeof intervalsbeing necessary inorder tounderstanditsconstruction.
Anotherkindof scale
known
as theEnharmonic
shouldalso be mentioned; it can be played on instruments ofthe
violincharacter,
and
canalsobe sungby
thevoice. Inthis scale, the soundsare closer togetherthan a semitone; it is,therefore, not used in
Harmony.
The
term enharmonic, atthe present day, isexclusivelyusedfora changeinthe
name
ofanote,such as C-sharpto Z)-flat, there beingno changeof pitch whatever.The
termkey
isemployed
to represent the notes of adiatonic scaletaken collectively. This term, however, has really a wider significance, for it comprises also certain
chromatic notes.
3.
A
diatonic'scaleisnamed
afterthe noteon which itcommences,
and
this note is called the key-note.Any
sound in music
may
be taken as a key-note. There aresevendifferent 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; allthe 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, theword
major being understood. It is also called the Natural,and
sometimes, theNormal
scale.THE MAJOR DIATONIC SCALE
The
semitones occurbetweenE
and
F,and
betweenB
and C. These areknown
as diatonic semitones, to dis-tinguishthem
from the chromatic semitone, which isex-pressed
by
the use of oneletter only, as, for instance,C
toC-sharp.
Musicalnotation
somewhat
fails, itwill be seen, to in-dicate the difference between the tonesand
semitones; allthe notes in the above scale being apparently equidistant.
The
construction of a scalemay
be exactly representedin the form of a ladder. In
the
accompanying
diagram theroundsof 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 andsemitone; 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-bleintotwo
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 thethird
and
fourth degrees in each case, the other spaces beingtones.Upper
Lower
Tetrachord Tetrachord
THE MAJOR DIATONIC SCALE 9
In the foregoingexample the tetrachordsare saidtobe
disjunct.
When
the last note of one tetrachord becomesthefirst ofthenext,the tetrachords are saidtobeconjunct.
If a scale is extended for
two
ormore
octaves, the succes-sive tetrachordswill occurdisjunctlyand
conjunctly, alter-nately, throughout.Although
any
alphabetical group offournotesmay
becalledatetrachord,yetthistermin
modern
musicinvariablysignifiesa tetrachord in whichthe notesoccurin theabove order, namely, tone, tone
and
semitone.5.
The
principal use ofthe tetrachordisin connection with the formation ofnew
scales.A
new
scalemay
be formedby
takingeither ofthe tetrachordsinthescale of C,and
addinganew
tetrachordtoit..The
new
tetrachordwill,in every ease, necessitate the use of anaccidental in order that thesemitone
may
occur in the right place; otherwisethescale
would
stillbeinthe keyof C.The
uppertetrachord inthescale ofC may
betakentoform the lower tetrachord in the scale ofG;
and
in order thatthenew
uppertetrachordshallbe properly constructed,itwillbenecessary forthe
F
to bemade
sharp.1
F-sharp,
now
beinganintegralpartofthescale,is placed immediately after the clef-signature,
and
becomes
known
as the key-signature, thereby in-dicating that everyF
in the keyisto be sharpened.The
upper tetrachord in the scale ofG
may
next betakentoformthelower tetrachordinthescale of D,
and
inaddingthe
new
uppertetrachord theC
willnow
havetobemade
sharp.The
key-signature in this case will therefore10 THE MAJOR DIATONIC SCALE
Inlikemanner,
by
taking theuppertetrachordofeachsuccessive
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),areknown
asthe*
sharp
'
scales.
6.
The
lowertetrachordinthescale ofC may
betakentoformtheuppertetrachordinthescaleofF;
and
in order thatthenew
lower tetrachordshallbe properly constructed,itwill be necessary forthe
B
to bemade
flat.B-flat,
now
beinganintegral partof the scale,isplacedin the key-signature.
The
lower tetrachord in the scale ofF
may
next be taken to form the upper tetrachord in the scale of B-flat,and
inaddingthenew
lower tetrachord theE
willnow
haveto be
made
flat.The
key-signature will therefore containtwo
flats.In likemanner,
by
taking the lower tetrachordof eachsuccessive
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 tablesshow
the signatures of all the sharpand
flat keys:THE MAOR DIATONIC SCALE 11
S
fcB-flat ^1-flat D-flat 6?-flat C-flat
It willbe seen that the relative positions ofthe sharps
and
flatsareinvariably thesame
inboththetrebleandbassstaffs. This accounts for the upper
G
having been chosenas the third sharpin the treble for the key of
A;
also forthelower
F
as thelast flat in thekey of C-flat.The
signatures ofC-sharpand
C-flat, especially, should becommitted tomemory,
forthey respectively contain thesignatures ofallthe other sharp
and
flatscales.The
signa-ture ofthekeyofE,forexample,willbeseenifthelastthreesharps aretaken
away
from the signature of C-sharp; andthe signatureof #-flat will beseen if the last four flatsare
taken
away
fromthe signature of C-flat.These signatures, C-sharp
and
C-flat,when
employedforthe 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 ofB
(with five sharps). When two scalesare thesamein pitch,but different inname, they aresaid to be 'en-harmonicequivalents.'
8.
The
te.trachordal system of scale constructionmay
(if desired) be continued beyond the scales of C-sharp and12 THE MAJOR DIATONIC SCALE
with eight sharps, (one double
and
six single sharps); after whichwould
come
D-sharp, with nine sharps;and
thenA*
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 asA
-double-flat; the next scale, D-double-flat, being also identically thesame
asthe scale ofC.The
following diagram illustrates the cyclical systemupon
whichthe scalesare constructed.G
Intheabovediagram, the sharpscalesareonthe outerside,and theflatscalesonthe innerside,ofthecircle.
By
means
ofthe tetrachords a closedegree ofrelation-ship
may
be said to existbetweenany
given scale and thescaleswhich immediatelyprecede
and
followit; thesescales, therefore, are said to be related tothe given scale,and
are called 'attendant '
THE MAJOR DIATONIC SCALE 13
The
scale ofC
iscalledthefirstscale, simply becauseit isthefirstto be considered; for in the theoryof scalecon-struction there is no first scale,
any more
than there is abeginningto a circle.
9.
Each
degree of the scale has a technicalname,
by
the use of which reference
may
bemade
to the functionsofany
given note, irrespective of its alphabetical name,and
regardless of the pitch of the key. For convenience in writing, these technicalnames
may
be representedby
symbols.
The
technicalnames
forthe degreesof the typi-cal scale of C, together with their symbols, are given inthe followingtable, which shouldbe read upwards.
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 ofC
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.
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
thatany
major scale
may
be formedby
tetrachordal constructionfromitsattendant 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
statewhy
theSubmedi-ant is so called.
7. Give the technical
name
of the note A,when
itoccursinthe 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 theenharmonicequiva-lent of this scale;
and
in each case mark, with a slur, the tetrachords.9.
By
theemployment
of accidentals,wherenecessary,convert the following passages into major scales, the first
16 THE MAJOR DIATONIC SCALE
10. Correct the following key-signatures:
() (*) ..() (<*) 00
11. Writethe key-signaturesoftnescales ofwhichthe
keynotesare:
hK
band
12.
Name
the scales in which the following intervals occur.(<*)
W
00 (4) () (/)13.
Name
(1) the subdominantand
(2) theleading-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 theflatsand
employingthecorrect key-signature; also insertthe symbols.
1
i*
16.
Add
the necessary clefsand
accidentals (butno
key-signature) to convert the followingmelody
into (1) thekeyof B,
and
(2) the keyof D-flat.CHAPTER
IIINTERVALS
10.
The
terms major (greater)and
minor (less), asappliedtothediatonicscales, require
some
explanation, forit
must
not be supposed thatby
their use it is intended toimplythat onescaleis greaterthananother. These terms referto theintervalwhichexistsbetween the key-note
and
thethird degree, or, to use the technicalnames, the Tonic
and
theMediant
ofthescale. Informertimes thescale ofA
-minor, for example,was
known
as 'the scale of
A
withthelesser third,' a designation
more
definite, perhaps, butcertainly less convenient than that which obtains at the present day.
Some
knowledge of intervals is thereforenecessary beforethe construction ofthe minor scalecan be thoroughly understood.
By
an interval is understood the difference in pitchbetween
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 measuredby
the number of semitoneswhichitcontains.
Every
intervalhasbothaparticularand
ageneralname
;thelatterbeinganordinalnumber,signifyingthe
number
of degrees, orletters,concerned; theformerbeingaqualifyingname, dependent
upon
thenumber
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 the18 INTERVALS
upper, are termed Normal.
The
normal intervals in thescale of C, withtheir respective names,
and
thenumber
ofsemitones contained in each, areasfollows:
INTERVALS 19
oflettersconcerned,
and
theneitherby
counting thenumber
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
toG
is a minor third; for since there are three letters concerned, viz., JS,F
and
G, it is athird;
and
since the intervalcontains three semitones, itisminor. Or, to applythenormal test, if
E
isregarded as atonic, the key will be
E
with four sharps,and
since thenormal or major third is(r-sharp,
E
toG
natural, being a semitone less, will be minor.The
intervalF
toD
is amajorsixth, for there aresix letters concerned,
and
F
toD
normal in the key of F.
The
intervalB
toA
is a minor seventh, for, in the key ofB
(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 aloneF
to B, which, being a semitone greater than perfect, issaid to beaugmented;
likewise allthe fifths are perfect, exceptB
to F, which, being a semitone kss thanperfect, 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-valsmay,
for convenience in writing, be representedby
symbols. Thus, forexample:
2+
indicatesa major second;3- " a minorthird;
4x
" anaugmented
fourth;20 INTERVALS
The
sign+
may
be regarded as representing normalintervals generally, so that
when
employed in conjunctionwitheither4or5, it
would
indicateaperfect interval. This signmay
often be omitted altogether, being taken forgranted,especially
when
referring toperfect intervals; justas thescale ofC-majorisoften calledthescale of C,theterm major being understood.
14.
The
following table comprises all the simple dia-tonic intervals; showingalsothesymbol employedforeach interval,thenumber
ofsemitones containedineachinterval,and
the frequency of occurrence of each interval inany
major
scale.NAME
INTERVALS 21
The
simplediatonic intervals inthekeyof C,the octave (and unison) excepted, together with their symbols, are asfollows: Seconds
^
2+ 2+ 2- 2+ 2+ 2+ 2-Thirds^i
-&-3+ 3- 3- 3+ 3+ 3-Fourths44
44+444
Fifths h?r22 INTERVALS
themajor third at (a)
may
be reduced in sizea chromaticsemitone, either as at (6) or as at (c). Care
must
betaken,insuchcases, never to change theletter
name
ofeithernoteofthe interval, for although theintervalsat (d)
and
at (e), are respectively thesame
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 noteC
are as follows:2- 21- 3- 3+ 4 4X
50 5 6- 6+ 7- 7+ 8
16. Theoretically speaking, all the major
and
perfect intervalsmay
beaugmented,by
being increasedachromatic semitone;and
all the minorand
perfect intervalsmay
be diminished,by
being decreased a chromatic semitone.Practicallyspeaking, however,
some
of these intervals, not-ably theaugmented
thirdand
the diminishedsixth, contain exactly thesame
number
ofsemitonesas aperfect interval,and
for this reason are not used in harmony.1 Intervals soderived are almost invariably chromatic; the
augmented
fourth
and
thediminished fifth, however,areambiguous, forthey 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
INTERVALS 23
The
chromatic intervalsabovethe given noteC
are as follows:IX 2X
-&-30 40 5X
6x 7o tk>
The
following example comprises all the simple inter-vals, bothdiatonicand
chromatic, asemployedinharmony
abovethe given note C. CHROMATIC by Diminution DIATONIC CHROMATIC byAugmentation Unisons Seconds Thirds Fourtlis Fifths Sixtlis jfl
24 INTERVALS
17.
Compound
intervals have thesame
qualifyingterms 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 weresimple 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 invertedwhen
thelowersoundis placed above the upper, or vice versa. Perfect in-tervals
when
inverted remain perfect; major intervalschangeto minor,
and
minorto major;augmented
intervalschange to diminished,
and
diminished to augmented.The
numerical
name
of an inverted intervalmay
be foundby
subtracting the numerical
name
ofthe (uninverted)intervalfromthe
number
nine. Thus, the inversions ofthenormalintervals in 11 willbe as follows:
-a
1- 6- 5 4 3- 2- 1
An
interval is said to be melodicwhen
thetwo
soundsareheardin succession,
and
harmonicwhen
theyareheard incombination.
19-3
19.
Harmonic
intervals are divided intotwo
namely, consonant,
when
the effect is pleasantand
dis-sonant,
when
otherwise.The
consonant intervals are sub-divided intotwo
classes, namely, perfect
and
imperfect; the perfectcon-sonant intervals comprising the fourth, the fifth
and
theoctave, includingalsothe unison; theimperfect, comprising