Alegant, Even Partitions in Twelve-Tone Music (MTS 1988)

7/28/2019 Alegant, Even Partitions in Twelve-Tone Music (MTS 1988)

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Society for Music Theory

The Even Partitions in Twelve-Tone MusicAuthor(s): Robert D. Morris and Brian AlegantSource: Music Theory Spectrum, Vol. 10, 10th Anniversary Issue (Spring, 1988), pp. 74-101Published by: University of California Press on behalf of the Society for Music TheoryStable URL: http://www.jstor.org/stable/745793 .

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T h e E v e n Partitions in Twelve-Tone M u s i c

RobertD.MorrisndBrianAlegantAny competent ntroductory ccountof twelve-tonetheory1

andanalysispointsout how divisionsof a row intohexachords,trichords, etrachords,and so forthcangenerateformaldiffer-entiation,fosterrichnessof interrelation,andimplementproc-esses of association n atwelve-tonecomposition.2Yet the divi-sion of a row into two or more successivepartsservesan evenmore basicfunction,that of facilitating ecallof thetwelve-toneunititself. Inpoint of fact, few musicians,whenpresentedwithan arbitrary tringof twelve pcs, find it an easytask to remem-ber the series accurately,even aftermanyhearings.Whenthesequence is articulatedas a successionof smallerunits, how-ever, the taskbecomesquitemanageable,especially f thepartsarerelatedby musically ntelligible ransformations.Therow snow a hierarchicstructure,a series of two or more "chunks,"each of whose lengthis below the limit of "the magicnumber

'It is assumed he readerhas some familiaritywiththe basic termsand con-ceptsof twelve-tone andatonaltheory.For a glossaryof suchtermssee RobertMorris, Compositionwith Pitch-Classes:A Theoryof CompositionalDesign(New Haven: Yale UniversityPress, 1987).John Rahn, BasicAtonal Theory(New York: Longman, 1981) also provides a good introduction o the samesubjectmatter.

2Probably he locus classicus of analyses that use partitioningas a form-differentiating unction is the discussion of Schoenberg'sKlavierstiick,opus33a in George Perle, SerialCompositionandAtonality:An Introduction o theMusic of Schoenberg, Berg, and Webern(Berkley: Universityof CaliforniaPress, 1962).

seven, plusor minus two."3Thismeansthatthe aural mpactofthe compositionalstructureof atwelve-tonepiece is relateddi-rectlyto those compositionalstrategies hathighlight he iden-tificationand transformation f segmentsandpcsetsof a row,or of othertwelve-pcunits.

It is therefore the force of analyticandcompositionalnter-est inwhat determinesaurallycogentmusicalrelations hat hasled to the many previousstudies of twelve-tonepartitioningnthe literature,especially nworkdealingwithsegmentalassoci-ation and combinatoriality.4CertainlyDonald Martino'spio-neering "The Source Set and its AggregateFormations"andthe recent articleby Steve Rouse, "HexachordsandTheirTri-chordal Generators:An Introduction," whichexpoundsandexpands topicsin the Martino)providea greatdeal of valuableinformation for composers and analystsalike.5 Such studies

3SeeG. A. Miller, "TheMagicNumberSeven, Plus or MinusTwo: SomeLimits on OurCapacity or InformationProcessing,"PsychologicalReview63(1956):81-97.

4Perhaps he earliest references to hexachordaland trichordalpartitioningare found in Milton Babbitt, "Some Aspects of Twelve-ToneComposition,"TheScoreand IMA Magazine12 (1955):53-61.5See Donald Martino, "The Source Set and Its Aggregate Formations,"Journalof Music Theory5/2 (1961): 224-73, and Steve Rouse, "Hexachordsand Their Trichordal Generators: An Introduction,"In Theory Only 8/8(1985):19-43.

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The Even PartitionsnTwelve-ToneMusic 75

tend to concentrateon the "source-set,"pcset, or "collection"as the basictheoreticentity togetherwith some basisfor chro-maticcompletionsuch as complementation.Nevertheless,theprocessof generatingaggregates rom "sourcesets" alsotendsto de-emphasizethe importof the partitionsper se, or at bestdeals with only some aspectsof even partitionsof twelve pcs.For hexachordalcombinatoriality he source-set approach ssuitable;but when other types of even or uneven combinato-rialityarecontemplated,the situationbecomesboth unwieldyandinelegant.

Whilepartof this problemstems froman "entity"versusa"transformational"pproach,a naive source set orientation otwelve-tonepartitioningdoes not specify adequately he aggre-gate context for any arbitrary et of disjoint pcsets. In the caseof hexachords,for example, two complementarypcsets neednot be members of the same hexachordalsource set. Thus,there areonly thirty-fivecomplementarypairingsof hexachor-dal source sets that partitionthe aggregate,whereasthere arefiftydifferentdistincthexachordal ourcesets-a fact noteasilydiscoveredby readingMartino's ource set tables. In the caseoftrichordal source sets, matters are more counter-intuitive.Take the rows X and Y derivedfrom the source set {037},the"majoror minor triad":

X=(037 A25 8B4 169)Y=(059 184 B36 27A) (A=10;B=11)

Both rows seem generatedfrom {037} n differentways;for in-stance, the firsthexachordsof the two rows are eachmembersof different all combinatorialhexachords. Yet both rows arederivedfrom the samepartitioningof transformationsf {037}.The partition n questionis{{037} {169} {25A} {48B}}.

Xis a shufflingandreorderingof the trichordsn the partitionwhile Yis a similarpermutationof the T6Iof the partition.6As its title implies, this papertakes a broadview of parti-tions. Althoughthe subjectwillbe those partitions hatdividethe aggregateevenly, the theoreticbackdrop s designedto ac-commodatea study of all partitions,even or uneven. On theway, we shall produce some elementarydefinitionsand rela-tions, outline some methods of generatinga list of even parti-tions with a computer, develop categories suggested by thecomputeroutput,discussaspectsof thepitchandtemporalpre-sentationsof partitions,andend witha few observationson ex-tendingthe work to allpartitionsof the aggregate.To illustratethe analyticscope of our theory, we shall analyseportions ofthe songcycleDu by Milton Babbitt.7GeneralDefinitions

DEFINITION. The aggregates the unorderedcollectionofall twelvepcs.DEFINITION.1. Apartition of the aggregate is any unorderedand disjointset of pcsets which in union comprisethe aggre-gate.

Examples:threepartitionsof the aggregate.A: {{0532} { 4AB} {1} {678} {9}}B: {{A} {354} {2} {B896} {107}}C: {{02468A} {13579B}}

6As we shallshowlater, the crossrelationexemplifiedbyrowsX and Y is ofatypethat hasgreat mportance n worksof Martino,PeterWestergaard,Bab-bitt andothers.7Du(1952)shares ts specialtrichord/hexachord tructurewithalmostall ofBabbitt'smusicfromthe 1950s. Babbitt'sown descriptionof suchcompositionmethods s found n hisarticle,"SinceSchoenberg,"Perspectives f New Music12/1,2(1973):3-28. See also note 32.



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76 MusicTheorySpectrum

DEFINITION.2. A partition-cardinality-listpea)is the listofthe cardinalityof the pcsets in a partitionof the aggregate.Apeais arrangednorder and is usuallynotated n "power"nota-tion.Examples:pcas.peaof A (above): 1 1 3 3 4 (or 12324 npower notation).pcaof C: 6 6 (or62).

DEFINITION.3. A partitionclass is the set of all partitionswithequalpeas.Examples: partitionsof the 48 partitionclass.D: {{0123} {456789AB}}E: {{0369} {124578AB}}D and E are members of the same partitionclass since theyboth have the same pea, 48. PartitionsA and B from abovesharemembership n a differentpartitionclass, designatedbythe pea 12324.

THEOREM2.4. There areseventy-sevenpartitionclasses.Thisimpliesthat there areseventy-sevenpeas;see AppendixAfor a list of these. (No proof is offered for Theorem2.4. sincethe 77 peas are enumeratedeasily by hand or by a computeralgorithmwhichthe authorswill senduponrequest.A generalproofof the number of classesfor an aggregateof n pcs is of ascope anddifficulty hat is inappropriatehere.)

DEFINITION 3. A mosaic8 isa setof partitionshat areequiva-lent undertranspositionand/or nversion(TnorTnI).

8Martino,"SourceSet," uses the term "mosaic" n the sense of our "parti-tion class";his "trichordalmosaics"are the members of our partitionclasses

PartitionsP andQ aremembersof the same mosaic if for eachpcsetp E P thereis a pcsetq E Q such thatq = Kp,whereKisTnor TnI.It follows that membersof a mosaic mustbe mem-berbof the same partitionclassbut not necessarilyvice versa.Forinstance,considerA and B from above rewritten or com-parison:

A: {{0523} {4AB} {678} {1} {9}}B: {{B698} {710} {543}{A} {2}}The pcs of A and B line upso that the pc pairsareTBIrelated;therefore the partitionsA and B areTBI-relatedandaremem-bers of the same mosaic. The partitionsD and E, however,while havingthe same pea, are not membersof the same mo-saic.The relationof the aggregate,partitionclasses, mosaics, andpartitions, s shown as a "tree" n Figure1.

DEFINITION3.2. A evenpartition s one whosepcsetsareallof the samecardinality.For example, partitionC above is even. By extensionwe canspeakof even andunevenpartitionclassesas well.

THEOREM.3. There are six even partitionclasses.The peasof these sixclassesfollow:

112 26 34 43 62 12

havinga pcaof 34. These termsare also found in AndrewMead, "SomeImpli-cationsof the PitchClass/OrderNumberIsomorphism nherent n the Twelve-Tone System,PartI," forthcoming n Perspectives f New Music. Mead's"mo-saics"areour "partitions"andhis "mosaicclasses"areour "mosaics."



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TheEvenPartitionsnTwelve-ToneMusic 77

Figure1u

(77) partition classes: / \ etc

mosaics: ec.

,4 / < i1\1\ /A1\partitions: * etc. *-- etc. *- et(-- '" et

As statedearlier,we will discussthe mosaicsthat aremem-bers of these six classes:the even mosaics.Methodsof Computing heEvenMosaics

We use two differentmethods andcomputingenvironmentsto enumerate andlistthe even mosaics. The programsgeneratesimilar ists of mosaicswhicharecomparedand crossrelatedasindicatedbelow.9One set of programs writtenbyMorris) s co-ded in a versionof FORTRAN that runs on an 8-bitKaypro-4computerunder the CPM operating system. The aggregate sdividedinto hexachords or tetrachords,and then into two orthreepcsets in all the unique possibilities.Eachdifferentparti-tion is transformedunderTnandTnIto find a "normal orm"

9Theoutput lists of mosaicswere sorted in numerouswaysto relate a mo-saic to its set of superpartitions.The result is more or less equivalent to"GraphsX andY" in Rouse, "Hexachords."Whilethe resulting ists would fillat leastone hundredpagesof output, crossrelationsbetweenmosaicsof differ-entpcasareeasilyassembledby storing helistsincomputer iles andsearchingthe files withsimplead hoc programs.

which uniquely represents each mosaic. The list of normalforms is sortedandanyduplicatesremoved.Theother setofprograms byAlegant)uses asimilarprocessbut is more generallyimplemented by recursion n the C pro-gramminganguage.The operatingsystemis VENIX (a UNIXlook-alike)runningon a 16-bitPRO-350computer.Twomutu-ally recursiveproceduresform the core of an algorithmde-signedto eliminatemosaicsthat arepermutationsandtransfor-mations of other mosaics.10The firstprocedurebeginswith a"universe"consistingof all twelve pcs andan n-pcsubset(thefirstpartof the partitionto be generated),calculates he com-plementof the subset, and then callsanotherprocedurewhichcalculates recursivelythe combinationsof the complement.Thisprocedurethen calls the first,passing he old complementas the "universe"and the currentcombinationas the new sub-set. Whenthe entireaggregate s generated,the resultingparti-tion is transformedunderTnandTnIto finda normalformforitsmosaic;as above, the resultant ist is sortedandduplicationsare removed.

The Output.Our first concern was to find out how manyuniqueeven mosaicsexist. The followingchartprovides he an-swer:

pcs # of mosaics112 126 55434 71343 29762 35121 1

1?Weare indebted to Professor Alexander Brinkman, of the EastmanSchool of Music, for composingthiselegantand efficientalgorithm.



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The thirty-five62 mosaicscorrespond o the thirty-fivepairsof complementaryhexachordalset-classesavailable n the Tn/TnIequivalencesystem (exemplifiedby Forte's"prime orms"or Rahn's "settypes").l1If the systemof set-classes s basedonT,n quivalencealone, there areforty-four omplementary ex-achordalset-classpairings out of eightydifferenthexachordalset-classes).Since these are equivalentto Hauer's"tropes,"12our enterpriserepresentsultimatelya generalizationof one ofthe veryfirstattemptsatconstructing twelve-tonesystem.Anotherprimarynterestwasto see what set-classeswerein-vokedbythe pcsetsin a mosaic. Sinceallpartitionsof a mosaicare related underTn and/or I (as are all pcsets in a set-class),eachpartitionof a mosaic retains its characteristic epresenta-tions of set-classes;but are such setsof representations nique?And moreover,what would the set-classesassociatedwith vari-ous mosaicsprovideasa meansof crossrelatingpartitionsromdifferentmosaics?In order to help answer such queries,the computeroutputlists both the set-classesof eachpcset in a mosaic aswell as theset-classesproducedbyanycombinationsunions)of themosa-ic's pcsets. For instance, for the trichordal 34) mosaic {{012}{568} 379} 4AB}}, the outputlists:the fourtrichords3-1[012],3-2[013],3-5[016], 3-8[026];13nd the six hexachordsproducedby taking the trichords in union in pairs, 6-2[012346], 6-

"See Allen Forte, The Structure f Atonal Music(New Haven: Yale Uni-versityPress, 1973);Morris, CompositionwithPitch-Classes; r Rahn, BasicAtonal Theory.12SeeJosef Hauer, Vom Melos zur Pauke (Vienna: Universal Edition,1925).'3Weuse a combinationof John Rahn'sand Allen Forte's names for set-classes. In "4-2[0124]","4-2"is Forte's contributionand "[0124]" s Rahn's.(The termset-class s synonymouswithset-typeor collectionclass; t denotes acollection of pcsets related by Tnand/orI.) When the context is clear, termssuch as trichord hatusuallystand for pcsetsare sometimesused in an abstractsense, as in "trichordal et-class"or "set-classcontaining richords."

2[012346], 6-12[012467], 6-17[012478],6-41[012368], and 6-43[012568]. The unions of trichordpairs are shown below;brackets ndicatecomplementaryhexachords.

{012}{012}{012}{568}{568}{379}

U {568}U {379}U {4AB}U {379}U {4AB}U {4AB}

E 6-43[012568]E 6-41[012368]E 6-2[012346]E 6-2[012346]E 6-12[012467]E 6-17[012478]

]

The use of Forte'snames for set-classespermitsus to omit (intheoutput)the identification f the unionsof pcsets n a mosaicwhosecardinality xceedssix, since in Forte'ssystema set-classof cardinality reater hansix hasthe same secondnumeraln itsname as its set-classcomplement.This meanswe mayinferthepresenceof set-classescontainingpcsets arger hansixbylook-ingat the listed set-classesandchanging heir namesfromn-mto (12-n)-m. Thus. in theexamplejustgiven,we knowthatthetrichords akenin threesproduceset-classes9-1[012345678], -2[012345679],9-5[012346789] nd9-8[01234678A].Properties f theEven Mosaics1. Degreeofsymmetry (Ds).

DEFINITION.1. The number of Tn or TnI operationsthatleave apartitionnvariant s thatpartition'sdegreeofsymmetry.The readershouldunderstandhat underanoperation hatpro-duces invariance he pcsetscontributingo the partition emainunaltered; he pcs in a pcset eithermapinto each other withinthe pcset or into the pcs of anotherpcset in the partition.Bydefinitionall partitionshave a degree of symmetryof at least



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The EvenPartitionsnTwelve-ToneMusic 79

one, due to invarianceunderidentity (To). Since mosaics arepartitionsrelatedby the same operationsthatmay produce n-variance, t isnatural o extendthedegreeof symmetry o mosa-ics.

THEOREM.2. The numberof distinctpartitionsN that com-prisea mosaic S is given by N = 24/DS.Example: partitionX: {{013} {245} {678} {9AB}}T5IX: {{542} {310} {BA9} {876}}Thus, DX = 2 (invarianceunderTo and T5I) and the mosaiccontainingX has twelve (24/2)distinctpartitions.2. Set-classdistributionsmongmosaics.

(a) Some mosaics are generatedfromone set-class.For in-stance, there are six mosaicsof pca 26 with a single generator.But not all diadic set-classesparticipate; he set-class2-4[04]cannotpartition he aggregate.Two of the six mosaics aregen-eratedby the sameic, namely c3:

{{03} {25} {47} {69} {8B} {A1}}{{03} {14} {25} {69} {7A} {8B}}The example shows that a set-class istingof a mosaic'spcsetsdoes not alwaysprovideadefinitive dentification f themosaic;two differentmosaicscan have partitionswhose pcsets are ofthe same set-classes and in the same number.We willreturn osuch cases below when we discussZ-relatedpartitions.Movingfromdiadsto larger-sizedpc-sets,ouroutputrevealsthat whichBabbitt and othershave observed: 1)alltrichord-setclassesexcept3-10[036],the diminished riad,cangeneratetheaggregatein 34 partitions; (2) seven tetrachordalset-classesgeneratethe aggregate;and(3) onlynon-Z-relatedhexachordsform 62partitionsgeneratedby a singlehexachordal et-class-

making twenty such mosaics. A more generalmeasure of set-classredundancys presented n definition9 below.(b) In contrast to generatinga partition rom one set-class,there aremanymosaicsgeneratedby combinationsof differentset-classeswithminimalduplication.That such mosaicsexist isnot surprisingconsideringthe number of hexachordal,tetra-chordal,or trichordal et-classes. In the case of the diads,how-ever, a partition ncludingall six ics would be quiteextraordi-nary, but there is no suchmosaic. There are quite a few nearmisses, however; for example, ten distinctmosaics are madeout of one membereach of 2-1, 2-2, 2-3, 2-4, plustwomembersof 2-5:

{{01}{{01}{{01}{{01}{{01}{{01}{{01}{{01}{{01}{{01}

{24} {38}{24} {38}{25} {37}{25} {38}{25} {48}{25} {68}{26} {58}{35} {48}{37} {68}{47} {68}

{59} {7A} {6B}}{69} {5A} {7B}}{49} {8A} {6B}}{79} {6A} {4B}}{79} {3A} {6B}}{49} {3A} {7B}}{79} {3A} {4B}}{29} {7A} {6B}}{49} {5A} {2B}}{29} {5A} {3B}}

(c) Among the 26partitionsare the cyclesof the pitch-classoperationscomposedfromTn,I and/orM.14 ome of theopera-tions that are involutions15 roduce six cycles whose contentgivesthe six diads of a partition.Among these are the cyclesofT6, {{06} {17}{28} {39}{4A}{5B}};the cyclesof TnIoperationswhosen isodd-such asTBI,{{OB} 1A} {29} 38} 47} 56}}; hecyclesof TnMwhere n is2, 6 orA-such asT2M,{{02}{17} 35}

14The c operationM is equivalentto Ms. MI is the same asM7.15An nvolution s anoperationthat whenrepeated wice results n the iden-tityoperator. In pc operations,TnI for any n) is aninvolution;TjITnI= To.



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{4A} {68}{9B}};and the cyclesof TnMIwheren is 3 or 9-suchasT3MI,{{03}{1A}{25}{47}{69}{8B}}.Thesemosaicsgive thefollowing c countsrespectively:6 6 6 6 6 6; 1 13 3 5 5;2 2 2 2 66;and 3 3 3 3 3 3. Since these diad collectionsare found npc-to-pccorrespondencesbetween rows that producecombinatorialityunderthe operationswhose cycles producedthe fourmosaics,we should expect to find the diads frequentlyin aggregate-preservingmusic. This is so, but it is ironicthat the 1 1 3 3 5 5diad collection derived from cyclesof TnIoperationsspecifiesno less than six differentmosaics:

{{01}{{01}{{01}{{01}{{01}{{01}

{23} {58} {49}{25} {47} {89}{25} {67} {49}{25} {78} {69}{34} {27} {69}{67} {58} {49}

{7A}{3A}{3A}{3A}{5A}{3A}

{6B}}{6B}}{8B}}{4B}}{8B}}{2B}} = cycles of T1l

Similarly,as mentioned above, the 3 3 3 3 3 3 ic collectionspecifiestwo differentmosaics,both different"cross-sections"of the T3operation:

{{03} {14} {25} {69} {7A} {8B}}{{03} {25} {47} {69} {1A} {8B}}3. Z-relatedmosaics. Theprecedingparagraphs ave alluded oan extensionof Forte'sZ-relation rompcset pairs o pcsetsthatcomprisecollectionsof partitions.16

DEFINITION 5.1. Two Z-relatedpartitionsare those whosepcsetsare of exactlythe same set-classesandin the samenum-ber but are not membersof the same mosaic, that is, they areunrelatedbyTnand/orI.

16SeeForte, Atonal Music.

The two partitionsgivenin section 2a above show a "Z-pair"ofpartitions.We can generalizethe Z-relation to mosaicsjust as the Z-relationbetweenpcsetsisgeneralized o set-classes n atonalsettheory.

DEFINITION.2. Z-related mosaics have the same generatingset-classesbutare distinct.The differencesbetween Z-relatedmosaicsareusuallydetectedwhen one producesthe variousunions of partsof apartition oform pcsets of higher cardinalities.The list of set-classespro-duced nthis mannerwillbe different or Z-relatedpartitions.17The Z-relation s not confined opairsof mosaicsonly (as inthe 62 case amongpcsets); the exampleof the ten-tupleof Z-relations among the 26 mosaics makes a striking counter-example.Fromsuch cases we see thatthe Z-relationpervades he uni-verse of partitionsandis not confined o only a few pairsof set-classes of a certain size. Indeed, in the 34 and 26 partitionclasses, the number of mosaics that are not involved in Z-relations s outnumberedbythose that are.This scontrastedbythe 62 and 43 partitionclasses-all of which aredevoidof Z-relations.Table1provides he distribution f Z-tuplesovertheevenpartitions.Before leavingthis subjectwe shouldpointout that the Z-relationfor pcsets is not exactly analogousto the Z-relationsamongpartitionsand mosaics. In the former,twopcsetsarere-latedbythe sameinterval-class ontent; n the latter,partitions

17SomeZ-relationsare the result of partial nvariancesunderM andMIpcoperators.Twodistinctmosaicsgeneratedby pcsetsof the same set-classeswillmap nto each other underM(orMI). Since the sameoperatorsalso forcesomeYyrelations(see below), one mightbe temptedto define mosaicmembershipunderTn,I, and/or M. While this would resultin fewer mosaics, it wouldim-plicitlyrelatepartitions hat do not "soundalike."



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Table 1. Distributionof Z-tuplesoverthe even partitions Table2. Z-relatedhexachordsamong34partitions26mosaics:

81 Z-duples35 Z-triples26 Z-quadruples6 Z-pentuples14Z-sextuples1 Z-octuples+ 2 Z-dectuples165 Z-relatedtuples

34 mosaics:147Z-duples26 Z-triples8 Z-quadruples+ 1Z-pentuple182Z-relatedtuples

162mosaics105mosaics104 mosaics30 mosaics84mosaics8 mosaics+ 20 mosaics513mosaics n Z-relations

+ 41non Z-relatedsingletons554 26 mosaics

294 mosaics78 mosaics24 mosaics+ 5 mosaics

401mosaics n Z-relations

mosaic{{013}{457}{26A} {89B}}{{016}{289} {45A} {37B}}{{025} {469}{18A} {37B}}{{027}{159}{38A} {46B}}{{036}{159} {47A}{28A}}

trichords hexachords2 2 2 12 10 10 1039 39 395 5 5 12 17 17743 43 437 7 7 12 24 24 24 46 46 469 9 9 12 26 26 26 48 48 48100 10 12 28 28 28 49 49 49

can have its pcsets associated to form three pairsof Z-relatedcomplementaryhexachords. Since the three instancesof thesametrichordmust be relatedby T4orT8,all other3-12mem-bersmust be found straddledacrossall threeof thesetrichords.Therefore, in all but one suchcase of trichordpairings n a 34partition, he3-12 andone of the others arepittedagainsta hex-achordwithouta 3-12, andthe complementaryhexachordsarenot of the sameset-class.Thefive mosaics hat illustrate hissit-uationare shown in Table 2.18A similarargument anbe madefor 26 partitions and their combinations into diminished-seventh chords(4-28[0368])and Z-relatedhexachords.19The Z-relation between some pairsof mosaics can be ex-plained as resulting from partitions related across mosaicboundariesbyTnMorTnMI.

+ 312 non Z-relatedsingletons713 34mosaics

have pcsets includedin the same set-classes n the same num-ber. (For instance, hexachordalset-classes can be Z-relatedwhile 62mosaicscannot.) Keepingthis distinctionnmind,vari-ous sets of mosaicscanprovide nsightto the (pcset)Z-relationamonghexachords.Any partitionof the aggregate hat is madeup of one 3-12[048]and three membersof any other trichord

4. Unions ofpartition pcsets. As in the case of the five partitionsjustshown andinthe topicsto follow,one can takeanynumber

18Thenumbers under the headings "trichords"and "hexachords" n Ta-ble 2 and other examples are the ordinalnumbers n Forte's set-classnames.Thus, the firstmosaic's"22 2 12 10 10 10 39 39 39" standsfor trichordal et-classes 3-2, 3-2, 3-2, 3-12 and hexachordal set-classes 6-10, 6-10, 6-10, 6-39,6-39,6-39.

19Unfortunately, omplementaryrelationsamongmosaicsdo not explainallaspectsof the Z-phenomena.See DavidLewin, "OnExtendedZ-Triples,"Theoryand Practice7/1 (1982):38-39.



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of pcsetsof a partition n union. In general,the numberof dif-ferent set-unions n a partitionclass differs oreachpca.Table 3shows the distributionof cardinalities or each even partitionclass:n is the cardinality f eachpcsetinthepartition lass;2n isthe cardinalityof pcsets taken in pairs,3n is the cardinalityoftriosof pcsets, etc.; the intersectionof a rowand column n thetablegives the number of pcsets in a partitionclasscontainingpcsets of size n (column) alone, in pairs, etc. (row). For in-stance,the 15 inthe "2n"rowand"26"columnof the table ndi-cates that there are 15waysof takingpairs(2n) of diads n a 26partitionclass.Likewise, the table shows that there are 4 waysof combining hree (3n) trichordsn a partitionclassof pca34.5. Similarityof mosaics.The existence of Z-related mosaics-or those that sharemanyset-classes-encourages us to set upasimilaritymeasure between pairsof mosaicsby countingtheirsharedset-classes.20 he set-classesof a mosaic are determinedby takinga partitionof the mosaic in questionandformingallunions of its pcsets, as described above. One then determinesthe set-class associatedwith each union. As anexample,we listin Table 4 the set-classesof two mosaics,A andB. Since eachlist of set-classes containscomplementary et-classes,to avoidredundancywe divide the numberof shared set-classesby twoand omit the 12-1(andits complement,the nullset-class).

DEFINITION6. The similarityrelation between two mosaicsA and B is given by the functionSIMOZ(A,B) whichequals(N-1)/2 where N is the numberof set-classessharedby bothAandB.

20Thiss in the spiritof JohnRahn'sTMEMBmeasure,designedto assignadegree of similaritybetween two set-classes by counting all set-classes ab-stractly ncludedby both. Given two mosaics,we count the set-classes n each.See JohnRahn, "RelatingSets," Perspectives f New Music 18/2(1980):485-502.

Table 3. Cardinalities f unions of partsof even partitions

26 34 43 62n 6 4 3 22n 15 6 3 13n 20 4 14n 15 15n 66n 1

Table4. Set classesof mosaicsA and BA: {{012} {567} {48A} {39B}}{012} E 3-1;{567} E 3-1;{48A} E 3-8;{39B} E 3-8;{012567} E 6-6;{01248A} E 6-21;{01239B} E 6-2;{01256748A39B} E

{56748A39B}{01248A39B}{01256739B}{01256748A}{48A39B}{56739B}{56748A}12-1

B: {{012} {567} {38A} {49B}}{012} E 3-1; {56738A49B}{567} E 3-1; {01238A49B}{38A} E 3-9; {01256749B}{49B} E 3-9 {01256738A}{012567} E 6-6; {38A49B}{01238A} E 6-9; {56749B}{01249B} E 6-8; {56738A}{01256738A49B} E 12-1

E 9-1E 9-1E 9-8E 9-8E 6-38E 6-21E 6-2

E 9-1E 9-1E 9-9E 9-9E 6-38E 6-9E 6-8



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TheEven PartitionsnTwelve-ToneMusic 83

The two mosaicsA andB of Table4 share3-1, 3-1, 9-1, 9-1,6-6,6-38, and12-1; herefore,Nis 7. Since(7-1)/2 = 3, mosaicsA and B have the similarity index of 3, written asSIMOZ(A,B) = 3.The Z-relationdoes not automatically mplymaximalsimi-laritysince there are many Z-tuples (not only Z-pairs)whichhavedifferentdegreesof SIMOZ.Forinstance,examinethe 4-tuple of Z-relations below, where each mosaic is composedfrom a 3-1[012],a 3-2[013],a 3-3[014],and a 3-4[015]:

S: {{012} {467} {589} {3AB}}T: {{012} {485} {679} {3AB}}U: {{012} {346} {59A} {78B }}V: {{012} {356} {489} {7AB}}

hexachords1 1 12 19 41 441 1 15 15 18 182 2 5 5 14 143 3 16 16 36 36Figure2 givesthe SIMOZrelation orpairsof these fourmosa-ics. We see that S and T are slightlymore similar ince S andTsharetwo instances of the hexachord6-1[012345]whereas theother fivepairings hare no hexachords.6. Inclusion relations. When the differencesbetween the Z-related mosaics of pca34 are considered,the variousunionsoftheir generating pcsets invoke implicitlymosaicsof 62. Thus,our examinationof the propertiesof mosaics eads to the com-parison of mosaics of different pcas. Two definitions areneeded:

DEFINITION 7.1. Partition S is literallyncludedn partition Tif each pcset in T canbe broken into disjointsubsets suchthattheresulting ubsets are those of S. (We sayS is asubpartitionfT and that Tis asuperpartitionf S.)As an example, note that the firstpartition,P, below includesthe second, Q:

P: {{0347} {128B} {569A}}Q: {{03} {18} {2B} {47} {5A} {69}}

It is important o note that literal nclusionrelatesuneventoevenpartitions.Forexample,if thepcaof S is34,T can havepca39;or if S haspca2242,Tcanhave 62.

DEFINITION 7.2. Mosaic S is abstractlyncludedn mosaic Tifa partitionmember of S is literally ncluded n a memberof T.Just as there are subpartitionsand superpartitions,here aresubmosaics andsupermosaics.For example, the mosaic of thepartitionP (above)includesa submosaic hat ncludes heparti-tion L, {{14} {29} {03} {58} {6B} {7A}}. This follows sinceL isthe T1of Q andQ C P.The inclusionsamongmosaicscanassertsurprising elationsthat have importantcompositional mplications.For instance,the mosaic of six diads taken from cycles of a TnI operationwhere n is odd (discussedabove) is included n 43mosaicsgen-eratedbyone or more all-combinatorialetrachordal et-classes

Figure2u

4 4

S V5 4 4

T



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aloneor withset-classes4-7, 4-17and/or4-20. The same mosaicis included in 62 mosaics generated by four of the six all-combinatorialset-classes. These inclusion properties distin-guishthisspecialmosaic fromits fiveZ-relatedmosaics.Below, we document heseinclusions n apartitionmade outof the cyclesof the T5Ioperationarrangedn the fifteenpair-wiseunions that form the tetrachords ited above.

{{05} {14} {23} {6B} {7A} {89}}E 4-7E4-10E 4-9E 4-23E 4-17

{1423}{146B}{147A}{1489}{236B}

E4-1E 4-23E 4-28E 4-20E 4-17

{237A} E 4-20{2389} E 4-9{6B7A} E 4-7{6B89} E4-10{7A89} E 4-1

Measures3 through5 of Schoenberg'sKlavierstuck, pus33a(Ex. 1), the firstpassage n which thecomposition's ombi-natorialrowpair spresented ncounterpoint, ealize his"odd-TnI" partition.The passage parcels out the pair of rows inblocksof eight pcs-two successivediadsfrom each row. Thefollowing shows pitch-class reduction of Example 1 (pc 0- B[ )21with the groups of all-combinatorial etrachords a-beled;the partition s basedon the cyclesof T5givenabove:

Selectedpassages romanotherpiecebySchoenberg, he Pi-ano Suite,opus 25, can be usedto show how supermosaicsandsubmosaics can control musical coherence and progression.The row of the Suite s (45716382B09A).Even a cursoryglanceat the Suitereveals thatSchoenbergdeploysthe row tetrachor-dallyso that each of its three non-overlappingordered tetra-chords can be reversed, overlappedor concatenatedwith theother two to formvarious linearaggregates.Therefore,in animportant ense, the row is a partiallyorderedset, beingactu-allya presentationof the partition{{1457}{2368} 09AB}},ex-ceptthatthe tetrachordalpcsetsareordered. The notation

{(4571) (6382) (B09A)}capturesmost clearlythe nature of Opus25'spc material.Thispartition s a memberof the mosaic made from set-classes4-1[0123],4-12[0236],and4-15[0146]; he "normal orm" of thismosaic is {{0123}{478A}{569B}}.Despite the tetrachordal ri-ginof the row, one shouldnot get the idea that the orderingofthe row is arbitrary.Firstof all, the row as a linearstringoftwelvepcsis used often in thepiece; secondly,thelinkingof thethreetetrachordsby diads as concatenated n the rowprovidesmore instancesof the set-classes n the tetrachordalmosaic asshown below.

4-12RT5IP: B 1 7 8 0 2 9 6 4 3 5 ARP: 64A9 538B 12074-23 4-10 4-14-1 4-10

4-15 4-1P: (457 1 6 3 8 2 B 0 9 A)

4-15 4-124-23

21Pc0 = C~ in all our pitch-class representationsof extant music unlessotherwisespecified.

The overlappedtetrachordssuggestthat a 26 view of the rowmightbe informative.The mosaic for the six diadsof the row,calledW,with its supermosaics f pca62,is providedbelow.

{0514}{0523}{056B}{057A}{0589}



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Example1. Schoenberg,Klavierstiick, pus33a,mm. 3-5.

? Copyright1929by UniversalEdition. Copyright enewed1956by GertrudeSchoenberg.Used by permissionof BelmontMusicPublishers.

W: {{01} {23} {78} {69} {4A} {5B}}Diads: 2-1, 2-1, 2-1, 2-3, 2-6, 2-6Hexachordalsupermosaics: 6-2(4x);6-5(4x);6-6,6-38;6-13,6-42;6-17(2x), 6-43(2x);6-28(2x), 6-49(2x)The reasonwe do not list the tetrachordalmosaicsthat in-clude Wis that we regardWas a submosaicof the row'stetra-chords;overlappingeither the first or last diad of one of thethree ordered tetrachordswith another yields a hexachordfound in one or more of the mosaicsthatinclude w. The multi-

ple instancesof 6-2, 6-5, 6-17, etc. in the supermosaicsof Wprove to be of great importanceto the piece. Those readerswishingto see how these diads and hexachordsgeneratelargescale pitch-classrelations in the Suite(andnot only withstate-ments of singlerows,butamongcombinationsof diadsandtet-rachordsof all eightrowsused in the work), mayreadMarthaHyde'srecentarticletreating hismatter.22

22SeeMarthaHyde, "MusicalForm and the Developmentof Schoenberg'sTwelve-ToneMethod,"Journalof Music Theory29/1(1985):85-143.

The abstractnclusionrelationscreatedby apartialorderingbetweenmosaicscanbe shownby a lattice.Fromthe combina-tionof pcsetsin partitionswe candeterminewhichmosaicsareincludedin which, as shownin Figure3. The lattice is incom-plete, for manyother34partitionsareincluded n B or C, andsinceallmosaicsareincluded n U andinclude N. On the otherhand,onlyA and D areincludedmutuallynB, or A isincludedmutually n B and C.The inspectionand comparisonof mosaiclatticesallowsusto generalizethe mosaicZ-relation.

DEFINITION. Two mosaicsS and Taresaid to be Yy-relatedif allthe mosaicsof the samepcay thatincludeS alsoincludeT.The reason we regard he Yy-relationas a generalization f theZ-relation s that if two mosaics of pca y areZ-related,theyareautomaticallyYyrelated.23

23Maximallyonnected musicalprogressioncan be developed by sequenc-ing partitionsalternatelyrelatedbythe Z- andYy-relation.Thistendsto gener-



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Figure3. Abstract nclusionrelationshipsamongmosaicsA, B,C, D, N, and UA: {{012} {345} {678} {9AB}}B: {{012345} {6789AB}}C: {{012678} {3459AB}}D: {{024} {68A} {135} {79B}}N: the null mosaic.U: the aggregatemosaic.{0123456789AB}.

class) operations;the two mosaics are related such that forevery partitionX in one, ToMX resides n the other.An exam-ple of a M/MI relatedpairof Z/Yy-mosaics s givenbelow;theactualpartitionsrepresenting he mosaics n the illustration rerelatedby T7MI.

mosaic{{014}{278} {59A} {36B}}{{014} {389}{56A} {27B}}

trichords345 11345 11hexachords11 17 19 40 43 4411 17 19 40 43 44

B C

N

Therearequitea few examplesof Y(62)-related4 mosaics.For instance,{{013}{247} {56A}{89B}}and{{013}{256}{78A}{49B}}are so related;both mosaicshave the same three hexa-chordalsupermosaicswhich are representedby the set-classcomplementpairs: 6-3[012356]and6-36[012347]; -10[013457]and6-39[023458]; -25[013568] nd6-47[012479].Ingeneral,of the 71334mosaics,therearethirty-nine asesof Y(62-relations.All of these casesassociatemosaics npairs-except one, which is a Y(62)-triple.Ten of the Y(62)-pairsrealsoZ-related,havinga maximalSIMOZindexof seven. Suchcasesshowthat even the completeset-class istof a mosaicmaystillbe insufficient o distinguisht froma Z-partner.The "mys-tery"underlyinghese cases is explainedbytheTnM/MI pitch-alize Babbitt's use of hexachordaland trichordal ource-sets as describedin"SinceSchoenberg."

Thetopicof abstractnclusionamongmosaicsnaturally pillsoverinto set-class associationbetweenandamongvariousmo-saics.As iswellknownandwill be reiteratedbelow,Babbitthascomposed music based on 34 mosaics that are abstractly n-cluded in six 62 supermosaics-the complementarypairs of"all-combinatorialexachords."An examinationof all 34 mo-saicsrevealsthat there areseven trichordalmosaicswhose su-permosaicsarerestricted o thesix hexachordalmosaics hatareall-combinatorial.f we widenthis setof hexachordalmosaics oinclude he complementarypairsof set-classes6-14[013458] nd6-30[013679],we add fourteenmore trichordalmosaicsto thelist.The twenty-onemosaics n questionare listedin Table5.The reasonwe singleout these twentymosaics s thatunderall combinationsof thepcoperationsofT, I, M and/oraspecialpc operatorQ, we canmapthe mosaics nto eachotherin twoclosed systems.24The mosaicson the list are labelledA or B,indicating he system to which they belong. The operationQmapspc 0 to pc 3, 3 to 6, 6 to 9, and9 backto 0; all otherpcsremain nvariant.Among the many other topicsrelatedto the associationofset-classesin mosaics, we brieflydefine a measureof the set-classredundancyof the supermosaics f a mosaic.

24Thats, a partition n one of these mosaicsmaybe mapped nto itself orinto anotherpartition n the same or different mosaic.



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Table 5. MosaicgroupsA andB, closed underTTOsandQmosaic trichords

{{012} {345}{678} {9AB}}{{012} {678}{459} {3AB}}{{013}{245} {679} {8AB}}{{013} {458} {679}{2AB}}{{014} {589} {67A} {23B}}{{013}{679} {58A} {24B}}{{024}{135}{68A} {79B}}{{012}{678}{35A} {49B}}{{015}{489} {23A} {67B}}{{015}{348} {29A} {67B}}{{015} {249}{38A} {67B}}{{027} {168} {35A} {49B}}{{024}{357} {469}{84B}}{{013} {679} {25A} {48B}}{{014} {358}{67A} {29B}}{{025} {479} {13A} {68B}}{{014} {259}{67A} {38B}}{{024} {159}{68A} {37B}}{{025} {149}{37A} {68B}}{{048}{159}{26A} {37B}} 1{{037} {169}{25A} {48B}} 1

DEFINITION. The set-class redundancy of the mosaic S,SCR(S), equalsthe numberof differentset-classesof all unionsof the pcsetsof S arrangedaccording o the cardinalities f theset-classes.We write

SCR(S) = slls2/s3/s4/s5/s6,

1 1 1 1 11 7 7 7 A1 1 4 4 1 1 7 71414 B2 2 2 2 1 1 8 8 30 30 B2 2 3 3 1 1 14 14 30 30 B3 3 3 3 1 120203030 B2 2 7 7 1 130303232 B6 6 6 6 1 132323535 B1 1 9 9 7 7 8 8 8 8 B4 4 4 4 7 7 88 20 20 B4 4 4 4 7 714141414 A4 4 9 9 7 7 14 14 32 32 B9 9 9 9 7 732323232 A6 6 6 6 8 8 8 835 35 A2 2 11 11 8 814143030 B3 3 7 7 8 814143030 B7 7 7 7 8 830303232 B3 3 11 11 14 14 14 14 30 30 B6 6 12 12 14 14 14 14 35 35 B7 7 11 11 14 1430303232 B2 12 12 12 20 20 20 20 35 35 A1 11 11 11 20 20 30 30 32 32 B

whereSn s the number of differentset-classes(withmembers)of cardinalityn produced by unions of the pcsets of partition(mosaic)S. Cardinalityn does not exceed 6 since the numberofdifferentset-classes of cardinality12-n is the same as that ofcardinalityn due to complementation.Obviously,where n islowerthan the smallestpcsetofS, Snwillnecessarily qualzero.The valueof sndependssomewhaton the S's degreeof symme-try. In general, the lower the values of sn, the fewer the set-

hexachords



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classes determinedby unions of the pcsetsof S, andthe greaterthe set-classredundancy.For ourpresent purposes,we need not examineallthe vari-eties of SCR. We willonly touch on the most redundantvaluesfor diadicpartitions.For instance,the unique partitionmadeout of disjoint c ls has the SCRof 0/1/0/3/0/3.Thisrepresentsavery limitedarrayof set-classes since s4 ands6 have maximalpossiblevalues of 15 and20, respectively.Other diadicmosaicsthat have the same SCR are thosegen-eratedfrom ics 3, 5, and 6. The latter,made out of tritones,isfound in the SchoenbergPiano Suite mentionedearlier,since

his use of the P and I rows withtheirT6transpositions ieldsaseries of tritones n equalorderpositionsof the transpositionalpairs.From the SCR just given, (0/1/0/3/0/3),we see thatonlythreetetrachords4-9[0167],4-25[0268],4-28[0369]), hree hex-achords (6-7[012678], 6-30[013679],6-35[02468A]), or threeeight-pcset-classes(8-9, 8-25, 8-28)can occuradjacentlywhenSchoenbergusesthe two tritone-relatedP or I rows nparallel.Another 26partitiongeneratedbyic-3,Z-related o the onementioned above, produces the SCR of 0/1/0/5/0/4which itshareswith mosaicsgeneratedbythefollowing: ix c 2s;three cls andthree ic 5s; fouric 3s and two ic 6s;etc.Our finalexample of a SCR, given with its mosaic below,providesa contrast o the redundant xamples ustcited.

{{01} {26} {58} {79} {3A} {4B}} 0/5/0/14/0/16ThePresentation f Partitions

A partition'saural dentity,firstas a unordered ollectionofunorderedsets of pcs, and second as a representation f a mo-saicor partitionclass, is highly dependenton its interpretationin pitchandtime. Suchinterpretations an be implementedbycompositionaldesigns25uch as combinatorial rrayswherethe25SeeMorris, CompositionwithPitch-Classes.

pcsof aset ofpartitionsare ordered n the horizontaland/orver-tical dimension. Sucharrays an be usedimmediatelyncompo-sition or can be the source of other arraysthat are so inter-preted. Placingeachpcsetof a partition n successivepositionsof a one-dimensionalarraymightbe one way of constructingsucha "sourcearray,"but one mightaskif therearemoreopti-malwaysof notatingapartitionn a sourcearray o that the lat-terdisplaysasmanyof the propertiesof the formeraspossible.Given the 34partitionJ= {{049} 27B} 158} 36A}},one canwrite a source array as follows:26

049 27B36A 158J's four trichordsare now presentedclock-wise"around" hepositionsof the source arrayso that they are cyclically oined.The chiefadvantageof thispresentations the ease in whichonecanreadoff the three 62partitionswhich ncludethispartition.These are given by the first versus second column{{03469A}{12578B}}, hetopversusbottom row{{02479B} 13568A}},andthe two diagonals{{014589} 2367AB}}.In contrast o this sourcearray, wo arrays or use in a com-positionalcontextare shown below:

049127BI36AI158 04927B36A158The two arrays espectively mplya linearand vertical nterpre-tationofJ so thatonlysomeof its62partitionsareavailablead-jacently.The{{04927B} 36A158}}partitionspresentedclearlyas the union of the first wotrichords ersus he unionof the last

26Meadand Rouse use such presentationsof 34mosaics as describedinMead, "SomeImplications,"or Rouse, "Hexachords."



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two; in the second "harmonic"array,the unionof the highesttwo trichordsversus the union of the lowest two presentsthesame hexachordalpartition.The other two hexachordalparti-tions are less easily presented, althoughthe union of the firstand last (or top andbottom)versusmiddletrichordpairsmightstill assertweaklya hexachordalpartition.Suchinside/outsidepresentationsare often invoked in tonal music where the so-pranoand bass of a four voice setting guide the voice leadingandharmonyof a phrase.As forthe foursuperpartitionsfJ ofpca 39, these can be realizedby some sortof phrasemarkerorbyother means of articulation uch asdynamicsor timbre.27

The situation is more complicatedwith source arraysof 26partitionsand their superpartitions.Each 26 partitionis in-cluded in fifteen43andin ten 62partitions,not to mentionthe48 and 2A partitionclasses.By denoting he sixdiadsof thepar-titionby the lettersa through we can use the followingarrayformatto present the tetrachordalcombinationsof the six di-ads:a b c (a b c)d e f (d e f)

The letters in parentheses indicate that the source arrayisjoined end to end, as if writtenon a cylinder.In this arraywefind that every tetrachord s the adjacentunion (either verti-cally,horizontallyordiagonally)of two diads.Ofcourse,notallof the43partitions hatinclude a six-diadpartitioncanbe adja-cently located. For instance, the partition{{a U e} {d U b}

27With he presentation of a 43partition, only three superpartitionsarefound-all of the uneven 48 peavariety.In the melodic or harmonic nterpre-tation of the three tetrachords,all thingsbeingequal, two of the three super-partitionswill be more prominent;the remainingsuperpartitionwill be pro-jected bythe moresubtleinside/outsidepresentation.Thusthe sourcearrayof43partitions s simplya triangular rrangement f the threetetrachordalmem-bers of these partitions.

{cUf}} cannotbe encircled n twodimensionswithoutwrappingthe arrayaroundvertically to form a "doughnut" urface).Asfor thetenhexachordal uperpartitions, narrangementuch as

a b c d e fd e f a b cwithits repetitionsof diads can capturealmost allof the 62su-perpartitionsas two non-intersecting"triangles" f threeadja-cent diads.In anycase, the possibilityof hiding(orhighlighting) ertainsuperpartitionsf apartitionasafunctionof the latter'spresen-tation in pitchand timeis of interestsince it can makepartitionpairswitha highSIMOZindexmoreor less similaron the "sur-face" of a composition.PartitionsandMosaics n Babbitt'sDu

A look at a swatch of musicfrom Babbitt'ssong cycle Dubrings ogethermanypointsconcerning hepresentationof par-titions andmosaicsand theirfunction n twelve-tonemusic.Wewill begin by examiningthe first four aggregatesof the sixthsongin the series, "Traum,"whosemusic s found in Example2. These four aggregatesmake up the first of four blocksthatcomprise "Traum."Each block has differentmaterials(tri-chords and hexachords)coordinatedby the same structure:four twelve-pc "lynes"28 f pcs relatedby twelve-toneopera-tions whichproducesimultaneoushexachordaland trichordalcombinatoriality.Thecompositionaldesignthatundergirdshefirstblockis foundinExample3.From a serialpointof view, the sequenceof fourhorizontallynescan be heard asrelatedrespectively romtopto bottom as

28Theermlynewascoined by MichaelKassler o denote anuninterpreted(monolinear)stringof pitch-classes.



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90 MusicTheory Spectrum

Example 2. Opening of "Traum" from Du by Milton BabbittVI. Traum

/ I_ nxtjkU)Pr) -3 -I >-Iu 'ru q-j- lDurch die Bui - sche win - den Ster-ne Au - gen tau - chen bla - ken

^itjr =iUr 7,v"^ ^iD r t7p 8> =< mPr3 /I PP PP PP3- --j--- -n 3-LJ4-

3b. *

? Copyright1957by Boelke-BomartInc., Hillsdale,New York. Used bypermis,

Example 3. The combinatorial array for Example 2Soprano:High piano:Midpiano:Lowpiano:

904 2B7 158 A63158 A63 904 2B72B7 904 A63 158A63 1 5 8 2B7 904

indicated by the following labels:29 P; r6P; TBIP; r6TBIP.30Each lyne is its own T6 retrograde. Continuing in this vein, therelations between the (ordered) non-intersecting hexachords in

'. *sion.

the lynes of the design are portrayed below: E is the first hexa-chord of the top lyne, (9042B7); F is the RT6 of E; G is theTBIE; and H is the RT5IE.

E FF EG HH G

29Theoperation of rotation on an ordered set is denoted by r,; the set isrotatedbys orderpositions.This means that its last s pcs aremoved(in order)to the beginningof the set: r4(0A6524)= (65240A).30TheoperationsTo, T,I, r6,and r6T,Iform a "four-group"which makesmanyother appearances n Du. In thiscase, n = B.

The same operations that relate E, F, G, and H-T6, TBI,T5I and R-conspire to produce similar relations among the tri-chords in the design. Here we write a square of sixteen instancesof four labels with A standing for the (904) in the compositionaldesign's upper lefthand position, B = TBIA = (2B7), C =RT5L4 = (158), and D = RT6A = (A63).

_ ---

I dL::



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A B CDC D A BB A D CD C B A

Sucha configuration f letterswhichpresents he samesetofelementsin eachof itsrows andcolumns s calleda latinsquare.Ignoring he serialderivationof the squareandthe internalor-deringof A, B, C, andD, we understandhe squareto containtheelementsof the pcsetsof a 34partition.Infact,the partitionis identicalto J above generatedby the set-class3-11[037].Wewill continueto call thispartitionJ and the squareS. S containseight presentationsofJ, four vertical thehigh,middle,and owpiano lines and the sopranomelody) and four horizontal(thefouraggregates).The designof the square mplements he pre-sentationof all the 62superpartitions f J. {{A U B}{C U D}},whichwe will label H1, is foundsuccessively n the lynes. The{{A U C}{B U D}} partition,calledH2, is found in the top twoversusbottom two positionsin each columnof Sand is realizedin the pairingof the middle andlowpianoversusthe high pianoandsoprano. The lasthexachordal uperpartition f J, H3, {{AU D} {B U C}}is presentednot by adjacencies n S, butby theinside/outsideassociationof the top and bottomversusmiddlepositionsof each columnof S.Not only is H3 not adjacentin S, but the realizationof S inthe music furtherhides H3 from audition. Since the sopranosings n the samerangeas the lynefollowedbythe middlerangeof the piano, the mostobvioustimbral/registraldjacenciesare:(1) high piano; (2) low piano; and (3) sopranowith middlepi-ano. This set of associations urtherstrengthens he firstof thethreehexachordpartitions,H1, whichpits, as it were, A andBagainstC andD.To continueour examination of the realizationof S, a dia-gramof the temporalandregistraldeploymentof pcsin the de-

sign sgiveninExample4. The four ynesofpcsarepresented nthree registersand are writtenalignedandoverlappedas theyoccur in the music. Solid vertical lines indicate aggregateboundaries,dotted verticallines show the placementof mea-sures,and dashes show sustainedpcs.LikeJ, which sa memberof a mosaicgeneratedsolely by 3-11[037),the three hexachor-dalsuperpartitions(H1, H2, andH3)are the membersof threedistinctmosaics,each generatedfrom its own singlehexachor-dal set-class: H1 is generated by 6-32[024579]; H2 by 6-20[014589];andH3 by 6-30[013679].Thesediffering et-classesdistinguish anditssuperpartitionsrom each other.Moreover,as mosaicmembers,they help to disassociate hispassagefromothersin Du which are based on the same or different3 mosa-ics included n the mosaicsof H1, H2, orH3found elsewhere nthe composition.To continue, trichordC is presentedalone in the first mea-sure, announcing he set-classthatgeneratesJ;H1is presentedclearlyby the first two aggregates.The sopranoand middlepi-ano exchange trichordsA and B between the aggregateswhilethe outerpianovoices exchangetrichordsC andD. The afore-mentioned middle piano/voiceassociation makes the 6-32 ofthispartitionquiteaudible. The H3 partition s presentedtem-porallyin the last aggregateof the music(andin Ex. 4). Herethe combinationof trichordsA andD isgiven byalternatinghepcs of each in the low piano and voice; this is contrastedandfollowedby a similaralternationof the notes of trichordsB andCin the highand midpiano. Both pairsof trichords,andespe-ciallythe last,project clearlyH3'scharacteristic-30 set-class-the "Petrushka" hord-type.The H2partitions slightlyharderto hear;although ts6-20set-classesare found n the middleandlow piano in the firstaggregate (accompanying he voice), theconnection between the midpiano and voice andthe trichordalStimmtauschust discussedtendsto cover a clearexpositionofH2. Perhaps he contour "dive"to Et 3 in the secondaggregatein the middle piano (see Ex. 2), which connects A~ and Cq,couldbe enlisted to make a case fordirectlyhearinga6-20hexa-



20/29


Example 4. Pitch-classrealizationin pitch (register)andtime (succession)of Example3 in Example2

high pn.

sopranomidpn.low pn.

highpn.sopranomidpn.low pn.

I i15- - I8 1 A-631 -9041 I Ii I II I I1 9-0-4-4 2B-77 1-5 8i I I'I 2-B7- 9-04 A63I I Ii ! -A 6-3- 15-8 -2 7-i t I 2B 7-i iti ~~~~~~~~~~~~~~~i

2-B-7- Iii .A 6-3 l

1-5-8 19 0. t9-0 4~~~~~~~~~~~~~1.

chord,buttheEb2, hangingover from the firstaggregate, endsto obscure thisrelationship.Before completingouranalysis,we call attention o anotherpresentationalaspectof the firstblockof Du thatis broughtupby the Eb just mentioned.Why does Babbitt let pcs fromoneaggregatesustaininto the next? While such a practicemildlydisturbs he chromaticsaturationof a passage(as do repeatednotes and other inequalities n the durationof pcs), these extrapcshelpmarkthe aggregateboundaries. n eachcase, theEbin

aggregate2 and the El andAb in aggregate3 form vertical n-stancesof set-class3-5[016],whichcontainsa tritone-an inter-val foundneither n the 3-lls thatmakeupthepcsetsofJ norinanyothergeneratorof atrichordalpartitionused in Du. In theface of multiple nstancesof A, B, C, andD in thisblock,suchboundarieshelp indicatewhichtrichordsparticipate n whichaggregates.In additionto the local articulations f J, H1, andH3, thereare other partitionspresentedby the music of the firstblock.



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Thecomposite temporalsequenceofpcsin the fourthaggregatepresentsthe following yne:

9A0 643 21B 578whichis partitionedto show its articulationof a different mo-saic, generatedsolely by set-class3-2[013].The relationsof or-der andpitch-class ransformation reidentical o thoserelatingtrichordsA, B, C, and D. The hexachordalmosaics hat ncludethis partition are generated by 6-30, 6-1[012345], and 6-8[023457]. The hexachordal presentation of this aggregateforces the inclusionof bothJ and thispartitionnH3. As mightbe expectedin a workof Babbitt,the 3-2mosaichaslong rangesignificance.31Another partitionof apparentlymore local significance sprojectedbytherhythmicarticulation f the soprano yneinthelast two aggregatesof Example2. The diadicarticulation f thetext setting "Augen tauchen blaken" is continuedin the so-prano'scontribution o the first two aggregatesof block2; thetwelvepcsform a linearaggregate, hereby nvokinga diadic26partition.Sucha distribution f pcsconflictswith the trichordalstructure of blocks 1 and 2-causing a sort of "partitionhemiola"-and therefore links the two blocks.One reasonforthe link is to display musically he similaritybetween the parti-tions thatgeneratethe twoblocks.Thesecondblockhasexactlythesamelatin-squaretructureas thefirst,exceptthat tsgener-ating partition,calledK, is basedon the set-class3-7[025].K'shexachordalsuperpartitionsare members of three mosaics,generated rom6-8, 6-30,and6-32.SIMOZ(J,K) = 6, the max-imum possible index being 7. We now understandbetter thereason forthe clearpresentationsof 6-32 and 6-30 nblock1, asthese are the intersectingset-classesbetween the blocks. And

31There re whole blocks of Du based on mosaicsgenerated by 3-2, mostnotably n the opening, middle, andlastsongs.

justas 6-20wassuppressedn block1, 6-8 is suppressedbysimi-lar means in block 2.Examples5a and 5bprovidethesquaresand their associatedpartitionsandmosaics for the third and fourthblocks. Like thefirst two blocks, these blocks are based on a similarstructure,however they are more complex--with four distinctpartitions-and they are not latinsquares.Since the partitionsformpairsrelatedby Tn and/orI, there are actuallyonly twomosaicsarticulated.MosaicsL1 andL2, found in block3, arebased on the trichordal et-class3-4[015]andare alsoZ-related;

block 4's Z-relatedpartitions,N1 andN2, are generatedby 3-11[037], hegeneratorof block 1. This return o 3-11 does roundout thesongbutdoes notachieve closuresince none of the eighttrichords n block 4 are the same as those in block 1. The greatercomplexityof the latterblock also arguesagainstcompleteclo-sure;specifically,while N1 and J are membersof the samemo-saic,N2, sinceit is Z-relatedto N1, is not relatedbystandardpcoperations oJ. Furthermore, he realizationsof blocks1 and4differin the positionsof the 6-32 and6-20 hexachordal uper-partitions;6-32, whileoriginallyn a lyne, is betweenlynesandvice versa for 6-20. Since 6-20 wasrepressed n block1, itsobvi-ous presencein block4 makesthe end of "Traum" ound a bitdifferent rom itsbeginning.Example 6a displaysthe similaritybetween the partitions(mosaics)that generatethe four blocks of "Traum."Example6bprovidesanoverview of the complete song bylisting he fourblocksin order andshowingtheir SIMOZ indices. We can seethat the very strongconnectionswithin the firsttwo blocksarenot reiteratedby similarbonds between the last two, and thatthejuncturebetween the middletwoblocks s the weakestcon-nectionamongallblocks andmosaics.Two moreissuesremainbefore we leave Du. How does ourpartitionapproach o "Traum"help us understand he form ofthe entire cycle, and what kind of presentationaldetail doesBabbittemployto helpthe eardiscover hisform?A discussionof each song's partitionstructurewould take too muchspace



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Example5a. Partitionsandmosaics orblock3soprano:high piano:midpiano:lowpiano:

abc def g hba dcf e hg

Example5b. Partitionsandmosaics orblock4soprano:high piano:midpiano:lowpiano:

s t u vwxyzt s v utsvuxwzy

a = (780)b = (43B) = TBIac = (912)= RT9Iad = (A65) = TAa

L1 = {{780} {43B} {59A} {621}}3-4 3-4 3-4 3-4;superpartitions:-7L2 = {{780} {43B} {912} {A65}}3-4 3-4 3-4 3-4;superpartitions:-6

T6L2e = (59A) = T5Iaf= (621) = RT6ag = (B04) = T4ah = (873) = T3Ia

6-7 6-8 6-8 6-20 6-20.6-9 6-9 6-20 6-20 6-38.

N1

T8N1

N2s = (803)t = (B74) = T7Isu = (691) = RT9Isv = (52A) = T2s

N1 = {{803} {B74} {A15} {962}}3-11 3-11 3-11 3-11; superpartitions:6-2032 6-32.N2 = {{803} {B74} {691} {52A}}3-11 3-11 3-11 3-11; superpartitions:6-2033 6-50.

T6N2w = (A15) = RT1Isx = (962) = RT6sy = (48B) = Tgsz = (730) = T3Is

6-20 6-30 6-30 6-

6-20 6-29 6-33 6-

here, but the relevanthighlightsof such an investigationaregiven in Example 7.32 The example shows the nine sections(sevensongsandtwo piano interludes)of Du with the trichordsthat generateeach of the combinatorialblocks of each of the

32Itis interestingto compareDu with a slightlyearliercomposition, TheWidow'sLament nSpringtime 1950), whichmayhave been an earlierdraftofthe types of structural rossrelations we examinein thispaper. Like Du, TheWidow'sLament asettingof the poem byWilliamCarlosWilliams orsopranoandpiano) makesuse of fourall-interval welve-tone rows which createsimul-taneous hexachordal and trichordalcombinatoriality.These rows begin fourlynes that form48 trichordalaggregateswhicharegrouped nto six sections of

Example6a. SIMOZ indicesfor mosaics oundin "Traum"SIMOZJKL2N1N2

J K L1 L2 N1 N27 4 1 1 7 54 7 1 0 2 01 1 7 5 1 11 0 5 7 1 17 2 1 1 7 55 0 1 1 5 7

T4L1



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TheEven PartitionsnTwelve-ToneMusic 95

sections. As one can see, only five trichordalset-classes (3-2[013], 3-3[014], 3-4[015], 3-7[0251and 3-11[037])are used inthe composition;thereforeonly 3 mosaics that contain thesefive trichordsare usedto generatethe blocks.The opening and closingsongs of Du, "Weidersehen"and"Schwermut,"eparated rom theinternal ongsbypiano nter-ludes, are each based on their own ordered hexachord thatplays the same role as E in "Traum."Example7 shows thesehexachords written under the partition information for"Weidersehen"and "Schwermut."From these hexachords sderivedallof the trichordalmaterialof the entirecycle, a com-mon procedure n Babbitt'scompositionsof the 1950s. The ex-

Example 6b. Succession"Traum"of SIMOZ indices for blocks of

1lock:

partition:

2 3 41 L1 1 N1J 4 K

L2 1 N21

eight aggregateseach. Each of these six sections is subdivided urther nto twoRTnI-related locks, the firstof which,calledS, is shown below.Lyne 1Lyne 2Lyne 3Lyne4

5428AB013976

730 691 8AB691 730 54248B 52A 97652A 48B 013

The pc representationof S reveals severalcharacteristicswhichare main-tained throughouteach of the eleven remainingblocks: (1) each lyne is RT6-invariant; 2) lynes 1+ 2 and lynes 3+ 4 preserveboth trichordalcontent andorder;and (3) the entire blockis generatedfrom two trichordal et-classes.Babbittuses only four trichords n the lynear pitch-classarrayof the Wid-ows Lament:3-2[013],3-4[015], 3-7[025], and 3-11[037]. (As in Du, the set-classesare obtainedby imbricating uccessively he trichordsof the first ynearhexachord, (542730), thus yielding (524), (247), (273), and(730).)Each tri-chord s presentedat leastonce witheveryother trichord aveone combination(3-4with3-7), and the pairsof trichordgeneratorschangefrom section to sec-tion so that the beginningof each of the poem's sixstanzas s announcedby acorresponding hangein that section'sgenerating richordal et-classes.The generatingtrichordsof the six blocks'lynear pc structure n the Wid-ow's Lament are shownbelow.

Section, Stanza 1 2 3 4 5Trichords:3-: 2/11 2/7 4/11 7/11 2/462/11

ampleshows how the five trichordalpartitiongeneratorsareim-bricated n each hexachord-so that four of the five trichordsare imbricatedin each. In addition, the difference betweenthese two germinalhexachords s crucialto the composition'sform. The content of the second hexachord(a memberof 6-32[024579]) is either the MI or TBM of that of the first (6-1[012345]),which means thatthe imbricated richordsmightbeinvariantand/ormappedinto each otherby operations nclud-ingM. Set-class3-4 is the onlyM/MI nvariant richordwhile 3-2pairswith 3-7 and3-3pairswith3-11. Another ook at the exam-ple showsthat3-11 is onlyusedinthe second half of the piece-startingwiththe passageof "Traum"we havejust analyzed.ItsM-correspondent -3 is usedonly upto (and ncluding) he thirdsong "Wankelmut."The songs between "Wankelmut"and"Traum" therefore use only 3-2, 3-7 and 3-4. Of these,"Verzweifelt," he middlesection of the wholework, sbuiltex-clusively rom 3-2and3-7, the other trichordalM-pair.This de-signreflects he differencesbetween thegeneratinghexachords;the firstgerminalhexachord excludes 3-11 and includes3-3,while the otherhexachorddoes the reverse.



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Example7. The overalldesignof DuI:Weidersehen3-23-7(305124)305 = 3-7

051 = 3-4512 = 3-3124 = 3-2

pianointerlude II:Wankelmut3-2 3-23-3 3-43-4 3-7

IV: V:Verzweifelt Allmacht

3-3 3-4 3-2 3-3

VI:Traum3-11 3-7 3-4 3-11 3-73-2

(2740B9)274 = 3-7740 = 3-11

40B = 3-4OB9 = 3-2

The positionof trichordalpartitions uggeststhat the entirepiece mighthave a large-scaleretrogradeM/MI33ymmetry,atleast withrespectto the mosaics'trichordal ontent. This is al-

33In he ensuingdiscussionwhenwe speakof hearing hisM/MIsymmetry,we do not mean that the compositionwill be heardas "balanced"around ts

center.After all, M and MIpreserveneither all intervalsnor set-classmember-ship. We do mean that the structural ymmetrywill be heardas a progressofrelations, some of which are preserved, others altered. For instance, the 3-3[014]set-class willdisappearafter the thirdsong to be replaced aterbythe 3-11[037]sonoritiesstartingat "Traum."If the partitionsused in Du were une-ven, the M/MI symmetrywould also produce a palindromic pattern of pcdistribution n the arrays.

III:Begegnung3-4 3-2 3-43-7

3-23-7

pianointerlude3-4 3-7 3-43-7

VII:Schwermut

3-4 3-43-7 3-73-11



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mostso; the sole exceptionis in thepianointerludes, he firstofwhichhasa blockbasedonmosaicscontaining3-2, 3-3, and3-4,while the second interludehas a correspondingblockbasedon3-7 and3-4.34What isvery interestingabout this deviation romexact M/MIsymmetry s thatthe two piano interludesarespe-ciallyrelated to the middlesong by a kind of twelve-tonestruc-turethat is relativelyrarein Babbitt'smusic: tetrachordal om-binatoriality.Returning to the two germinal hexachordsin the outersongs, one sees that they are not actuallyorderrelated underMI orTBM.It is quitecurious hat underRMI the first 305124)becomes (427B09) which contrastswith the second germinalhexachord(2740B9)but only withrespectto totalordering, orthe order of the four imbricatedset-classes s the same in boththe RMI of the first and in the second. Of course, this "nearmiss" reflects the deviation from exact M/MI symmetryof thewhole piece, but more importantly,t impliesthatthe structureof the blocks of mosaics sindependentof theorderof thepcsetsused in the blocks.We shall say more about the permutational aspects ofblocks, mosaics, and twelve-tone music later, but the secondquestionfrom above still remainsunanswered;what cues doesDu offer the listener to recognize the M/MI symmetriesthatcontrolits form? To attemptto answer his,we first ook at thepitch-classdesignof the center section of the whole piece, thesong "Verzweifelt."

Example8 gives the two blocks of "Verzweifelt,"each con-sistingof threelynes. The exampleis laid out to show that eachlyne is derivedfroma mosaicgeneratedby3-2[013]or3-7[025].Nevertheless, the lynes fit togetherto form tetrachordal om-binatorialitywhich is shownby vertical dotted lines. Fromthe

34The hared 3-4mightbe meant to single3-4 outbecauseof itsM/MI nvar-iance. Of course, the middle song is the obvious placeto featureM/MI nvari-ance, and while 3-2 and3-7 (M/MIpairs)are used there, 3-4 is not found.

mosaicpointof view, the two blocks areM/MI-inverses f eachother: in the firstblock, the sopranosingsthe 3-2 mosaic whilethe pianoplaystwo 3-7mosaics; n the secondblock, thegener-ators of the mosaics arereversed. Once againexactsymmetrysavoided;while the mosaics or all sixlynesare either denticalorM/MIequivalent,the dispositionof the mosaics n the blocks sdifferent.Thusthe two blocks aredifferentiatedbytwo distincttetrachordalarrays.If the four-against-three onflict between trichordalandte-trachordalpartition n "Verzweifelt"reminds he readerof the26/34 "hemiola" in "Traum,"the path of associationsfrom"Traum" o the whole has been laid. The diadicpartitionof thesoprano'shexachord n the firsttwo aggregatesof the secondblock of "Traum," 9072 B4), occurs as (2B 49 07) at the com-mencement of the second block of "Verzweifelt."The parti-tioning slightlyobscures he fact that the outer trichordsare ex-changed between the two hexachords. The "AB-- BA"permutation hat connectsthe two hexachordsvia trichordshasbeen featuredin most of the blocksof Du, andis analogous othewayinwhich two (ordered)trichordsarecomposed o makehexachords,orhexachordsarecomposedto make linearaggre-gates. A second equallysignificantconnection between thesehexachords akes us to the beginningof the lastsong,wherethediadicpartitioningof the soprano(2740B9), the samepcsas inthe other hexachords,conflicts with the 34 mosaicof the so-prano's lyne. These three hexachordsalso refer back to theopeningof "Traum"andthe soprano's 9042B7),whicharticu-latesthe firstoccurrenceof a mosaic based on 3-11 in the cycle.To summarize hese connections,the partitionhemiola andcontentof the soprano'shexachordat the beginningof the sec-ond block of "Traum"connects both to the central section ofthepiece andto its lastsong. This firstorderedhexachordof thesopranoof the last song is identicalin content to the sopranohexachordat the beginningof "Traum"and is responsible orthe introductionof the new sonority n thathexachord, he 3-11trichord.The comparisonof the openingof Du withthesefocal



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Example8. Tetrachordal ombinatorialityn "Verzweifelt" romDu

Soprano:highpiano:midpiano:lowpiano:

013 8 AB 54 2 976 2B4I II I96B A: 18 27 4 503274 5 03 96 B A18 A87I 35656

hexachords, heircontexts,and the otherpartitions ndmosaicsof the entirepiece, eventually eads one first o intimate, hen tohear,the controllingM/MIsymmetryof the whole.Permutation ndMosaics

Our discussionof the form of Du, aside from the initialde-scriptionof orderingrelationsamong the lynes, hexachords,and trichordsof the firstblock of "Traum,"has been accom-plishedwithouta laboriousstudyof the permutationalelationsamong all pc segments of each block. While a permutationalviewof this work is hardly rrelevant o its richnessandvitality,our aim hasbeen to showthat a partitionapproach o a combi-natorial welve-tone workshowsclearlyandsaliently ts hierar-chicalorganization.In fact, it mightbe interesting o pushourapproach o the point where we could assert that the internalorderingof the pcsetsof a partitionand the orderingof the pc-sets asassignments o positionsof a combinatorial rray s inde-pendentof the local orderingof the music.We could view theaggregatesuccessionof an arrayas a kind of foregroundwhichistransformed ntoanintelligible urfaceby pc orderings.These"local"orderingsareusedprimarilyo relate andproject heTnandTnIrelatedpcsetsin the arraypositions,so that the big pic-

9 07 16 3 A58I IB935 1241 24OB9I II II II I01B9 3516 124

1[ 24 A8 l7 0B9

ture of this work isdependentontrichordal artitionsof the ag-gregate.35This view canbe defendedby showing hatthe latinsquaresof Du (orthe morecomplicateddesignsof Examples5aand5b)can be filled by any pcsets that follow the letter label assign-ments and produce aggregates. Any of the n! assignments orthe n pcsetsof a partition o the n letter labelsin the arrayarelegal.Inthe trichordalatinsquares,any34partition an be cho-sen; with the more complex diagramsusingeight distinct tri-chords,the only constraint s thattwo distinct3 partitionsareused andthattheirmosaicsmustshareatleastonesupermosaic.Babbitt'suse of Z-relatedpartitions mosaics)is a specialcasethat allows all the members of the resultingarray o be of thesame set-types (in one or the other mosaic). Babbittrestrictshimselffurtherso that the hexachordal upermosaicsncludeatleast one hexachord from the set of six "all-combinatorialsource-sets."This allows the special ntricacyof orderrelations

35The ndependence of the (unordered) partitionsfrom the serial lynesmight seem to contradict Babbitt'sdescriptionof the twelve-tone system as"permutational" as opposed to the "combinationalsystems"of the past).However, the partitionsare ordered andpermutedwhenthey areplaced n ar-raysand the arraysare interpretedmusically.



27/29


amongthe lynesandtheirsub-lynes n the array hatmarks hecompositionsof Babbitt's first period.36Other sets of hexa-chordscanprovidesimilarorderrelations, f desired.As the readermust realizeby now, the four-by-fourarrayscan be formed out of unevenpartitionssuch as 1236,2324,or2242. Even partitionsof less than four pcsets can be assigned;forexample, in a 4-by-4 atinsquare,the three tetrachords f a43partitioncan assignedto three of the four letter labelsandanull set to the last label. Generalizingurther o arrayshat mea-surem-by-n,combinationsof two partitionscomposedrespec-tivelyof m and n (orless)pcsetscan formverticaland horizontalaggregatecompletionin the array,providedtheir mosaicsaremembersof the same supermosaicpartitions.Toward heSeventy-Seven artitionsof theAggregate

As we have just implied,the move from the even partitionsto the unevenis anaturalgeneralization f compositionalssuesin aggregate preservingcompositionaldesignsin twelve-tonemusic. ThepathtakenbyBabbitt n the late 1950s, rom his firstperiodinto his second, is well exemplifiedbythe comparisonofthe structures f hispiano piece, Partitionswhichsharesa simi-lar kind of trichordal tructurewithDu) with those of anotherpiano piece, Post Partitions.37 he latterwork is a series of ag-gregatesformed from the combinationof twelve-lynes,each a

36Babbitt'sirstperiod includes the hexachordal/trichordal ieces, worksup to circa 1959. The works employing all-partitionarrays,such as RelataI(1960), mark the second period. A thirdstyle period begins in the late 1970swith the introductionof "superarrays." ee AndrewMead, "RecentDevelop-ments in the Music of Milton Babbitt," The Musical Quarterly70/3 (1984):310-331, andWilliamLake, "The Architectureof a SuperarrayComposition:MiltonBabbitt'sStringQuartetNo. 5," Perspectives f NewMusic24/2(1986):88-111.

37SincePartitionsbequeaths its germinalhexachordal yne to Post Parti-tions, the title of the laterpiece serves to announce ts compositionaldeviationfrom the former andtherebyheraldBabbitt'ssecondperiod.

concatenationof complementaryordered hexachords.Each ofthe aggregateshasa differentpartitionwithrespectto register;there aresix registers,each containing wo lynes.38Sincethereareexactly fifty-eightpartitions hatcontainfrom one to sixpc-sets, inclusive,there are fifty-eightaggregates n the composi-tion. Thismeans that the set-classesrepresentedbytheimbrica-ted sets in the generating hexachord of Post Partitionscontribute o the partsof the mosaicsrepresentedby the regis-tralaggregates.Even from the necessarily imiteddescription ustprovided,it is clear that the enumerationand inclusionof the differentmosaicsof each partitionclasswould shed much lighton the"all-partition class]" compositions of Babbitt'ssecond styleperiod.Worksof composerscontemporaryo Babbittaswell asearliercomposerswould be well servedby a generaltheoryofpartitions.Forinstance,the combinatorial owpairof Schoen-berg'sPiano Concerto s given below with its generating2242partition El = pc 0).P: 07B219 35A684T5IP: 5A3648 207B91{{19} {48} {027B} {356A}}

Among the topicsto be explored n atheoryof the seventy-seven partitionsare the enumeration39nd inclusion of parti-tions, the role of invarianceunderclassicalas well as arbitrarypc operations,and the cross-sections f partitions.This ast sub-jectwouldgeneralizenaturallyhe "partition emiolas"and the

38For ach pairof lynes sharing he same register, the two lynes are hexa-chordallycombinatorialandaredistinguished rom eachotherbyarticulation:legato (long) versus staccato. The use of a six-lyneall partitionarrayof fifty-eightaggregates s foundin Babbitt'sArie da Capo.39A number of ways to calculate the number of mosaics for each partitionclass exist in combinatorymathematics.As partof theirprocesssuchmethodscalculate he variousdegreesof symmetry oreachpcsetunder he twenty-fourpc operations.



28/29


Example 9. First three aggregates of Post Partitions by Milton Babbitt~=90 [~ 5----- -J =go | s 3!a

sempre8 higher x ,fIA~~~~~~~~~~~~ps.> 1 qI I

f 3ffff fPfPPp 3Bff3 p----3--ppp P,p p l ffff 7

fff7f lsempre8 lowerf f fffff

8--- p ff_. fL--- 5---JTo be performedimmediatelyafterPARTITIONS FOR PIANO (1957) or independently.Accidentalsaffectonly those notes they immediately precede.Pedalsshould be employed to secure the indicated durations.? 1975by C. F. Peters Corporation.Used by permission.

f 7 1fff; b



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two-dimensionalarraysof pcscited in ouranalysisof "Traum."An example of such cross-sections s found at the opening ofPostPartitions,wherethe twelvelynesof eachof thefirstsevenaggregatesof the piece arealigned n diads so thatonly two pcsare struckat once. Example9 showsthe first hreeaggregatesofPostPartitions.

The two-note chords n eachaggregateof Example9 (andinthe next four) are presentationsof the partitionQ: {{2B}{08}{37} {16}{A5} {94}}. Thus the diadsof Q are located "across"each partitionof the seven aggregates.The firstaggregatean-nounces Q with its registral partitionof pca 26. The mosaicwhichincludesQ, {{03}{16}{48}{29}{5A}{7B}} s Z-relatedtotwo other mosaics. Babbitt immediately distinguishesQ fromthese otherZ-relatedmosaicsby partitioningemporally hepcsto provide an articulationof a supermosaicthat includes Quniquely;the two hexachordalsets in questionare (of course)all-combinatorial-members of 6-20[014589].

AppendixThePcasof the77 PartitionClassesoftheAggregate*CB1A293847566A12921831822741732651642632

5225434391382127312722164126321623521254215321423153224222

4322348147213631362212541353212523142212432124322142433213223

71562145314522134214432134231233133222123241266165215431542214

*A= 10;B = 11;C = 12.

322143231325125174216321632215241441832172316319221821A1C


Alegant, Even Partitions in Twelve-Tone Music (MTS 1988)

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