Hasty, An Intervallic Definition of Set Class (JMT 1987)

7/28/2019 Hasty, An Intervallic Definition of Set Class (JMT 1987)

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Yale University Department of Music

An Intervallic Definition of Set ClassAuthor(s): Christopher F. HastySource: Journal of Music Theory, Vol. 31, No. 2 (Autumn, 1987), pp. 183-204Published by: Duke University Press on behalf of the Yale University Department of MusicStable URL: http://www.jstor.org/stable/843707 .

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AN INTERVALLICDEFINITIONOF SET CLASS

Christopher. Hasty

During hepasttwo decadesa largebodyof post-tonal ompositionhasbecomeincreasinglyntelligible rom the theoreticalpointof view,largelythroughhe workof Allen Forte.Theconceptof pitch-class et andthesys-tematicnvestigationf the relations f setshaveenabled hetheoryof post-tonal musicto free itself fromboththeanachronismf tonalparadigms ndthe superficiality f motivicanalysis.Doubtless,the success of the theoryof unordered ets in providing basis for the studyof 20thcenturymusicstemsin partfrom the explicitnessandgeneralityof its fundamental on-cepts. These virtueshave permittedmany scholars to contribute o thedevelopmentf thetheoryboth n its systematic laboration nd n its rangeof application.In thepresentessayI shallexaminecritically he notionof set class (orcollectionclass), a basicconceptof post-tonal heorywhich I believe hasyet to findadequatedefinition. have chosen a briefexcerpt rom the firstsong,"Wiedereshen,"f MiltonBabbitt'sycle,Du, to provide he materialfor the examples n this essay.Since Babbitt s responsible or so muchofthe conceptualstructure hat the followingdiscussionengages, it seemsappropriatehat a work of his shouldprovidethe startingpoint for ourinvestigation.'The first measureof "Wiedereshen"Ex. la) maybe taken to present

183

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=4 mm3p 3 1.IppDein Schrei - ten bebt

p mp pI3-4 m 34

b.

8(+12) 7(+24) 4(+24) 11(+12A

Example 1

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consecutivelyhreeformsof set-class3-4; E6, E, Ab ollowedbythetrans-positionally elated ormG, A6, C followedbytheinversionC, B, G. I wishtoexamine arefullywhatthis assertionandotherssimilar o it canbetakento mean.Consideringhefirsttwo instancesof set-class3-4, we mightsaythat oindicate hese structuresmpliesthat on the thirdbeatof this examplewecan heara newsonority,differentiatedomehow romthefirstsonority,yetsounding n some waylike it. Differentiations necessary f a set is to beconceivedof as an entitywithdistinctpropertieshatmaybe transformedor comparedwith otherentities. This determinate,hing-likeview seemsgenerallyaccorded o the notion of pitch-classset. Likenessheremaybetaken o mean"composed f the same or similar ntervals."n the presentcase the intervalclasses found in each of the two trichordsare the same

while the registral ntervalsare similar(Ex. lb). Differentiation ndlike-ness are reciprocal erms of structure hatinteractn very complex ways.Forinstance,couldnot the intervallic imilaritywe have notedincline usto hear a continuity n the intervallic ealm rather hana segregationntotwo discreteentities?The intervalclass 4, whichtransposes he firsttri-chord ntothe second,is an intervalhatappearswithineach trichordpos-sibly making he distinctionbetweenthemless plausible.Suchquestionsaretheprovinceof segmentationr structuralormation ndarebeyond hescopeof thepresent nquirywhich is notdirectedoward nanalysisof thiswork.Forourpurposest will be assumed hat hepitch-class etsindicatedin Example a aredifferentiated,utonomous tructures.This assumptionwill allowour attention o be focusedexclusivelyupon questionsof simi-larityandequivalence.The conceptof set class has achieved ts powerfulgenerality hroughprogressive bstraction.Whatdefinesa set class mustsurelybe the inter-vallic relationsamong its pitch-classconstituents,not the pitch classesthemselves(alreadyhighly abstracted)nor their temporalorder nor themyriadother possible relationsand qualitieswhich inhere in an actualmusicalstructure.Theseabstractions, owever,are notpurenegationsbe-cause theexcludedqualitiescanbe recovered o gaugethedegreeof trans-formation setundergoes r thesimilarityof various ormsof thesame setclass. Specificcharacteristicshat are omittedfrom the definitionof anequivalence lass are notdeniedexistencebutrather ake on the characterof variable erms.In this waythe conceptof set class while existingon adifferentplane fromany of its concretemusicalmanifestations olds alltheseas possibilities.Moreover,he notion of set class, since it rests on aseries of abstractions, an to some extentorganize hesepossibilities.Forexample,while pitch-classset has eliminateda determination f temporalordering romits elements,such a determinationmayin factintensify herelationof members f thesameset class. Assuminghat here s somesortof correspondence etweenthe pitchclasses of membersof the same set

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class andthat hiscorrespondences essential o the"equivalence"f thesesets, the temporalorderingof pitchclassesmayin varyingdegreesmatchthiscorrespondence. hus hecategory f temporal rdercan take ts mean-ing from the more abstract et class. Wecouldin principleeven establisha scale of possible orderings romthe maximum o minimumsimilaritybased on set-classcorrespondence.Likewisemanyother propertiesab-stracted utof a pitch-class etmay,operatinginglyor inconcertwithoneanother, ind a principleof organizationn therelationsof pitch-class ets.The generalizationswe maketo createthe conceptof set class are in thissensenevercompletelydetached rom the particulars. uchgeneralizationmay n factbe a steptoward moresystematic onnectionof theseparticu-lars.And,thatsuch connectionsarepossibleclearlyshowstheproductive-ness of the conceptandits potentialas an organizingprinciple.

Anelementary uestion hatI have ried ocircumventhusfarmustnowbeaddressed.What s theprincipleof equivalencehatallows he formationof the conceptof set class?Tobeginansweringhis question et us returnto the openingof DU. It was assumed hat there aretwo instancesof thesingleset class 3-4 found n the beginningof this passage.If we knowinwhatwaytheyare considered quivalentwe will understand hat s meantbysetclass.Thecustomary nswer o thisquestion s that hesepitch-classsets are transpositionallyelated.The operationof transpositionor pos-siblyof inversion) ppliedexhaustivelyoeachelementof a givensetyieldsa setof thesameclass.However,his accountwhenpressedwill notprovidea definitionof set class. Wemightsaythatin the case of transpositionallyrelated ets eachpitch n one of two so related ets is relatedbya specifiedinterval modulo12) to a single pitchin the otherset, and thuswhateverstructureharacterizesrdefines he onesetis replicatedn the second.Buttounderstandheconceptof setclass,we must ocatethe fundamentalimi-larity t seeksto establish.Thischaracteristictructurewill itselfhavesomebearingon the way we interpret ransformationalperations?Fromtheoperational ointof view one mightargue,absurdly,hat sinceall theele-mentsof one set sharepreciselythe propertyof beingrelatedby a singleinterval mod 12)to the membersof another et-a propertynot sharedbyany pitch outsideof this set-they are unifiedas a structure.Fromthisdefinitionwe mustassumethatthe structural roperties hat definea setclass come intobeingonly when a set is compared o anotherset of thesameclassor perhaps hata singleinstanceof a set canacquire hisprop-ertyby beingtransposedntoitself at t = 0. I bringin this argument nlyto indicate he fundamentalroblemof circularity:et-classmustbe givenbeforeoperationscan be performedwhich can be claimedto determinemembershipn a set class.In attemptingo defineset class it will be usefulto pursue urtherprob-lems involving ranspositionndoperationsn general.It is statedwithoutdispute hroughouthe theoreticaliteraturehat ranspositionnd nversion186



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are relations, hat sets are "relatedby"theseoperationsor mappings.Wemay reasonablyasklwhat sort of relationships impliedhere. Operationsmaybe viewedfromthe logical, axiomaticperspective s eternalrelation-shipsexisting ndependentlyf any particularmusical)manifestation.Themathematicalperation f additionmaybe regarded s a universalpropertyof number,a relationcompletelydetached romany activity.On the otherhand,additionmaybe regarded s a realoperation r humanactivitywhichmathematicssymbolizes-the linguistic developmentof the concept ofnumber ertainlypointsto this interpretation. owever his issuemightbeargued ormathematics, theoryof musicwhichhopesto comprehendhestructure f soundingmusic will haveto considerwhether ts operations rrelationsare divorced from acts. If we say that two sets are relatedbytransposition efinedas a one-to-onemapping,we maybe asserting,forexample,that there is a functionalrelationship stablishedbetween theintervalof transpositionnd thecorresponding itchesand thatthis can insomewaybe detectednthesensuousmedium.Or wemaysimplybe assert-ing a formalconnectionbetween ranspositionwhatever angeof musicalpossibilitiesthis may encompass)and arithmetic.One might point to amusical instance n which a demonstrable perationstronglycontradictshearingor intuition,butthis would in no wayimpugn he logical validityof theoperation.Such a divergenceof theexperiential nd the logicalwillnevercast doubt on the integrityof the groupstructuregoverning dealtransformations.3If we wish to bringthesetwoquiteproperlydistinctworlds ogether,aswe arecompelled o do in attempting musicalanalysis,thenrelationsoroperationsmustgive up some of theiruniversality ndtimelessness o takeon thecharacter f actualconstructions,hat s, theywill come to be viewedas structuringather hanas immanentlytructured. ollowing hispathweare led to ask whatsort of act is, for example, ransposition.n responsewe shouldfirstrecognize hattheremaybe manyresultsof such anopera-tion. Thusin transposing 12-tone et we canpointto variouspatterns fchange,such as pitch-class nvariance r orderinversions,and even sys-tematicallydescribesuchphenomena,but theseresultsof theoperation renotthe operationtself-it mightin fact be possibleto performa differentoperation ieldingthe samesequenceof pitchclasses fromwhichperspec-tive we could view the patternof transposition elationsas a result.Thedifference etweensuch"equivalent"perationswill havesignificantmeth-odologicalconsequencesoranytheorywhichattemptso applyits opera-tions to the analysisof music. (Note for examplecases where identicalresultscanbe reachedbythedifferent perations f transpositionr inver-sion appliedto inversionally ymmetrical ets. Suchpossibilitiesmayen-rich thestructure f themathematicalroupbutpose seriousproblems ormusicalanalysis f the notionof equivalence s strictlymaintained.)Sinceoperations recustomarily iventhe formof one-to-onemappings

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theactof relating orrespondingairsof pitches s suggested.Presumably,eachpitchof one set is somehowconnected o a single pitchof another etbya fixed interval. n orderto sensethe set-classequivalence f suchcol-lectionswe musttherefore ensethisconnection;hat s, we musthearthisfixedinterval. n Example2a two trichordsromExample a are extractedandpresentedn a simplified ormretaining nly thequalitiesof pitchandtemporalorder. In Examples2 and 3 intervalsare labeledby the numberof semitonesmeasuringhe"distance"modulo12 fromthe firstnoteto thesecond.Plusand minussignsindicatedirection.In parentheses reshownoctavedisplacements.)Herethepitchesof the secondtrichord rearranged(in a perhapsartificial nterpretationf Ex. la) to expose an identityoforderamongcorrespondingranspositionallyelatedelements."Evenwiththehelpof thisorderingt is quitedifficult o imagine hatoursenseof thesimilarity f thesetwo sonoritiesmustarisefrom heperception f thethreebracketedntervals, uchthathearing hethree nstancesof the sameinter-val type-E-AL, Ak-C,EL-G-we arriveat an intuitionof equivalence.Myquestioning f theauralvalidityof theseone-to-one orrespondencess notmeantas an indictment f theoperationalmodelof transpositions a map-pingbutratheras a test of its limits.In Example2c the replication f setclass is perceptually ery immediateand one easily senses the tranposi-tionalrelationship. hisperceptionmaybe strengthenednpartbecause heinterval f transposition, , is not confusedwithanyintervalwithin hetri-chordsand because the intervalof three semitones,being the smallestintervalpresented,is not challengedby other intervalsfor relationbyregistralproximity.Yet, the interval,8 (modulo12), appearingboth be-tweenthe sets(EL-G) ndwithin hemmayto someextentweaken herela-tionshipin question.Example2d rectifiesthis problem.Becauseof thenecessarycomplexityof musicalphenomena,heseexamplesdo not satis-factorily solatethe specificfactorswhich contributeo the transpositionalrelationship;f one feature s changedall relationships rechanged.Thisis of coursethe markof wholeness.Nevertheless,as one listensto Exam-ples 2a, b, c, and d, a particular ort of connectionemerges,absentinExample2a andquitestrong n Example2d.If inquiringvery brieflyandsuperficially fterour experiencehas notsettledthis issue it surelyhas illuminatedt to someextent.Eventhe fewobservationswe havemade in connectionwith Example2 indicatethattransposition s an actualoperationconnectingpitchesis a special andtherefore"abstractable"ttribute f the relationsof sets of the sameclass.Liketemporal rder,durationpattern,andso forth,theoperation f trans-positionis a structural ossibilityheld by the conceptof set class. Thescalararrangementf Examples2a-drepresents small selectionfromthewhole rangeof suchpossibilities.Evenif the possibility ortranspositions an attributeof the set ratherthanthe foundationor groundof the set, it is nonethelessa necessary188



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a. b. +4

intervals:8(-12), +7(+24)1-7(-24)I+4(+24),-5 -8(-12), +7(+24)1-7 1 8(-12), +7(+24).+3 d. +2

+3 +3 +2 +2

-8(-12) +7(+24)-8 -8(-12), +7(+24) -8(-12), +7(+24) -10 -8(-12), +7(+24)e. f.+3

V-+3 -3

-8 +7 -8 +53-4 3-11 3-4 3-1

Example2a. x y b. x y

S117 3a

(+24) 8(+12) C+2)11 11 11 1S7(+24) 7(+24) 17(+24) 7(+12)8(+12) 8(+12 8(+12) 8yC. X d. x 3(+12)

3-3

S7(+24) 7(+24) 8(+24) 11 7(+24) 11(+12)7(+48)8(+12) 7(+24) 8(+12) 8(+24) "Example3 189



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attribute;veryset thatwe call a memberof a set-classcan be transformedintoanyothermemberof the class by transpositionor transpositionou-pledwithinversion).Froma different ointof viewlet us nowinquirewhythe operationof transposition-evenwhen conceivedaxiomatically, partfromanactualevent- is incapable f servingas a basis for the definition fset class.Wemustbeginwitha briefexcursusbydistinguishinghree ypesof pitch-intervalicquivalence: -Pitch intervals andi' areJ-equivalentif andonly if i = i' in exactsemitones;hencethe interval-3 semitonesis notequivalento +15 semitonesor to -9 semitones.K-Pitch intervalsi andi' areK-equivalentf andonly if the followingconditionsa) andb)arebothsatisfied: ) the difference f i and ', measuredn exactsemitones,is divisibleby 12 (numbers"divisibleby 12"are0, ? 12, ? 24, and soforth),andb) i andi' have hesamesign;that s, both arepositiveor botharenegative.Here +3 semitonesbecomesequivalento +15 or +27 semi-tones;thus we speak,forexample,of parallel ifthsrarely inding t neces-saryto makea distinctionbetweenfifths andtwelfths.L-Pitch intervalsi and i' are L-equivalentf and only if conditiona) above is satisfied.Through he registralnversionof twopitches(i.e., invertinghe relationsof "above"nd"below")n addition o allowingdisplacement y octaveoroctavemultiple, he interval+3 semitonescan be regarded s equivalento-9 semitones.Whileseverelyrestrictingts applicationo the bass, clas-sical tonalmusicallows hissortof variabilityooccurquite reely.Anothersort of equivalence,"interval-class"M), will be considered aterwhen weinvestigatenterval n moredetail.In each of the Examples3a-d the secondtrichord,Y, is "relatedo"or"transformedrom"hefirst,X, bytheoperation f transpositionwhere hevalueof the operator is 3, or moresuccinctly,Y = T (X,3). Since theoperations performed n pitchclasses and results n pitchclasses, therearea vastnumber f possiblepitchrealizations.f we endowthetransposi-tion operatorwith more intervallic electivityso thatit can discriminateamong he three ypesof intervalistedabove,we find hatunlesstheopera-tor has a fixed numericalvalue the degreeandtype of intervallic hangeeffectedby the operationcannotbe predicted.ConsiderExamples3a, b,andc. For thetranspositions 3 on the lowerstave n Example3a a trans-position +15 (+3 displacedupwardby an octave)has been substitutednExample3b. The resultingchange n the intervallic tructure f the set isshownbynumeralsbeloweach trichord.Comparinghetwosets in Exam-ple 3b we see that one interval +11)remains he same and two intervalsare alteredby octavedisplacement.By applyingthese same intervalsoftranspositiono differentpitchesof X as in Example3c we can generateaset of moredissimilarntervallic tructure.Comparedo setX, setY of Ex-ample3c exhibits achof thethreedifferentypesof intervallic elationshipslistedabove.Conversely,Example3d appliesthe threetypesof intervalastransposition peratorsoproducen Y a structureerysimilar o thatof X.190



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Now it maybe objected hattheoperationof transpositions employedin the customarynotionof set-classequivalenceavoidsany involvementwithsuch issues since theoperationdealsonlywithabstractpitchclasses.Certainlyhis is true. There is no questionof contradictionn this usage,butonly a questionof the relationof this model to the phenomenont ismodeling.The abovedemonstrationan shed some lighton this relation-ship.It shouldbe recalled hat n Example3 a-d thelack of correspondencebetween ntervalsof transposition nd the structure f the transformedetdid not occurbecausea too abstract r toogeneralkindof interval-type asunfairlyasked to be responsible or finerdiscriminations-thedistinctionof 3 interval ypeswas givenbothto the transposition peratorand to thecomparisons f sets. The reasonfor the failedcorrespondences that theoperationof transpositionakesplace from outsidethe contextof the set.It is amapping f singlepairsof pitchclasses(orpossiblyof pitches)with-out any regard or the relations o pitchesoutside the pair.This atomismproduces he analyticdifficulties een in Examples3 a-d since in ordertodescribe the intervallic tructureof a set we have to regard he set as awhole.Perhaps fundamentalroblemherearisesfrom he distinctionbetweenpitch and interval.We may regarda set as a collection or aggregateofpitchesor pitchclasses,butto view a set as a whole we turnourattentionfromthe collectionof objects(orpitches) o thetotalityof relationsamongthose objects. Traditionally hese relations are expressed as intervals(though t shouldbe rememberedrom the list above hatthis is a complexsort of relationshipnvolving everal ypesof interval).Moving romset toset class, we assert some similarityamongall the membersof the class,something ommon o all suchsets andonlysuchsets.Surely, hiscommonpropertyhas nothingdirectlyto do with the pitches(or pitchclasses) assuchbut ratherwiththesimilarity f intervallic elationsamong hepitchesof setsbelonging o the sameequivalence lass. Nevertheless, hecustom-arymeans of expressing his similarity s through he operationsof trans-position and inversionappliedto individualpitch classes mappingeachelementof a set into thecorresponding itchclass of an equivalent et. Assuggested arlier, hepossibilityof performinguchmappingspresupposesanalready ormed et; it does notcreateor found heset itself. Sincemap-pingcan replicatea founded et we mightthereforebe inclined o saythatthisreplication reates heequivalence lass;however, uch anexplanationleaves the natureof the set itself undetermined;romthis perspective heset couldbe describedsimplyas an aggregate f pitchclasses.

Inanattemptounderstandheinternal tructure f thepitch-class et weare led to consider he totalityof intervals ormedamongthepitchclassesof the set. Theconceptof interval-classectorrepresentshisholisticunder-standingof set structure. n the case of set-class 3-4 the intervalvector,[100110],hows nthe formof anarrayheuniquentervallictructure f this191



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set class: there s one instanceeach of interval lasses 1, 4, and5 formedamongwhateverpitchclasses mightconstitutemembersof this set class.It is theseintervallic elations hat formthe set as a structuralwhole, andit is the identityof these relationsamong.various

collections of pitcheswhichpromptsus to regard hese collectionsas similaror in some sense"equivalent."While the interval ectordisplays heinternal tructure f theset, it makesno referenceo the pitchclassesamongwhichthe intervallicrelationsaredistributed. n thissense thepitch-class epresentationndtheintervalvectorrepresentationomplementone another.But this is not aproductive omplementarity,ince the two principlesexcludeone anotherso completely; nsteadof creatinga universe hrough heirmutuality, achlaysclaimto the whole.Theintervallic elations, hen,arewhat makea set out of the constituentpitches:priorto the set thereare no "constituent"itches.Once formedasconstituents,hesepitchesmay henbe subjectedotransformations.pply-ing the sametransformationo each of the pitchesof the set presumablyguarantees imilarity f theoriginal o the transformedproduct?The inter-valvectorproves ts usefulnesswhenwe wishtoconsider hesetas awhole;forexample,when we wishto determine he invariants etween wo trans-positionallyrelated sets or when we wish to comparesets of differentclasses or therelationof a set class to the set class of its complement.Ontheotherhand,whenwe attend o transformationsf set classes our under-standingof wholechanges n an attempto comprehendhe totalityof thetransformationshence the attentiongiven the structure f the operationsthemselves.Since the operationsare translatednto a modulararithmetictheirsystematicnter-relationshipan be verypowerfullydescribedusingthe notion of the mathematicalroup.Theidealreciprocity f the twoperspectives utlinedabove s seriouslycompromised ytheanomalyof theZ-related airsof set classes.0 n thesecases sets which cannotbe relatedbytranspositionr inversionperverselypresent he same intervalvector,andas yet therehas been no "canonicaltransformation"oundwhich canmapeach of thesethirty-eightet-classesinto itscorrespondentntervallicallyquivalentmate.'Evenif sucha trans-formationwereto be found,thereis reason o doubtthat this wouldrein-state the intervalvectorto the role of determininget-classequivalence.Sucha reinterpretationouldonly be possibleif the new transformationscould be systematicallyeconciledwiththeoperations f transpositionndinversion. Even if such a solution were found theremightstill be someobjectionto the.perceptualobstacles presentedby this special sort ofequivalence. t is doubtlessa consideration f these two factors, he sys-tematicandtheperceptual,whichhas, in the faceof theanomolycausedbytheZ relation,resultedntheabandonmentf interval ectorrather han heabandonmentf operationsnthedefinition f set class.This is an unfortu-nateloss; particularlyf, as I havearguedabove,the intervallic tructure192



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of a set is the essentialpropertycompared o which the operations hatrelatea set to others of its class can be viewedas epiphenomena.This isnot to denythatsincetransformationsrenecessarily ntimatelyiedto thestructures hey transform, he operationalperspectivecan reveala vastrange of propertiesof the twelve-toneuniverse. The susceptibilityoftwelve-tone perationso systematizationas enabledmanyscholars ocon-tribute o the creationof a very potenttheoretical nstrument.And yet,divorced rom tscomplementaryerspectiveheoperational iewbecomesmisleadingwhen it demands o be taken or thewhole. Thisview,becauseit is atomisticandtakes he elementsof a setto bepoints,simplequantities,will tend oward eification,owardhething-like,hequantitativendawayfrom herelational,hequalitative ndperhapsheless sharplydetermined.In order to begin bridging his difference shall attempt o establishanintervallicdefinitionof set class.A setmaybeunderstood ot as a givencollectionoraggregate f pitchesor pitchclasses,but as a genuinewhole definedbythetotalityof intervallicrelationsamongits constituent ones. There is a reciprocityor dialecticunitingpitchand interval:musical nterval s therelationof pitchesandsois clearly dependentupon pitch;conversely,pitchesare dependentuponintervalor relation f theyare to be regarded s constituents r tones. (Aswe beginconsidering he relationof pitchesit maybe appropriateo usethe word"tone."Thisusagehas theadvantagef shiftingour attentionromtheacoustical owardhe musicalandallowsusto circumventhe distinctionmarkedby pitchandpitchclasswhenever hisdistinctions notuseful.)Asnotedearlier,"interval"s acomplexconceptcomprising everal ypes.Wedistinguishedourtypes, arrangedn orderof increasinggeneralityor ab-stractness: ,K, L, M. Whilethere s apparently othing opreventus fromtakinganyof these intervallic elations o definethe set, thereis reason obeginour investigation y givingpreferenceo the mostgeneralor inclu-sive; thusif the relationL holds so must the relationK, but the reverse snot true.

On the basis of the foregoing, he intervalvectorwould seem to be anexcellentrepresentationf set structure-it lists all the intervals ormedamong he set'sconstituents, singthe mostgeneral nterpretationf inter-val, the "interval lass"(our M). The usefulnessof this theoreticalcon-struction s, however, imitedby its suppression f the importance f theconstituent lementsof the set overagainst he relationof theseelements.Thisdifficultybecomesapparentwhen we look morecloselyat theconceptof "interval lass."Let us begin by inquiring urther nto what the notion of interval n-volves. In Example4a two tonesareheard n relation o one another.Wearenot concernednowwiththeirtemporalorderof presentation-Acouldprecedeor followC, theycould be playedsimultaneously, r other tonesmight ntervene.Likewisewe arenotconcernedwith heirrelativedurations,193



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a. b. -Register C. -Registralorder d. -RegistralorderA -Register -Register

A=-3 A=-15 -3 A--+9 -3 A=3C=+3 C=+15= +3 C=-9 +3 C=3J K L MExample 4

C (1186)[146] .--C (11 5)[135]C#(1I" ) [135] -* C# (1 10 6) [126]E (10 4)[234] -------- D# 2 8) [234]F# (2 5 6)[256] G (4 7 6) [456]

4-Z15 [111111] 4-Z29 [111111]

b. set class 4-Z15 (10 3 4)0 10 3 4

0 il i2 ... in-1 -0 0 10 3 4-10 2 0 5 6-0. 0-0 i-0 i2-0 .. in-1-0 -3 9 7 0 1O-i 0-i1 il -il i2-i1 . in-1-i -4 8 6 11 0-i2 0-i2 il-i2 i2-i2 ?

* in--i2

-in-1 0-in-1 i -in-1 i2-in-1 ? in-1 in-1c. D (1 4 6)

. .. C# (11 3 5)SA#(2 9 8)4-Z15 G# (10 7 6)

d. 1 9 1(46) (2)9(8)

9 7 (10)7(6)Example 5194



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loudnessor timbre.Theirrelationas tones is restrictedo the sole domainof interval.We may call the interval hey form"3 semitones" r "minorthird,"but shouldnot forgetthat in naminga single intervalwe are stillpresentedwith two tones, C soundingthree semitones above A and AsoundinghreesemitonesbelowC. C asa tonehasacquiredherelationshipor, we might say,the quality+3. It is this qualitythat makes of the pitchC (c.524cps)a tone. This samequalitycan be heard n anyisolatedpairoftones threesemitonesapart.A similarqualitycan be heardwhentheinter-val is expandedby an octaveas in Example4b. In this example"minusregister"meansthatexpansionor contraction yanoctaveor octavemulti-ple is nota distinctive eature, o long as theA remainsbelowthe C. Oursense of the similarityof an intervaland its inversion s representednExample4c. Here the categoryof registralorderor the distinctionabove/below is mitigated o thatthequalityof A, -3, is equatedwiththequality+9. Noticethat -3 is not equatedwith +3. This distinctions eliminatedin Example4d, an interpretationorrespondingo the notionof intervalclass and based on a typeof inversional quivalence.Theoperational er-spectivedoubtless nforms his view of intervallic quivalence-theactionof movingbythreesemitonescan be carriedout in anupward r downwarddirection;or, moreabstractly,he purearithmetic peration f subtractionyieldsa quantitywhichdoes notowe its existenceas number o thisopera-tion.Whilethisunderstandingf interval rovesveryvaluablen somecon-texts, it createsdifficultiesfor the definitionof set by renouncing hecomplementarityf pitchand ntervaln favorof thehegemonyof the latter.As Example illustrates,he advance n abstractionmarkedbyExample dis the eliminationof the distinctionbetweenthe two pitchclasses-A andC areassigned hesamequalityand arethusundifferentiated.she intervalvectorheightenshisseparationf pitchand ntervalbytallying heinterval-class contentof a set in sucha wayas to makeit impossible o infer fromthislist a pitch-class epresentationf theset. That s to say, nterval ectorcan be derivedfroma set of pitchclasses but not vice versa.Itis, however, ossible orepresentheintervallictructure f a setwhilepreservingherelationpitch/intervalr tone.Forthis twoalterationso theinterval-vector odelareneeded.First,ourrepresentationf intervalmustdistinguishbetweenthe two tones. While each of the representationsnExamples a-c dothis,we will forthepresent ake hemostgeneral,Exam-ple 4c, fora model. (Forthe sakeof economywe will eliminateplus andminussigns fromrepresentationsf interval ypeL, substitutingor nega-tiveintegershecomplementary ositive ntegers.Thus,in Example c, theintervallic ttribute f the tone A will be called9 andthatof C willbe called3.) Second,ourlist of intervalswill refer n turn o each of thetonesof theset. Example5a showssucha representationpplied o twosetclasses, theZ-relatedetrachords-Z15and4-Z29. Afterthe lettername of eachtone(whichcouldof coursebe rewritten s aninteger)arelistedall theintervals195



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L that one formswith theremainingonesof the set. (Thesenumerals reratherarbitrarilyistedin theascendingorderof theircorrespondingnter-val classsimply o facilitate ertain omparisons.)ThusC forms heinterval11withCI, 8 withE, and6 withFR. Recall romEx. 4 that11here means11semitonesabove,1 semitonebelowor anyoctavemultipleof theseinter-vals.)Againforeconomy, n thisrepresentationhe intervallic omparisonof a pitchclass with itself has been omitted.Interval-classquivalents ftheserelationsare listedin bracketsn Example5a to indicatea similaritybetweenthe Z-related ets. Note thattwo tonesof 4-Z29 (C andDI) areconstituted f thesame nterval-classelations saretwo tonesof 4-Z15(C#andE). But since interval ypesaremixedin each case thereis no morereason orpositinganequivalence elationhere thantherewouldbe in theparallel"operational"ituations hownin Examples2e andf.

While sucha representationf set class is rathercumbersome,t con-tains a greatdealof structuralnformationndcanfor certainpurposesbeabbreviated. inceeachtoneis defined n relation o thewhole, thequalityof anytone can define he set. Forexample, inceC#in relation o the othermembersof the set assumesthe intervallicqualities1, 9, 7, we can, fromthe perspectiveof Ct, generate he set by supplying he complementsoftheseintervals:11,3, and5 (C, E andF#). Similarly,he set maybe trans-posedbytransposing nyof its constituents ndreconstitutinghe interval-lic relations f thatconstituent.From heintervallic ssociations f anytoneof a set theintervallic ssociations f theremainingonescanbe calculatedbysubtractingn turneachmemberof thegivenstringof intervallic ssocia-tionsfromeverymemberof the string.Thisprocesscanbe representedntheformof a matrix n whichcolumnsandrowscontainunorderedollec-tions as in Example5b. Herethe variablein stands oran intervallic sso-ciationof anunspecifiedpitchO representinghe interval hatpitchformswith itself. Thecardinality f the set is representedythevariablen. Thussinceanyof the setsof intervallic ssociations anserveto generateall theothers,everyset class canbe represented y a singlestringof n-1integers(O beingunderstood)wheren is the cardinality f the set class.This view of set class distinguishes nversionallyrelatedforms butclearlyallowsfor a specialequivalence elationbasedon the principleofintervallicnversion.Inversion rom this perspectiveappearsas a ratherEscheresquedialecticof "inside" nd"outside."Comparinghe set 4-Z15of Example5a withthe inverted ormshown n Example5c, notethatallintervallic elationsareinverted.Example5d illustrateshissituation romthe pointof view of the individual one:the intervalsby whichthe otherelementsof the setconstitute particularone becomein the invertedormthe intervalshroughwhichthe"corresponding"onecontributesothecon-stitution f theothers.This"inversion"f intervallic elationships, otsim-ply of interval, s also illustratedn the matricesof Example5b whereinversionof the set resultsin the exchangeof columnsand rows. From196



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thisperspectivewe can abstract rom inversion he concommitanteaturesof "directionality"up/down)and axis which, like transposition,mayormaynot have a structural unction n an actualmusicalinstance.In Example5 intervalswere calculatedaccording o our most generalmodel, L. We can discriminatemore specificintervallicqualitiesby em-ploying he relationsK andJ to representets. In Example6 a series of sixcomparisonss made. In each case a different ormY of set class 3-4 iscomparedo thefixedformX While all the trichords n Example6b dis-playthe same L-intervallicssociations,only Examples6a, b andc displaythe sameK-intervallicssociations,and,of these, only Example6a main-tainsthe same J intervals.K andJ maybe considered ub-classesof setequivalence, nd indeed hequalitative ifferences ositedbythese distinc-tionsare,Ibelieve,perceptually uiteclear(that s, to the extent hatExam-ples 6b and c as a groupcanbe heard o differfromExample6a and thatExamples6d, e andf as a groupcan be heard o differfromExamples6a,b and c as a group).The variousformsbelonging o anyof theseequiva-lence classes are of courseundifferentiateds membersof that class. Butsince we aredealingherewiththree evelsof abstractiont maybe possiblefor the membersof one of the classes to be orderedaccording o theirdegreeof similarityas measuredbya less abstractnterval ype. Thus,forexample,whileExamples b andc areundifferentiatedntheirK-intervallicassociations, richordY in Example6b shares wo J entrieswithX whereasExample6c sharesnone.It is notclear howsimilarity houldbe defined n all cases. Presumablythe "orderof octavedisplacement"the numberof octavesby which anintervals expanded r contracted)wouldbe neededforcomparisons.Evenwiththisrefinementwe cannotorder, orexample,Examples6fY and6gYin theirsimilarityo X, nordoes thisdistinction ppearo havemuchauraljustification.While sucha rankingof similaritydoes not seem capableofexhausting ll thepossibleregistral ormsof a set andmayin some casesbe of uncertainanalyticvalue, nevertheless t does pointto an importantfeatureof ourdefinitionof set class-that rather hansimplyabstractingsinglecharacteristicroman actualset of tones, the concept s capableoforganizingo someextent heparticularnstances t generalizes.Thiscapa-bility is of considerablemethodological aluesince it createsa systematicwhole fromamongthe possibleforms whicha set class can assume.Theanalyticresultis that we can comparedifferent ormsof a set class andjudgeto someextent heirdegreeof relatedness, nd,mostimportantly,hatthisjudgment s based on the conceptthat unites themas a class.For an example et us return o the firstmeasureof DU Againwe willassume hatthis musiccanbe segmented s three nstancesof set class 3-4labelledX, Y andZ in Example7. In thisexample hethree nterval ypesare listedtogether n the forme ((L) (K) (J)) wheree is an element of aset. In comparingsets, intervalliccorrespondences an be considered

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L: E (11 7) G(11 7)E(1 8) AS4(18)Ab(4 5) C(4 5)EI(ll 31) G(11 31) J: G(11 19*) J: G (23* 55*)7 E (-11 20) Ab(-ll 20) Ab(-ll 8*) Ab(-23*32*)f A?(-20-31) C(-20-31) j b C(-8*-19*) C(-32*-55a - Ib . C.

x y x y x y

K: E6(11 7) G(11 7) K: G(11 7) K: G (11 7)BE(-118) Ab(-ll 8) AbI(-1 8) Ab(-118)A (-8 -7) C (-8 -7) C (-8 -7) C (-8 -7)

J: G(-1* 31) J: G(23*"-5*) J: G(-1* -5*)J:G(-13*-29*)Ab(l* 32*) Al(-23* -28*) Ab(l* -4*) Ab(13*-16*)C(-32* -31) C (28* 5*) C (4* 5*) C (16* 29*)Sd. e. f. g.

x y x y x y y

K: G (-1* 7) K: G(11 -5*) K: G(-1* -5*) K: G(-1* -5*)A6(1* 8) Ab(-ll -4*) A,(l* -4*) Ab(l* -4*)C(-8 -7) C (4* 5*) C (4* 5*) C (4* 5*)*indicates lack of correspondence to X

Example 6

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"stronger"r morespecificor concreteas theyappear owardheleft of thearray.As we saw in Example6e, X andY are not closely related.If werecognize nversional quivalenceof sets then, whileY andX shareno J-intervallic ssociations,Z hasone suchintervaln commonwith each.Thestructural ignificanceof this sort of analyticobservationwill of coursedependupona muchbroader ontext hanwe haveexposedhere,butsincethevariousdomainsrepresent iscretestructuralealms, t is advantageousto describe heir"internal"rganization eforeattemptingo describe heirinteraction.10 he structural ossibilitiesof anydomainwill of courseberealized(or suppressed)n varyingdegrees by interactionwith otherdo-mains.Forexample,notice thattheJ-intervallic onnectionof set Z whichis least similar o the intervalsof sets X and Y-G (-16), B (16)-is rhyth-micallyandtimbrallymost clearly exposedin the second half of measure1 of DU (See Ex. 1). Indeedthere are manyfactorswhich in this firstmeasuresubvert he relationships hown in Example7.Intheaboveexampleswe haveexplored heconceptof set classthroughcomparisons f different itchsets.Inthislight,setclassappears s ausefulanalyticinstrument or disclosingsimilarities,both obviousand hidden,amongdifferentmomentsof a composition.Thecomparisonsmaybe inter-preted o reveal ransformationsf setsor to indicatedegreesof relatednessamongsets of the same class. There s, however,anotherandperhapspri-marymeaningof theconceptwhichhasemerged n thisinquiryand whichwe should now considermoreexplicitly.Theformation f a classmaybe thoughtof "extensively"s theaggrega-tion of objectswhich share some characteristic.The comparisonof suchobjectsreveals heunifyingcharacteristic hichencompasses llparticularrepresentations. his interpretationeems to be favoredby the operationalview.Herethere s a tendencyo requirea comparison f sets forthe estab-lishmentof theclass. Thus to speakof anyrepresentativef theclass refer-ence must be made to a particularorm("prime orm"or "representativeform") ather han o a general ormwhichincludesall themembersof theclass. Inpublished nalysesoneoftenencounters arious ortsof "priority,""referential"ollections termswhichlackingsystematicdefinitionnever-theless showanattemptoestablish omeprivilegedormoutsideandabovetheproliferationf particulars. heneedforcomparisons ndthus theneedforreplication fforms hroughwhichcomparisonsanbemaderaisesques-tionsconcerninghestructuralignificanceandperhaps ventheexistence)of a singleinstanceof a setclass. Thepositionoutlinedaboveand itsatten-dantdifficultiestemfromtheabsenceof a priordefinition f that"commoncharacteristic"hichunifiestheparticularepresentationsndthusdefinesset class. ErnstCassirer tatesthisgeneralproblemveryclearly:

. . it is evidenthatbeforeonecanproceedogroup heelements f classand ndicate hemextensively yenumeration,decisionmustbe madeas199



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(RI) (RI)

x y z

L K JEb 111 7) ( 7) (1 31) )x E (1 8) ( 8) (! 20) )AM,(

(4 5) ( -8 -7)" i( -20-31))G ((117) ( 11 -5)

.(23 9)

y Ab( (1 8) ( -1 -4) (-23 28))C ( (4 5) (4 5) , \(28 ) )mI)IC ( (1 5) (-1 (?5) ( )z B ((11 4) ( 4) ( 16)G ( (8 7) -4 5) ( -16 5))(dotted ines ndicatenverse elations)

Example7

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to which lements re oberegardedsbelongingo theclass:and hisques-tioncanbe answerednlyon thebasisof aclassconceptnthe"intensional"senseof the word.What eemsto holdtogetherhemembers nitedn theclass s that heyall meeta certain onditionwhichcanbeformulatedngen-eral terms.Andnowtheaggregatetselfno longerappears s a meresumof individuals,ut s defined ythisverycondition,whosemeaningwecangraspand stateby itself,withouthavingo askin howmany ndividualstis realized,or even whether t is realizedn anyindividual t all."Therepresentationf set4-Z15shown n Example5a is intended o pro-vide such a generalformulation.The fourstringsof integersare commonto all (non-inversionallyelated) ormsof 4-Z15.(Recallthatthis set classcouldbe represented yanyoneof thesestringssincetheremaining tringscan be derived romit.) Here there is no privileged orm,no replicawithwhichto compare he set. While the termset class implies n its extensiona limitless numberof forms,it is misleading o view theprimary unctionof the conceptas unitingall thesepossibleparticulars s a collective.Setclass maybe viewedperhapsmoresignificantly s an interpretationf thetotalityof intervallic elationships resentedby a groupof tones. Strictlyspeaking,a pitch(or pitch-class) et is a purelynegativeabstraction. t isa set of pitchesfromwhich all otherdeterminations-order, uration,andso forth-have been removed includingregister f a pitch-classset). It istheconceptof setclass whichrelates he set of pitches,and it is such rela-tionship hatmakes onesof thepitches.A set consideredn thelightof thestructure rovided yset classmaybe regarded s the "tonal" ontextof thepitches.Suchtonesmayenter nto all sortsof otherrelationships, ut theirdefinitionas tones is dependentexclusively uponthe intervallicrelation-shipswhichcharacterizehe set. The functionof set class thusappearsasa meansof penetratinghe intervallic tructure however his maybe de-fined)of a particular roupof tones.Thattheremaybe othergroupswhichpossess the samecharacteristicsis from this perspectivenot of primary nterest.Where our interest isdirected owardhe structure f a particularet we can use the term"set"to mean"representativef a set class."12Viewed n this way,the abstractconceptof setclassdefines herangeof possible(intervallic)elations romwhich a particularet is realized.Set class is thus a theoreticalnstrumentfor describing"tonal" elationships.This formulation pens our investigationo the crucialanalyticques-tions of what,in a particularmusicalcontext,theserelationshipsmightbe,howtheyareconstituted, ndwhatrhythmic nd formal unctions heymayserve.HereI hopeto have aid a moresecuregroundworkortheinvestiga-tionof thesehighlycomplex ssues whichfarexceedthescopeof thepres-ent essay.3The analytic andalso systematic)mplicationsof the descrip-tionof setclass I haveproposed uggestnumerous venues orfuture tudy.

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Familiarproblemsof similarity,nclusion,andcomplementationould bereconsideredn lightof an intervallicdefinitionof set class. Comparisonsof setsrepresentingifferent lasseswill revealvaryingdegreesof similar-ity amongconstituentpitches.Setsof quitedifferent otal intervallic om-position may presentclose similaritiesamongsome of their constituents.Similarly,a consideration f the specificintervallic tructure f pitchcol-lectionsmightshowconsiderable isparitybetweensetsof the same class.While the aboveperspectivewill put intoquestion he significanceof thecomplement elation, t couldprovidea measureof chromatic aturation-the chromaticwouldbe completedor closed to the extentall twelvepitchclassesassume dentical ntervallic ssociations. believe one of the moreinteresting amifications f the conceptof set class sketchedaboveis theemphasisgiven to individual ones in the contextof sets. This changeofperspectivemightin factpermitsome degreeof rapprochementetweentheoriesof post-tonalmusic andtheoriesof tonalmusic,properly o called.

k o ? t %%Ise*

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NOTES1. I wish to thank Allen Forteand David Lewin for their sensitive readingof an earlydraftof this manuscriptand for their manyvaluablesuggestions.2. Lewinexplicitlyavoidsthis problem n the followingdefinition:"By'canonical rans-formation' shall mean an operationon pc'swhichis understood n a giventheoreticalcontext,to transform the totalcontentof) any pcset intoa pcset which is acceptedas'similar'inthat context."(my emphasis). ("Forte'sntervalVector,My IntervalFunc-tion, andRegener'sCommon-NoteFunction," ournalofMusic Theory21(1977): 195.)3. I shall frequentlyreferthe discussion of set class to aspectsof 12-tone heory,particu-

larly the notion of closure under the operationsof transpositionand inversion. Whilea treatmentof unorderedsets does not necessarily imply a connection with 12-tonetheory, there is some methodological advantage o unitingthese two theoretical do-mains in the presentdiscussion, especially since many of the basic conceptsof thetheoryof unordered ets arederived from serial theory.I believe some of the conclu-sions of thisessay havespecial implicationsfor the analysisof 12-tonemusic (consid-ered as a "combinationalystem"),and I intendto pursuethis topic in a futurestudy.4. The simplificationof Examplela shown in example2a may be worthconsideringinmore detail. Ourorderingof the second trichordmightbe justifiedon the groundthatAbsounds beforeC and the G follows C at the dynamic level of mp. In the case ofExample la the transpositionaloperationE - Abrefersonly to the second attackofAband yet the firstAbmust be heardas the second if we are to maintain hat orderis preservedbetween these two "instancesof 3-4." I drawthe reader'sattention o thisproblemonly becausethe ironyof this sort of situationhasnotbeensufficientlyappre-ciated in many analyses.5. An analogycan be drawn o geometryif we substitute rianglefor trichord.Giventwoangles we can constructany similar replica-of any size and anywherein space. Orwe can start romthreepointsof a Cartesianmetric andoperateon the resultant igure(moving it about in space, changingits size, and so forth)by performingarithmeticoperationsuponthe threepoint coordinates,except thatthese operationsmust be solimited thatthe resultant iguremaintains he same angles; hence, a squaringof quan-tities is, for example, disallowed.6. It shouldbe noted here thatForte,in "ATheoryof Set-Complexes or Music"Journalof Music Theory8 (1964): 136-83, initially attempted o base the notionof set classon intervallic relations using the interval-class vector devised by Martino ("TheSource-Set and Its AggregateFormations,"Journalof Music Theory5 (1961):224-73.). Because of the problemsintroducedby the Z relation Fortesubsequentlyaban-doned interval as the determinantof set-class membership.A differentintervallicrepresentation f set class was proposed by RichardChrisman n "ATheoryof Axis-Tonalityfor Twentieth-CenturyMusic" (Ph.D. diss., Yale University, 1969). Thisrepresentationwhich Chrismancalls the "successive ntervalarray,"ater endorsedbyEric Regener under the name "intervalnotation"("On Allen Forte'sTheory ofChords,"Perspectivesof NewMusic 13/1(1974-5): 191-212),succeeds in distinguish-ing Z-related sets (as does Forte's"primeform"from which it is derived)but doesnot explain the correspondenceof interval vectors. I believe these difficulties stemfrom the notionof intervalclass (or, alternatively,"directed nterval") nd shall laterproposea refinement o the interval-classvectorwhich excludesthe equivalenceof Z-relatedsets.

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7. In an importantrticleconcerning et-classoperations nd intervallic elations,RobertMorrisdiscusses ransformationshichmapmanyof these Z-relatedets.Throughnilluminatingnvestigationf thesystematicifficultiesf theZ-relation,Morris rrives t aset-groupystem, G(vz), hat ollapses llZ-relatedairs hrougha setoftransformationalperations. sMorris oints ut,thisaccommodationesultsin anextraordinaryegree f abstraction,educing,orexample, orte'siftydistincthexachordlasses oonly hree roups.Morris, SetGroups,Complementation,ndMappings mongPitch-Classets,"Journal f MusicTheory 6 (1982):101-44.)8. Exampled representsdifferentortof relationhan hatof the series n Exs.a-c.Since his newperspectiveppliedo Ex. c resultsn Ex. d, we couldapply t alsoto Exs.aandbtobringhe otalnumberf intervalypes o6 (2 x 3)-two interpreta-tionsof three undamentalypes.However,ince tannihilateshedistinctionetweenthepitches rpitch lasseswhich"form"he nterval,nterval lass(ourM)doesnotstrictly elong o thesetrepresentationamproposing. otealsothat"directednter-val,"while it closelyresemblesnterval-type, defines ntervaln one "direction"ratherhen n bothsimultaneouslys doestypeL.9. Theselection f the tonesG, AbandC to composeY is essentially rbitrary,incewe areconcernednlywiththeautonomousntervallictructuresf thesetsX andY.Thepossiblentervallicelations etween he setsof eachpairand hecorrespon-denceof common ubsetswillnotconcern s heresince hesearereallyquestions falternativeegmentationsf the examples.Thusif we admit ntervallicelationsacross heboundariesf thetrichordsrconsider ubsetswithin heseboundaries ewill have o considermore hanone set class.This couldof coursebe done,but itis preferableo avoid uchcomplicationst thisstageof our nvestigation.point hisoutto assure hereaderhat heartificialimplicityf theseexamples oesnotreflecta necessaryimitation f theconcept et class.10. Fora moredetailedreatmentf this ssueseemy"SegmentationndProcessnPost-TonalMusic,"MusicTheory pectrum (1981): 4-73.11. ErnstCassirer,ThePhilosophy f Symbolic orms,vol. 3, ThePhenomenologyfKnowledge,rans.RalphManheimNewHaven:YaleUniversity ress,1957),pp.294-295.12. This useof the term"set" orrespondso AllenForte's racticen TheStructurefAtonalMusic NewHaven:YaleUniversity ress,1973). n hiscriticismf thisusage,Regenerop.cit., p. 194) gnoresheratherntricateelationfset and et classI havetried o sketchabove.13. Elsewherehaveattemptedoconfronthese ssues n somedetail. refer hereaderin particularo '"ATheoryof SegmentationevelopedromLateWorks f StefanWolpe"Ph.D.diss., YaleUniversity,978)n which heconcept f intervallicsso-ciation s applied nalytically.

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Documents

Hasty, An Intervallic Definition of Set Class (JMT 1987)