jackendoff, r., & lerdahl, f. (1981). generative music theory and its relation to psychology....

7/27/2019 Jackendoff, R., & Lerdahl, F. (1981). Generative Music Theory and Its Relation to Psychology. Journal of Music The

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Yale University Department of Music

Generative Music Theory and Its Relation to PsychologyAuthor(s): Ray Jackendoff and Fred LerdahlSource: Journal of Music Theory, Vol. 25, No. 1, 25th Anniversary Issue (Spring, 1981), pp.45-90Published by: Duke University Press on behalf of the Yale University Department of MusicStable URL: http://www.jstor.org/stable/843466 .

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GENERATIVEMUSIC THEORYAND ITS

RELATIONTO PSYCHOLOGY

RayJackendoff ndFredLerdahl

For a numberof yearswe have been developinga formalized heoryof tonal music,based n parton the goalsandmethodology,but not the.substance,of generative inguistics.' In additionto beingof interestforwhat it says about music, the theory also has certainproperties hatrecommend t as a model for moretraditional ssuesin psychology.Thepresentarticle describesa representative ragmentof the music theoryand then shows how it bears on the theoryof visualperception.GOALSOFA GENERATIVE HEORYOFTONALMUSIC

The overall goal of the theory is to account for the musical intui-tions that a listenerexperiences n a particularonal musical diom. By"tonal" we meanmusicwhose pitch organizationncludesone specifiedpitch, the tonic, which determinesrelative consonanceanddissonance.By "musical ntuitions"we mean the unconsciousprinciplesby whichthe listenerexperienced n the idiom organizeswhat he hears,beyondthe simple registeringof such surface features as pitch, duration,vol-ume, attack envelope, and timbre.It shouldbe made clear that we takemusical intuitions as largelydistinctfromanythingthe listenerhasbeentaught formally.Thetheoryis thus explicitly psychological, n that it is

45

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concerned not with the organizationof music in and of itself, butwith the organizationhat the listener s capableof hearing.Like other music theories, our theory developsan analyticsystemwithin which relationshipsheardby the listenercan be expressed.Un-like other theories,however,our theory poses a furtherquestion:if ananalysismodels what organization he listenerhears,what is it that thelistener knows that has enabled him to arriveat this organization?Toanswerthis question,we have tried to developa formalmusicalgram-mar that models the listener's connection between the presentedmusi-cal surfaceand the organizationor organizationshe attributesto themusic. The grammar akes the form of a system of rulesthat assignsanalysesto pieces. By contrast,previous approacheshave left it to theanalyst's ntuitionto decidehow to fit an analysis o a particularpiece.

Readers acquainted with generativelinguistics will recognize thesimilarityof our researchgoalsto those of the study of language.2Lin-guistic theory is an attempt to describethe linguistic intuitions of anative speakerof a human language.Generative inguisticsseeks thisdescriptionin terms of a formalgrammarwhich models the speaker'sknowledge of his language. Because many people have thought ofusing generativelinguistics as a model for music theory, it is worthpointingout what we regardas the significantparallel: he combinationof psychologicalconcernsandthe formalnatureof the theory. Formal-ism alone is to us uninteresting,except insofaras it servesto expresspsychologically nterestinggeneralizationsand to makeempirical ssuesmoreprecise.Previousattempts to apply linguisticsmethodology to music haveproven relatively uninteresting because they attempted to translatelinguistictheory, more or less literally, into musicalterms,lookingformusical parts of speech, for instance, or deep structures,or trans-formations,or semantics.3 We believe that a generativemusic theory,beyond the general principlesstated above, must be sought in purelymusical terms, uninfected by the substance of linguistic theory. Ifsubstantiveparallelsbetween the two theoriesemerge(as they in facthave in a number of areas), they are to be regardedas simply an un-expected bonus. We are concernedabove all with developinga theoryof music.Another recurringmistakemade in previousattemptsto apply gen-erativelinguisticsto music is to regarda linguisticgrammar s a deviceto manufacture grammaticalsentences. Under this interpretation,amusical grammarshould be an algorithm that composes pieces ofmusic.4 In stating his criticismsand apprehensionsof a music theorybased on generativeinguistics,Babbitthas sucha conceptionin mind.sHowever, t was pointedout by ChomskyandMiller,andit has been anunquestionedassumptionof actualresearch n linguistics,that what is46



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really of interest in a generativegrammars the structure t assignstosentences, not which stringsof words are or are not grammatical en-tences.6 Ourtheoryof musicis thereforebasedon structural onsidera-tions; it reflects the importanceof structureby concerning tself notwith the compositionof pieces but with assigning tructures o alreadyexistingpieces.This view of grammareads to a crucial difference between the re-search paradigmsof linguisticsand music theory, revealinga way inwhich music is not very much like languageat all. In a linguisticgram-mar,perhaps he most importantdistinction s grammaticality-whetheror not a givenstringof wordsis a sentenceof the languagen question.A subsidiarydistinctionis ambiguity:whethera givenstring s assignedtwo or more structureswith differentmeanings. n music,on the otherhand, grammaticalityper se playsa far less importantrole, sincealmostany passageof music is potentiallyvastlyambiguous-thatis, it is mucheasier to construemusic in a multiplicityof ways becausemusicis nottied to specific meaningsand functionsas is language. n a sense,musicis pure structure, o be "playedwith" within certainbounds.Thustheinterestingmusical issuesusually concernwhat is the most coherentorpreferredway to hear a passage.Musicalgrammarmust be able to ex-pressthese preferencesamonginterpretations, functionthat is largelyabsentfromgenerative inguistictheory.

Our music theory specifically addressesthose aspects of musicalstructure that are hierarchical n nature. It is therefore not directlyconcerned with undoubtedly important matters such as thematicdevelopment, although these often play a role in analysis.We haveidentified four distinct hierarchical tructureswhich aresimultaneouslyimposed on a musical passage to form its structuraldescription(oranalysis):groupingstrucutre,metrical structure,time-spanreduction,and prolongationalreduction.Briefly,groupingstrucutredescribes hesegmentationof the music into motives,phrases,and sections. Metricalstructuredescribesthe regular,hierarchicalpatternof beats which thelistener attributesto the music. Both reductions,while using differentcriteria,ascribedegreesof relative mportance o all pitch-events notesor chords)of a passage. n time-spanreduction, mportance s measuredwith respect to the other pitch-events n the same time-span,where atime-span s a rhythmicunit constructed from aninteractionof group-ing and metrical structure.Prolongationalreduction also develops ahierarchyof pitch stability, but in ratherdifferentterms.It emphasizesthe connections among pitch-events,establishing heir continuity andprogression,their movement toward tension or relaxation. It is thiscomponentof our theory that correspondsmost closely to a Schenker-ian analysis.Elsewherewe describeall four componentsof musical structure n47



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detail.' Here we will restrictourselves o a discussionof grouping truc-ture. This is musicallythe simplestof the four structures;t is also sys-tematically the most basic, in that the other componentsmake heavyuse of it in their rules.Moreover, he groupingcomponentappears o beof most generalpsychological nterest, for the grammarhat describesgrouping structure seems to consist largely of generalconditions forauditorypatternperceptionthat have farbroaderapplications han formusic alone. As such, the rules for grouping eem to be idiom-indepen-dent-that is, a listenerneeds to know relatively ittle about a musicalidiomin orderto assigngrouping tructure o piecesin that idiom.Like the othercomponentsof themusicalgrammar,hegrouping om-ponent consists of two sets of rules. GroupingWell-Formedness ules(GWFRs)establishthe formal structureof groupingpatternsand theirrelationshipto the string of pitch-eventsthat form a piece; GroupingPreferenceRules (GPRs)establishwhichof the formallypossiblestruc-tures assignedto a piece correspond o the listener'sactualintuitions.Before discussinggroupingstructure,we must make an importantcaveat. At the theory'spresentstageof development,we aretreatingallmusic as essentiallyhomophonic;that is, we assumethat a single group-ing analysis suffices for all voices of a piece. Ourtheory is thereforeinadequate for more contrapuntaltonal music where this conditiondoes not obtain. We consideran extensionof the theoryto accountforpolyphonic music to be of great importance.However,we will notattemptto treatsuch music hereexcept by approximation.GROUPINGWELL-FORMEDNESSULES

This section defines the formalnotion "group"by statingthe condi-tions that all possible groupingstructuresmust satisfy. We use slursbeneaththe musicalnotation to represent he groupingof a piece. Ex-ample1 showsthe grouping or the firstfour barsof melodyin Mozart'sSymphonyin G minor,K. 550.8The first ruledefinesthe basic notion of a group.GWFR1. Any contiguoussequenceof pitch events (or drumbeats,and so forth) can constitute a group, and only contiguous se-quencescanconstitutea group.

GWFR1 permitsall groupsof the sort designatedn Example1 andpre-vents certain configurations from being designated as groups. Forexample, it prevents all the eighth notes in Example 1 from beingdesignated ogetheras a group,or the firstsix occurrencesof the pitchD. (This contiguity condition is what makesthe slur notation a viablerepresentationof groupingintuitions; if there could be discontinuousgroups,some othernotationwouldhave to be devised.)48



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LI -

Example1

u U

J

G2k- - - ,2 ..k.G k

Example2



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The second rule expressesthe intuition that a piece is heard as awhole rather hanasmerelya sequenceof events.GWFR2. A piece constitutesa group.The third rule providesthe possibilityof embeddinggroups,evidentin Example1.GWFR3. A groupmaycontainsmallergroups.The next two rules state conditions for the embeddingof groupswithingroups.GWFR4. Ifa groupG1 containspartof a groupG2, it mustcontainall of G2.

This ruleprohibits groupinganalyses n whichgroups ntersect,suchasthose shown in Example2. G1 in these examplescontainspartof G2,but not all of it. On the otherhand,all the groups n Example1 satisfyGWFR4, resulting n an orderly embeddingof groups.There are casesin tonal music in which an experienced istenerhas intuitions that vio-late GWFR4. Weturn to these phenomena n a moment.The second condition for embedding s perhaps ess intuitivelyobvi-ous than the other GWFRs.It is however formally necessaryfor thederivationof the time-spanreduction.GWFR5. If a group G1 containsa smallergroup G2, G1 must beexhaustivelypartitioned nto smallergroups.

GWFR 5 prohibits groupingstructures ike Example 3a, b, in whichpart of G1 is contained neither in G2 nor in G3. Note howeverthatGWFR 5 does not prohibit groupingstructures ike those shown inExample 4, in which one subsidiarygroupof Gxis further subdividedand the other is not. In fact such a situationoccurs n Example1 andisextremelycommon.Let us now consider the discrepancybetween the predictionsofGWFR4 and certainactualmusical ntuitionsaccordingo whichgroupsoverlap.Example5 presentsa typical case. In this example,one hearsthe beginningsof the third and fifth measuresas belonging simulta-neously to two intersectinggroups,at various evels of grouping.Thissituationis a violation of GWFR4, since there aregroups hat containpartbut not all of other groups.Such situtationsarecommonin tonalmusic, especially in developmentalpieces such as sonata-formmove-ments, where they provide a sense of continuity: overlapsat majorgroup boundariesprevent a piece from reachinga point of rhythmiccompletion.In additionto true overlap, n which an eventor sequenceof eventsis sharedby two adjoininggroups, there is another overlapsituation50



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a. \G2 _ G

b. G2 G3G1

G23 G3G1 j

Example3

kGExample4

51



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more accurately describedas elision. ConsiderExample 6. The firstgroup,markedpiano, is interruptedby the newfortissimogroup.One'ssense is not that the downbeat s shared,asif the firstgroupwere heardas Example7a. A more accuratedescriptionof the intuitionis that thelast eventof Example7b is elidedby the fortissimo.A total abandonmentof GWFR4 would additionallypredict theexistence of groupingstructuressuch as those in Example8, which donot in fact occur in music. In Example 8a the intermediate level ofgroupingbisects one element of the smallest evel of grouping;n Ex-ample 8b the intermediate evel of groupingoverlapsand the smallestlevel does not; in Example 8c the smallgroups overlapand the inter-mediateones do not.In order to make those and only those modifications appropriateto grouping structures (that is, to permit the groupingsshown inExamples6 and 7, but not those in Example 8), we proposeto dis-tinguishtwo formalstepsin describing piece's grouping tructure.Thefirst, underlyinggroupingstructure, s describedcompletely by meansof the groupingWFRsgiven above;that is, it containsno overlapsorelisions.The second step, surfacegroupingstructure,containsthe over-laps and elisions actuallyobserved.These two stepsareidenticalexceptwhere the surfacegroupingstructurecontainsan overlapor elision.Atpoints of overlap,the underlyinggroupingstructureresolvesthe over-lappedeventinto two occurrencesof the sameevent,one in eachgroup.At elisions, the underlyingstructurecontainsthe event understoodasbeingelided.The theory requiresa transformationalule to state the exact rela-tionship between underlyingand surfacegroupingstructure.Consideroverlapfirst. The essential idea is that adjacentgroupsin underlyingstructure that have identical events at their common boundarymayoverlap those identical events in the surface structure.To make thismore precise, safeguardsmust be added to the rule to ensurethat allgroupsmeeting at a boundaryare overlapped n exactly the sameway.Such a condition preventsthe rule from creatingsituationslike Ex-ample8b, c in which overlappings not uniform from one level to thenext. Here is the rule; the last two conditions are the desiredsafe-guards.

GROUPINGOVERLAP RULE. Given a well-formedunderlyinggroupingstructureG as describedby GWFR1-5, containingtwoadjacentgroupsg1 andg2 such that1. g endswith event e1;2. g2 beginswith evente2; and3. el e2;then a well-formedsurfacegroupingstructureG' may beformed,identicalto G except that

52



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A l l e g r o

or k71 F - ttF-7- f

,-------!J

Example5. Mozart:SonatinaK.279, I

p r tT-I r rr --

(( ' ' .. ... . ' - r. ' ' r " ' 'Example6. Hayden:SymphonyNo. 104, I, mm. 29-33

a.

__. , ,_I

fpI r rFr_ , r elided-4

t-... ..... . J ..

Example753



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K J 'L ) K )

a. k, )kI )

b. k,Y

k )

C"\ JLrk )

Example8

el e2 e

underlying surface

Example9

54



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1. it contains one event e' whereG had the sequencee1e2;2. e' =1e = e2;3. allgroupsendingwith e1 in G end with e' in G';4. allgroups beginningwith e2 in G beginwith e' in G'.When notating surface groupingstructure,we designategroupingoverlapsby overlapping lursbeneaththe music. Whennotatingunder-lying groupingstructure,we use bracesto join events that come to beoverlappedn the surface.Thesenotationsare shown in Example9.The formal rule for elision is almost identical to that for overlap.The only differencelies in the relationships f the boundaryeventse1,e2, and e'. For the type of elision illustratedin Example 6, e1, theunderlyingevent to be elided, is harmonicallybut not totally identicalto e2; typically, it hasa lowerdynamicanda smallerpitchrangeas well.The corresponding vent in surfacegrouping tructure,e', is identicaltoe2. The descriptionof a groupingstructurecontainingan elision thuscontainsin its underlyinggroupingstructurea descriptionof the intui-tively elided event.The rules for overlapandelisionhave the desiredeffect of expandingthe class of well-formedgroupingstructuresto include the counter-examplesto GWFR4. In doing so, they expressthe musical ntuitionsbehind these counterexamples,and they restrictthe predictedrangeof

counterexampleso two specificrelatedtypes.If one were to look for analoguesof overlapand elision rules inlinguistic theory, two different parallelsmight come to mind. First,with respect to their place in the formal description,they resemblelinguistic syntactic transformations, n that they increasethe class ofwell-formed structures by applying certain optional distortions tounderlyingstructures.However, n theirsubstance, hey do not particu-larly resemblesyntactic transformations,n that the distortionstheyintroducedo not include such thingsas movement of constitutents(asin the passiveor subject-auxiliarynversion ransformations f English).Rather, their effects are most like those of highly local phonologicalrulesthat delete or assimilatematerialat wordboundaries, or example,the processthat resultsin the pronunciationof only one slightlyelon-gatedd in the middle of the phrasedeadduck.To sum up, the five well-formedness ulesplus the transformationsdefine a class of grouping structures that can be associated with asequenceof pitch-events,but which arenot specifiedin any directwayby the physicalsignal,as pitches and durationsare.Thus to the extentthat groupingstructures rulycorrespond o a listener's ntuitions,theyrepresentpart of what the listenerbringsto the perceptionof music.This will become clearer as we discuss the preferencerules in the nexttwo sections.

55



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PERCEPTUALOTIVATIONORTHEPREFERENCEULEFORMALISMWhile the GWFRsexclude as ill-formedcertaingroupingssuch asthose in Examples2 and 3, they do not precludethe assignmentofgroupingstructuressuch as those in Example10a, b to the openingofthe G minor Symphony. Althoughthese groupingsconformto GWFR1-5, they do not, we trust, correspondto anyone's intuition of thepassage'sactual grouping.One might attempt to deal with this prob-lem by refiningthe well-formednessand transformational ules;but inpractice we have found such an approachcounterproductive.A dif-ferent type of rule turns out to be more appropriate.Wecall this typeof rulea preferencerule, for reasons hat will soonbe obvious.To begin to justify this second ruletype, we observe hat nothinginthe GWFRs tatedin the previoussectionrefersto the actualcontent ofthe music;these rules describeonly formal,not substantive,conditionson grouping configurations.In order to distinguishExample 1 fromExample 10, it is necessaryto appeal to conditions that referto thespecificmusic. In workingout these conditions,we findthat a numberof different factors within the music affect perceivedgrouping,andthat these factors may either reinforce or conflict with each other.Whenthey reinforceeach other,stronggrouping ntuitionsresult;whenthey conflict, the listenerhasambiguousorvague ntuitions.Some simple experiments that compare musical groupingwith avisual analogue suggest the general principlesbehind groupingpref-erence rules. Intuitionsabout the visualgroupingof collectionsof smallshapeswere explored in detail by gestalt psychologists9. In Example11a, for instance, the left-handand middle circlesgrouptogetherandthe right-handcircle is perceivedas separate; hat is, the field is mostnaturallyseen as two circlesto the left of one circle. In Example1b,on the otherhand,the middle andright-hand irclesare seenas groupedtogether and the left-hand circle is separate.The principlebehind this

groupingobviously involvesrelativedistance:the circlesthat are closertogether tend to form a visual group. The groupingeffect can beenhanced by exaggerating he difference of distances,as in Example12a; it can be weakenedby reducingthe disparity,as in Example12b.If the middle circle is equidistantfrom the outercircles,as in Example12c, no particular roupingntuitionemergesat all.Similareffects exist in the groupingof musicalevents. Consider herhythmsin Example 13. The perceptionsabout grouping or these fiveexamplesare auditoryanalogues o the visualperceptions n Examples11-12. The first two notes of Example13a grouptogether(that is, theexampleis heardas two notes followedby one note); the last two notesgroup together in 13b; the groupingof the first two in 13c is verystrong, n 13d relativelyweak;13e has no particularperceivedgrouping.These examplesshow that on an elementary evel the relative ntervals56



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-...

Example10i)

b)rT--% ir L L , I + i+

: ,. _ . . , r + ,__

Example 1 a. O O O b. O

Example 2 a. OO O b. O O O

Example1l3)r) b) c) d)'



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of time between attack points of musical events strongly influencegroupingperception.Examiningsimple visual perception again, we see that like shapestendto be grouped ogether.In Example14a, the middleshapetendstoform a groupwith the two left-handshapes,since they are all squares;in Example14b the middleshapegroupswith the two right-hand hapes,sincethey are all circles.Similarly,equally spaced notes will group by likeness of pitch. InExample 15a the middle note is groupedmost naturallywith the two-left hand notes and in Example 15b with the two right-handnotes(assumingall notes are playedwith the same articulationandstressandare free of harmonicimplications, since these factors can also affectgrouping ntuitions). Considerablyweakereffects areproducedby mak-ing the middle note not identical in pitch to the outer pitches, butcloser to one than the other,as in Example16a,b. If the middlepitch isequidistant from the outer pitches, as in Example 16c, groupingintuitionsareindeterminate.These examples have demonstratedtwo basic principlesof visualand auditorygrouping:groupsare perceived n termsof the proximityand similarity of the elements available to be grouped. In each casegreaterdisparity n the field producesstrongergrouping ntuitions,andgreateruniformity throughout he field producesweaker ntuitions.

Let us next consider fields in which both principles apply. InExample 17a, the principlesof proximity and similarityreinforceeachother, since the two circlesare close togetherandthe threesquaresareclose together; the resulting grouping intuition is quite strong. InExample 17b, however, one of the squares s nearthe circles,so thatthe principlesof proximity and similarityare in conflict. The resultingintuition is ambiguous:one can see the middle squareas partof eitherthe left or rightgroup (it may even spontaneouslyswitch, in a fashionfamiliarfrom other visuallyambiguousconfigurations uch as the well-known Neckercube). As the middlesquare s movedstill furtherto theleft, as in Example 17c, the principle of proximity exerts an evenstrongereffect and succeeds in over-riding he principleof similarity;intuition now clearly groups it with the left-hand group,though onemay still sense some conflict. Parallel musical examples appear inExample18.Three important properties of the principles of grouping haveemerged. First, intuitions about grouping are of variable strength,depending on the degree to which individual grouping principlesapply. Second, different grouping principlescan either reinforceeachother, resulting n stronger udgments,or conflict, resulting n weakorambiguous udgments.Third,one principlemay over-rideanotherwhenthe intuitions they would individuallyproduce are in conflict. The58



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Example14 a. O O 0 b. 0

Example15a. b.

1 - . - - - - - --

Example16a. b.

Example17 a.O 0 O O O b.O 0 "0 130

Example18a. b.

-f-e1 , f- --T --- .. .



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formal system of preference rules for musical perceptionwhich wedevelop in the next section possessesthese properties.The term"pref-erence rule" is chosen because the rules establish not inflexibledecisions about structure,but relativepreferencesamonga number oflogically possible analyses.We have illustrated with elementaryvisualexamples as well as musical ones in order to show that the preferencerule formalism is not an arbitrarydevice, invented solely to makemusicalanalyseswork out properly.It is ratheran empiricalhypothesisabout the nature of humanperception.Wereturn to this general ssuetowardthe end of this paper.With this background,we turnto statingpreferencerules for musicalgrouping n some detail.

GROUPINGREFERENCEULESTwo types of evidence in a musical surface can determinewhatgrouping an experienced listener hears. The first is local detail-thepatternsof attack, articulation,dynamics,and registration hat lead toperception of groupboundaries.The second type of evidence involvesmore global considerationssuch as symmetry and motivic, thematic,rhythmic, or harmonic parallelism.We explore these two types ofevidence n turn.Local Detail Rules. There are three principlesof groupingthatinvolveonly local evidence. The first is quite simple.GPR1. Stronglyavoidgroups containinga singleevent.

Perhapsthe descriptive ntent of the rule would be clearer f the rulewere stated as "Musical ntuition stronglyavoidschoosing analyses nwhichthere is a groupcontaininga singleevent" or "Onestronglytendsnot to hear single events as groups." Readers who may be initiallyuncomfortablewith our formulationmay find suchparaphraseselpfulas they continuethroughthe rules.The consequence of GPR 1 is that any single pitch-eventin thenormalflow of music will be groupedwith one or moreadjacentevents.GPR 1 is overriddenonly if a pitch-eventis strongly isolated fromadjacentevents, or if it has an independentmotivic function. Undertheformercondition, GPR 1 is overriddenby another of the rules of localdetail, which we will state in a moment. Under the latter condition,GPR1is overriddenby the preferenceruleof parallelism,o be statedasGPR 6. But the rarityof clearlysensedsingle-notegroupsattests to thestrength of GPR 1 as a factor in determiningmusical intuition. (Anexampleof an isolated note functioningas a group s the fortissimoCin m. 17 of the finaleof Beethoven'sEighthSymphony.An exampleofa single element functioning motivically occursat m. 210 of the first60



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movementof Beethoven'sFifth Symphony:in the precedingmeasures,elements of the motive are progressivelydeleted until one eventstandsfor the originalmotive.)An alternativeformulation of GPR 1 is somewhat more general.Evidence or it will appear ater on.GPR 1 (Alternative orm).Avoid analyseswith very smallgroups-the smaller,the less preferable.The effect of this versionis to prohibitsingle-notegroups except withvery strong evidence, and to prohibit two-note groups except withrelatively trongevidence.By three or four-notegroups, ts effect wouldbe imperceptible.Put more generally,this rule prevents segmentationinto groupsfrom becomingtoo fussy:verysmall-scale roupingpercep-tions tend to be marginal.The secondpreferenceruleinvolving ocal detail is an elaboratedandmore explicit form of the principle of proximity discussed in theprevioussection. It detects breaksin the musical low that are heardasboundaries between groups. Consider the unmetered examples inExample 19. All else being equal, the first threenotes in eachexampleare heard as a group,and the last two areheard as a group.In eachcasethe carat beneath the examplemarksa discontinuitybetweenthe thirdand fourth notes: the thrid note is in some sense closer to the secondnote, and the fourth is closer to the fifth note, than the third andfourth are to each other. In Example19a the discontinuity s a break na slur;in 19b it is a rest;in 19c it is a relativelygreaterntervalof timebetween attack points. (The examples of proximity in the precedingsection involveda combinationof the last two of these.)In orderto state the preferenceruleexplicitly,we must focus on thewaythemusicgets from one note to the next-the transitions romnoteto note-and pick out those transitions hat aremore distinctivethanthe surroundingones. These more distinctivetransitionsare the onesthat intuitionwill favor asgroupboundaries.To locate distinctivetransitions, he rule considersfourconsecutivenotes at a time, designatedas n1 throughn4. The sequence of fournotes contains threetransitions: rom the first note to the second,fromthe second to the third, and from the third to the fourth. The middletransition,n2n3, is distinctive if it differs from both adjacenttransi-tions in particularrespects. If it is distinctive, it may be heard as aboundarybetweena groupendingwith n2 andone beginningwith n3-.There are two ways we measure the distancebetween two notes:from the end of the first note to the beginningof the next, and fromthe beginningof the first note to the beginningof the next. Both ofthese ways of measuringdistance contribute to grouping udgments.The former is relevant when an unslurred ransition s surroundedby

61



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slurred ransitions,as in Example19a, or whena transitioncontainingarest is surroundedby transitionswithout rests,as in 19b. The latter isrelevant when a long note is surroundedby two shortnotes, as in 19c.Thus the rule of proximity has two cases, designatedas (a) and (b)below.

GPR 2 (Proximity). Considera sequence of four notes n1n2n3n4.All else being equal, the transitionn2n3 maybe heardas a groupboundary f(a) (Slur/RestRule) The intervalof timefrom the end of n2 tothe beginningof n3 is greaterthan thatfrom the end of n1 ton2 and that from the end of n3 to the beginningof n4;or if

(b) (Attack Point Rule) The intervalof time between the attackpoints of n2 and n3 is greater than that between the attackpoints of nj and n2 and that between the attack points of n3andn4.It is important to see exactly what this rule says. It applies inExample 19 to mark a potential group boundarywhere the carat ismarked.However, t does not apply in cases suchas Example20. Con-siderExample20a. Thereare two unslurred ransitions,each of whichmightbe interpretedas a potentialboundary.But sinceneither of thesetransitionsis surroundedby unslurred ransitions,as GPR2a requires,the conditions for the rule are not met, and no potential groupboundaryis assigned.This consequenceof the rule corresponds o theintuition that groupingin Example 20a is far less definite than inExample 19a, where GPR 2a genuinely applies. Example 20b,c areparallel,illustratingthe nonapplicationof GPR2 when rests and longnotes areinvolved.Another rule of local detail is a more complete version of theprincipleof similarityillustratedin the previoussection. 21a-d showfour cases of this principle; he first of these correspondso the earlierexamples of similarity.As in Example 19, the distinctivetransitionsare heard between the third and fourth notes. Whatmakes the transi-tions distinctive in these cases is changein (a) register,(b) dynamics,(c) pattern of articulation,(d) length of notes. We state the rule in afashionparallel o GPR2:GPR 3 (Change).Considera sequenceof four notes n1n2n3n4. Allelse being equal, the transitionn2n3 may be heardas a group

boundary fa. (RegisterRule) The transitionn2n3 involvesa greaterchange npitch than both n n2 and n3n4; or ifb. (Dynamics Rule) The transition n2n3 involves a change indynamicsandn1n2 andn3n4 do not; or if62



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ON

Example19a b. c.

A F -i A I

Example20b. c

--.-" I!didzS- I __

Example21 a. b. c.

Example22 o.b. c.

p ,"



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c. (Articulation Rule) The transitionn2n3 involvesa changeinarticulationand n1n2 and n3n4 do not; or ifd. (LengthRule) n2 and n3 areof differentlengthsand bothpairsn, -n2 and n3-n4 do not differin length.(One might add further cases to deal with such things as changeintimbreor instrumentation.)Againthis rule relies on a transitionbeingdistinctivewith respecttothe transitionson both sides. Example22, like Example20, illustratescases where transitions are distinctive with respectonly to the transi-tion on one side;grouping ntuitions areagainmuch less securethaninExample21.As observed n the previoussection, the variouscases of GPRs2 and3 may reinforce each other, producinga stronger enseof boundary,asin Example 23a. Alternatively,different cases of the rulesmay comeinto conflict, as in Example23b-d. Here,eachcarat s labeledwith thecases of GPRs 2 and 3 that apply. In Example 23b,c,d, there isevidence for a group boundary between both the second and thirdnotes and between the third and fourth notes. However, GPR 1prohibitsboth being groupboundariesat once, since that would resultin the third note alone constituting a group. Thus only one of thetransitionsmay be a groupboundary,and the evidence is conflicting.This predictionby the formaltheory corresponds o the intuitionthatthe grouping judgment is less secure in Example 23b,c,d than inExample 23a, where all the evidence favors a single position for thegroupboundary.Although judgments are weaker for Example 23b,c,d than forExample 23a, they are not totally indeterminate.Close considerationsuggests that one probably hears a boundary in Example 23b andExample23d after the secondnote, andin Example23c after the thirdnote (though contextual considerations such as parallelismcouldeasily alter these judgments if they occurredwithin a piece). Theseintuitions can be reflected in the theory by adjustingthe relativestrengthsof the different cases of GPRs 2 and 3, so that in these con-figurations the Slur/Rest Rule (GPR 2a) overridesthe Attack PointRule (GPR 2b), the Slur/Rest Rule overrides he RegisterRule (GPR3a), and the DynamicsRule(GPR3b) overrides he Attack Point Rule.In general, all cases of GPR 3, with the possible exception of theDynamics Rule, appearto have weakereffects than GPR2. As in theexamplesin the second sectionjudgmentsshouldchangedependingonthe degree to which differentconditions are satisfied.For example,ifthe G in Example 23b is lengthenedto four quarters, ncreasingthedisparityin time between attacks, the evidence for the Attack PointRule becomes strongerthan the evidencefor the Slur/RestRule, andthe G is heardasgroupedwith the E andthe F.64



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0NUl

Example23O. c b. C.v - A A A Af 20 20 2b 3a 203(t,5b

Example2412 3 45 6 7 8 9 1 112 13

V A AIA A2b 2b 2o 3a 20 23d 2b

Example25 ~C

Example26Ji. b\

JJ~J~jstt



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In order to make the theory fully predictive, t might be desirableactually to assign each rule a numericaldegree of strength,and toassignvarious situations a degreeof strengthas evidencefor particularrules. Thenin eachsituation,the influence of a particular ulewouldbenumericallymeasuredas the productof the rule's ntrinsicstrengthandthe strengthof evidencefor the rule at that point; the most "natural"judgmentwould be the analysiswith the highesttotal numericalvaluefrom all rule applications. However, we will not attempt such aquantification here, in part for principled reasons to be discussedbelow. Our theory is neverthelesspredictive,even at its presentlevel,insofar as it identifiespointswhere the rules areandare not in conflict;this will often be sufficient to carry the musical analysisquite far.Furthermore,the construction of simple artificial examples such asthose in Example 23 can serve as a helpful guide to the relativestrengthsof variousrules,and these judgmentscan then be appliedtomore complex cases. We will often appealto this-methodologywhennecessaryrather hantryingto quantifyrulestrengths.Before statingthe remaininggroupingpreferencerules, let us applyGPRs 1, 2, and 3 to the opening of Mozart'sG Minor Symphony.Besides illustratinga number of different applicationsof these rules,this exercise will help show what further rules are needed.Example24repeatsthe Mozartfragment.For convenience, he notes are numberedabove the staff. Below the staff, all applicationsof GPRs2 and 3 arelisted as in Example23.Considerfirst the sequence from notes 2 to 5 in Example24. Thetime between attack points of 2 and 3 is an eighth;so is that between4 and 5, while that between 3 and 4 is a quarter.Therefore he condi-tions of the Attack Point Rule(GPR 2b) are met and a potentialgroupboundary is markedat transition3-4. Similarconsiderationsmotivateall the rule applicationsmarked. On the other hand, one might betempted to think that the Slur/RestRule (GPR 2a) would marka po-tential boundarybetween 2 and 3. However,since there is no slurattransition 3-4, the conditions for the rule are not met and no boun-daryis marked.Next observe that, with three exceptions, the potential groupboundariesmarkedby GPRs2 and 3 in Example24 correspond o theintuitively perceivedgroup boundariesdesignatedin Example 1. Theexceptions are at transitions 8-9, 9-10, and 18-19, where the rulesmark potential boundariesbut none are perceived.Transition9-10 iseasily disposedof. BecauseGPR 1 stronglyprefersthat note 10 alonenot form a group,a boundarymust not be perceivedat both 9-10 and10-11. Thus one of the rule applicationsmust override the other.Example23c shows a similarconflict between GPRs 2a and 3a. Therethe former rule overrides he latter, even with a relativelyshorterrest.66



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Hence, GPR2a should predominatehere too. Furtherweightis put on10-11 by the Attack Point Rule (GPR 2b). Hence the applicationofGPR 3a at 9-10 is easilyoverridden.Let us ignore transitions 8-9 and 18-19 for the moment (we willreturn to them shortly). The remaining ransitionsmarkedby GPRs2and 3 are exactly the groupboundariesof the lowest level of groupingin Example 1: 3-4, 6-7, 10-11, 13-14, and 16-17. Thus the bizarregrouping analysis in Example 10a, though permitted by the well-formednessrules, is shown by GPRs 1-3 to be a highly non-preferredgrouping or the passage.However,consideragainthe analysisin Example10b, which hasallthe low-levelboundaries n the rightplace,but whoselargerboundariesare intuitively incorrect.GPRs 1-3 do not suffice to preferExample1over Example 10b, since they deal only with placement of groupboundaries and not with the organization of higher-levelgroups.Further preferencerules must be developed in the formal theory toexpressthis aspectof the listener's ntuition.

Organization f higher-level rouping. Beyond the local detailrules,a number of differentprinciplesreinforceeach other in the analysisofthe higher-levelgroupsin Example 1. The firstof these dependson thefact that the largesttime intervalbetween attacks, as well as the onlyrest,areattransition10-11. Thistransition s heardasa groupboundaryat the largest level internal to the passage.The most generalform ofof this principlecan be statedas GPR4.GPR 4 (Intensification Rule). Wherethe effects picked out byGPRs2-3 are relativelymorepronounced,a relatively arger evelgroupboundarymay beplaced.

A simple example that isolates the effects of GPR 4 from otherpreference rules is Example 25, which is heard with the indicatedgrouping.GPRs 2a and2b (Slur/RestandAttackPoint Rules)correctlymark all the group boundaries in Example 25, but they say nothingabout the second level of grouping,which consists of three groupsfollowed by two groups.GPR4, however,notes that there is a rest atthe end of the third small group, strongly intensifying the effects ofGPR2 at that particularransition.It is this morestronglymarked ran-sition that is responsible or the second level of grouping.A second principleinvolved in the higher-level roupingof Example1 is a general preference for symmetry in the grouping structure,independentof the musicalcontent:

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GPR 5 (Symmetry Rule).Prefergroupinganalysesthatmost closelyapproach the ideal subdivision of groups into two parts ofequallength.GPR 5 alone is involved n the higher-level roupingof Example26a, inwhich the smallergroupsare furthergrouped wo and two ratherthan,say, one andthree.In Example 26b, where there are six small groups, GPR5 cannotapply in the ideal fashion. The ideal can be achievedin the relationbetween the small and middle-levelgroups,or in the relation betweenthe middle-level and large groups, but not both. The result is anambiguous middle-level grouping, shown as analyses i and ii inExample 26b. In a real piece, the ambiguity may be resolved, forexample, by metrical or harmonic considerations, but then theresult is not due solely to GPR 5. In general,it is the impossibilityof fully satisfying GPR 5 in ternarygroupingsituations that makessuchgroupings omewhat ess stablethanbinarygroupings.In Example 24, GPR 5 has two sorts of effects. First, it reinforcesGPR 4 (IntensificationRule) in marking ransition10-11 as a higher-level boundary, since this divides the passage into two equal parts.Second, each of the resultingintermediate-levelgroups contain threegroups, the first two of which are two quarternotes in durationandthe third of which is four quarternotes. GPR 5 thereforegroupsthefirst two together into a group four quarter notes in duration,producingthe ideal subdivisionof all groups.(Note that GPR 5 doesnot requireall groupsto be subdivided n the sameway: it is irrelevantto GPR 5 that the first and third four-quarter-notegroups aresubdivided,but that the secondandfourtharenot.)In additionto GPRs4 and 5, a thirdimportantprinciple s involvedin the higher-levelgroupingof Example24: the motivicparallelisms fevents 1-10 and 11-20. We can isolate the effects of this principle npassages such as Example 27a,b. Other things being equal, one'sintuition is that Example27a is groupedin threesandExample27b infours. Since both examples have uniform motion, articulation, anddynamics, the groupingpreference rules make no prediction abouttheir grouping.Hence a furtherpreferencerule, GPR 6, is necessaryto describe hese intuitions.

GPR6 (ParallelismRule). Wherewo or moresegmentsof the musiccan be construedas parallel,they preferablyorm parallelpartsofgroups.The application of GPR 6 to Example 27 is obvious: the maximalparallelism s achieved if the "motive"is three notes long in Example27a andfournotes long in Example27b.

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aN

Example27 O. _r _rtJ~~, -

Example28

Example29 .e,-. -i--:-- + ~6- ... .....



27/47

However, considerExample 28, in which a contraryarticulation sappliedto Example27a. One's ntuitionis that a grouping nto threes isnow relativelyunnatural.The theory accountsfor this as follows: theSlur/RestRuleplacesratherstrongpotentialgroupboundariesnExam-ple 28 after every fourth note, where the slursarebroken; hus eachofthe three note segmentsso obvious in Example27a comes to have aninternalgroupboundaryin a differentplace. Since motivic parallelismrequires, among other things, parallelinternalgrouping,Example 28cannot be segmentedinto three-note parallelgroups as easily as canExample 27a. Thus GPR 6 either is overriddenby the Slur/RestRuleor actuallyfailsto apply.GPR6 says specificallythat parallelpassagesshouldbe analyzedasformingparallelpartsof groups,ratherthan entire groups.It is statedthis way in order to deal with the quite common situation in whichgroupsbegin in parallelfashion and divergesomewhere n the middle,often in order for the second group to make a cadential formula. Aclear case is Example 29, the opening of Beethoven's QuartetOpus18, No. 1. GPR 6, reinforcedby the Slur/RestRule, analyzesthe firstfour measuresas two two-measuregroups.Thefifth measureresemblesthe first and third, but at that point the similarityends. If GPR 6demandedtotal parallelism, t could not make use of the similarityofmm. 1, 3, and 5. But as we have chosen to state the ruleabove,it canuse this parallelismo help establishgrouping.In the Mozartexample (Example24), GPR6 has two effects. First,it reinforcesthe Intensificationand Symmetry Rules in assigning hemajor grouping division at the middle of the passage.Second, recallthat GPR 2a (Slur/Rest Rule) marks a possible group boundary attransitions 8-9 and 18-19, and that these group boundarieswereintuitively incorrect.Consider ransition8-9; 18-19 is treatedsimilarly.GPR 6 is implicatedin the suppressionof this potentialboundary bydetectingthe parallelismbetween the sequencesof events at 1-3, 4-6,and 7-9. If a group boundary appearedat transition8-9, parallelismwould require t at 2-3 and 5-6 as well. But, in violationof GPR1, thiswould in turn make notes 3 and 6 form single-notegroups.Hence theonly way to preserveparallelism s to suppressthe possible groupboundaryat 8-9; indirectly,then, GPR 6 here overrides he Slur/RestRule.The ParallelismRule is not only important in establishing nter-mediate-levelgroupingssuch as those in the previousexamples: it isthe majorfactor in all large-scalegrouping.For example, it recognizesthe parallelismbetween the exposition and recapitulationof a sonatamovement, and assigns to them parallel groupingsat a large level,establishingmajorstructuralboundariesn the movement.Finally, a seventh preference rule concerns primarilylarge-scale70



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grouping.Different choices in sectionalizationof a piece often resultin interesting differences in the time-spanand prolongationalreduc-tions, and it is often the case that the choice cannot be made purelyon the basis of grouping evidence. Rather, the choice of preferredgroupingmust involve the relativestability of the resultingreductions.Without a full account of the reductions,we obviously cannotjustifysuch a preferencerulehere,but we state it for completeness.

GPR 7 (Time-spanand ProlongationalStability).Prefera groupingstructurewhich resultsin morestable time-spanand/orprolonga-tionalreductions.Havingstated the systemof GPRs,we proceednow to a few remarkson the notion of parallelismwhich was mentionedin GPR 6 andon thenature of the formalismwhich we usedin statingour rules.Remarks on parallelism.The importanceof parallelism n musicalstructurecannot be overestimated.The more parallelismone detects,the more coherent an analysis becomes and the less independentinformationmust be processedandretained.However,ourformulationof GPR 6 still leaves a greatdeal to intuition in its use of thelocution"parallel."Two identical passagescertainlyare parallel,but how differentcan

they be before they no longer seem parallel?Among the factors in-volved in parallelismare similarityof rhythm, internalgrouping,andpitchcontour. Whereone passage s an ornamentedor simplifiedversionof another,similarityof correspondingevels of the time-spanreductionmust also be invoked. Hereknowledgeof the idiom is often required odecide what counts as ornamentationandsimplification.It appears hata set of preferencerules for parallelismmust be developed,the mosthighly reinforced case of which is identity. Butbeyondthis we arenotprepared.togo, and we feel that our failureto flesh out the notion ofparallelism s a seriousgap in our attempt to formulatea fully explicittheory of musical understanding.For the present we must rely onintuitive judgments to deal with this area of analysis in which thetheory cannotmakepredictions.The problemof parallelism,however,is not at all specific to musictheory: it is a specialcase of the more general problemof how peoplerecognizesimilaritiesof any sort, for example similaritiesamongfaces.This relationof the musicalproblemto the moregeneralpsychologicalproblemhas two consequences.On the one hand, we may take somecomfort in the realizationthat our unsolvedproblem s reallyonly oneaspect of a largerand morebasicunsolvedproblem.On the otherhand,the hope of developinga solution to the musicalproblemthroughthepreference rule formalismsuggeststhat such a formalismmay be of

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more widespreadapplicability n psychological heory. Recognizing hegeneralityof the problem may providea point of attack for a broadertheoryof cognition.Remarkson formalism. Some readersmay be puzzledby our asser-tion that GWFRs 1-5 and GPRs 1-7 constitute a formal theory ofmusical grouping.There are perhapstwo respectsin which our theorydoes not conform to the popular stereotype. First, the rules arecouched in ordinary English, not in a mathematical or quasi-mathematical anguage.Second, even if the rules were translated ntosome sort of mathematicalterms, they would not be sufficient toprovide a foolproof algorithm for constructinga grouping analysisfrom a givenmusicalsurface.Thisseemsan appropriateplaceto defendour theoryagainstsuchpossiblecriticisms.The first point is easily refuted. As we mentionedat the outset, weare interested in stating as precisely as possiblethe factors leadingtointuitive musicaljudgments.Mathematicization f the rules rather hanprecise statement in Englishis useful only insofar as it allows moreinteresting or more precise predictions. Consider for example theGWFRs,which together define a class of hierarchicalgroupingstruc-tures connected in a simple way to musical surfaces. One couldpresumably ranslatethese rules into the mathematical anguageof set

theory or network theory. But it would serve little purpose; sincethere are no particularly nterestingtheorems about sets or networksthat bear on musical problems,no empiricalcontent would be addedby such a translation.In fact, such formalismwould only clutter ourexposition with symbolic formulas that would obscurethe argument.Hence we have chosen to state our rules in ordinaryEnglish,but withsufficient precisionthat their consequences,both for and againstthetheory, aremadeas clear as possible.Thesecondargument dvancedagainst he theoryis moresubstantive.The reason that the rulesfail to producea definitiveanalysis s that wehave not completely characterizedwhat happenswhen two preferencerules come into conflict. Sometimes the outcome is a vague orambiguousintuition; sometimes one rule overrides he other, resultingin anunambiguousudgmentanyway.Abovewe suggested he possibilityof quantifying rule strengths, so that in a conflicting situation ajudgmentcould be determinednumerically.A few remarksareneededto defendour decisionnot to attemptsuch a refinement.First, as we pointed out earlier,our main concernhereis to identifythe factors relevant to musical ntuition,andto learnhow these factorsinteract to produce the richnessof musical perception. To presentacomplex set of computations nvolvingnumericalvaluesof ruleapplica-tionswouldhaveburdenedourexpositionwith too muchdetailthat didnot involvepurelymusicalor psychological ssues.72



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But our decision was not merely methodological.A little reflectionsuggeststhat the assignmentof numericalvaluesto rule applications snot as simple a task as one might at first think. PatrickWinston,indeveloping a computer programfor certain aspects of visual patternrecognition,uses proceduresnot unlikepreferencerules."1 Because hecomputer must make a judgment, Winstonassignsrulesstrengthsandsets threshold values which must be attained in order to achieve apositive judgment. Winston himself notices the artificiality of thissolution, for it allows only positive and negative judgments, notambiguousor vagueones. Moreover, he choice of thresholdvalues s toa certain extent arbitrary: hould a valuebe, for instance,68 or 72? Asimple numerical solution of this sort creates an illusion of precisionthat is simplyabsentfromthe data.

A moreformidableproblem ies with the need for preferencerulestobalance local and global considerations.While t is not hardto imaginenumericallybalancing, or instance,the length of a restagainst he sizeof an adjacentchangein pitch, it is much moredifficultto balancethestrengthof a parallelism gainst he break n a slur.Part of the difficultylies in the presentobscurityof the notion of parallelism,but partalsolies in confusion about how to compare parallelism o anythingelse.Evenworse s the difficultyof balancing nter-component onsiderationssuchas those introducedby GPR7, the ruleof Time-span ndProlonga-tional Stability.Howmuch local instability n grouping,or loss of paral-lelism, is one to tolerate in orderto producemore favorableresults inthe reductions? Evidently, if we are to quantify strength of ruleapplication, nothing short of a global measure of stability over allaspects of the structuraldescriptionwill be satisfactory.Thus,we feel,it would be foolish to attempt to quantify local rules of groupingwithout a far better understandingof how these rules interact withother ruleswhoseeffects arein manyways not comparable.To sum up, we acknowledgeit as a failingthat our theory cannotprovide a computable procedure for determining musical analyses.However,achievingcomputability n a meaningfulway requiresa betterunderstandingof many difficult musicaland psychologicalissues thanexists at present,so we have felt uninclinedto pursueit. In the mean-time,wehaveattemptedto makethe theoryas predictiveas possible, bystating rulesclearlyandcarefullyfollowingthroughtheirconsequences,avoiding ad hoc adjustmentsto make analyses work out the way wewant. Webelievethat the insightsthe theoryhasbeenable to affordaresufficientjustificationfor this methodology.Theperformer's nfluenceon preferredhearing.The performerof apiece of music, in choosingan interpretation, s in effect decidinghowhe hears the piece and how he wantsit to be heard.Amongthe aspectsof an interpretationwill be a preferredanalysis (either conscious or

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unconscious) of the piece with respectto the grammaticaldimensionsaddressedby our theory. Since groupingstructure s the most cruciallink between the musical surface and the more background evels oftime-spanand prolongationalreductions,the perceptionof grouping sone of the moreimportantvariables he performercanmanipulate.A performer'sprincipalmeansof influencing groupingperception sin the execution of local details, which affect the hearer's choice oflow-level groupingboundariesthrough GPRs2 and 3 (the local detailrules) and of largerboundariesthrough GPR 4 (Intensification). Forexample, consider the beginningof the MozartPiano SonataK. 331,whichappearsn Example30 withtwo possible groupings. The musicalsurface is in conflict between these two groupings.Since the longestdurationbetween attacks is after the quarternote, local detail favorsgrouping (b). But maximal motivic parallelism avors grouping(a) (ifthe piece beganwith an upbeateighth, parallelismwould favorgroup-ing (b)). The variationsthat follow take advantageof this groupingambiguity, tipping the balance in favor of grouping(b) in Variations1, 2, and 5, andin favor of grouping a) in Variations3, 4, and 6.A performerwho wishes to emphasizegrouping(a) will sustainthequarternote all the way to the eighth andboth shortenthe eighthanddiminish ts volume. He thereby creates the most prominentbreakandchangein dynamicsat the bar line, thusenhancing he effects of GPRs2 and 3. On the otherhand,a performerwho choosesgrouping b) willshorten the quarter, eavinga slightpauseafterit, andsustainthe eighthupto thenext note. The effect of GPRs2 and3 is then relativelygreaterbeforethe eighthnote and less after it.A second, less noticeablealterationcouldinvolve a slightshift in theattack point of the eighth note, playing t a little earlyfor grouping a)and a little late for grouping(b). This slight change in attack-pointdistancealso affectspreferred rouping hrough ts influenceon GPR2.Subtle variationsin articulation such as these are typical of thestrategiesperformersuse to influenceperceivedgrouping.However, t isimportant to emphasizethat the performer'sconscious awarenessofthese strategiesoften does not go beyond "phrasingt this way ratherthanthatway"; hat is, in largepartthese strategiesare learnedand usedunconsciously. In making explicit the effect of such strategies onmusical cognition, we have suggested how our theory potentiallyaddressesssuesrelevant o performanceproblems.

TWOMOREEXAMPLESTo supportthe claim that the rules of groupingare not limited to asingle style, we analyze the grouping structure of the opening ofStravinsky'sThreePieces for ClarinetSolo (Ex. 31). As in the examplefrom Mozart's G minor Symphony, each note is numbered and

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iiul

Example30

b. J\ )b.c _J k . J k

Example31(Clarinet in A)sempre p e molto tranquillo, MMJ =52

(3d) (2a) (20)(30) (2b) ,,, "

. -'-- 3o0) --'19 20 21 22 23 24 25 26 27

(2o) (2a)13d) (2b)(3g)



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applications of GPRs 2 and 3 are marked at appropriate ransitions.We assume that the breath mark tells the performer,in effect, toproduce a groupingboundary by using one or both of the strategiesjustdiscussed: hortening he precedingnote andleavinga space,provid-ing evidencefor the Slur/RestRule (GPR2a), andperhaps engtheningthe time betweenattackpoints, providing videncefor the Attack PointRule (GPR 2b). Inadditionto ruleapplications, he exampleshowsthesmallest evelsof groupingpredictedby the rules.The dotted lines in Example31 requiresome explanation.Considerfirst transition 2-3; although there is weak evidence for a groupingboundaryat this point due to the changein note values,one tends tohear events 1-4 groupedtogether and to suppress he smallergroups.Above we suggestedan alternativeversion of GPR 1: Avoid analyseswith very smallgroups-the smaller,the less preferable.Thisversionofthe rule would say that the weak evidence at transition 2-3 isinsufficient to establisha groupboundarybecauseof the shortnessofthe resultinggroups.At transition 9-10 no local evidencesupportsa groupboundary,butparallelismwith transition 2-3 and its context would argue for aboundaryif one were chosen at 2-3. Similar(though weaker)parallel-ism, plus the change n register, ndicatea boundaryat 15-16; finally,anumberof relativelyweakrulesapplyat transition 18-19. Placementofa group boundaryat each of these points resultsin one or more two-note groups, which the revisedGPR 1 attempts to avoid. The overalleffect of the revisedGPR 1, then, is to suppress,or at least make lesssalient,all groupsrepresentedby dotted linesin Example31.On the other hand, at all other marked transitions, there areapplicationsof the more influential preferencerulesof proximity. Ingeneral,these rule applicationscause little difficulty. However,at onepoint they do lead to a two-event group, notes 12-13. The analysisretains this group for two reasons: first, the local evidence for aboundary at transition 11-12 is relativelystrong, and second, group1-4 followed by group 5-7 is parallel motivically to group 8-11followed by group 12-13. Thus, strong local evidence and motivicparallelismboth supporta groupingboundaryat transition 11-12 andoverride the preference of the revised GPR 1 against the two-notegroup12-13.The result of local evidence interactingwith GPR 1, then, is toestablish he small-scale rouping ndicatedby solid linesin Example31.In attempting to establish larger-levelgrouping,we first observe thatmotivic parallelismof the groups beginningat 1, 8, and, to a lesserextent, 14 and 21 favors larger level boundariesat transitions7-8,13-14, and 20-21. In addition, the strongestlocal rule applications nthe passageare at transitions7-8 and 13-14; the breathat 20-21 also76



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establishes t as a relativelystrongapplicationof GPR2a (the Slur/RestRule). So far, then, GPRs 6 (Parallelism)and 4 (Intensification)suggest he grouping n Example32.There are two possible ways to construct still larger groups.Symmetry(GPR 5) suggests he grouping n Example33a. On the otherhand, transition 7-8 has the strongestapplicationof GPR2 in the pas-sage, because of its rest and the preceding ong note; thus Intensifica-tion favors the groupingin Example 33b, where this transitionis themost important grouping boundary.Moreover,the strongestmotivicparallelism n the passageobtainsbetween events 1-4 and 8-11; sincethe rule of parallelismprefersthese to be parallelpartsof groups,thisrule too favors the groupingin Example 33b. (If, in addition, purelybinary grouping is desiredin Example 33b, to minimally satisfy theSymmetryrule, the relativelystrongmotivic parallelismbetween 8-10and 14-16 favors an additional group, including events 14-27, asshownin Example33c.)The choice between Examples33a and 33b is the first point wherethe preferencerules result in an ambiguous grouping.We personallyfavor Example 33b, treatingthe second large group, in effect, as anextended repetitionof the first. The resultingasymmetry s character-istic of the piece's style, a style which deliberatelyavoids symmetry,thus blocking the maximalreinforcementof preferencerules. Thatis,the differencebetween this style andMozart'swith respectto groupingis not grammaticalper se, but residesin what structures he composerusingthe grammar hooses to build.In the two previousexamples, the groupingpreferencerules haveencountered at least minor conflicts. Consideran examplein whichthepreferencerules encounter no conflicts and strongly reinforce eachother at all points. Such an examplewould havestronglymarkedgroupboundaries; the major group boundaries would be more stronglymarkedthan the minor ones; the piece would be totally symmetrical,haveonly binarysubdivisionsof groups,andwould displayconsiderableparallelismamong groups. The theory predicts that the groupingofsuch a passagewould be completely obvious. Example34, partof theanonymous 15th-centuryFrenchinstrumentalpieceDit le Bourgignon,representssuch a case. As usual, applications of GPRs 2 and 3 aremarkedat relevant ransitions.This example deserves ittle comment. The repetitionof phrases s,of course, the strongestform of parallelism.The smallestgroups oinby twos with adjacent groups of equal length; these intermediategroups againgroupby twos with groupsof equallength. Furthermore,the intermediate-levelboundaries are marked both by rests and bygreater duration between attack points, whereas the less importantboundariesare markedonly by the latter distinction, and to a lesser

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1 -4 5 -7 8- 11 12-13 14--20 21 -24 25 -27L J L - J

Example32

a. 1-4 5 - 7 8 - 11 12-13 14 - - 20 21 - 24 25 - 27

b. 1 - 4 5 - 7 8 - 11 12-13 14 - - 20 21 - 24 25 - 27

c. 1- 4 5 - 7 8 - 11 12-13 14 - - 20 21 -24 25-27

Example33

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rrr

SF..2b .

- - - - - -

L___L_ . _2, 2, -_ 2 . _ 2b_

Example34

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degree. Thus the rules of Intensification,Symmetry, and Parallelismare simultaneously satisfied by the grouping suggested by local evi-dence;thereis no questionof ambiguityor vagueness.Many folksongs and nursery rhymes also exhibit this sort ofregularity n the applicationof groupingpreferencerules. Piecesof thissort are often thought of as having"stereotypical"grouping tructure,which meansin terms of the presenttheory maximalreinforcementofgroupingpreferencerules. And here lies a dangerfor research.Someattempts at a generativegrammarof music have treated such stereo-typical grouping tructuresas basic andassumed hey could be extendedto more complex structures.12 But if our theory is correct,the stereo-typical structuresreveallittle, since they representthe confluence of agreat number of interactingfactors whose individualeffects thereforecannot be identified. It is essential to investigatemore sophisticatedexamples from the very beginningin orderto arriveat any notion ofwhatis goingon.

Closingremarkson musicalgrammar.Thiscompletesourexpositionof the rules for groupingstructure.To sum up, the rulesfall into twosubcomponents.Thewell-formedness ules describea set of hierarchicalstructuresand certainaspectsof their connectionto the musicalsurface.This subcomponentalso containstransformationaluleswhichaccountfor certain permissible violations of the normal hierarchy. Thepreference uleschoose,from allpossiblewell-formed tructuresassignedto a given musical surface, those which are most highly preferredby the listener.These rules can be divided nto local detailrules(GPRs1, 2, and 3) and ruleswith more global conditions(GPRs4, 5, 6, and7).The other three components of the musical grammar-metricalstructure,time-spanreduction,and prolongational eduction-displayasimilarabstractorganization.In addition,there arevariousrulesystemswhich link together the structuresassignedby the four components-both well-formedness ulesand preferencerules (of which GPR7 is anexample).Theeffect is a highlyinteractivesystemin which ocalchoicesmade in one componentof the musicalstructurecanhaveramificationsin all components.These effects have muchto do with the richnessandsubtletyof musicalcognition.Since the groupingcomponent provesto be independentof style, itis worth mentioninga few areaswhereidiom-specificrulesdo appear nthe musical grammar,differentiatingClassicaltonal music from othergenres.One place is in the metricalwell-formedness ules,whichdefinethe possible meters in the idiom. Another place is in the rules thatdefine scale structure and establish a hierarchyof consonance anddissonancerelative to the tonic; these rules are intimately involvedinconstructingboth reductions. Third, Classicaltonal music appearsto80



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have a preferencerule that establishesa normativeform for the struc-ture of prolongational eductions.Thisrule,whichis relatedto Schenk-er's notion of Ursatz,influencesgreatlythe way tension andreleasearestructured,from the largestto the smallestlevels of structure.Certainother idioms appeareither to lack this rule or to have a differentnor-mativeprolongational-structure.Other differences among idioms, such as overall thematic form,motivic vocabulary,andtimbraleffects, concern eithernon-hierarchicalaspects of structure or differentchoices in how the musicalgrammarsexploited. Thus the claim that musicalgrammars basicallyunchangedfrom Bach to Brahms parallels the claim that English grammar sbasicallyunchanged rom Shakespeareo the present;what has changedis the artisticuse madeof the grammar.

We close the paper by discussingthe relevanceof this generativemusictheoryto othercognitivedomains.RELEVANCE TO LINGUISTICS AND PSYCHOLOGY

As we promisedat the beginning, he fragmentof a generativemusictheory presentedhere does not look much like generative inguistics.Thedifference nmethodologybetween the two theories s symptomatic:where linguistic theory is highly concerned with grammaticality,musictheory is moreheavilyconcernedwith preferenceamofiga considerablenumber of competing well-formed (that is, grammatical)structures.The closest analogue to linguistic grammaticality n the theory ofgroupingis well-formednessof groupingstructure. The rules defininggroupingwell-formednessresemblelinguistic rules in that they eitherestablish a nested segmentation of the surface string (like phrasestructurerules in syntax) or characterizepermissibledistortionsof thehierarchicalstructure (like transformations).This suggests that thebulk of a linguistic grammarconsists of well-formednessrules-fromthe phonology throughthe syntax to the semantics.Eventhe lexiconcan be considereda part of the well-formedness omponent, in that itestablisheswell-formedmatchingsbetweenphonological,syntactic,andsemantic form at the terminal nodes of hierarchicalstructures. Thewell-formednessrules for music, even consideringall four componentsof the musicalgrammar, ardlyapproach he linguisticwell-formednessrules in complexity. This reflects the muchgreaterrole of grammatical-ity in language han in music.The question thus arisesas to whetherlinguistictheory containsanyrule systems comparable o preferencerules in music.Webelievethat anumber of phenomenahave been discussed n the linguisticliteraturethat have appropriatepropertiesto be governedby preferencerules.One of these is the notion of markednessconventions: conditionsthat establishpreferred orms for rules of grammar,but whichmaybe

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violated by particulargrammars.A second area is parsingstrategies:heuristic proceduresfor assigningsyntactic structuresto potentiallyambiguous strings of words in the course of processinga sentence.Theseare perhapsclosest in spiritto the preferencerulesof the musicalgrammar.A third area is in the theory of pragmatics,which is con-cernedwith the relation between a sentence'sliteralmeaningand howthe sentence can be used in a verbal or nonverbal context. Thebalancing of conflicting preferences,characteristicof preferenceruleinteractions,appearsquite clearly, for example, in the applicationofGrice'sconversationalmaxims.13 Finally, the use of preferencerules aspart of the internal structureof word meaningsappears o accountfor"family resemblance" and stereotype phenomena that have plaguedstandardaccounts. 4 Inshort, the notion of preferencerulesmay proveuseful in the treatment of several disparate aspects of linguisticcompetence that have resisted formalizationalong more traditionallines.

Turning to psychology, recall the parallels between visual andauditory grouping pointed out above. That such parallelsexist wasobserved by Wertheimer,Koffka, and K6hler;is K6hler conjecturedthat they are a result of similarphysiologicalorganization n the twodomains.Lashleyalso arguedthat essentiallysimilarmentalrepresenta-tions serve for both spatially and temporally sequenced memory.16However,noneof these psychologistssaid little systematicallyaboutthestructureof temporalorganizationperse, beyond enumerationof somehighly interestinganecdotalevidence.Havingdevelopeda formaltheoryfor one type of temporalorganization,we are in a positionto turn theparallelismon its head and see what the theory of musical groupinghasto offer to the study of visualperception.In Koffka's detailedand wide-rangingnvestigationof the organiza-tion of the visual field, there appearexamples (p. 135) almost exactlyanalogous to our illustrationsof visual grouping.17 Koffka suggeststhree generalcharacteristics re involved n thesejudgments:(1) greaterproximity and greater similarity enhance visual groupingjudgments;(2) judgments are less certainwhen the principlescome into conflict;(3) undercertainconditions,one principlecanoverride he other. Theseare the exact characteristicswe have used to motivate the preference-rule formalism. Similarobservationswith a numberof otherprinciplesoccur throughout Koffka's discussion of the perception of shapes,figure-groundopposition, and three-dimensionality.He demonstrates,in addition to proximity and similarity,the influence of such factorsas symmetry, continuation of lines along regularcurves,ability of apatternto enclosespace,and(significantly) he viewer's ntention.Koffka's work is intended as a demonstrationof the fundamentalclaim of gestalt psychology: perception,like other mentalactivity,is a82



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dynamicprocessof organization,n whichall elementsof the perceptualfield may be implicated n the organizationof any particularpart.He isat painsto point out and prove two crucialaspectsof this claim. First,perception is not simply a product of what is in the environment:the viewer plays an active, though possibly unconscious, part indeterminingwhat he perceives. Second, the totality of the field asperceivedcannot be built up piecemeal,as a mereaccumulationof theperceptionof its partseach taken in isolation.The general underlying principle that forms the basis of thegestalt account of perception was formulatedby Wertheimeras theLawofPrdgnanz.Koffkadescribes t as the following:

Psychologicalorganizationwill alwaysbe as "good"as the prevailingconditions allow. In this definition the term"good"is undefined.Itembraces such properties as regularity, symmetry,

jackendoff, r., & lerdahl, f. (1981). generative music theory and its relation to psychology....

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