Review of Composition With Pitch-Classes [PDF]

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Review of Robert D. Morris, Composition with Pitch-Classes Composition with Pitch-Classes: A Theory of Compositional Design by Robert Morris Review by: Andrew Mead Perspectives of New Music, Vol. 29, No. 1 (Winter, 1991), pp. 264-310 Published by: Perspectives of New Music Stable URL: http://www.jstor.org/stable/833081 . Accessed: 24/06/2014 21:43 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp

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REVIEWOFROBERT D. MORRIS,

COMPOSITIONWITH PITCH-CLASSES

ANDREWMEAD MOST FASCINATING aspectsof Robert Morris's book, with Pitch-Classes: A Design,residein Composition Theory ofCompositional whatitdoes notsay.Moreto thepoint,whatitdoes haveto say,and ithasa greatdeal to sayto composersand theorists alike,takeson an evengreater in of what the author has chosen not to address.Thisis no significance light accident:hisactionin leavingcertaintopicsto thereader'simagination is as criticalan act of responsibility towardsmusicand its makingas are his extensivetheoretical but he has takena daringriskin doing formulations, so because he demandsof his audienceas greatan act of responsibility towardstheirreadingas his textembodiestowardsits subject.Lest this seem too forbidding an assertionat the outsetof what is unabashedlya celebration of an important contribution to theliterature, let me hastento add that the whole edificeis drivenby the author'senormousenergy,

SOME

OF THE

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enthusiasm and evidentrelishin makingmusicon manylevelsand ofmany kinds. While I believethatits most profoundmessagesemergeby inference, Morris'svolumeis nevertheless of a whole hostof explicitin itstreatment each with a nimbus of and enticesignificant issues,surrounding suggestion mentforthe engagedreader.In the followingI shallattemptto demonstratethe usefulness of a fewof his primaryideasand techniquesforboth and composition analysis,and takea fewtentative stepsthrougha coupleof themanydoorwayslefttemptingly thetext.I shallclose open throughout what strike me as Morris's most by discussing urgentand criticallessons, thoseimpliedbywhatis not said. BeforetracingsomeofMorris'sideasthroughthevolume,I mustdigress to mentionthe unfortunate stateof the book's production.While briefly theverynatureofitssubject,not to mentionthedegreeofformalism with whichit is approached,mustmake the book a challengeto the reader, manyof the hurdlesone is facedwithare the resultof insufficient proofofa criticalnature, reading.The book is strewnwithmisprints, frequently and radically manyof the chartsand figuresincomprehensible, rendering the of numerous and changing meaning passages.Figures examplesarealso oftencrowdedon thepage,increasing thedifficulty of mastering themore abstractpointsof the argumentby illustration. This is regrettable as the book demandsand rewardspreciseunderstanding, and offersthe reader enoughof a challengewithoutimposinga simultaneousseriesof comprehensiontestsbasedon recognizing wherethetexthasgone offtherails.

Severalreviewsof the volumehavealreadyappeared,containingdetailed summariesof the book's chapters.lFor thisreason,I shallprovideonlya verybriefoverview.The openingpagesof ChapterOne terselyrevealthe author'smotivationfordevelopinga theoryof compositionwith pitch classes. In his view, "the realizationin the firsthalfof the twentieth of pitchorganizations thatdo not involvetonal centuryof the possibility centerswas a revolutionary This realizationproduceda (1). development" of of one set which reduced theimportance of variety responses, essentially in structure or considered it the of means pitch composition, merely the surface. Another set of led to the carrying sounding responses explorationof nontonalpitchrelationsin new typesof musicalcontinuity. Morris is appreciative ofa widevarietyofmusicalapproachesand wishesto address his book to as broad a compositionalaudienceas possible,so he avoids he overtlyallyinghimselfwithanyparticular stylistic camp. Nevertheless, statesit is his view "that the pitch-class relationswill servecomposersof ofpitchmanystripesifwe concernourselvesnot onlywiththegeneration classmaterial,but withthenatureof pitchitself"(2).

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The attitudeimplicithere,'know theterrainon whichyou wishto act,' informsthe book as a whole. This healthyecumenicismcontinuesin the descriptionof what a theoryof compositionin today's musicalclimate mightbe: "... a compositionaltheoryforsomethinglargerthana small segmentof today'smusicneeds to be explicitand generalat once, since, although it may be designed to help generate a certain species of music.. ., it mustnot specify, in anybut a rathertentative manner,the of the His clearest statement of purpose, music" (3). stylistic component ofbothwhatis and is notsaid oftheimportance and theone mostrevealing in thebook followsshortly:"What is neededis a theoryat a higherlevelof whichpermitstheoriesof thespecificand reductivekindto generalization be invented.. . . [A] theoryof compositionaldesign that can satisfy structuralrequirements functions,uses, and aesarisingfromdifferent theticsof musicconsistsofa setof toolsand methodsforconstructing and interpreting compositional plans" (3). In otherwords,itis notMorris'saimto tellus how to writemusic,or to tellus how musicgoes; his goal is to outlinewaysof thinkingabout the materialsof music,so thatwe can make a seriesof responsiblechoices, based on an informedunderstanding of the implications of theirconseat levels. He chooses to do this more quences many by implicationand than didacticism. His ultimate mode for this, example by demonstrating the compositional is more an embodiment of the more fundamental design, attitudeof compositionalresponsibility than it is a unique solution to of but in sets tools forbuildingcompositional his writingmusic, deriving he shows us in the dialectic of musical inventionand responsibility designs action. The remainder of the openingchapteris designedto whetour appetite forall thatis to followbydemonstrating thepowerof histoolsof transformationand the flexibility of interpretation availablethroughhis compositional designs. By takinga particularly simplemindedbut nevertheless believable and musically fragment tweakingit in variousways, Morris createsprogressively more and more complexand suggestivepassagesof music.The realizations ofthetransformations arequitediversedespitetheir and of none them could be considereda brevity, although satisfyingly each carries with it sufficient musical cues to invite completecomposition, one to imaginelongerworksofwhichit mightbe a part. In keepingwithhisagendaofproviding thereaderwithtoolsformaking Morris does not now choices, responsible simplyshow us how to do what he just did.2 In fact,ifthatis all thatwe as readerswishto know,we shall haveto takea verydeep breathat thispointand hold it forquite a while. The nextthreechapters,in manywaysthe heartof the book, gradually buildup fromthe mostbasicconsiderations of musicalspace and perceptionthe toolsand materials adumbratedbyChapterOne's sleight-of-hand

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of the particular sorts (or -ear!)so thatwe can understandthesignificance It would be all too easyto lay ofchoicesmade,as wellas theirmechanisms. out a seriesoftechniquesforgenerating stuff,to be rendered compositional or surfacegesture,but at "musical" throughthe alchemyof orchestration everyturn the author is concernedthat we be able to believe in his structures as audiblethroughthe musicalsurface.A criticalpointwhichI shalldeal within moredetaillateris thatwhileMorrisis overtlyconcerned thata structure maybe heard,he avoidsissuesofconstrualin orderto keep histheoryas open as possible. ChapterTwo takesan enormousstepbackwardsfromthecompositional particularsof the opening chapterto consideraspectsof variouspitch we space,or c-space, spaces. In the most basicpitchspace defined,contour can onlytellifan elementis thesameas, or is higheror lowerthan,another element.Ways of indexingintervals,or measuringand comparingthe distancesbetweenelements,are not availablein c-space.Intervals,defined as ratiosbetween frequencies,are used to definemore specifiedpitch spaces, with unequal minimalintervals(u-space),or with equal minimal of a modularintervalin a givenspace intervals(p-space).The availability ofthatspace,m-space motivatesthegenerationofa modularrepresentation forp-space.3The body of the book is concerned foru-spaces,andpc-space modularreductionof theequalwithtwelve-pitch-class space,thefamiliar chromatic the octave. One mightwonderabout tempered pitchspace by thedegreeofattentionafforded kindsof pitchspacesat thispoint different in thebook, but ifwe understand twelve-pitch-class spaceas an instanceof a varietyofpossiblepitchand pitch-class we can spaces, generalizemuchof what follows to musical systemsvery differentfrom those that are discussed. Much of the restof ChapterTwo containsthe introduction of certain within and tools and the chroconcepts c-space p-space, equal-tempered maticgamut.A discussionof contoursin c-space,theresulting patternsof and down a formed of elements in time, ordered up by string c-space introducesa tool of immenseimportto the entiretext,the comparison matrix.In one guise or another,employedwithcontours,pitches,pitch a classes,and operations,such matricesprovidethe meansof investigating widerangeofusefulmusicalproperties.4 The examination ofcontourspace is followedby a consideration of chromaticp-space,in whichthe notions developedin contourspace are expandedby the additionof recognizable intervals.A particularnotion of intervalclass appropriateto p-space emergesas theabsolutevalueofthedistancebetweentwo pitches;intervals attainpositiveor negativesignsdependingon theorderofpitchesin time. Orderedpitchesareexpressedas segmentsor cycles,thelatterequivalentto thatarewrappedaroundon themselves. The matricesintroduced segments in contourspacenow containnumerical values,and areused fora originally

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betweenunorderedsets, varietyof purposes,to examinethe intersection or to searchfororderedcollectionsheld in common betweensegments and/orcycles.A discussionof equivalencerelationsamongsets of pitches coversa varietyof criteria,ultimatelyoffering those generatedby the oftransposition combination and inversion (Tn) (T7/). ChapterThree is a giddily-paced grandtourthroughmuchof the most up-to-datetheoryof unorderedand orderedsetsin the twelve-pitch-class and expandingupon theworkof a hostof theorists.5 space,summarizing All of what was developedin ChapterTwo is translatedand extended, particularly comparisonmatricesof varioussorts.It closeswithsummaries of severaldifferent measuresforcomparingsegments,ordered similarity of classes. This be treatedas strings pitch chaptercan perhapsmostusefully and review. As the author in his Preface,althoughone synthesis suggests could use the textas an introduction to the ideas, "it would be fareasier firstto encounterthebasicconceptsand termsin anothertext" (xi). ChapterFour is in manywaysthe most strikingsectionof the book. ofthematerials oftwelve-pitch-class Turningfroman examination spaceto a studyof the operationsthatrelatethem,Morrisdevelopsat lengththe group-theoretical propertiesof the twelve-toneoperators(TTOs). These he has earlierdefinedas thefamiliar Tnand T,I, in additionto multiplication by 5 and 7 mod 12, T,M and T,MI. These lattertwo represent the circle-of-fifths which the circle of the fifths for chromatic operation, swaps the distribution of intervaltypes,whileexchanging their scale,preserving locations.Two importantnotionsemergefromthe discussion,and combineto extendtheidea of transformational operationson the pitchclasses in a wealthofsuggestive directions.Byexpressing theTTOs as setsofcyclic Morris allows us to consider them as special intervalpermutations, of cases a much wider of range preserving possibleoperationswhichthemselvesmightbe made musicallyvividunder certaincircumstances. Secthe relations the TTOs in of terms he ondly,bydescribing among agroup, is able to invokeideasfromgrouptheoryto describerelations amongsmaller collectionsofTTOs and collectionsofthosecollectionsinvolving subgroups, cosetsand (informally) The veryabstractnatureof this automorphisms.6 sectionof the book underscoresthe diversityof domains to which it insights. providesdeep-structural Five thearrayas a usefulexpressionof thecoordinaintroduces Chapter tionof pitch-class relationsin two dimensions.7 For convenience,thetwo defaultdimensionsare consideredregister and time,but as Morrissubsein hisfinalchapter,theseareonlya singlepairingofa quentlydemonstrates muchwiderrangeofpossibilities. Morrisintroducesa numberoftermsfor thecontentsoflocations,rowsand columnsofarrays, as well characterizing as a numberof techniquesforgenerating arrayswithparticular properties, theorderingpotentialofpartially orderedsets(posets), embedinvestigating sizes and ding arraysin largerarrays,and combiningarraysof different

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types.ChapterFive's techniquesmightseem at timesmoread hoc than thesphere whathas precededthem,but it is herethatwe beginto re-enter of actionadumbratedbythebodyof thefirstchapter;we are beginningto deal withthe instantial, we are beginningto exertour creativewillwithin the universewe have been charting:we are beginningto compose. No matterhow generalthingsmightstill appear at this point of Morris's discussion,we are, in ChapterFive, morepreparedto thinkabout how a piece mightgo, and so we are tryingout tools thatcould be usefulin way.The veryqualityofthespecific,ofthe makingitgo in someparticular ad hoc, willemergeas partoftheoverarching messageofthetext,and is in itselfa signalabout thenatureof composition. ChapterSix is an annotatedcatalogueof eightof the author'scompositional designs,coveringa wide range of premisesand structures,and how thetoolsand techniquesgeneratedin theprecedingchapillustrating tersmaybe wieldedin a varietyofcombinations. These designsareprefaced of theirstructures, by some aestheticcharacterizations entailingissuesof and hierarchy. All of the compositional coherence, closure, saturation, designs are illustrated as arrays, witha detailedaccountof theirconstruction and a certainamountofcommentary on how theymighteffectively be realizedas actualpiecesof music.One valueof thedesignsin ChapterSix is thatthey of the techniquesof the allow us to appreciatethe waysthe interaction precedingchapterscan generaterich and complexmusicalcontinuities. Unlike the examples in Chapter One, these compositionaldesigns, althoughbrief,are much more elaboratelyworkedout, and may more readilybe construedas substantial portionsof compositions,ifnot indeed as shortbut satisfying self-contained pieces. intothethickets ofquestionsunderChapterSevenmakesseveralforays the realization of the ideas of compositionaldesignsby generalizing lying the variouspitchand pitch-class spacesexploredin ChapterTwo to other dimensionsof music. Severaldifferent dimensions sequentially-orderable are explored for their potentialto project orderingsof pitch classes, includingdynamics,aspectsof timbre,spatiallocation,envelope,duration and the like.Not surprisingly, thisleads to a consideration of the possible between and and the concomitant pitchspaces analogies temporalspaces, modulartemporalspaces thatmaybe determinedmetrically. One of the of a of this is the of sorts structures consequences description rhythmic Milton Babbitt's to underlying time-pointtechnique,generalized moduli of any size.8 Needlessto say,the brevityof the discussionprecludesan exhaustiveexaminationof the topic,but the authoropens up a realmfor futureexploration. The book is roundedout withextensivenotesto thechapters,a handy a bibliography, and a reasonablycompleteindex,in additionto glossary, threeveryusefulappendices,consisting ofa set-classtablefortwelve-pitchclassspace,a listofthecyclesoftheTTOs, and a listof theautomorphism

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classesofsubgroupsof thegroupof theTTOs. Listsat theopeningof the book providea compendiumof the abbreviations and functionsfoundin thetext,as wellas themathematical symbolsemployed. Withouta doubt,thevolumeis verydemanding,and whilemostof its is straightforward, readersmayfindcertainaspectsof thetext presentation Some readers be in problematic. may putoffbythehighdegreeofformality thepresentation, whileothersmoreaccustomedto thisstyleofwriting may be disappontedbytheoccasionallapsefromformality or ellipsisin technical termsborrowedfrommathematics. As we noted above, the author that it to is easier come to his text suggests preparedwitha knowledgeof the basic conceptsand terms,but it would also help the readerto be familiar withsome of the analyticaland theoreticalramifications of those ideasas well.As theauthoris carefulto pointout, manyoftheideasin the book have theirroots in the work of a varietyof composers,and a withthe particularcompositionalconcernsof those musicians familiarity helpsplaceMorris'sworkin a largercontext.An awarenessof the criteria the structures and compositionalrealizationsof Milton Babunderlying bitt'sall-partition theverydifferarrays,forinstance,helpsone understand ent natureof certainaspectsof Morris'sideasabout the construction and shorton specificillustracompositionof arrays.The book is deliberately tionsoutsideits immediatepurview,and it helps to have a sense of the and analytical worldit so interestingly compositional augments. Froma pedagogicalpointof view,I would havefoundit usefulto have had a detailedaccountof some of the transformations used in the initial chapterplaced at the beginningof Chapter6, the section on specific compositionaldesigns.While it is reasonablyeasy to find the relevant sectionsof the textretrospectively, I believethata second inspectionof thesetransformations in lightof the body of the textwould makea good summationin preparationforthe more extensivecombinationsof techniquesfoundin thesamplecompositional designs.

ChaptersTwo throughFour,whatI considerto be thetheoretical bodyof the book, containsome verypowerfultools usefulboth foranalysisand fromothersourcesbutattain composition.Some ofthesetoolsarefamiliar theirparticularpowerfromthe overarching conceptionthatlinksthem; othersgaintheirstrength fromthewaystheyprovideus withnew insights into particularmusicalrealms.Runninglikea threadthroughthe entire book is the idea of the comparisonmatrix,a tool forinvestigating the relationsbetweenentitiesin a varietyof domains.These matricesare first introducedin ChapterTwo as a wayof characterizing contours,but they areswiftly to and then to generalized pitches, pitchclasses,bothas setsand

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withPitch-Classes Morris,Composition

orderedsegmentsand cycles.Example1 containssome samplematrices, basedon contour,pitchsegments, and pitchsets. (a)

COM 1 2 4 3 5

(b)

ip{Pa,P}

(d) T matrix

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(c)

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sum

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1

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1 2 5

2 3 6

5 7 10 6 6 8 11 7 9 11 14 10

EXAMPLE

1

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a comparisonmatrixfora contour,expressing the Examplela illustrates between or of events. +, 0, relationship, anypair c-space Examplelb the of elements la in as as be reinterprets pitches p-space; may seen, the and minus have been the plus signs supplementedwithvaluesexpressing measureofthedistancebetweenelements,whicharethemselves orderedin time.9 In both matrices,the diagonalsfromupper leftto lower right representintervals(or in the case of la, directions)between different distancesof order positions; the centraldiagonal representsintervals betweenelementsat the same orderposition,the nextout represent the intervalsbetween adjacentpositions,the next representthose between everyotherposition,and so forth.ExampleIc treatstheunorderedcollectionof thesegmentin lb, by removingthe directional theplus indicators, and minussigns.Becauseeverypitch-interval classis represented twice,the or lower-left cornerof the arraybecomesa catalogueof interupper-right valscontainedin thecollection. how thesamesortsof matricescan be used ExamplesId and le illustrate to investigate theintersecton betweendifferent pitchsegments.Id, calleda T matrix,or transposition matrix,listsall oftheintervals fromtheelements of the segment(or set, forthat matter)in the column to those of the in two ways.The segment(or set) acrossthe top. This can be interpreted of a interval in the matrix the number ofpitchesin multiplicity given yields common betweenthe horizontalcollectionand the transposition of the verticalcollectionby thatvalue. If the collectionsare ordered,the matrix can tellus whethertheintersection willbe similarly ordered,eitheras is or underretrograde, by whetherthe givenintervalvaluesforma diagonalin thematrix,and whichdirectionthediagonalruns. WhiletheTmatrixyieldsa catalogueofinvariance betweentwo arbitrary sets or segmentstransformed withregardto each otherundertransposition, the I matrix,or inversionmatrix,yieldsan analogouscataloguefor two arbitrary setsor segmentstransformed withregardto eachotherunder inversionand transposition. The latteris illustrated in Examplele. The valuesin the firstcatalogueare formedby subtracting the valuesof the columnarsetor segmentfromthevaluesof thehorizontalset or segment, whilethevaluesofthesecondcatalogueareformedbysummingthevalues of the columnarand horizontalsets or segments.Althoughthese two methodsforconstructing thematricesreinforce our intuitions about transthe positionand inversion(thevaluesof thefirstcatalogueare,effectively, differences betweenorderedpitches,while the valuesof the second are indexnumbersofinversion), theirdistinction tendsto obscuretheidentical formof the matricesas additiontables.10As Morrisin effect underlying on a set or pointsout, T matricesare equivalentto I matricesperformed ofthesetor segmentwe wishto compareitwith. segmentand theinversion This mightseem a finicky pointto make,but it will allow us to see the

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betweenthesematricesand matricesformedwithstringsof relationship TTOs. we can reduceall of theprecedingmatricesbymod 12 Not surprisingly, in pc-space.The familiar row to examinerelations amongsetsand segments on a particular tableis, needlessto say,an I matrixperformed orderingof the twelvepitchclassescomparedwithitself,and containsboth thelisting ofall of itsrowtransformations undertheclassicaltwelve-tone transformaand all the information about embedded ordered segments tions, mutually to explorecomplementation as well. Morrismakesuse ofsuchmatrices and invariance in ChapterThree." and suggestive extensionof these What is perhapsthe mostinteresting matricesoccursin ChapterFour.'2 Here, Morrisis discussingstringsof TTOs, and wishesto comparea seriesof operatorswithanotherseriesof operators.ForanypairofoperatorsFmand G,, therewillbe someoperator thatwillmap one onto theother,and anotherthatwilldo thereverse.We can figureout thoseoperatorsas follows:ifZGn = F,, thenZ = FmG,-' (see Morris,page 139). Thus, ifwe havea set or stringof operators,IFI (read ), and a set or stringof operators/G/,we can derivea matrixof the operatorsthatmap the membersof IGI onto IFI by placing/F/in our horizontalposition,and the inversesof IGI in our column,and combiningthemas F,Gn-l.This,ofcourseis analogousto the formof the T matrixwe discussedabove in whichwe treatedit as an I of thesecondset we wishedto examine. matrixof a set and the inversion The foregoing is illustrated in Example2. Let IFI = < ToT

T T7TI T3>

Let /G/= < T6 T7 T3I T > /GI/-= To T6 T5 T3I T7

T,

T71 T81 I T T, TA T7 TI6 T6

T5

T7

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T1

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TA

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EXAMPLE

TBI

3

T8

2

By lookingat matricesofoperatorsin thismanner,we can beginto see all sortsof analogiesbetweenoperatorsand theirrelationsand pitchesor pitchclassesand theirrelations.We can readthevaluesin the bodyof the

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matrixin Example2 as the intervals(in a moregeneralsense) fromthe elements(theoperations)of/G/to theelementsof IFI.3 It is possiblefrom this to generalizeto an equivalenceof intervalclass among operators, simplybycreatingequivalenceclassesbased on pairsof inverseoperations. we can findthe samesortsof interas Morrisdemonstrates, Additionally, sectionand orderedinvariancebetweensets and stringsof operatorsand theirtransformations as we did betweensets and segmentsof pitchesor If we consider a pitch-class classes. segmentto be the output of a pitch stringofoperationson a singlepitchclass,theanalogiesbecomeveryvivid. Morrissuggestsas muchon page 138. The preceding further comparisons suggeststhatwe can makeinteresting ofdifferent kindsofmusicalentitiesrelatedbythechain ofsetsor segments of operatorsthatgeneratethem.This has implications fora wide rangeof in Generalized so extensively Musical musics,as David Lewin hasillustrated 4 We shallexamineone particularanalytical Intervals and Transformations. exampleto give a senseof the preceding,and thensuggestsome possible to pursue. directions compositional The openingsectionofArnoldSchonberg'sPianoConcerto,Op. 42, up to itsfirstmajorthematicrecapitulation at measure133, proceedsthrough all twelvecombinatorial its of rowclass quartets inversionally-combinatorial in a seriesoftranspositions to the successive intervals of therow equivalent of its initialaggregate.'5Example3 liststhe operatorsand the measure numbersatwhichtheirassociatedcombinatorial family appears,alongwith themelodyof theopeningaggregate.

Transpositions:

To

Measure:

1

T7 46

Tp 54

T2 63

T 74

86

EXAMPLE

103-106

TA

T T

T T8

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107

117

122

126

3

For variousreasons,interval-class 2s play a special role, both in the melodyat theoutsetand in thewayit is broughtbackin measure133. As may be seen, thereare only two instancesof this intervalclass found in therow-class's intervalpattern,but both receiveinteresting segmentally twistsin the openingmelody.The firstrepresents the firstchangein the rhythmic patternof the opening,whilethe second is partof an interior repetition,which introducesstill anotherinstanceof the intervalclass, derivednon-segmentally. While the firstdyad in question representsa change,it stillcan be readas a diminutionof the patternshortrhythmic longthathas groupeddiscretedyadsin the melodyup to thatpoint.The

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withthe repeatedAb, allowsthesecondsegmentalintervalcontinuation, class2 to be projectedin thesamerhythmic patternand metricalplacement as thefirst. also allowsus The repetition of Ab and thebreakbeforeitslastiteration to comparetheopeningdyadof thepiece,Eb-Bb,withtheresulting dyad, Ab-Db. One readingof this,whichis profitableboth locallyand in the the is to hear the two as relatedby inversion(theyrepresent long-range, combinatorial initialdyadsof a row and its inversional complement),but theymayalso be usefullyconstruedas unordereddyadsrelatedbyintervalclass2. One of the manyvaluablewaysof hearingthe openingmelodyof thedyad 2 dyadssurrounding theConcertois to hearthetwo interval-class 2 relationship withtheopeningEb. Ab-Dbalertingus to Db's interval-class realizationthat the firstthree (This mightpermitus the retrospective elementsof the melodyare thentransposedbythe same intervalclassand is vivid forthenextthreeelements.Whetherthislastobservation reordered at thispoint in our hearingis less crucial,as it will become vividin our hearinglateron.) Turningnow to the firstsectionof the Concertoas a whole, we shall theanalogousrelations investigate bythe amongthosepassagesrepresented In been associatedwiththe noteswe have discussing. this transpositions is "T121,"or thetransposicase the"intervalclass" we shallbe considering membersoftherowclass tionbyeitherT2 or TIo. As Example4 illustrates, of trichords in one relatedin thismannercontaina segmentalintersection of theirhexachords,an obvious resultof the last observationwe made above. When we look at the passagesin question,we can see thatthey thisinvariance relationin vividways.The passagesareillustrated exemplify in Example5. P:

3 A 25

T2P:

5 0 417 6 2 8 A3

4 016 8 1 9 B 7

EXAMPLE

B 1 9

4

The firstpassagein questionactuallyjams the two transpositional areas to invoke the not unlike contraction of the together rhythmic relationship, theanalogousdyadat theopening.The relationship as it is invokedin the next passagerefersback to the initialtrichordof the Concerto. We are encouragedto hear the connectionin severalways. It is immediately demonstratedin the followingpair of passages,in which the motivic sequenceused to linktheopeningwiththepassagein questionis reversed. This, of course,echoes our initialhearingof the analogousspots in the openingmelody.The passagein questionalso invitescomparisonwiththe

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ofNewMusic Perspectives

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with Pitch-Classes Morris, Composition

277

initiations of openingthrougha chainof associationsbased on recognizing will As be we the two of halves the remembered, compared spans. opening thattheyhad similarinitiations. in phrasebasedon a recognition Similarly, of the continuation the we would wind the hearing melody up comparing pair of phrasesrepresented by the initialrow and thatwhichfollows,a inversionof the same combinatorial retrograde quartet.Not surprisingly, follows that with a of Schonberg pair pair phrasesbased on the inversion and theretrograde fromthequartet,inviting us to comparetheopeningof the workwith the openingof this second pair of phrases.This spot is markedby the introduction of a new motivicdispositionwhichdoes not makea seriousreturnin theremainder ofthissectionoftheConcertountil that the T21 relationshipwith the opening. precisely passage invoking Thus, througha varietyof means, we are led to hear similarsorts of betweenanalogousportionsof stringsof dissimilar entities, relationships basedon thesimilarities oftheirtransformational structure.16 The precedingis merelyan anecdotalillustration drawnfroma compositionofextraordinary but it hints some at the ofthewaysMorris's richness, meansof describing relationsamongdisparatesortsof itemspermitsus to generalizecertainmusicallyinteresting concepts.We shall pursuethis a littlefurtherby examiningthe implications of anothernotion normally associatedwith rows or segments,applied to stringsof operators.One compositionally suggestive conceptthatMorrisdiscussesin the book but has also publishedon elsewhereis the multiple-order-function row,those of classes that contain the same of ordered interval orderings pitch sequence classesat different setsof ordernumbers,to withintheclassicaltranformabased on tions.'7 Such rowshavean obviouspotentialforcross-reference orderedsegments.The same conceptmaybe appliedto stringsof operaresults. tors,withinteresting For thefollowing well-known row,the examplesI shalluse a particularly so-calledMallalieurow,namedafteritsdiscoverer, mathematician Pohlman Mallalieu.18This row,illustrated in Example6, has thepropertythatall of itstranspositions areembeddedin itat different order-number intervals. To withintheTTOs, it is unique in the twelve-tone universe.It is also an allintervalrow. 0 1 4 2 9 5 B 3 8 A7 1

2 4

5

3

5

A 8

2

3

EXAMPLE

6

0 1 4 2 9 5 B 3 8 A7

6

0

6 6

6: THE MALLALIEU

4 1

9

B

9 4

8 3

B

ROW AND ITS EMBEDDINGS

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6 etc.

7 7

etc.

7

etc.

Perspectives ofNew Music

278

If we replacethe elementsof the row with transpositionoperators its successiveintervals,we can generatea simplestringof representing oftheoriginalrow.Whatmightthismean operatorswithall theproperties if we run a collectionthroughtheseriesofoperators, First of musically? all, each successivecollectionwill be relatedto the precedingcollectionby a different interval.19 This could be exploitedin variousinteresting ways.For we select the hexachord if our is a could collection hexachord, example, index class by the propertiesof intersection providedby its interval-class M-transformation and its Hexachord class (6-1) [012345] (interval vector). class(6-32) [024579] aretheuniquepairofhexachordtypeswitha unique ofone or ofinterval classes.Thus, eachsuccessivetransposition multiplicity the otherof these hexachordsthroughthe initialseven operatorsof the intersection with the precedinghexachord stringwould yielda different based on the class of the intervalof transposition.Furthermore,the multipleorderfunctionof the stringof operatorswould permitthe same sortsof relationships foundbetweenadjacenthexachordsto be articulated betweenhexachords could spans.These relationships separatedbydifferent a briefillustrabe articulated in a wealthofmusicalways.Example7 offers tion. In this particularchart,I have additionallyorderedthe successive of the unorderedcollectionto articulatecertainsortsof transpositions relationsbetweenand amonghexachordsboth adjacentand at a distance. To

T4

T1

T2

T9

T5

TB

J III - (1 I I Ln I I II 532410 652431 976854 245367 9AOB12 56978A B12034

A.

.~

1

I

652431

x

1

~. III

III

245367

56978A 56978A

978654 EXAMPLE

7

Anotherapproachwould be to selectcollectiontypesfora maximal of interval-class distribution.For example,the two M-related similarity pairsofZ-relatedhexachordtypes,(6-10) [013457] /(6-39) [023458] and (6-46) [012469] / (6-24) [013468] have the interval-classindices < 333321> and < 233331> respectively. interval Thus, forfourdifferent classesrelatingtranspositional pairs,a pairof collectionsfromanyone of

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with Pitch-Classes Morris, Composition

279

The compositheseset typeswould alwayshavea trichordofintersection. tionalpotentialhereis ratherdifferent fromtheprecedingexample,in that as the indexfordifferentiating insteadof degreeof intersection transpositionaldistances,changeof trichordtypeor order-number positionwould if we run be the measure.Things can become particularly interesting a of hexachords series of through pairs transpositions complementary As may be seen in Example8, each pair of hexachordal simultaneously. threetimeseach in the pairsrelatedbythefourintervalclassesrepresented of a trichordal mosaic.20These intervalvectorrepresents the reorientation mosaicsmustcontainthe hexachordalpairat least twice,and thatfound betweenpairsrelatedbyT4 mustalso containitthreetimes.21The example detailstheseand otherpointsof interest. To

|T,

T4

T2

T9

T5

T,

45789B 6A0123

235679 48AB01

9A0124 B35678

5689A0 7B1234

B()2346 1!;789A

(6-10) [013457]/ (6-39) [023458]: 013457 2689AB I

124568 379AB0 , v

9A0 124 568 37B '(6-10) [013457]/ (6-39) [023458]

(6-24) [013468] All horizontaland verticalpairingsof trichordsin the mosaicsare membersof

(6-10) [013457]/(6-39) [023458]. EXAMPLE

8: DIAGONAL

HEXACHORDAL

MOSAICS.

So far,our exampleshaveonlydealtwithstringsof operatorsrelatedby and thustheirstructures aredirectly withthe commensurate transposition, sortsofstructures foundin pitch-class space. However,interesting patterns

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280

Perspectives ofNewMusic

Q)

X X X

'C

uC

a,Q 'C

'C

'C

XN

0

KK

0

K cl\ r4

...

11.

'C

'C

~

C

X

fr < 0C

0

0

X

X X *-

X .

.

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withPitch-Classes Morris, Composition

281

can be introducedin stringsof operatorsrelatedby I and M as well. We shallgiveonlyone briefexamplehere,as a glimpsethroughthedoor intoa a sequenceof operatorsthat realmworthexploring.Example9 illustrates the similar to Mallalieu sequence, althoughit only possessesproperties containsfourdistinctoperators.The operations,however,are not limited to transposition. *

*

*

The fouroperatorsemployedin Example9 are relatedto each otheras a that group,a way of describingkindsof fundamental algebraicstructures underliesmuchofthebook.22Althoughnottheonlytheoristto use group provided theoryto describerelationsin music,Morrishas nevertheless some interesting new insightsabout its applicationthatare helpfuland to composerand analystalike.23We shallsamplejust a few. suggestive The group foundin the previousexampleis itselfa four-element subgroupof the groupformedby the TTOs. As mustbe the case, it is selfcontained;i.e., anycombinationof the elementsof the group mustproduce an elementof thesubgroup.Anothertypeof four-element subgroup of the TTOs is foundin the pitch-classoperatorsrelatingthe rows of ArnoldSchonberg'sSuite Op. 25 forpiano. Both typesare illustrated in Example10. ToP:

4

5

7

1

6

3

8

2

B

0

9

A

T6P:

A

B

1

7

0

9

2

8

5

6

3

4

T8IP:

4

3

1

7

2

5

0

6

9

8

B

A

T2IP:

A

9

7

1

8

B

6

0

3

2

5

4

EXAMPLE

ToP:

0

1

4

T6P:

6

7

A

T3P:

3

2

B

T9P:

9

8

5

10: ARNOLD

SCHONBERG,

OP.

25

One immediateconsequenceof the group structureof the two sets of operatorsthat Morris utilizes extensivelyis visiblein the columns of Example10. As maybe seen,thetwo subgroupsofoperatorspartitionthe chromatic totalityintobatchesof pitchclasses,each batchalwaysand only

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ofNewMusic Perspectives

282

associatedwith itselfin each subgroup. In general,group-determined equivalenceclassesare foundto be of greatuse throughoutChapterFour and subsequently, fora numberof purposes.The factthatthe two setsof classes two different fundamental oftwelve equivalence represent partitions revealsthat the two subgroups,althoughof the same order, are not structurally equivalent.24 As Morrisdemonstrates, the equivalenceclassesgeneratedby a group allow us to comparedifferent structures generatedby the same group,or find different but througha given conversely, pathways analogously-related structure a of Milton Babbitt's comtrichordal generatedby group.Many sorts of these the positionsemploy relationshipsby maintaining same of groups operatorsthroughouta series of trichordalarraysbased on different trichordtypes.This is illustrated in Example11. To:

0-1

4

0-1

T6:

6-

A

6-75

6-74

6-

T3I:

3-2B

3-2

325

3-2

T9I:

9-8

5 EXAMPLE

(BY COLLECTION,

9-8

B

A

0-1

9-8

11: TRICHORDAL WITHOUT

A

B

0-1

9-8

5

4

ARRAYS

REGARD TO ORDER)

A subgroupof a largergroupof operatorswill (informally) allow us to partitionthe largergroupinto a numberof discretebatchesof operators, eachofwhichhasa structure equivalentto thesubgroup.These batchesare called cosets,and Morrisshows how it is possibleto describeboth what underthe happensto themas a whole and to theirelementsindividually operatorsof the subgroupitself.This has some immediateanalyticaluse. Forexample,we maythinkoftheunderlying structure ofa blockofone of Milton Babbitt'stwelve-partall-partition arraysas constructedfroma collectiontransformed T and I operasinglehexachordal byall twenty-four tions.25The block of six combinatorialpairs of rows distributesthese the TTO group operationsas a subgroupand its fivecosetspartitioning definedby T and I. Transforming theblockbyone oftheoperationsofthe and each subgroupwill map the subgroupand itscosetsonto themselves otherin particular We can of ways. perform analogoustypes mappingsby thenotionofautomorphism, whicheffectively reorients us within invoking thelargergroupin termsof thestructure of thesubgroupand itscosets.26 Thus, to returnto our example,we can invertthe arrayby anotherodd indexnumberand come up witha similarsortof mappingof thecontents ofthepartitions ofthelargergroup.The factthatthemappingsoflocations

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with Pitch-Classes Morris, Composition

283

ofoperatorsin theblockarefundamentally different underinversion byan in structure of the even indexnumberis nicelycapturedby the difference two four-element subgroupsfound in Example 10: the mappingsnow a different representcompletely underlying subgroupand cosetstructure.27 The precedingis illustrated in Example12. P = {9, A, B, 0, 1, 2} IGIP:

TBI/G/P:

T9I/G/P:

ToIIG/P:

To T5I

T6 TBI

TBI

T5I

T9I

T3I

T6

To

T4

TA

TOI T7

T1

T1 T6I

T7 ToI

TAI

T4I

Ts

TB

T8I T3

T2I T9

TBI T6

T5s TO

T2 T7I

T8 T1I

T91

T31

T7I T2

T1I T8

TAI T5

T4I TB

T3 T81

T9 T21

T81 T3

T21 T9

T1

T7

T4

TA

T4

TA

T71 T2

T1I T8

T5s To

TBI T6

T81 T3

T2I T9

T5 TAI

TB T41

TJ

TO T7

T4I

TAI

TB

T5

T91

T31

T1

T3I

T7I

T2

TJ

T9

T1I

T8

EXAMPLE 12: ARRAY BLOCK BY SUBGROUP AND COSETS

One of the morecompositionally suggestiveaspectsof Morris'sdiscussionofgroupsofoperatorsis hisinvestigation ofthewaysnon-TTOs can be combinedwithTTOs to producemusicallyinteresting transformations. I shallillustrate a fewof thesein combinationwithsome arraystructures to showtheirpotentialusefulness. For thefollowing I willuse thetwelve-tone rowillustrated in Example13. P:

2

3

1

6

A

5

7

4

8

EXAMPLE 13

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B

9

0

ofNewMusic Perspectives

284

I shallbe concernedwiththreenon-TTOs, definedas follows: 0:

(0) (2) (4) (6) (8) (A) (1 3 5 7 9 B)

J: (A 4) (B 5) (0 6) (1) (2) (3) (7) (8) (9) Q:

(8 4 0) (B 7 3) (1) (2) (5) (6) (9) (A)

Put moreinformally, O transposesone whole-tonecollectionby T2 while the other J mapsa memberof (6-7) [012678] onto itself constant; holding while T6 its constant;andQ mapsa memberof (6by holding complement onto while itself T8 20) [014589] by holdingitscomplementconstant.O concatenated sixtimeswillreturnto To,Q does so at threeiterations, while J is itsown inverse.28 Whenwe applyeachoftheseoperationsto theparticular rowP foundin some and Example12, interesting compositionally suggestive thingsoccur. OP mapsP onto RT2IP, whilethe othertwo operationstakeP out of its classicalrowclass.29However,P is orderedso thatitsfourdiscretesegmental trichordsforma mosaic consistingof one each of the trichordtypes [012], [013], [014], and [015]. There are fourdistinctmosaicclassescontainingone each ofthesefourtrichordtypes,and a look at theimagesofP underJandQ revealsthattheirdiscretesegmentaltrichords formrepresentativesof the remainingtwo mosaicclasses.30We can dramatizethe differences amongthethreemosaicclassesbylistingthehexachordtypeseach includes.While the various transformations of P change the ways the trichordscombineto formthe aggregate,the internalorderof the individualtrichords in each instanceis maintainedto withinI and R31 All of theabove is illustrated in Example14. As we noted,a singleapplicationof O to therowP in Example13 yields the equivalentimage to the TTO RT2I applied to P, the retrograde inversion at indexnumber2. This is not to saythattheoperationO is the equivalentto the operationRT2I. A secondapplicationof O, in contrast, takesus out of the classicalrow-classof P, but due to the natureof the hexachordtype,(6-14) [013458], anynumberof applications of segmental the operationwillpreservethe classof thesegmentalhexachords.32 Additionalapplications of O to P carryus throughtwo membersof a thirdrow of our originalrow class, and class, equivalentto the M-transformation eventuallyback throughanothermemberof our second row classto our in Example15. As maybe observed,each pointoforigin.This is illustrated of the threeorderings so derivedcontainsa different mosaicof fourdifferenttrichordtypes. So farwe haveonlydealtwithO appliedto P. A quickinspectionreveals thattheapplicationof O to T1P willgeneratean orderingwhichis a TTO transformation of two applicationsof O to P, our secondrow classin the

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wi'th Morris, Pitch-Classes28 Composi'tion

OP:

2 5 3 +3 -2

JP.

2 3

QP:

2 B 1 -3 +2

+1

-2

A

6

7

if

+4 6

4

9

+4 -3

-5 +4

4 B 17A -

3-

A 5 +4 -5

3 0 -3+4

8

285

i BO -2 +1

815 4

9 6 +4 -3 7 9 8 +2 -1

TrichordalMosaics: OP:

253 16A7 948 1IiBO

(6-14) [013458] 6-2) [012346] (6-5) [012367]

JP:

231 104B 7A8 1 596

-'(6-1) [012345] p6-15) [012458] (6-18) [012578]

QP:

[013478] /(6-44) [012569] 2Bl1 6A5 ~-(6-19) 304 1798 ~(6-12) [012467]! (6-41) [012368] (-1) [012345] EXAMPLE 14

OOP:

2

3

1

6

AS5

O'P:

2

5

3

6

A

7

Q2p:

2

7

5

6

A

9

2

9

7

6

A

B

04P:

2

B

96

O5P:

2

1

B

7

Al1 6

A

3

4

8

B

9

4

8

1

B

B

4

8

3

1

1

4

8

5

3

3

4

8

7

5

5

4

9

7

9

8

Trichordtypesof Q2P and O05P:[013], [037], [014], [025] Trichordtypesof 03P and 04P: [025], [037], [OlS], [027] EXAMPLE 15

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0-~

0--

286

ofNewMusic Perspectives

precedingdiscussion.O appliedto T2P keepsus withinP's rowclass,while OT3P putsus in thesecondrowclass.We can generalizethisto saythatO of P will yieldmembersof P's row class, appliedto even transpositions whileO appliedto odd transpositions will createmembersof a new row class. This is a directconsequenceof the effectof the even and odd T we mayobservethatthe operatorsappliedto thecyclesofO. Consequently, inverseof O, reversing the directionof mappingin O's cycles,has the of P willyieldmembers oppositeeffect:O-1 appliedto odd transpositions ofP's rowclass,whileO-1 appliedto eventranspositions ofP sendus into thenew rowclass. The combinationof O and I similarly splitsour originalrow classinto two parts,suchthatodd transpositions ofIP combinedwithO staywithin therowclass,whileeven transpositions pushus intothesecondrowclass; theinverseof O havingtheexpectedcomplementary effect.The operation hastheeffect ofpartitioning thegroupof T andI operatorsintoa subgroup ofordertwelveand itscoset.OperationsJ andQ also partition therowclass of P into distinctbatchesof a subgroupand its cosets,each subgroupor coset with its distinctrow class of images.The numberof distinctrow classesis determinedby the invariance of thecyclesof the nonproperties TTOs undertheTTOs. This is summarizedin Example16. OP: 2

5

3

6

A

7

9

4

8

9

B

0

OT1P: 5

4

2

9

1

6

8

7

B

0

A

3

JP: 2

3

1

0

4

B

7

A

8

5

9

6

JT1P: 3

A

2

7

5

0

8

B

9

6

4

1

JT2P: A

B

3

8

6

7

9

0

4

1

5

2

2

B

1

6

A

5

3

0

4

7

9

8

QT1P: B

0

2

3

7

6

4

5

9

8

A

1

QT2P: 0

5

B

4

8

3

9

6

A

1

7

2

5

6

0

9

1

4

A

3

7

2

8

B

EXAMPLE

16

QP:

QT3P:

We arejust scratching thesurfaceof themusicalworldgeneratedby the made from the TTOs P. As group plusJ,Q and O appliedto theordering mosaic class suggestedabove, P, JP, and QP each representa different thesamefourtrichord containing types,and all containthesametrichordal of operationsin our largegroupwillhave orderings.Not all combinations

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with Pitch-Classes Morris, Composition

287

thisproperty, butitis interesting to notethatwithcertaincombinations of sets of operatorsit is possible to generatesimilarfamiliesof different trichords.This is illustrated in Example17. The sharedhexachordaland trichordal familiesprovidea wealthof paths typesamongthesedifferent areasofthiscircumscribed world.The differthroughhighlydifferentiated ent hexachordtypesinvolvedrangefromthe familiar groundof the first order all-combinatorial hexachords,throughthe T-combinatorialhexachordtype,(6-14) [013458] of P, to the strangeworldof the Z-related pair,(6-19) [013478] and (6-44) [012569]. -1P: 2 1 B -1 -2

6 A 3 +4 +5

5 4 8 -1 +4

9 7 0 -2 +5

OJT3P: 1 0 A -1 -2

B 3 8 +4 +5

4 9 7 +5 -2

2 6 5 +4 -1

8 1 B +5 -2

4 0 7 -4 -5

9 A 6 +1 -4

5 3 2 -2 -1

03P:

2 9 7 -5 -2

6 A B +4 +1

1 4 8 +3 +4

5 3 0 -2 -3

JO3P:

2 9 7 -5 -2

0 4 5 +4 +1

1 A 8 -3 -2

B 3 6 +4 +3

3QP:

2 5 7 +3 +2

6 A B +4 +1

9 0 4 +3 +4

1 3 8 +2 +5

QT2IO-'P:

EXAMPLE

17

We can gain a more specificsense of how thesenon-TTO operations of mightfunctioncompositionally by consideringsome transformations that have been constructed to take of the arrays specially advantage operations' properties.Example18 consistsof a briefarray(labeledA), built frommembersof our originalrow class,and threetransformations of A. The originalarraycontainssix membersof P's row class, relatedby a subgroupofTTOs thatwillpreservethecyclesofO andQ. This means,for A by Q to yieldthe same sequence of example,thatwe can transform articulated rowclass,but one sharing partitions by membersof a different the trichordal of P's row class.Additionally, orderingproperties by transA by O, we can generatethe same sequenceof partitionswitha forming

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288

ofNewMusic Perspectives

different set of membersof P's row class, embeddedin the arrayin a different way fromA. A comparisonof the intervalsand collectionsat variousspotsin A and OA willshow thisvividly.However,therowswere selectedso thattheircollectiveimagesunderO and underRT2I carefully would be equivalent.Thus, we can constructstillanothertransformation, whilecontainingthesame RT2IA, thatwillreversethe orderof partitions rows as OA. We could exploitthesecompositionalopportunitiesin any numberofways. A: T,P: RTsIP: RT9IP: RT4P: TiP: RT8P:

23 58691

A07B

4 BA07

1 3869524

48B 23 90 308B92 A576 1

2B 54691

A837

16A53 0

0 7A83

1 B469520

047 2B 98 B84792 A536 1

25 786B3

A091

36A79 4

4 1A09

3 586B724

16A57 4

90 A

1524A3867

8574031629B A

QA: 98 A

15207B463

45308B16297 A

OA: a b c d e f

481 25 BO 5081B2 A796 3

BO A

372415869

879405362B1 A

RT2IA: a c b f e d

25 786B3A091 4 4

35081B2A796

36A BO 25 8794 1

79481 A 05362B

3724

BO 15869

1 A09586B

A 7243

EXAMPLE 18

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289

with Pitch-Classes Morris, Composition

from Example19 containsa second briefarray(labeledB), constructed membersofP's rowclassrelatedbyoperationsthatpreservethecyclesofj, and the transformation of B by J. Under J, all of the rows of B are transformed intomembersofanothersinglerowclass,as maybe seenin the example.The row classso invokedis thatwhichreplicatesthe trichordal of P, as notedabove. This propertyhas a specialconsequencein orderings B, as therowsused also exchangetrichordal contents,in two quartets.The discretetrichords ofa singlelynein B aredistributed amongthefourlynes ofitsquartetinJB,and viceversa.33Thus theentiretrichordal contentsof B is preserved and redistributed inJB. B: ToP: T6P: T7IP: TIIP:

897 5

231

6A

4 BAO 7

RT3P: 30 968 RT9P: 75 RT4IP: 4 RTAIP: A1B26

57 619203B8

3

2 B7A819 4 514 2 809B 6 A 3 504

04B1A25 869 73

48B90 A7 52 6 31

36 41 5 AOB 2 978

JB: 897 B 36 A 41520

231 A 546 908 7B

04

B7 01926358

7

3

2 B1A 8695 3

574819 2 0 B6A

A 4

6A5142B 809 73

A8596 47 B2 O 31

30 A1 B 465 2 978

EXAMPLE 19

But stillmoreinteresting are available.The two quartetsof possibilities rowsin B each belongto a different batchofrowsin P's rowclassbasedon theirimagesunder0, so thatbyapplyingO to B and itsinverse,0-1, to B, we can turneach of the two quartetsof rowsinto membersof eitherrow class under0. By furthermanipulating one of the resultingarrayswith TTOs, it is possibleto map the row contentsof one onto the other.The

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290

Perspectives ofNew Music

locaresultingpairof arrayscontainthe same rows,but in two different in tions,and the same partitions,but in reverseorder.This is illustrated Example20. OB:

a b c

d e

f

g h

8B9 7 50 4 A3126

253

6A

4 1AO B68 97

79 63B20518

9

5

2 734 80B1 5

19A83B 2 6 704

4 A

0413A27 86B 95

481B0 A9 72 6 53

56 43 7 A01 2 B98

RTB4O-'B:

f

e g h a b d c

50 A3 2 8B9 1 746

B6873 49 12 5 AO

219A83B 657 04

42 780B16A5 3 9

95

A01

0

4

3 B98

6A7948 1 5 3B2

1 3A2 86B7 0

756 24

467 2 BO 3 518A9

EXAMPLE 20

The possibilities forinteresting of non-TTOs compositional exploitation in combination withthemorefamiliar are endless,but operations virtually we willrestrain ourselvesto two moreexamplesto suggesttheoportunities availablefromcompoundsoftheoperations,bothTT and non-TT, in our extendedgroup. As we mentionedabove, it is possible,by compounding operators,to createadditionalfamiliesof rowslinkedby sharedtrichordal embeddedin different orderings typesofmosaics.Whenwe applyO toJB, we generatean arrayonce againconsistingof two quartetsof rows.The rowsofone quartetaremembersoftherowclassinvokedbyrotatingJP six while those of the second positions(order-number transposition by T6), containthesamesetoforderedtrichord typesas OT1P, embeddedin (6-8) [023457] type hexachords. Example 21 contains the array,which redistributes the trichordalcontentsof those foundin Example20 in a manneranalogousto thatfoundin Example19.

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OJB: 8B9 1 56 A 43720

253 A 746 B08 91

04

19 03B26578

9

5

2 13A 86B7 5

79483B 2 0 16A

A

6A73421 80B 95

4

A87B6 49 12 0 53

50 A3 1 467 2 B98

EXAMPLE 21

And just to bringthe discussionfull circle,we shall offerarrayA, transformed firstby0-1, and thenadditionally byan (inner)automorphism ofQ, to show how our firstarraycan participate in the secondtrichordal both by meansof the (6-14) [013458] hexachords,and withthe family, Z-relatedpair.This is illustrated in Example22.

21 3867B A059 4 9A05

B6A35 4

489 21 70 B 108972 A356 1867324 B

70 A

B32491865

835401B6279 A

T2IQT2IO- A: 61 78AB3 2059 4 9205

3A275 4

489 61 BO 3 1089B6 275A 18AB764 3

BO 2

3764918A5

8754013A6B9 2

EXAMPLE 22

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*

*

*

One of the greatvalues of Composition is the way the withPitch-Classes authorprovidesglimpsesofrealmsforfutureexploration. As thepreceding is richwithposmightsuggest,the worldof non-TTO transformations but it is but one of vistas from a traversal of the text. sibility, many spied Anotherarea ripe forfurtherinvestigation is the conceptof chainsofset classes,specialstringsof overlappedmembersof a givenset class. Morris it withhexchains,and illustrates providesan algorithmforconstructing achordsoverlappedby trichords(90-98). Althoughchainsneed not be limitedin this way, our examplesshall also deal with hexachordsand trichords.Example23 containsa couple of samplechains,the latterof whichis therowoftheSchonbergViolinConcerto. To T8 To/ (6-14) [013458]: 03714851013148B 79014581037 T8I To T4 TBI To (6-18) [012578]: 9A31B461017182519A3 TAI TB EXAMPLE23: SCHONBERG,OP. 36

While Morris'sdiscussionfocuseson chainsof membersof a singleset to extendthe idea to chainsinvolving type,it is possibleand interesting membersof morethanone set type. Doing so, one maycreatepathways of set types,based on theirsharedsubsets.It is also througha repertoire to considertherestrictions interesting chains,a imposedbysuperimposing studycloselylinkedto the studyof relationsamongmosaicsof different of types.Such an approachcan yieldinteresting insightsintothestructure twelve-tone rows,althoughitis byno meanslimitedto twelve-tone theory. rows,each consideredin termsofits Example24 containstwo twelve-tone threetrichordal chains.The firstis drawnfromMorris'sown Piano Concerto,while the lattersuperimposestwo shortchainsbased on the hexachordalpair (6-19) [013478] and (6-44) [012569] used in the preceding discussion.34 A further extensionto theidea of chainsimposeslimitations oforderon theoverlappedcollections.Considerthepitch-class cycleHand itstransformationsunderO fromabove,foundin Example25. The readerwillnotice thatthe hexachordwe haveused is fromP, the row used in the previous discussionof non-TTOs, and thatthe two hexachordsofP are themselves

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with Pitch-Classes Morris, Composition

(6-28) [013569]: To (6-49) [013479]:

293

T5

TA

561901131471AB128156190 T5

To

TA

TBI To (6-27) [013469]: (5)691013417AIB285169 T5 T4I RobertMorris,Piano Concerto (6-19) [013478] /(6-44) [012569] TO 061237B1841A519 Tol TA

(0)62137B814A15190 TAI EXAMPLE 24

H: OH: 02H: 03H: 04H: 05H:

: : : : : :

+1 -2 +5 +4-5-3 +4-3-5 +3-2+3 +5-2 +1 +4 -1 +5 -5 -2 -1 +4 + 1 +3 -3-2 -3 +4+3 +1 -1 -1 - 2 -5+4+5

03Hrl T8Hr5 Hrl OH T803Hr504H 25316A71523116A1532167Al2B916A11B29176AIB92161AI253... RT4IO4Hr3,etc. (r= rotation) EXAMPLE 25

Different relatedunder0, as was implicitin the foregoing. rotainversely tionsofH underdifferent numbersof applicationsof O yieldmultiplesof different of a set of trichordtypes.We can constructa chainof orderings different ofH bymeansoftheseduplications, transformations as illustrated in Example25.

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chaincontains Sincetheset typeofH is invariant under0, the resulting butone settype.However,we need notlimitourselvesthisway.Consider, forexample,the pitch-classcycleG, its transformations under0, and its rotations.As may be seen in Example26, we have constructeda more of G underO areequivalent compactworld,in thatcertaintransformations to rotationsof othertransformations of G underO and theTTOs. We can also see, that,as in Example25, thereare sharedorderingsof the various discretetrichords ofG and itstransformations, of allowingtheconstruction varioussetsof chainsmovingthroughthehexachordtypesgeneratedby O appliedto G. G: OG: 02G: 03G:

04G: 05G:

< 532410->: < 752430->: < 972450->: : < 1B2490->: < 3124B0->: G

-2-1 +2-3-1 +5 - 2- 3+2- 1- 3- 5 - 2 - 5 + 2 + 1 -5-3 -2+5+2+3+5-1 -2+3+2+5+3 +1 -2+1 +2-5 +1 +3 T202Gr

(6-1) [012345]--(6-8) [023457]--(6-32) [024579]

etc.

53214102B91467158917A01 532... T,0I4Gr2

etc.

TJOGr5 TI05Gr4 etc. 142136BIA81109BIA7215301142... etc. TB03Gr3 EXAMPLE 26

These chainscan be combinedin trichordal arrayswhichcan be further and to reflect thecontentsofthe ordered,bylynepair byquartet, variously hexachords of thelynes,as wellas availableorderings fromthetransformationsof G. This is illustrated in Example27. Once again,we can see how Morris'sideascontaintheseedsfora wealthof compositional growth.

Morrisgenerates hisvastedificefroma generalconcernwith"the natureof itself." His extensivedevelopmentsin the body of the book arise pitch fromhisinitialinvestigation ofpitchspacesofdifferent kinds,distinguished in by the propertiesof theirintervals.Intervalsare definedspecifically

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532

4

1 0 12 B 9

A 01B23146

4

6

75

B 9

7 64

9 8 5

8 9

6

7 A

0 A 7 3 5 8

5 1 0 A

EXAMPLE

8 9

7 A

A 7 6

8

0

9B2

1 4

3 2 B

64

5 3

1

27

termsof pitch,as the ratiosof theirfrequencies.It is not untilthe final of otherdimensionsof musicand their chapter,and some considerations withthe pitchdomain,thatthe notion of interval possiblerelationships we can see how manyof thetools becomesmoregeneral.Retrospectively, maybe generalizedbeyondthe pitchand pitch-class developedpreviously to contemplate the implications of Morris's domains,but it is interesting moregeneralnotionofinterval ideasin lightofa slightly fromtheoutset.35 Morrisopens his studyof pitchspaceswithc-space,a linearly-ordered areundefined,althoughlocationsin itappear pitchspacein whichintervals to be quantized, and "each pitch is the intersubjective correlateof [a] From this he his tools for frequency"(23). develops dealingwithcontour. More definitionbecomesavailablewiththe additionof measurableintervals.Two morerefinedpitchspacesaredescribedas follows:"... [W]hen different distancesbetweenpitchescan be perceivedand are attendedto, we can classifypitch-spacesas eitheru-spaces,havingdifferent intervals betweentheirsucessiveadjacentpitches,orp-spaces, havingequal intervals betweenadjacentpitches" (23). From his p- and certainu-spaces,as we mentionedabove, he derivesmodularspaces, based on the selectionof modularintervals. Anotherconsiderationof the same issues mighttake the following as a linearly-ordered approach.We can definec-spacesomewhatdifferently continuumin which intervalis undefined,and in which, forany two distinctelementsof c-space,thereis at leastone elementbetweenthem. (This is similarto the "densitytheorem"forrationalnumbers.)Givena seriesoflocationsin c-space,we can orderthemwithregardto each other, but we haveno tools formeasuringtheirrelativedistances:as in Morris's c-space,we only know whethera givenlocationis higher,lower,or the same as anothergivenlocation.Thus we maystilluse Morris'stools for dealingwithcontourofspecificcollectionsoflocationsin thisspace;we are thatthespace is not discretely simplyasserting quantized.

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We can quantize c-spaceinto a second space, step space or s-space,in whichelementsare distinct;in otherwords,givenanyset of elementsin c-space,we are awareof how manyelementsof the setsare betweenany pairofelements,althoughwe arenot concernedwithanyabsolutemeasure of distancebetweenelements.In s-spacewe can defineintervalsas the numberofsteps,or elements,betweentwo elements.S-spacestillallowsus to measurecontour,butadditionally allowsus to measureordinalintervals, and thusdefinetheordinal-interval-preserving operations,and theirconseSince in intervals are determined quences. s-space onlybynumbersofsteps, can be broken into we care to: modularity mustbe s-space anymodularity determined other criteria than the itself. by space By construingthe pitchcontinuumin termsof s-space,and usingthe measureof frequencyratiosas an additionalway of characterizing the intervals betweenelementsof s-space,we can generatep-spaces,in which the ratiosare equal, and u-spaces,in whichtheyare not. We can use the oftheintervalmeasuresofu-spaceto determinepossiblemoduproperties lar reductionson s-space; p-space,like s-space,can be reducedto any we use octaveequivalenceas a familarbitrary modularity. Conventionally, iarmodularity on p-space,butthisis basedon bothperceptualand cultural ofthespace. criteria,ratherthanon thestructure about the domain in termsof the interaction between Thinking pitch and other of measures interval some results,and s-space yields interesting the abstractnatureof itsdefinitionhas ramifications in otherdomainsas well. A greatdeal of our thinkingin pitch involvesthe simultaneous invocationof different spaces. For example,the invocationof s-spacein diatonicmusic allows us to recognizethe equivalenceof intervalsand collectionsthatare distinctin our familiar equal-tempered p-space,as well as in the spacesobtainedbyothertuningsystems.Our comprehension of diatonicsequences,forexample,depends on the recognitionof s-space equivalencesprojectedagainstthe intervallic inequalitiesof the diatonic space. Conversely,the particularinequalitiesof measurein the diatonic spaceallowus to identify uniquelytheelementsofitss-space,as modularly reducedby octaveequivalence.The interplay betweenthe two spaces is crucialto our understanding of tonaland modalmusic. fordealingwith eventsin the pitch S-space gives us some flexibility continuum.The locationsin c-spaceof elementsof an s-space,underthe rightmusicalconditions,need not be fixedto specificfrequencies.By movesbetween hearingc-spacethrougha givens-space,we can distinguish elementsofthes-spacefromornamental bendsor chromatic inflections ofa singleelementof thes-spacein thecontinuum.This is useful,forinstance, fordealingwith vibratoor blue notes in jazz.36 When such inflections become sufficiently quantized,we can createthe sortof associationsthat allow us to construean inflectedelementof a givens-spaceas a different

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elementof thesame s-spaceshiftedagainstthec-spacecontinuum.This is whatwe do when,forexample,we referto #4 tonicizing5 fora precisely briefpassage.Much of the richnessof tonal musicinvolvessuch multiple of pitcheventsin termsof different interpretations spaces.37 We need not limitour use of s-spaceto tonality.By consideringthe octatoniccollection,a memberof (8-28) [0134679A],as an eight-element modular s-space, with the octave as the modulus, we can construct TTOs applied sequencesof elementsrelatedto each otherby the familiar mod 8, with the sortsof interval-preserving are familiar that properties from the chromaticworld. The intervalspreserved,however,are the of the octatoniccollection'ss-space,so thatwhenwe realizeour intervals examplesas notes in p-space,we wind up with different-sized p-space intervals fora givens-spaceinterval.Nevertheless, it is not hardto learnto heartheinterval-preserving of thetransformations in theapproproperties when are reinforced contour priates-space,especially they by preservation.38This is illustrated in Example28. Mod 8 s-space:

P: 0 1 2 7 3 4 6 5 T6IP: 6 5 4 7 3 2 0 1 RT2IP: 5 4 6 7 3 0 1 2

Mod 12p-space:

P: 0 2 3 B 5 6 9 8 "P'P: 9 8 6 B 5 3 0 2 R'"IP: 8 6 9 B 5 0 2 3

P:

r'r'p:

P:

RCr,p:

EXAMPLE 28

The interactionof different spaces need not be limitedto the pitch domain. It is not hard to imaginethe possibilities of interplay between "beat space" and "tempo space" in the temporaldomain,forexample. in suchworksas theVariations for ElliottCarter'soverlappedaccelerations Orchestraand theDouble Concertotakeadvantageofour abilityto follow a stringof "equal" beatsthrougha changingtempo,whilerecognizing the

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ofthattempoin a multipleof theaccelerated beats. returnoftheinitiation in Example29.39 This is illustrated J.=J=80--

accel.

J=80

J=96

J=115

J=139

J=166

etc.

-

J=201

r rrIrrrr rrr ' r r-rrr r r r-r-3

3

3

EXAMPLE 29

We can followa beatspacethroughabrupttempochangesas well. In the following simpleillustration (Example30), we mayconsidereach attackas an elementin a beat space equivalentto an s-spacein the temporalcontinuum. In this space, we can recognizea modularityincurredby the changesin absolutedurationof bundlesof beats. However,we can also durationsmeasuredagainstan absolute recognizethevariousbeats'relative tempo.The secondpartof theexamplesuperimposestwo stringsof beats to show theirinteractions. When the beat-spacemoduliare reinforced by othermusicaldimensions,it is possibleto followboth beatspacesat once, and further comparethemin a moregeneraltempo-space.40

3rr r' r 43 4 _

3

2 4

j 98 8

3

Fr rr vp V

EXAMPLE

'r

rrpr

30

The ramifications of the precedingdiscussionextendbeyondthe confinesof thisreview,and would includethe interaction of spacesin more thanone musicaldomain. One need simplyremember, forexample,the rowsto see how waysordernumberand pitchclassinteractin twelve-tone

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createequivalencein another.41 equivalencein one domainmightmusically Such a notionis alreadyimplicitin thosewritings on twelve-tone musicin whichsegmentalcollectionsreceivespecialattention.42 One ofthevaluesofMorris'swork,and one ofthereasonshe takestime to presenta varietyof pitch spaces beforeconcentrating his effortson chromaticpitch-class is it is that and to extend worthwhile space, possible all of his formulations into different or different interdomains, virtually of those domains. pretations *

*

*

As I suggestedat theoutset,the book is as interesting forwhatit chooses not to addressas forwhatit does address.Conspicuouslyabsentfromthe textis anyextensivediscussionof how we mayconstruethe structures we aredealingwith.This is not to saythattheauthoris unconcerned withthe issue.He is at painsto show us how to projectthe materials of hisdesigns in a varietyof ways,and his discussionof variousp-spacesand criteriafor betweenboth orderedand unorderedsetsdemonrecognizingsimilarity stratehis desireto erecthis structures on a sound perceptualbase. But Morrisrathercourageouslydependson his readers'imaginations forfindvarious means to their hear his more elaborate construcing waythrough tions.Althoughthe authortakesthe riskof losinghis readers,by leaving the point open he invitesus to considerthe myriadways in which his structures maybe construed. The compositionaldesignin Example31 allowsus to raisethe issueof construalin a ratherdramaticway.The heartofthisdesign,thelowerseven lynes,replicatescertainfeaturesof Morris'sDesign VI (265-69). It is constructedfroma singlesegmentclass of seven elements.Its fifteen columnsall containmembersof thesegment-class's setclass,and represent the possiblepartitionof sevenelementsinto sevenor fewerparts.Unlike Morris'sdesign,however,thisparticular portionof thedesignin Example 31 is basedon an orderingofthediatoniccollection.43 We maycompose out thisportionof Example31 as a designin itself. Becauseof theubiquityof thediatoniccollectionin itslynesand columns, it would not be hardto realizethe designas a canonicpathwaythrougha numberofclosely-related tonalkeys.If we allowourselvesto add a layerof or other supportingvoices,compositionally by instruments distinguished a musicalsurfaceembodyingthe design means,we could easilyfabricate that would encourageus to hear it fromthe perspectiveof tonality. Withoutthesupportoftonalvoice-leading cues,we couldstillcomposethe designto accommodatethe sortsof interval-duplicating patternsbetween associatedwith such arrays, lynesand columnsthatare more frequently whilethenatureofthegenerating set typemightstillencouragea listening associated with strategy pandiatonicism.

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5 3 1 8 A

1A 6 3

470B

9 52

692 Partitions: lls15 Top: Bottom: 43 All: 4315 A037?5

8

68941?

Top: Bottom: All:

5 52 522

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68B

8A? 6 13?

074 ---9A5 2

5? 4? 2 B

25 9A 307

213 421 42214

41 322 43221

312

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58A1

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B O 4 7 2 96

213 231 2414

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32 3212 322212

321 3313

41 215 4216

221 17

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-470B9250

(5) 7 75

EXAMPIE31

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92 74?

5?

6 B 1

49 B 2 7?

312 2213 32215

22 31 32

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The upperportionof thedesignin Example31 mayitselfbe considered separatelyfromthe whole design.It consistsof fivelynes,based on an columnscontain unorderedset type,the pentatoniccollection.Its fifteen offiveelementsinto two completebatchesof thesevenpossiblepartitions at mostfivelynes,withan additionalsingle-lyne of thecollecpresentation tion. As withthe lowerdesign,its realizationcould encouragea formof settype. hearingassociatedwiththepropertyof itsgenerating If we takethetwo designstogether,however,we findthattheresulting columnsforma set of distinctpartitionsof the twelve-toneaggregate, encouragingstill another mode of construal.The resultingcomposite designcould be realizedbytwo timbrally-distinct ensembles,in sucha way thatwould allow us to followeach ensemblein termsof the perceptual set type,whilehearing associatedwithits constituent familiarly strategies thewholewiththeperceptualstrategies associatedwithaggregate hearing. A compositionusingtransformations of the designin Example31 could shiftamong the design'svariouscomponents,changingour modes of construalfrompassageto passage. By not takingup the issueof construal,Morrisdoes not abandonit: on thecontrary, he makesita livelyone forthosewho wishto use histoolsin a way.As theprecedingexampleshouldsuggest,histoolsarenot responsible tied to one compositionalsystemor another,but maybe used to rigidly makea widerangeof music,evenmusicwhoseunderlying areto strategies playwithour modesofconstrual. In orderto preservetheecumenicism of hisdiscussion,Morrisalso stays prettygeneralabout the wayscompositionaldesignsmay be realizedas actual pieces. Once again, this is not to say that he avoids the issue of compositionalrealization.As we suggestedabove, he spends his final thematerials ofhisdesigns,and he chapterinvestigating waysofprojecting thepartial additionally providesseveraltoolsin ChapterFiveforinspecting theircomposition. orderingsof arraycolumns,as a meansof controlling But thesetools and techniquesprimarily deal withrealizingdetails.How thesedetailsmaybe marshalledinto a seriesof interlocked that strategies makesup the narrativeof a piece is leftalmost entirelyto the reader's This has led one reviewerto a verynarrowconclusionabout imagination. the natureof the musicone mightcomposeusingMorris'stools. Indeed, the attributesthat Morrisgenerallyascribesto his designs-coherence, closureand saturation-might lead one to imaginesuch musicas "relaBut theseattributes of his designsare tional,ratherthangoal-oriented."44 the backgroundagainstwhich the specificpathsof a composition,be it or not,mayunfold.45 Creatingthesepathsis partoftheactof goal-oriented with a compositional composing design. Fromtheforegoing, it shouldnot be concluded,however,thatcompositional designsare indifferent to the pathwayswe can compose through

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them.Compositionaldesignsmaycarrya wealthof narrative possibilities, and a changeof orientationtowardthe designcan frequently changethe natureofthosepossibilities. The musicofMiltonBabbittdemonstrates this well. Duringthepasttwenty-five particularly years,he haswrittena wealth ofmusicbasedon a relatively smallnumberofall-partition arrays, compositionaldesignswithveryparticularproperties.46 Many of Babbitt'spieces consistof a numberof concatenations of a given array,transformed by certainTTOs. Each sectionre-orientsthe design,providinga different on how themusicmaymovethroughit,and compositions with perspective sections to moveus throughthe multiple generally employoverallstrategies variousre-orientations in a singlecomplexgesture.47That Babbitt'scomafford a numberofpathways throughthemmaybe heard positionaldesigns in the varietyof his compositions,but thatthe designsimpose certain interestinglimitationson those pathwaysemerges from the family resemblances of the same amongthose worksemployingtransformations design.48

Compositionaldesignsare clearlymorethanstaticobjects,but theyare less than complete pieces. As Morris states, "Compositional designs are ... akin to figuredbass in Baroque continuo parts or the chord symbolsused in lead sheetsin jazz . . ." (4). As JosephDubiel suggestsin his reviewof thevolume,we are invitedto thinkof compositional designs in a contextofperfonmance. This specialqualityof compositionaldesigns, theirimminence, theirairof beingin theprocessofbecoming music,can be stillbettercapturedifwe changeour orientation in a mannersuggestedby the subtitleof the book. "A Theoryof CompositionalDesign" implies designas action,ratherthan object. In effect,those objectswe have discussedas compositionaldesignsare cross-sections of processes,snapshots a in moment the of an action,thecreationof a piece freezing performance of music. When we read Morris'svolume in thislight,thereemergesfromthe book an attitudeabout compositionand musicmakingin generalthat, whilebyno meansnew,is nevertheless ofgreatvalueevenin repetition: the satisfactions to be had from music are the rewards of an greatest making active,informed with its materials. At of the engagement everystage game, a makerofmusicmustmakechoices,whetherthatpersonis makinga new a performance ofan existingwork,or is engagedin composition,preparing some musical event as a listener. Each stageis a creativeact,and traversing thegreatestflights of the imagination are sustainedby a thoroughknowledge of theworldin whichone flies. The keyto thisattitudeis embodiedin thefinallineofthelastchapter,a "maximusuallyattributed to Schonberg:craftis imagination!"(312), but fromtheoutsetthetextrevelsin thedynamicinteraction betweenimagination and an awarenessof one's constraints at everylevel. Thus we can

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understand thatMorris'sdescriptions ofdifferent pitchspacesat theoutset of ChapterTwo are not merelya gesturetowardscompleteness preceding the "real" subject,twelve-note pc-space;rather,we can see the latteras a particularchoice withina largerrealm,and so generalizeall of his techworldswe careto. niquesbackto theotherpossiblepitchand pitch-class the tools of Four and Five Similarly, ChaptersThree, provideus with of in the choices selection of collectional materials,the conways making structionof orderedpatternswithparticular properties,and the compositionof arraysthattakeadvantageof theseproperties.Once again,at every availablewe careto exploit, stepwe mustchoose whichof thepossibilities and in whatmanner.These choices,of course,can be shapedby intuitive compositionalimpulses,but an awarenessof the natureand rangeof the availableto us at anystepcan itselfinform our creativedesires, possibilities and inspirefutureendeavors.His compositionaldesignsare in effecttools thatallowus to see thesechoicesbeingmade. Composinga particular pathwaythrougha compositionaldesignis sima continuation of the seriesof choiceswe make at everylevel, and ply of the same dialectic ofinformation and imagination, oftechnique partakes and intuition.But suchchoicesarenot absolutes;thereis no one rightway of makingmusic,and everychoicewe makeinvitesus to returnanother time and explorethe road not taken. Choices like these underlieevery or to aspect of music. There are endlessways to hear a performance, a as there are to a realize just perform composition, myriadways compositionaldesign,to exploitthe properties of a collection,and so forth,down to our fundamental choicesof the elementsof musicitself.Our constant senseofmusic'sfreshness, our feelings of replenishment as we returnagain and again to those works, performances,or fields of compositional endeavorwe hold dear,havetheirsourcein thismultiplicity. Morrishelps us celebratemusic'srichnessbyproviding us withtoolsto makeinformed, creativechoiceson manylevels,and by inference, on all levelsof musical engagement.

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NOTES 1. These include,in orderof theirappearance,reviewsbyMichaelCherlin (Integral2 [1988]:179-200), JohnRoeder (Music Theory Spectrum 11, no. 2 [Fall 1989]:240-51), JohnPeel (Journal 33, ofMusic Theory no. 2 [Fall 1989]:400-16) and JosephDubiel (Journal ofMusicological Research, #10,nos. 1 and 2 [1990]: 47-80). 2. Morrisdoes, however,providereferences in hisown endnotesto those in his or in his text, places previouspublicationsthat detail the in his techniquesemployed ChapterOne. 3. There is a slighterrorin Morris's discussionof the relationship betweena modularintervalin a u-spaceand the unique intervallic contextsof the resulting m-space'spitchclasses(Endnote5, Chapter Two). This will be true only forthose m-spacesresultingfroma modular intervalof a u-space that cannot be divided by another modular intervalof that u-space. Thus the octatonic collection, fromtheoctavereductionof a u-space althoughan m-spaceresulting made of alternating whole and halfsteps, does not containunique intervallic contextsforitselements.Usingthe minorthird,however, would createa modularspace forsuch a u-space with the desired property. 4. Morrisis consistently carefulto referto the originsand othertreatmentsof theoreticalideas throughouthis text. Such matricesare exploredin depthin Bo Alphonce,"The InvarianceMatrix,"Ph.D. 1974. diss.,YaleUniversity, 5. The listincludes,but is not limitedto, MiltonBabbitt,AllenForte, David Lewin, George Perle,JohnRahn, and Daniel Starr,and the chapteralso takes up a numberof issues found in Morris'sown previouspublications. 6. More formally, the typesof transformations Morrisis interestedin are referred to as inner investigating (see, forexample, automorphisms the definition of automorphisms and innerautomorphisms provided in I. N. Herstein,Topicsin Algebra[Lexington,Toronto:XeroxCol1964], 57). My thanksto JohnRahnforpointingout legePublishing, thiserrorto me. 7. The termarrayin this contexthas its originsin GodfreyWinham, "CompositionwithArrays,"Perspectives ofNewMusic 9, no. 1 (FallWinter1970):90-112,as Morrisnotes.

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8. Babbitt'sown account of the time-pointsystemmay be foundin Structure and theElectronic MiltonBabbitt,"Twelve-ToneRhythmic no. New Music 1 Medium,"Perspectives 1, (Fall 1962):49-79. of 9. I have adopted Morris's notationalconventionsin this reviewto In Example1, middleC is interpreted facilitate cross-reference. as 0, and the notesabove and below are writtenas integerswith + or In laterexamplesin pc-space,I haveused 0 forC, 1 signs,respectively. forCO/Db,and so forth,withA and B representing Bband B. 10. Indexnumbersarethoseconstantsformedbythesumsofpitchclasses arounda particularinversional axis. They are equivalentto the subin the notational form TJ. The termoriginatedin Babbitt, script "Twelve-ToneRhythmic Structure." 11. Morrishas also done so in the past; see, forinstance,RobertMorris, "ReviewofBasicAtonalTheory byJohnRahn," Music Theory Spectrum 4 (1982):138-54. 12. Thisaspectofthebook hasalso receivedattention in MichaelCherlin's review. 13. This moregeneralwayof thinking about intervals derivesfromDavid MusicalIntervals and Transformations Lewin, Generalized (New Haven and London: YaleUniversity Press,1987). 14. See also David Lewin, "Transformational Techniquesin Atonal and Other Music Theories," Perspectives New Music 21, nos. 1 and 2 of (Fall-Winter1982, Spring-Summer 1983):312-71. 15. This is notedin WilliamRothstein,"Linear Structure in theTwelveTone System:An Analysisof Donald Martino'sPianississimo," Journal 24, no. 2 (Fall 1980):129-65. ofMusic Theory 16. Such ideas are centralto Lewin, "Transformational Techniques"and Generalized MusicalIntervals. A moreextendedanalysisofthisportion of the Concerto may be found in Andrew Mead, "Twelve-Tone Organizational Strategies:An AnalyticalSampler,"Integral4 (1990): 93-169. 17. See RobertMorris,"On theGenerationof Multiple-Order-Function Twelve-Tone Rows," Journalof Music Theory21, no. 2 (Fall with 1977):238-62, as well as theappropriate portionsof Composition Pitch-Classes. 18. This rowis discussedin RobertMorrisand Daniel Starr,"The Structure of All-IntervalSeries,"JournalofMusic Theory18, no. 2 (Fall 1974):364-89, and by Babbitt, Lewin, Morris, and Richmond

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Browne in "Maximally Scrambled Twelve-Tone Sets: A Serial Forum," In Theory Only2, no. 7 (October1978):8-20. Some of its in havebeen exploredmorerecently and their ramifications properties ofthePitch-Class/Order-Number AndrewMead, "Some Implications IsomorphismInherentin the Twelve-ToneSystem,Part Two: The MallalieuComplex: Its Extensionsand RelatedRows," Perspectives of NewMusic27, no. 1 (Winter1989):180-233. in contextsinvolving the 19. RobertMorrishas madesimilarobservations of MichiganSchool of musicof Bart6kin lecturesat the University 1990. Music,February, 20. The termmosaicwas firstused in Donald Martino,"The SourceSet and its AggregateFormations,"Journalof Music Theory5, no. 2 (November1961):224-73. A mosaic is an unorderedcollectionof collectionsthatpartitiontheaggregate. discreteunorderedpitch-class in Andrew of the have been studied Mead, "Some Implications They Inherent in the Twelve-Tone Pitch-Class/Order-Number Isomorphism of New Music 26, no. 2 (Summer System:Part One," Perspectives Robert Morris and Brian Alegant,"The Even and 1988):96-163, Partitions in Twelve-Tone Music," Music TheorySpectrum10 (1988):74-101. 21. This trichordalmosaicwill be of the type discussedon page 89 of withPitch-Classes. These trichordal mosaics,containinga Composition of and three members another trichordal set [048] typenot containing an ic 4, transpositionally will relatedby T4, alwaysgeneratethree membersof thesameZ-relatedpairof hexachords, whichin turncan Z ofthewhole-tonepairings onlybe membersofthose pairsconsisting [0248] / [02] //[0246] / [04], or [0248] /[06] //[0268] / [04]. These are,in additionto thetwo pairsfoundabove, (6-4) [012456] /(6-37) [012348]and theirM/MI-images(6-26) [013578] /(6-48) [012579], as wellas thetwo MIMI self-imaging pairs,(6-17) [012478] / (6-43) includes the all-trichord hexachordtypein (6-17)) [012568] (which and (6-28) [013569] /(6-49) [013479]. All of thesehexachordal pairs produce interestingresultswhen run throughthe patternfound above, based on the preserved and changed contents of their intersections. 22. Morris'sdefinitionreadsas follows:"A mathematical structure consistingof a binaryoperation,calledo, and a set ofelements.To havea group structure,forany elementsa and b in the group, a o b is a memberof the set (thisis knownas closure).In addition,thereis an elementi, suchthatforanya in theset,a o i = a (i is calledtheidentity foreach elementa in the set, thereis an element).Furthermore,

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of a) such thata o a' equals i, the identity elementa' (calledthe inverse a ? (b ? c) = (a o b) ? element.A finalrequirement is thatofassociativity: c" (342). 23. See Lewin,Generalized MusicalIntervals, or MiltonBabbitt,"TwelveTone Invariantsas CompositionalDeterminants,"MusicalQuarterly 46 (1960):246-59. 24. They are, in fact,membersof the two automorphism classesof subof the the and Tn TnIoperators,as illustrated groups groupemploying in Morris's"AppendixThree" (322). 25. This does not, nor is it intendedto, accountforthe orderof the collections,but merelyreflectsthe distributionof all six distinct hexachordal mosaicsbypairin Babbitt'sall-partition arraysusingfirstorderall-combinatorial hexachords.For a lengthieraccountof these arrays,see Milton Babbitt,"Since Schoenberg,"Perspectives of New Music 12, nos. 1 and 2 (Fall-Winter1973, Spring-Summer 1974):328; Milton Babbitt,WordsaboutMusic, editedby StephenDembski and JosephN. Straus(Madison:University ofWisconsinPress,1987), "Three on Milton 98-104; JosephDubiel, Babbitt,"Perspectives Essays ofNewMusic 28, no. 2 (Summer1990): 216-61; and AndrewMead, "About About Time'sTime: A Surveyof Milton Babbitt'sRecent Practice,"Perspectives Rhythmic ofNewMusic 25, nos. 1 and 2 (Summerand Winter1987):182-235. 26. The automorphisms thatMorrisis concernedabout areall oftheform LGL-1whereG and L are membersof thegroup(/GI)in question.As Morrispointsout (168), an automorphism of thistype(usuallycalled an innerautomorphism) willmap a subgroupof IGI eitheronto itself or anothersubgroupof /G/. 27. Compare with Examples4-16 and 4-17 on pages 102 and 103 of AboutMusic. Babbitt,Words 28. These are equivalentto non-TTOs employedin RobertMorris,"Set and MappingsAmongPitch-ClassSets," Groups,Complementation, Journal 26, no. 1 (Spring1982):101-44. ofMusic Theory 29. A rowclassis theyieldof a groupof operatorsappliedto a particular ofthetwelvepcs. A classicalrowclassincludesallvaluesof Tn ordering and T,I, plusR (or plus To and TBI in theorder-number domain),the familiar of forms forty-eight Schonberg'spractice. 30. See Morris and Alegant, "Even Partitions" and Mead, "Some Implications,"PartsOne and Two. The fourthmosaicclassmaybe

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representedas follows, with the contained hexachord types as illustrated: 012 356 7AB 489

,-(6-3) [012356] / (6-36) [012347] (6-16) [014568] (6-3) [012356] /(6-36) [012347]

The duplicationof the pairof Z-relatedhexachordtypesreadilysuggestssome interesting strategiesthat mightbe employedwith this fourthmosaic class containingthese fourtrichordtypes. (See, for example, measures 27-28 in Arnold Schonberg's Fourth String Quartet.)RobertMorrishasgenerateda listofall thepossibletrichord mosaicclasses. 31. Directedinterval classesarenotatedherein italicswith + and - signs to indicatedirection. 32. The hexachordtype containsa [024] trichordin one whole-tone collection,and a [048] trichordin the other. As thereis only one hexachordtype,(6-14) [013458], withthisstructure, of applications O to a memberofitwillsimplymap itwithintheequivalenceclass(see Morris,page 88). 33. The termlyneoriginatedwithMichaelKassleras a termto describe ofpitchclasses.I use it,ratherthanthe musically uninterpreted strings word row,in orderto distinguishbetweenstrandsof an arrayand twelve-tone rows. 34. The rowof Morris'sConcertocontainsa numberof interesting properties based on inversionalinvarianceat both hexachordaland pentachords-plus-dyadic partitions.For additionaldiscussionabout multiple embeddings,see Robert Morris, "Set-Type Saturation AmongTwelve-ToneRows," Perspectives ofNewMusic22, nos. 1 and 2 (Fall-Winter1983, Spring-Summer 1984):187-217. 35. Much ofwhatfollowsowes a greatdebtto Lewin,Generalized Musical the issues of as well as the Intervals, interval, including conceptsof the interaction of multiplecriteriain one domain,or the interaction of criteriain morethanone domain.Similarissuesarediscussedin John of OrderingOf and In Rahn, "On Pitchor Rhythm:Interpretations Pitchand Time," Perspectives New Music no. 2 (Spring-Summer 13, of 1975): 182-203. 36. Or withthesortsof slidingintonationinducedbycertainapplications ofmusicaficta, or byfatigueor ebulliencein a cappella choirs.

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37. J. K. Randall,in "Three Lecturesto Scientists,"Perspectives ofNew Music 5, no. 2 (Spring-Summer 1967):124-40, reprintedin Perspectiveson Contemporary Music Theory, edited by BenjaminBoretz and EdwardT. Cone (New York:W. W. Nortonand Co., 1972), 116-28, of thisin a discussionof providesa particularly elegantdemonstration a passagein Chopin's PreludeNo. 10, measures9-16. Otherdiscussionsoftonalpitchspacesmaybe foundin Lewin,Generalized Musical forexample,in hisExamples2.1.1, 2.1.3, 2.1.4, 2.1.5, etc., Intervals, and in Fred Lerdahl,"CognitiveConstrainson CompositionalSysProcesses in Music, edited by JohnA. Sloboda tems," in Generative Clarendon Press,1988), 231-59; and "Tonal PitchSpace," (Oxford: MusicPerception no. 3 (1988):315-50. 5, 38. For moreon the octatoniccollection,see Lewin, Generalized Musical Intervals, AppendixB: "Non-CommutativeOctatonicGIS [Generalized Interval Systems] Structures;More on Simply Transitive Groups,"251-53. 39. These passagesare discussedbyJonathan Bernard,in "The Evolution ofElliottCarter'sRhythmic Practice,"Perspectives ofNewMusic26, no. 2 (Summer1988):164-203; and byDavid Schiffin TheMusicofElliott Carter(London: EulenburgBooks, 1983). JonathanBernardquotes Carter,from"Music and theTime Screen,"in The Wrtitings ofElliott Carter,editedby Else Stone and Kurt Stone (Bloomington:Indiana Press,1977):346, on thepassagein question:". . . [W]hile University each of these notationalsystemssounds as if it were continuinga the beat has returnedto thespeed of thatof the regularacceleration, firstbeat of thesix-measure scheme." 40. We can invokestillothercriteriaforidentifying elementsof a beat themfromtheirsubdivisions.Recognitionof space,or distinguishing beatssubdividedby two or three,forexample,allowsus to construe like spansof different lengths,suchas thosefoundin timesignatures 5/ or 7/8, as members ofa beat-space.This can haveinteresting implicationsin a wealthof music,fromBorodin(the Scherzoof the Third instancesof duple and Symphony)to Dave Brubeck(the different in BlueRondoa la Turk).A farmoreelaboratespace triplesubdivisions Musical dealingwith similarissues is found in Lewin, Generalized Intervals,Chapter 4, "Generalized IntervalSystems(3): A NonCommutative GIS," 60-81. 41. Anotherinstanceoftheinteraction betweendomainsmaybe foundin Musical Interals, in his exampleof a "directLewin, Generalized productGIS," an analysisof a portion of the last movementof

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Webern'sPiano Variations,Op. 27. involvingpitchintervalscoordinatedwithtimespans. 42. See, forexample,Martha Hyde, Schoenberg's Twelve-Tone Harmony: The Suite Op. 29 and the Compositional Sketches (Ann Arbor: UMI ResearchPress,1982). 43. The designis different fromMorris'sin a numberof otherwaysas well. Instead of havingexactlytwo segmentsper lyne, this design variesfromone to three.The designis also not cleanlyfinished,in that one of itssegmentshangsofftheend bythreeelements.This could be takencareof simplyby loppingof the offending elements(not a very or the solution) by following designby the appropriate satisfactory retrogradeinversion,and doublingan elementover a border.This would produce a design containingthe completeset of partitions twice,and a duplicationof the openingand closingpartition,4.3, at the midpoint.No doubt with a littleadditionalfiddlingone could constructa similarsort of design without the deficienciesin the presentone. 44. JohnRoeder,250. 45. For thatmatter,Morris'stoolscould be used to maximizedifferentiation in the use of collections,etc., in a compositionaldesign.Related in the musicof ElliottCarter,as practicesare employedextensively detailedin Schiff, MusicofElliottCarterand Bernard,"Evolution." 46. See Babbitt,Words AboutMusic, Dubiel, "Three Essays,"and Mead, "About About,"as well as StephenArnoldand GrahamHair, "An Introductionand a Study:StringQuartetNo. 3," Perspectives ofNew Music14, no. 2/15,no. 1 (1976):61-80; JohnPeeland CherylCramer, "Correspondencesand Associationsin Milton Babbitt'sReflections," Perspectives of New Music 26, no. 1 (Winter1988):144-207; and William Lake, "The Architectureof a SuperarrayComposition: MiltonBabbitt'sStringQuartetNo. 5," Perspectives ofNewMusic 24, no. 2 (Spring-Summer 1986):88-111foraccountsofsomeofBabbitt's arraysand theirproperties. 47. The effectof reorientation in a singleworkis eloquentlyillustrated with regardto Babbitt'sA Solo Requiemin the second of Joseph Dubiel's "Three Essays." 48. This was demonstrated in Ciro Scotto'spresentation to theSocietyof MusicTheoryAnnualConference, 1987. Baltimore,

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