Cognitive Constraints on Compositional Systems

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Contemporary Music Review1992, Vol . 6 , Par t 2 , pp . 97-121Photocopying permitted by license only9 1992 Harwood Academic Publishers GmbHPrinted in the United Kingdom

ognitive onstraints on om positiona l System sF r e d L e r d a h l tColumbia University, New York CityIn th i s a rt i cl e I exp lo re the r e l a t ions h ip be tw een com pos ing and l i s t en ing . I beg in wi th a p rob lemat i cs to ry , d r aw s ome gene ra l conc lu s ions , in t roduce r e l evan t concep t s f rom Lerdah l and J ackendof f(1983) and r e l a t ed work , p ropos e s ome cogn i t ive cons t r a in t s on comp os i t iona l s y stems , d i s cussp i t ch s pace , and exp la in why s e r i a l ( o r 12 - tone ) o rgan iza t ions a r e cogn i t ive ly opaque . M os t o fthes e top ic s des e rve fu l l e r t r ea tmen t th an i s g iven in thes e pages . M y con ce rn he r e i s ju s t to lay ou t abas ic , i f wide - r ang ing , a rgume n t .I a m n o t i n t e r e s te d i n p a s s i n g j u d g e m e n t o n t h e c o m p o s e r s a n d c o m p o s i ti o n s t h a t a re m e n t i o n e d ,pa r t i cu la r ly no t on the r em arkab le work by Bou lez tha t I u s e a s a r ep res en ta t ive example . The th ru s to f my a r gum en t i s p s ycho log ica l r a the r tha n aes the t i c . Bu t s ince aes the t i c i ss ues inev i t ab ly imp ingeon the d i s cus sion , I t r ea t them b r i e fly a t t he end .KEY WORDS cogn i t ive cons t r a in t s, compos i t iona l s y s tems , mus ica l g r amm ar , p i t ch s pace ,ser ia l ism.

idden organization in Le Marteau sans MaftreB oul ez s Le Marteau sans Maftre (1954) was w ide ly hai led as a ma s terp iece of pos t -war ser ia l i sm. Yet nobody could f igure out , much less hear , how the p iece wasserial . From hints in Boulez (1963), Kob lyak ov (1977) at las t de ter mi ne d that i t 9was indeed ser ia l , though in an id iosyncrat ic way. In the in ter im l i s teners madewh a t s ense t hey cou l d o f t he p i ece i n ways un re l a t ed t o i ts cons truc t ion . Nor hasKob l yako v s dec i pherm en t subseq uen t l y change d ho w t he p i ece i s hea rd .Meanwhi l e mos t composer s have d i s ca rded s e r i a l i sm , wi t h t he r esu l t t ha tKob lyako v s cont r ibut ion has cause d bare ly a r ipp le of profess ion al in teres t . Theserial organizat ion of Le Marteau wo uld appe ar , 30 years l a ter , to be i r re levant.This s tory i s , o r should be , d i s turb ing . There i s a huge gap here betweencompos i t i ona l sy s t em and cogn i zed resul t. H ow can t hi s be?On e mi gh t su ppo se t ha t t he i mpene t rab i l it y o f Le Marteau s serial organizat ionis du e to insuff ic ien t exposu re . Af ter al l, the p iece w as innovat ive; l i s teners m us tbecome accus tomed to novel s t imul i . Such has been the t rad i t ional defence ofne w a r t i n the f ace of i ncomprehe ns i on . One mi gh t r e f i ne th i s v i ew by po i n t i ngout that there is l i t t le repet i t ion in Le Marteau. T h e l ac k o f r e d u n d a n c y p e r h a p sov erw hel ms the l i s tener s pro cess ing capaci t i es. Co mpreh ens ib i l i ty , then , i sa rguab l y a conseque nce bo t h o f t he deg ree o f cond i t ion i ng t o the mat e r i al s and* In J . Sloboda, ed., Generative Processes in Music Oxford Univ ers i ty Press , 1988.

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98 FredLerdahlo f t h e n u m b e r a n d v a r i e t y o f e v e n t s p e r u n i t t i m e . ( Is i t a c o i n c i d e n c e th a t LeMarteau w a s c o m p o s e d i n t h e h e y d a y o f b e h a v i o u r i s m a n d i n f o r m a t i o n t h e o ry ? )B u t t h i s e x p l a n a t i o n i s i n a d e q u a t e . F o r o n e t h i n g , c o m p e t e n t l i s t e n e r s t o LeMarteau e v e n a f t e r m a n y h e a r i n g s , s t i l l c a n n o t e v e n b e g i n t o h e a r i t s s e r i a lo r g a n i z a t i o n . F o r m a n y p a s s a g e s t h e y c a n n o t e v e n te ll i f w r o n g p i t c h e s o rr h y t h m s h a v e b e e n p l a y e d . T h e p i e c e is h a r d t o le a r n b y e a r i n a sp e ci fi c s e n s e;i t s d e t a i l s h a v e a so m e wh a t s t a t i s t i c a l q u a l i t y . C o n d i t i o n i n g , i n sh o r t , d o e s n o tsu f f i c e . F o r a n o t h e r t h i n g , Le Marteau d o e s n o t f e el s tr u c t u r a l ly c o m p l e x i n t h ew a y , f o r e x a m p l e , t h a t c o m p o s i t i o n s b y B e e t h o v e n o r S c h o e n b e r g d o . V a s tn u m b e r s o f n o n r e d u n d a n t e v e n t s f l y b y , b u t t h e e f f ec t is o f a s m o o t h s h e e n o fp r e t t y s o u n d s . T h e l i s t e n e r ' s p r o c e s s i n g c a p a c i t i e s , i n s h o r t , a r e n o to v e r w h e l m e d .

T h is is n o t t o d e n y t h e i n f l u en c e o f e x p o s u r e a n d r e d u n d a n c y o n c o m p r e h e n -s i b i l i t y . B u t t h e se f a c t o r s d o n o t g o f a r e n o u g h ; t h e y d o n o t a d d r e s s i s su e s o fs p e ci fi c o r g a n i z a t i o n . C o g n i t i v e p s y c h o l o g y h a s s h o w n i n r e c e n t d e c a d e s t h a th u m a n s s t r u c t u re s t im u l i i n c e r ta i n w a y s r a t h e r th a n o t h e r s . C o m p r e h e n s i o nt a k e s p l ac e w h e n t h e p e r c e i v e r is a bl e to a s si g n a p r e c is e m e n t a l r e p r e s e n t a t i o nt o wh a t i s p e r c e i v e d . No t a ll s t i m u l i, h o w e v e r , f a c il it a te t h e f o r m a t i o n o f am e n t a l r e p r e s e n t a t i o n . C o m p r e h e n s i o n r e q u i r e s a d e g r e e o f e c o lo g i ca l f it b e -t w e e n t h e s t i m u l u s a n d t h e m e n t a l c a p ab i li ti e s o f t h e p e r c e i v e r .E x p e r i e n c e d l i s t e n e r s d o n o t f i n d Le Marteau t o t a l l y i n c o m p r e h e n s i b l e , b u tn e i t h e r , I w o u l d a r g u e , d o t h e y a s s i g n t o i t a d e t a i l e d m e n t a l r e p r e s e n t a t i o n .T h i s is w h y t h e d e t a il s o f t h e p i e c e a r e h a r d t o le a r n a n d w h y t h e p i e c e d o e s n o ti n t h e e n d f e e l c o m p l e x . T h e s e ri a l o r g a n i z a t i o n t h a t K o b l y a k o v f o u n d is o p a q u et o s u c h s t r u c t u r in g .O f c o u r s e a m u s i c i a n o f B o u l e z ' s c a li br e w o u l d n o t u s e a c o m p o s i t i o n a l s y s t e mw i t h o u t d r a w i n g c r u ci a ll y u p o n h is m u s i c a l i n t u i t i o n a n d e x p e r i e n c e .M u s i c - g e n e r a t i n g a l g o r i t h m s a l o n e h a v e a l w a y s p r o d u c e d p r i m i t i v e o u t p u t s ; n o te n o u g h is k n o w n a b o u t m u s i c al c o m p o s i t i o n a n d c o g n i t i o n f o r t h e m t o su c c ee d .B o u l e z h a d t h e i n t e ll e c tu a l ly le s s a m b i t i o u s g o a l o f d e v e l o p i n g a s y s t e m t h a tc o u l d j u s t p r o d u c e a q u a n t i t y o f m u s i c a l m a t e r i al h a v i n g a c e rt a i n c o n s is t e n c y .H e t h e n s h a p e d h i s m a t e r i al m o r e o r l es s i n t u i ti v e l y , u s i n g b o t h h i s e a r a n dv a r i o u s u n a c k n o w l e d g e d c o n st r ai n ts . I n s o d o i n g , h e l i s te n e d m u c h a s a n o t h e rl i s te n e r m i g h t . T h e o r g a n i z a t i o n d e c i p h e r e d b y K o b l y a k o v w a s j u s t a m e a n s ,a n d n o t t h e o n l y o n e , t o w a r d s a n a r t i s t i c e n d .T h e d e g r e e t o w h i c h Le Marteau is c o m p r e h e n s i b l e , t h e n , d e p e n d s n o t o n i tss e ri al o r g a n i z a t i o n b u t o n w h a t t h e c o m p o s e r a d d e d t o t h a t o r g a n i z a t i o n . O n t h eo t h e r h a n d , t h e s er ia l p r o c e d u r e s p r o f o u n d l y i n f l u e n c e d th e s t i m u l u s s t r u c t u r e ,l e a d i n g t o a s i t u a t i o n i n w h i c h t h e l i s t e n e r c a n n o t f o r m a d e t a i l e d m e n t a lr e p r e s e n t a t i o n o f t h e m u s i c . T h e r e s u l t is a p ie c e t h at s o u n d s p a r t l y p a t t e r n e da n d p a r t l y s t o c h a s t i c . tt In Structures Book I (1952), Bo ulez flirted with algorithmic composition , and in Pli selon pli(1958-61) he incorporated aleatoric elements; so Le Marteau lies en routein h is dialectic of determinismand chance (see Boulez 1964). His interest in algorithmic composition continues to the extent that fora while there was a research group at IRCAM pursuing it. For six months in 1981 I was part of thatgroup, though in my project the purp ose of the computer program was n ot to compose but to assistcomposers in composing, much as a w ord p rocessor assists writers (see a brief description in Lerdahl1985). Many of the ideas expressed in this essay grew out of that experience, for which I remaindeeply grateful.


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ognitive onstraints 99ompositional and listening grammars

I t wi l l be usefu l to pu t these observa t ions in schemat ic fo rm. Le t me in t roducethe no t ion o f a musical grammar a l imi ted se t o f ru les tha t can genera te indef i -n i t e ly l a rge se t s o f mus ica l eve n t s an d /o r the i r s t ruc tu ra l descr ip t ions . Such agrarm ma r can t ake the fo rm of (a) o r (b ) in F igure 1 . (The rec tang les s t an d fo rse t s o f ru les , the e l l ipses fo r inpu t s and ou t pu t s o f ru les .) In (a ), the ru le sy s temg en e ra t e s t h e ev en t s o f a p a s s ag e o r p iece , an d i n th e p ro ces s p ro v i d es s o m e s o r to f spec i f ica t ion ( the inpu t o rgan iza t ion ) . In (b ), the sequ ence o f eve n t s is t akenas g iven , an d the ro le o f the ru le sy s tem i s to ass ign a s t ruc tu ra l descr ip t ion tot h e s eq u en ce .O u r d i s cu s s i o n o f Le Marteau sugg es t s tha t there a re two k inds o f mus ica lg rammar a t work here . The f i r s t i s the compositional grammar consc ious lyem p l o y ed b y Bo u l ez , t h a t g en e ra t ed b o t h t h e ev en t s o f t h e p i ece an d t h ei r s er ia lo rgan iza t ion as d i scovere d by Koblyakov . Th i s g ram ma r i s an ins tance o f (a ) inFigure 1 . The second k ind i s the listening grammar mo re o r l e s s u n co n s c i o u s l yem p l o y ed b y au d it o r s, t h a t g en e ra t e s men t a l r ep re s en t a t i o n s o f t h e mu s ic . T h i sgrammar i s an ins tance o f (b ) in F igure 1 ; the even t s themselves a re p resen tw h e n Le Marteau i s p layed .

a I Se o ru esSequenceof~~ e v e n t s

b) CSequence f e v e n ts ~ -~ Setof rules ~-~Structural description-~ ~igure I

Fi g u re 2 s u mmar i ze s t h e s e r emark s . T h e d i ag ram s h o w s t h a t t h e co mp o s -i ti o na l g r ammar g en e ra t e s t h e s eq u en ce o f ev en t s an d t h e m an n e r i n w h i ch t h eyare spec i f i ed . Only the sequence o f even t s , however , i s ava i l ab le as inpu t to thel i s t en ing g rammar : the l i s t ener hears the acous t i c s igna l , no t i t s compos i t iona lspec i f i ca t ion . The l i s t en ing g rammar then genera tes the menta l represen ta t ionthat com pr i ses the hear d s t ruc tu re o f the p iece . No te tha t F igure 2 incorpor -ates bo th (a) and (b) of Figure 1 .Th i s accoun t i s com pl ica ted by the fac t tha t , as no te d a bove , B oulez c rea ted LeMarteau n o t o n l y t h ro u g h s er ia l p ro ce d u re s b u t t h ro u g h h is o w n i n n e r l i s ten in g .In the p rocess he fo l lowed cons t ra in t s tha t , whi le opera t ing on the sequence o fev en t s p ro d u ced b y t h e co mp o s i t i o n a l g r ammar , u t i l i z ed p r i n c i p l e s f ro m t h el i s t en ing g rammar . Th i s more e labora te p ic tu re i s d rawn in Figure 3 .


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1 red Lerdahl

ompositionalgrammar ~ ~ i ~ p u t

igure 2

}= eardtructureL,,.... structura l description) J

ompositionalgrammarIntuitiveconstraints

Listening 1grammar j= "~eard structurek,,,.. (St uctural des cript ion )~)

igure 3Observe that none of the arrows in the diagram are inpu ts to the compos-itional grammar . This component is inaccessible to the rest of the system.Hence it becomes quite possible for the compositional gram mar to be unre-lated to the other rules, the listening grammar and intuitive constraints . Ifthis happen s, the input organization will bear no relation to the heardstructure . Here, then, lies the gap between compositional system and cognizedresult with which we began.This situation exists not only for e Mar teau but for much of contemporarymusic. I could have illustrated just as well with works by Babbitt, Carter, Nono,

Stockhausen, or Xenakis. This gap is a fundamental problem of contemporarymusic. It divorces me thod from intuition. Composers are faced with the unplea-sant alternative of working with private codes or with no compositional gram-mar at all. Private codes remain idiosyncratic, competing against other privatecodes and creating no larger continui ty - so that, for example, 30 years later theserial organiza tion of e Marteau becomes irrelevant even to other composers.

atural and artificial compositional grammarsWhere does a compositional grammar come from? The answer varies, but a fewgeneralizations may be helpful. Let us distinguish between a natural and anartificial compositional grammar . A natur al grammar arises spon taneousl y ina musical culture. An artificial grammar is the conscious invention of an indi-vidual or group with in a culture. The two mix fruitfully in a complex and long-


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ognitive onstraints 101lived musical culture such as that of Western tonality. A natural grammar willdomi nate in a culture emphasiz ing improvisation and en couraging active partici-pation of the community in all the varieties of musical behaviour. An artificialgrammar will tend to dominate in a culture that utilizes musical notation, that isself-conscious, and that separates musical activity into composer, performer,and listener.The gap between compositional and listening grammars arises only whe n thecompositional grammar is artificial , wh en there is a split between productio nand consumption. Such a gap, incidentally, cannot arise so easily in humanlanguage. People must communicate; a member of a culture must master alinguistic grammar common to both speaking and hearing. But music hasprimarily an aesthetic function and need not communicate its specified struc-ture. Hidd en musical organizations can and do appear.A natural compositional grammar depends on the listening grammar as asource. Otherwise the various musical functions could not evolve in such asponta neous and unified fashion. An artificial compositional grammar, on theother hand , can have a variety of sources - metaphysical, numerical, historical,or whatever . It can be desirable for an artificial gram mar to grow out of a naturalgrammar; think, for example, of the salutary role that Fux (1725) played in thehistory of tonality. The trouble starts only when the artificial grammar losestouch with the listening grammar.In the Western tradition the trouble began with the exhaust ion of tonality atthe tu rn of the century. Any th ing became possible. Faced with chaos, com-posers reacted by inventing their own compositional grammars. Within anavant-garde aesthetic it became possible to believe that one's own new systemwas the wave of the future. Boulez's genera tion was the last to believe this. To ayounger generation these systems have come to seem merely arbitrary. Theavant-garde has withered away, and all methods and styles are available to thepoint of confusion.One can react to this situation by giving up on compositional grammars andrelying solely on ear and habit - on the intuitive constra ints of Figure 3. Butcompos ing is too difficult for such a solution; there are too ma ny possibilities. Orone can react by reverting to earlier natu ral styles, a move that cond emns oneto a parasitic relationship with the past. Both reactions are common these days,and both are motivated by the desire to avoid the gap between composingprocedure and what is heard.My own reaction as a composer has been less to avoid than to confront thisgap. These were my initial groun d rules: (1) a compositional grammar is necess-ary; (2) it need not be nostalgic; (3) our musical culture is too fragmented andself-conscious for a natural grammar to emerge; but (4) an artificial grammarunrespon sive to musical listening is unacceptable. It was inevitable, I concluded,tha t early attempts at artificial grammars - say, f rom the 1920s to the 1950s -were defective in their relation to listening. Not enough was known aboutmusical cognition; the basic questions had not even been framed. Beginningaround 1970, however, a new perspective became possible through the simul-taneous decline of the avant-garde and rise of cognitive psychology.Contemp orary music had lost its way. What other f ounda tion was there to turnto than the nature of musical unde rsta ndin g itself? In other words, I decided thata compositional grammar must be based on the listening grammar. Figure 4


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102 FredLerdahlillustrates this by adding a dashed arrow to the flow chart of Figure 3. Inprinciple this arrow integrates the compositional grammar into the overallframework, rather than leaving it isolated from input.But for this proposal to have substance, a great deal mus t be kno wn about thelistening grammar. Hence it became necessary to develop a detailed theory ofmusical cognition (Lerdahl and Jackendoff 1983). Such a theory, I reasoned,could provide the basis for artificial compositional grammars that could beintellectually complex yet spontaneously accessible to mental representation.The commonali ty of compositional and listening grammars could produce a richyet transparent music.

h e o r e t i c a l o v e r v i e wThis is not the place to describe the specifics of Jackendoff's and my theory. Ipropose only to outline some of its general features. The reader should bear inmind, however, that the theory is concrete and detailed.Our theory models musical listening within the framework of some standardmethodological idealizations. Assuming (1) an experienced listener, (2) thepsychoacoustical organiza tion of the physical signal into a musical surfacecomprising a sequence of discrete events, and (3) a final-state knowled ge of thesequence, the theory predicts structural descriptions from musical surfaces bymeans of a set of rules that ideally corresponds to the listening grammar ofFigures 2-4. See Figure 5; and note tha t the theor y is genera tive in the sense of(b) in Figure 1. The predict ions of the rules are testable by introspection and, inprinciple, by experiment.tAs a practical matter the theory focuses on Classica} tonal music, but most ofthe rules appear to have a more general psychological basis an d therefore to beapplicable to other idioms as well. The extent of the universality of the rules is anempirical question. Again for practical reasons, the rules assign structural de-scriptions only to the hierarchical aspects of musical structure, neglecting asso-ciational dimensions such as motivic processes and timbral relations (but seeLerdahl 1987). By hierar chy is mea nt the strict nest ing of elements or regionsin relation to other elements or regions. The theory claims that, if the signalpermits, the listener unconscious ly infers four types of hierarchical structurefrom a musical surface: grouping structure or the segmentation of the musicalflow into units such as motives, phrases, and sections; metrical structure or thepattern of periodically recurring strong and weak beats associated with thesurface; time-span reduction or the relative structural importance of events asheard within contextually established rhythmic units; and prolongational re-duction or th e perceived pattern of tension an d relaxation among events atvarious levels of structure. Both kinds of reduction are described by structuraltrees. The prolongational component incorporates aspects of Schenker's (1935)theory.f Deli6ge (1985) has experimentally verified - and, in a few cases, improved upon - the localgrouping preference rules of the theory. Todd (1985) has used the time-span reductionalcomponentof the theory to model expressive timingat cadences in performance. Palmer and Krumhansl(1987)have found empirical support for predictions made by the metrical and time-span components.


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ognitive onstraints 1 3

_ .~J CompositionalgrammarIntuitiveconstraints

Listening [ _~ . ~e a r d structuregrammar I K,,~tructuraldescription)L..,)igure 4

Musical urface ~ Rule ystem ~ Structural escription(Sequence f even ts )S [ (Listening rammar) ~--',,.~Heard structure) Jigure 5

For the most part these four hierarchies interact as shown in Figure 6. Fromthe grouping and metrical structures the listener forms the rhythmic units, ortime-span segmentation over which the dominatin g-subordi nating relationshipsof time-span reduct ion take place; and from the time-span reduct ion the listenerin turn projects the tensing-relaxing hierarchy of prolongat ional reduction. Thusthe mapping from musical surface to prolongational structure is indirect. Inaddition, both kinds of reduction depend for their operation on stability con-ditions among pitch configurations as considered out of time . These con-ditions, which also can be described hierarchically (Krumhansl 1983; Bharucha1984b), are internal schemata induced from previously heard musical surfacesand br ough t to bear on the in-time even t sequences described by the reductions.I Grouping ~ ) ~ lstructure

Time-span I Time-span ProlongationatI structureetrical segmentati~ I reducti~ 1~ riductl~Stability 1conditions

igure 6For each of the four organizations there is a set of well-formedness rules thatdefines the conditions for hierarchy. Well-formed structures can be modified in

limited ways by transformational rules. These two rules types are abstract in thatthey only define formal possibilities. By contrast, a third rule type, preferencerules registers particular aspects of pre sent ed musical surfaces and selects whichwell-formed or transformed structures in fact apply to those surfaces. The


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104 FredLerdahlpreference rules, which do the bulk of the work of the theory, do not generatecategorically right or wrong analyses but ins tead predict descriptions as more orless coherent. When the preference rules reinforce one another, the analysis isstable and the passage in question is jud ged as stereotypical: wh en they conflict,the analysis is unstable and the passage is judged as ambiguous or vague.In summ ary , the th eory asserts that (1) the listener has at his or her disposal anumber of hierarchical organizations possessing certain formal properties, and(2) the listener unconsciously attempts to assign the most stable overall descrip-tion (or descriptions) by means of preferential principles that activate andinteract in re sponse to specific features of musical surfaces.

o n s t r a i n t s o n e v e n t s e q u e n c e sWhat does Jackendoff's and my theory have to say about closing the gapbetween compositional system and cognized result? Let me explore this ques-tion by proposing, in this and the following two sections, a number of psycho-logically plausible constraints on compositional gramm ars. These constraintswill give substance to the dashed arrow in Figure 4. Then I will use theconstraints to explain why serial organizations are no t easily learnable.We begin with a pr esupposition of the theory, nam ely that the musical surfacebreaks down into individual events. Certain recent musical developments (pio-neered for instance by Ligeti) have tended to blur distinctions between events.Sensuously attractive though this blurring may be, it inhibits the inference ofstructure.

Constraint 1: The musical surface mus t be capable of being parsed intoa sequence of discrete events.tIt is hardly fortuitous that our theory concentrates on hierarchies. Most ofhu ma n cognition relies on hierarchical structuring (Miller et al 1960; Simon 1962;Neisser 1967). Studies in music psycholog y have indicated that the absence ofperceived hierarchy substantially reduces the listener's ability to learn andremember structure from musical surfaces (Deutsch 1982). It does not suffice forthe input organiza tion to be structured hierarchically; such in fact is the case for

Le Marteau (Koblyakov 1977). It is the relationship to the listening grammar thatmatters.Constraint 2: The musical surface must be available for hierarchicalstructuring by the listening grammar.

As suggested above, there are four ways that the listening grammar canstructure event sequences hierarchically. Let us return to Figure 6 and considerwhat is needed at the musical surface for these organizations to come into play.Segmentation into groups is accomplished at local levels largely by detectionof distinctive transitions in the musical flow. A distinctive transition correspondst This formulationskirts the issue of simultaneous sequences of discrete events, an area that hasbeen researched as auditorystream segregation by Bregmanand his associates (see McAdamsandBregman 1979 for a review).


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Cognitive Constraints 105at a l ess f ine-gra ined level to the pr incip les by which events themselves areperceived as d i scre te ra ther than cont inuous . I t i s a change in some musicald ime ns ion , such as greater d i s tance in a t tack poin t s or sh i ft in dyna mics , t imbre ,or reg i s ter , wi th respect to an immediate context that i s re la t ively invar ian t .From this i t fol lows that constant change wil l not give r ise to sal ient dis t inct ivet rans it ions; nor of course w i ll no chang e a t all . Babbit t and Reich hav e s om eth ingin com mo n af ter a ll . The s teady -s ta te qual i ty of bo t h thei r mus ics i s due to apauci ty of d i s t inct ive trans i t ions , henc e of gro upin g boun dar ies .

Constraint 3: The es t ab l i shmen t o f l oca l g roup i ng bo undar i e s r equ i resthe p resen ce of sa l i ent d i s t inctive t rans i t ions a t the mus ical surface .The o ther fac tor creat ing local groups i s repet i t ion or , more general ly , parallelism of mus ical un i t s . R epet i t ion is an o bsess ion of minimal i sm. But wi th outd i s t inct ive t rans i t ions such grou ps prod uc e only f l at h ierarch ies ; that i s, therepe t it i ons a re no t hea rd w i t h i n l a rge r g roups t o any de p t h o f embe dd i ng .Gr oup s of gro ups ten d to ari se ra ther th rou gh the re inforc ing act ion of d i s tinc-t ive t rans i t ions an d para l le l isms , of ten sup ple me nte d b y a th i rd pr incip le ofsymmetry ( the app roxim ately equal d iv i s ion of a larger t ime span) . At g loballevel s para l le l i sm becomes the overr id ing grouping pr incip le : l i s t eners t ry tohear para lle l passa ges in para lle l p laces in the overa ll s t ructure .

Constraint 4: Project ion of group s , especia l ly a t larger level s , dep en dson s ym me t ry a nd on t he es t ab l i shmen t o f mus i ca l pa ra l le l isms .

Much con t empora ry mus i c avo i ds symmet ry and pa ra l l e l i sm . Indeed , i n t he1950s and 1960s l it era l repet i t ion w as wid ely reg arde d as aes thet ica l ly inexcus-able . But sym me try a nd especia l ly para lle l i sm are bas ic ingredien ts of anycomplex grouping s t ructure . The re l i ance on these pr incip les in Class ica l tonalmus ic and in var ious e thnic mu s ics i s no t nec essar i ly a s ign of l ack of invent ionbu t i s an es sen t ia l t echn i que fo r t he c rea t ion o f deep l y em be dd ed g roups .We turn no w to met rica l s tructure . A wel l - form ed met re i s a looping pa t tern ofequid i s ta n t b eat s occ urr ing a t mul t ip le h ierarch ical l evels . A beat a t any met r ica llevel is fel t to be s t ro ng i f i t i s also a be at at the next larger level . De pe nd in g onthe mus ical id iom, the cr i t er ion of equid i s tance may be loosened , bu t no t to theextent of abandoning a l l sense of per iodic i ty . A mus ical id iom typical ly hasavai lable a l imi ted re per to ry of wel l - form ed met r ica l poss ib il i ti es .W he n hea r ing a mus ical surface , the li s tener seeks an opt imal fi t be tw ee n theaccents perceiv ed d i rect ly f rom the s ignal and the rep er tory of avai lab le met r ica ls t ructures . These phenomenal accents ar i se f rom local s t resses such as s forzandos ,changing bass notes and chords , contextual ly long events , and so for th . Insofaras poss ib le , the onset s of ph eno me na l accents are a l igned wi th re la t ively s t rongbeats . In o ther words , the l i s tener looks for maximal per iodic i ty in the mus icalsignal.Grea t r egu la r i ty i n ph enom ena l accen t s p roduces a rhy thmi c qua l it y o f s t ab -i li ty and squarenes s . In mu ch rhy t hmi ca l l y v it a l mus i c , howe ver , m any pheno -menal accents are forced to a l ign wi th weak beat s , c reat ing cross -accents orsyncop at ions . But i f there i s very l it tl e regular i ty to the phe no me na l accents , thel i s tener ma y no t be ab le to infer any met r ica l s t ructure . Mo st mus ic i s metr ica l ,


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and the immediate location of events is established largely in relation to this orthat strong or weak beat. An inability to assign a metrical grid weakens theprecision of location of events, resulting in a quality of suspe nded rhyt hm.Constraint 5: The establishment of a metrical structure requires adegree of regularity in the placement of phen omen al accents.

Much contempor ary music, even if notated traditionally, avoids regularity ofphenomenal accents and does not give rise to a sense of metre. Complicatedrhythms can cancel as well as create structure. An interesting case is Carter'spractice of establishing two or more simultaneous tempos. Each such tempocorresponds to a single metrical level, or pulsation, bu t usuall y does not evoke ahierarchy of strong and weak beats. Carter contracts metrical depth in order tofocus on multiple speeds. However, two or three simultaneous tempos cannotbe taken in as readily as the four or five metrical levels that are commonplace in amarch or waltz. Musical cognition has a bias for hierarchical organization.Simultaneous tempos instead produce in dep end ent organizations competing forattention. The difficulties in atten ding to more th an one such organization arewell known (Cherry 1953) and are ameliorated only if the vertical correspon-dences are coherent (Sloboda 1985), a condition that only partly obtains in thiscase. The situation is analogous to tha t of harmonically controlled versus uncon-trolled polyphony.The grouping and metrical structure come together (via well-formednessrules) to form the time-span segmentation. Details aside, it should be intuitivelyobvious that events are heard and thought of within such units. We normallyspeak of an event at small levels in terms of this or that beat, at larger levels interms of this or that phrase or section. If the grouping or metrical structure isimpoverished, then so too is the time-span segmentation.

Constraint 6: A complex time-span segmentation depends on theprojection of complex grouping and metrical structures.Various passages of Le Marteau in which no metrical structure is apparent andthe groups are not deeply embedded, provide a contrary illustration. Because of

the consequent lack of hierarchical time-span segmentation, the sense of thesepassages is of a flurry of events followed by a pause, then another flurry andanother pause, like beads on a string. Each flurry tends to become one complexevent rather than an organized sequence of events. The phrase is replaced bya sonorous object extended in time. (This is one way of explaining why , despiteits generally fast tempos, Le Marteau feels slow.)The notion of an extended sonorous object filled with inner movement is acompelling one, especially in light of recent developments in computer music.Here lies an unexpected link between Le Marteau and Boulez's recent work,Rdpons. But from the perspective of musical cognition, one must ask how theinner m ovem ent of an object receives structure. We have returne d to Constraint1, the issue of musical surfaces as discrete events. If the listener can assign anyrich internal structure to the sonorous object, it must be perceived as consistingof subo bjec ts'. But if the subobjects are to be experienced as aspects of oneobject, the y canno t be too discriminable or allow salient structuring.


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Cognitive Constraints 107T h e t i me- s p an s eg men t a t i o n mu s t b e h i g h l y s t ru c t u red fo r a co mp l ex ev en th i e r a r ch y t o emerg e - s o me t h i n g n o t ach i ev ed , an d p e rh ap s n o t t r i ed fo r , i n LeMarteau Rdpons o r i n d eed mo s t co n t em p o ra ry mu s ic . Si nce ev en t s i n mu s i care usua l ly p i t ch even t s , even t h ie ra rch ies a re normal ly p i t ch h ie ra rch ies . Auni f i ed p i t ch h ie ra rchy i s an ins tance o f tona l i ty (b road ly def ined) , the s t ab i li tycond i t ions fo r which wi l l be d i scussed in the nex t two sec t ions . For now le t usa s s u m e s tab il it y co n d i t i o n s an d r ev i ew t h e p ro j ec t i o n o f ev en t h i e r a r ch ie s f ro mmus ica l su r faces .T h e t w o k i n d s o f r ed u c t i o n a r e c o mp l eme n t a ry d es c r i p ti o n s o f ev en t h ie r ar -ch ies . In the t ime-s pan reduc t ion , the l i s t ener comp ares the re la tive s t ab il i ty o feven t s wi th in each t ime-span segment . The l ess s t ab le even t in a segment i sjudged as subord ina te to , o r as an e labora t ion o f , the more s t ab le even t . Th i s

proc ess co n t inues recurs ive ly f rom loca l to g loba l l eve l s o f the t im e-span seg-me nta t io n un t i l an en t i re t ime-s pan t ree is bu i l t up . The re la tive s t ab il i ty o fev en t s w i t h in a s p an i s d e t e rm i n ed p a r t ly b y rh y t h mi c p o s i ti o n , f o r ex amp l ew h e t h e r an ev en t o ccu r s o n a s t ro n g b ea t o r c ad en ces a g ro u p . Bu t t h e ma i nde te rm inan t o f dom ina t ion o r subo rd ina t ion wi th in a spa n i s the se t o f s t abi l itycond i t ions .Constraint 7: T h e p ro j ec t io n o f a t ime- s p an t r ee d e p en d s o n a co mp l ext ime-span segmenta t ion in con junc t ion wi th a se t o f s t ab i l i tycond i t ions .

A li tt le though t -exp er im ent wi ll demo ns t ra te the necess i ty o f bo th the t ime-span segm enta t io n a nd the s t ab il i ty cond i t ions fo r the hear ing o f an even th ie ra rchy . Imagine the p i t ches o f Le Marteau p l u g g ed i n t o t h e rh y t h ms o fBee t h o v en ' s F i f t h Sy mp h o n y . T h e h i e r a r ch i ca l t i me - s p an s eg men t a t i o n w o u l dla rge ly be e rased by the l ack o f any conco mi tan t p i t ch a r t icu la tion , and , in anycase , the absence o f s t ab il i ty cond i t ions wo uld p rev en t the in ference o f any p i t chh i e r a rch y . O r , co n v e r s e l y , i mag i n e t h e p i t ch es o f t h e F if th Sy m p h o n y p l u g g edi n t o t h e rh y t h ms o f Le Marteau. The potent ial ly hierarchical p i tch s t ructurew o u l d b e g a rb l ed b y t h e l a ck o f s u p p o r t i v e t i me- s p an s eg men t a t i o n ( i n c l u d i n gthe absenc e o f any met r i ca l s t ruc tu re o r pa t t e r ned g ro up in g s t ruc tu re) . In shor t ,r h y t h m an d p i tch mu s t w o rk i n co n ce r t if t h e r e is to b e an y t i me- s p an r ed u c t io n .The p ro longat iona l reduc t ion der ives f rom the t ime-span t ree and , aga in , f romthe s t ab i l ity cond i t ions . Al tho ugh th i s is the mo s t impo r tan t o f the wa ys inwh ich even t s re la te to one ano ther , i ts descr ip t ion i s ra ther t echn ica l andabs t rac t , a nd so wil l be avo id ed he re . Suf f ice i t t o say tha t in th is co m po ne n teve n t s conne c t h iera rch ica l ly in t e rms o f p rogr ess ion and con t inu i ty , c rea t ingprolongational regions tha t ac t in s t ruc tu ra l coun terpo in t to the t ime-sp an segm en-ta t ion ( see Lerdah l and Jackendo ff 1983 fo r a de ta i l ed t rea tme nt ) .

Constraint 8: The p ro jec t ion o f a p ro longat iona l h ' ee de pe nd s on acor re spon ding t ime-sp an t ree in con junc t ion w i th a se t o f s t ab il i tycond i t ions .Be fo re co n t i n u i n g , i t w o u l d b e w e l l t o emp h as i ze h o w i n t e rd ep en d en t t h econs t ra in t s a re tha t ha ve be en l i s t ed so fa r . Mos t o f the m cal l on the i r imm edia teprede cesso rs . Thus a h ie ra rchy o f eve n t s (C ons t ra in t 2 ) requ i res a mus ica l


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surface that has been parsed into events (Constraint 1); the recursive segmen-tation of the surface into rhythmic units (Constraint 6) cannot take place withoutthe establishment of grouping and metrical structures (Constraints 3-5); therelative structural importance of events (Constraint 7) cannot be assessed with-out the time-span segmentation (Constraint 6); and the representation of howevents depart and return (Constraint 8) deman ds an evaluation of their functionin the rhythmic structure (Constraint 7). The requirement of projecting eventhierarchies to the listener has brought a great deal in its wake.

o n s t r a i n t s o n u n d e r l y i n g m a t e r i a l sWe turn now to the stability conditions underlying the two reductions.Generally, a stability condit ion says, Musical context aside, this structure isjud ged as more stable tha n that. Stability conditions interact like preferencerules, sometimes reinforcing and somet imes conflicting with one another. InClassical tonality, for example, a triad in root position is more stable than a triadin first inversion, and a I chord is more stable than a V chord. These conditionsreinforce one anoth er w he n the jud gement is between a I and a V 6, but conflictwhen the jud gem ent is betw een a I~ a V. In actual music such conflicts areresolved by context: on the assumption that the two events come up forcomparison in the same unit (time-span segment or prolongational region), thereductional preference rules decide which event is more stable on the basis ofrhythmic position, linear function, and other factors.Stability conditions are cont ingent on the basic materials and properties out ofwhich the event structure of a piece is made. In Classical tonal music, forinstance, the basic materials include the diatonic scale, which has certain inter-nal and transpositional properties that can project judgements of relative stab-ility (of which more below). Musical idioms vary in these respects. It is possiblethough rare for the basic materials not to be used in a way that gives rise tostability conditions: one can imagine a tr eatme nt of the diatonic scale where allscale degrees are functionally equivalent and the circle of fifths is not employed .Similarly, both Bach and Schoenberg used the well-tempered chromatic scale,but only the former carved it up so as to induce clear judgements of relativestability among pitch configurations; the latter instead sought a noncentricmusical universe in which the twelve tones . . . are related only with oneanother (Schoenberg 1941). However, since learning and m emor y dep end onhierarchical structuring (Constraint 2), and since event hierarchies depend onstability conditions (Constraints 7 and 8), it follows that stability conditions arecognitively advantageous.What are the constraints on stability conditions? First, there must be a fixedcollection of sonic elemen ts (normally pitches; but see Lerdahl 1987). Theassumption of such collections might go unremarked except that in electronicmusic they are fairly rare. Whe n there is a virtual inf initude of sonic possibilities,it is hard for a composer to select and stick with a limited collection.Non-selection, however, means no syntax.

Constraint : Stability conditions m us t operate on a fixed collection ofelements.


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Cognitive Constraints 109A fixed collection should not be thought of as completely rigid. First, anelement of a collection need only be perceived categorically (Burns and Ward1982); otherwise a slightly mis tun ed pit ch would cause havoc. Second, as long asthe categories remain the points of reference, there can be embellishing orsliding within or between categories (as in microtonal inflexions in Ind ian musicor string glissandos in Western music).The el ements of a collection can be placed along a dime nsio n to form a scale.The intervals of a scale have general size limitations; they mus t be large en oug hfor adjacent elements to be easily discernible, but not so large as to use upexcessive space along the continuum.

Constraint 1 : Intervals between elements of a collection arrangedalong a scale should fall with in a certain range of magni tude .Constraints 9 and 10 are analogous to Constraint I (the requirement of discreteevents). Both cases provide the building blocks for further organization,Constraints 9 and 10 for stability conditions and Constraint 1 for event hierar-chies. A fixed collection with moderate interval sizes allows other stabilityconditions to emerge. In surveying some of these, let us assume that thecollections are pitch collections - as opposed, say, to drums or synthesizedtimbres.Most pitch collections take advantage of the 2:1 frequency ratio of the octave,so that in one respect pitches reduce to pitch classes . There are 88 pitches onthe piano but only 12 pitch classes. Thus, in addition to giving a recurring

structure to the overall collection, the octave decreases to a more manageablesize the mem ory load for elements of the collection.Constraint 11: A pitch collection should recur at the octave to producepitch classes.

The octave can be divided up in inn umerable ways. A co mmon route has beento make a collection beginnin g with o ther small-ratio intervals and progressinggradua lly to intervals with larger ratios. From the present perspective, this routeis advantag eous because the resultant intervals provide a broad an d gradua tedpalette of sensory consonance and dissonance. Sensory consonance and disso-nance can in turn form the basis for musical consonance and dissonance, wherein a general sense consonance is equivalent to stability and dissonance toinstability. Thus a seventh in Classical tonal music resolves to a sixth not just outof cultural convention but because the syntactical resolution is supported bysensory experience.The claim is not that pitch configurations exhibiting relative sensory conso-nance m ust necessarily be more stable in musical contexts than those exhibitingrelative sensory dissonance. The reverse can occur contextually, and shows upin the reductions where appropriate. Rather the claim is that the stabilityconditions will be relatively ineffectual unless they are supported by sensoryconsonance and d issonance. Stability conditions in which a seventh is suppose dto be more stable than a sixth would contradict sensory experience and wouldnot lead to event hierarchies of any depth of embedding.Strictly speaking, these remarks hold for fundamentals with harmonic par-tials, such as produce d by the voice and by all inst ruments in which the source is


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110 FredLerdahlperiodically excited. For intervals with very small ratios, the partials of therespective fundamenta ls are largely reinforcing, creating the sensat ion of conso-nance. Sensory dissonance, or rough ness , arises whe n frequencies (whetherof fundamentals or of partials) interfere within a critical bandwidth, whichextends in mos t musical registers from a fraction of a semi tone to a little less thana minor th ird (Plomp and Levelt 1965). The critical bandwidt h is a consequenceof the resolving power of the hearing mechanism in the peripheral auditorysystem. One could also invoke pattern-recognizing templates of the centralauditory system, such as.proposed in Terhard.t (1974); these for present pur-poses lead to the same conclusion.

Constraint 2: There mus t be a str ong psychoacoustical basis forstability conditions. For pitch collections, this entails intervals thatproceed gradually from very small to comparatively large frequencyratios.

However, as is well known, small interval ratios'- whether in just orPythagorean tuning - present problems for the transposition of intervals andchords. A reading of Partch (1949) migh t additiona lly suggest tha t the numb er ofpitches and intervals in such a collection can become alarmingly large. Analternative route that avoids these problems is to divide the octave into equalintervals. The collection then permits equivalence under transposition, and thepitches and intervals to be learned are limited.Constraint 3: Division of the octave into equal parts facilitates trans-position and reduces memo ry load.

But these advantages are offset by the fact that almost all equal divisions of theoctave lead to interval ratios that are complex and that do not easily round offcategorically to simple ratios. As a result the intervals give a small range ofsensory consonance and dissonance, and Constraint 12 is not satisfied.The 12-fold division of the octave (and to a lesser extent the 19-fold division;see Yasser 1932) is an exception; its intervals are sufficiently close to justintervals to be useful; that is, the deviations from just intonation are not largeenough to cause serious roughness from otherwise consonant intervals. Theslight loss in distinctions in sensory consonance and dissonance is compensatedfor by the possibility of unfettered transposition. The familiar chromatic scalehas presumably survived so well because it meets the demands of bothConstraints 12 and 13.At this po int two routes are available, one viewing the chromatic collection (orset) essentially as a whole, the other thinking of it as material for subsets thathave various musical and psychological properties. Let us review thesealternatives.One can evoke stability conditions directly from the chromatic set through aselection and patterning of different degrees of sensory consonance and disso-nance. For example, Debussy's and Bart6k's harmonies often modula teamong chord types of varying degrees of dissonance. Schoenberg was at leasttacitly occupied with such a notion both early and late in his career: early,notably in the First Chamber Symphony (1906), where the basic dissonant-to-


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Cognitive Constraints 111consonan t progression is from a chord built in fourths to a whole-tone chord to atriad; late, in the 12-tone works of the 1940s that include tonal features, particu-larly at cadential points. Hindemi th 's (1937) theory of harmonic fluctuation isbased on a similar idea, generalized on a supp osedly psychoacoustic foundati onto embrace all possible sonorities. It is profitable to view harmon ic progressionin the work of Machaut and other fourt eenth-cent ury composers in such a light.Much of my own work also fits in this category.The other rou te is taken by the Classical tonal system, in which the referentialsonority is always the triad. Various versions of the triad alone do not projectadequa te differences in sensory consonance and dissonance. Stability conditionsamong triads are instead achieved through the use of subsets of the chromaticset, notably the diatonic. (Historically, of course, the diatonic scale precedes thechromatic, and until Wagner it was musically paramount as well. It can beilluminating, however, to view the diatonic in terms of the chromatic.)Balzano (1980~ 1982) shows through a g roup-theoreti c analysi s that onlycertain equal-interval sets - the 12-fold among t hem - lead to subsets incorporat-ing the psychologically import ant criteria of intervallic uniqueness , scale-stepcoherence , and tr anspositional simplicity . All three criteria have to do withlocation in pitch space . The first criterion deman ds a uniq ue vector ofrelations for each pitch in a subset, so that the listener can orient himselfuna mbi guo usly in relation to the other pitches. For example, each scale degreeof the asymmetrical diatonic subset of the chromatic set has a non-duplicatingintervallic relation to the other scale degrees. In contrast, the octatonic subset(alternating half- and whole-steps), which has been wi dely used in this century,is not uniq ue , since its vector repeats four times in an octave (this is thecharm of impossibilities described in Messiaen 1944). The second criterionensures that the intervals of a scale proceed in an orderly manner, so that, forinstance, the same distance is not traversed by one step in one part of a scale butby two or more steps in ano the r part. The harmoni c minor scale, not to ment ionmore extreme examples, would thereby be disqualified, since the augmentedsecond between the sixth and seventh scale degrees is equal in distance to theminor thirds that els ewhere take two steps to traverse; this presumably explainswh y the scale is avoided melodically in much diatonic music. The third criterionensures orderly transposition of a subset. The diatonic subset, for example,drops one pitch and adds another as it transposes along the circle of fifths, sothat distance in transposition correlates with the number of pitches shared intwo diatonic scales.

Constraint 4: Assume pitch sets of n-fold equal divisions of theoctave. Then subsets t hat satisfy uniqueness , coherence, and sim-plicity will facilitate location with in the overall pitch space.Balzano surprisingly demonstrates that in the 12-fold chromatic set only thepentatonic and diatonic subsets satisfy all three criteria. He advances this ratherthan traditional psychoacoustical explanations as the reason for the cultural

ubiquity of these scales; and he recommends that the few other n-fold divisionsof the octave that permi t these features (such as the 20-fold set) be explored fortheir musical potential. However, none of the other candidates approximatesjust intonation. The real cognitive appeal of the chromatic set, I woul d argue, is


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112 FredLerdahlthat only here do Constraints 12, 13 and 14 all converge intact. The structuralfeatures of the pentatonic and diatonic subsets wo uld appear to be correspond-ingly special. (For discussion relating to this section, see Sloboda 1985; Dowlingand Harwood 1986.)

i t ch spaceI referred above to pitch space . I wan t no w to examine this notion with thepurp ose of extracting a few more const raints on stability conditions.A number of cognitive psychologists, notably Longuet-Higgins (1978) andShepard (1982), have propo sed multidime nsional representations for pitch re-lations.t The basic idea is that in a tonal framework pitches are cognized asrelatively close to or far from one another, and that these distances can berepresented geometrically. To take the simplest case, pitches are heard asproximate in terms of both pitch height and pitch class, so at least a two-dimensional representation is necessary. Another dimension can be added toaccount for fifth-relatedness; and so on. One can think of these representationsas internalized maps. Just as cities are close to or far from home and can best bereached via some roads rather than others, so it is with tonal relations.

Cons t ra in t 5 : Any but the most primitive stability conditions must besusceptible to multidimensional representation, where spatial dis-tance correlates with cognitive distance.

The geometric pitch spaces proposed in the psychological literature seempromising, bu t I have lingering doubts. First, granting the insights provided byBalzano (1982) and Shepard (1982), it remains odd that the chromatic collectionis taken as the basis for the diatonic music they seek to explain. A second andrelated point is that, in pursuit of geometric symmetry, these models tend tocome up with more regularity than diatonic music warrants. For exampleLonguet- Higgi ns's (1978) array has a major-third axis, a nd Shepa rd's (1982)double helix is made up to two strands of whole-tone scales. But only thesemitone an d perfect-fifth interval cycles are rel evant to diatonic music, which isin many ways asymmetrical. Other interval cycles do not begin to assumemusical significance until nineteenth-century chromaticism.These questions lead me to suggest a complementary pitch space that has areductional rather than a geometric format. This space is musically obvious yethas explanatory power. I take the lead from Deutsch a nd Feroe (1981), whospeak of differen t pitch alphabets - chiefly the chromatic, diatonic, and triadic- that the tonal listener highly overlearns from previously heard musicalsurfaces. This in itself is an important point, since without rep eated exposure tosuch regularities, listeners could not spontan eously organize new input in termsof them.t Actually, here is a long tradition n music heoryfor such models, datingback to Weber (1824) andhis eighteenth-centurypredecessors; see Werts (1983). Schoenberg's (1954) chart of regions, whichhas been cited in the psychological iterature, can be traced through Riemannback to Weber.


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ognitive onstraints 3Constraint 6: Levels of pitch space mu st be sufficiently available frommusical surfaces to be internali zed.

Deutsch a nd Feroe show that m emor y load is greatly reduced if tonal melodiesare structured by means of their alphabets. As their formalism suggests, suchalphabets are hierarchically related as in Figure 7: the octave level a elaboratesinto the perfect-fi fth level b, which elaborates into the triadic level c, whichelaborates into the diatonic level d, wh ich elaborates into the chromatic level e. Inaddition, pitches at higher levels in the hierarchy are more stable than pitchesthat do not appear until successively lower levels. This correlates withKru mha nsl 's (1979) experimental results. So Figure 7 includes a partial represen-tation of stability conditions.

Level a : C CLevel b : C G CLevel c : C E G CLevel d: C D E F G A B CLe vele: C D , D E, E F F; G A~ A B~ B C

igure 7

This pitch space circumvents the reservations mentioned above. First, theoverall orientation is not chromatic. Non-well-tempered distinctions can beincorporated merely by expandin g the content of level e, leaving the other levelsuntouched. Second, level e expresses an equal interval cycle and otherunwanted equal divisions of the octave are absent.As chords a nd diatonic collections change, so too must the hierarchy in Figure7. For example, in the framework of a IV chord in C major, levels a, b and cwould shift respectively to F-F, F-C-F, and F-A-C-F. If the diatonic collectionchanges to that of F major, level d would also shift, with B flat replacing B. Thestructure of this shifting pitch space demands further representation, perhapsgeometric in format, but we will not pursue this here.Each level in Figure 7 can be thought of as a scale with stepwise motionoccurring between adjacent members. An octave is a step away at level a, as is afifth at level b; an arpeggiat ion proceeds by step at level c; diatonic and chromaticprogressions are by step at levels d and e. A skip is a progression between twoelements that requires more than one step for traversal. This step/skip distinc-tion is important for establishing cognitive distance, and was implicit inConstraint 14 above.Figure 7 can further be seen as expressing preferred melodic routes via theGestalt laws of proximity and good continuation. A preferred route is one that is

complete , where completeness is defined as a stepwise progression at anylevel that begins and ends at elements represented at the next higher level. Thusa complete chromatic progression must begin and end on diatonic pitches, andso on. Anything else - a chromatic appoggiatura, for instance - is felt as


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114 FredLerdahlincomplete . General ly , incompleteness i s more acceptable a t the s tar t than a tt he end o f a p rog res s ion ; uns t ab l e e l emeDt~ mus t be subsequen t l y anchored(Bharucha 1984a).

Constraint 17: A reduct ional ly organized p i tch space i s needed toexpres s t he s t eps an d sk i ps by wh i ch cogn it ive d i s tance i s me asu r edand t o exp res s deg rees o f me l od i c compl e t enes s .The not io n of com pleten ess t i es in s ignif icantly wi th curre n t m us ic theor iesthat are otherwise diss imilar . Meyer 's (1973) and Narmour 's (1977) implicat ion-real iza tion the ory d ep en ds exact ly on the space in F igure 7: a s tepwise m ot ion a tany level impl ies the next s tep a t tha t l evel ; such an impl ica t ion can be rea l i sed or

not , as the case may be . A nd Schenk er ' s (1935) centra l conce pt ion of the Zug, ad i a ton i c p rog res s i on bo un de d b y p it ches o f t he p ro l onged ha rmo ny , f al ls ou t a sa specia l case of com pleten ess a t l evel s c and d in F igure 7 . Oth er Schenker ianconcepts (coupl ing , arpeggia t ion , and in i t i a l ascent , the Urlinie, theBassbrechung can be i l luminated in re la ted fash ion . Completeness appears to bethe basic voice-leading principle in Classical tonal music.Tw o general p sychological po in t s m ay be inse r ted here . F irs t, ev en i f - as a tthe f i f th , t r iadic, an d diatonic levels - intervals along a level are psych oac oust i -ca lly uneq ual , they are cogni t ively equal bec ause they are s teps a t tha t l evel.Second, b eca use ea ch of these l evel s cons t i tu tes a ca tegory in t erms of whic h thenext l evel i s unders tood , Mi l ler ' s (1956) res t r i c t ions on memory load ( sevenp l us o r m i nus t wo ) a re eas i l y me t , somet h i ng no t accompl i shed by t he s i ng l ecategory of the chrom at ic se t .Final ly, Figure 7 reflects certain psychoacoust ic facts . In general , a descentf rom level a to level e br ings increas ing sens ory d i ssonance. Level a g ives thevirtual pi tc h (or root) of the col lect ions in levels b an d c (Terh ardt 1978). Level bse rves a s t he ha rmon i c no rm fo r med i eva l mus i c and va r i ous e t hn i c mus i cs ,l evel c for Class ica l tonal mus ic . Level d prov ides the m elodic bas i s for manymusical id ioms, wi th l evel e of fer ing inf lec t ional poss ib i l i t i es . Thus betweenl evel s c and d t he re appear s a concep t ua l li ne s epara t i ng h a rm ony and mel ody .This d if feren t ia t ion ma y in par t s tem f rom the cri ti ca l ban dw idth , s ince for mo s treg i s ters s teps a t l evel c fal l ou t s ide the b an dw idt h and s teps a t l evel d fal l ins ide .

F igure 7 embodies a l l o f the cons t ra in t s on s tab i l i ty condi t ions enumerated inthe las t two sect ions : f ixed co l lec t ions wi th appropr ia te in terval s izes (Con-straints 9 and 10); octave equivalence at level a (Constraint 11); increasingsens ory d i ssona nce f rom level a to l evel d (Cons t ra in t 12); equal d iv i s ion of theoctav e at level e (Con straint 13); un iqu en ess , cohe renc e, an d s implici ty at level d(Cons t ra in t 14 - and , incidenta l ly , u n iq uen ess and coher ence a t l evel c); mul t id i-men s ional repre senta t io n express in g cogni t ive d i s tances (C ons t ra in t 15); l evelsof p i tch space that are eas i ly indu ced f rom a wide var ie ty of mus ical surfaces(Cons t ra in t 16); and s teps , sk ips , and degre es of melodic com plete ness a tmul t ip le red uct ional l evels (Cons t ra in t 17). The cu l tura l pers i s tence of the ma nymusical id ioms re la t ing to F igure 7 may be due to th i s convergence.Some of these cons t ra in t s seem to me b inding , o thers opt ional . Con s t ra in t s9-12 are essent ia l for the very ex i s tence of s tab i l i ty condi t ions . Cons t ra in t s13-17 , on the o th er han d, can be var iou s ly je t t i soned . The resu l t ing s tabi l itycondi t ions ma y be we aker , bu t they can s ti ll l ead to h ierarch ically r i ch mus ic . For


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Cognitive Constraints 115e x a m p l e , S o u t h I n d i a n m u s i c a p p r o x i m a t e s j u s t i n t o n a t i o n a n d d o e s n o t m o d u -l at e , t h e r e b y i g n o r i n g C o n s t r a i n t 1 3 a n d p a r t o f C o n s t r a i n t 14 ; D e b u s s y , B a r t6 k ,a n d o t h e r s h a v e d e v e l o p e d c o n s o n a n c e - d i s s o n a n c e p a t t e r n s d i r e c tl y f r o m th et o t a l c h r o m a t i c , t h e r e b y i g n o r i n g C o n s t r a i n t s 1 4 - 1 7 . I t w i l l b e f a sc i n a t i n g t od i s c o v e r h o w t h e n e w s o u n d m a t e r i a ls o f c o m p u t e r m u s i c w i ll b e a b le t o m e e tt h e se c o n s t r a i n t s . I n al l p r o b a b i l i t y t h e n e w m a t e r i a l w i ll b r i n g a d d i t i o n a lr e q u i r e m e n t s i n t o p l a y . M e a n w h i l e t h e r e r e m a i n s a m p l e l e e w a y w i t h i n t h ec h r o m a t i c c o l l e c ti o n a n d w i t h i n t h e c o n s t r a i n t s l i st e d a b o v e f o r m u s i c a s y e tu n i m a g i n e d .

ognitive opacity of serialismNo w we a r e i n a p o s i t i o n t o s e e wh y se r i a l o r g a n i z a t i o n s a r e i n a c c e s s i b l e t om e n t a l r e p r e s e n t a t i o n . R a t h e r t h a n e x p l o r e t h e s u b je c t t h r o u g h t h e i d i o s y n c ra t i cse r i a l ism o f Le Martea u I w i ll r e f e r to e l e m e n t a r y a s p e c t s o f t h e C l a s si c a l 1 2 - t o n esy s t e m , wh i c h h a s b e e n m o r e w i d e l y i n f l u e n t i a l ( S c h o e n b e r g 1 9 4 1 ) . I w i l l g i v et h r e e c a u s e s f o r s e ri a li s m ' s c o g n i t i v e o p a c i t y . tB e f o r e p r o c e e d i n g , I m u s t e m p h a s i z e t h a t th e i s s u e is n o t w h e t h e r s e ri alp i e c e s ar e g o o d o r b a d . A s w i t h t o n a l m u s i c , s o m e s e ri al p i e ce s a r e g o o d a n dm o s t a r e b a d . N o r a m I c l a i m i n g t h a t l i s t e n e r s i n f e r n o s t r u c t u r e a t all f r o mm u s i c a l s u r f a c e s c o m p o s e d w i t h s e ri al t e c h n i q u e s . W h a t l i s t e n er s i n f ac t i n f e rf r o m s u c h s u r f a c e s i s a n i n t e r e s t i n g q u e s t i o n , o n e t h a t d e s e r v e s t h e o r y a n de x p e r i m e n t i n i t s o wn r i g h t . B u t t h i s i s n o t t h e i s su e h e r e . T h e i s su e i s wh yc o m p e t e n t l i s t e n e r s d o n o t h e a r t o n e r o w s w h e n t h e y h e a r s e r i a l p i e c e s .T h e f i r s t r e a so n i s t h a t s e r i a l i sm i s a p e r m u t a t i o n a l r a t h e r t h a n e l a b o r a t i o n a ls y s t e m . P i t c h r e l a ti o n s i n v i r t u al l y a ll n a t u r a l c o m p o s i t i o n a l g r a m m a r s a r ee l a b o r a t i o n a l . T a k e f o r e x a m p l e t h e p i t c h s e q u e n c e B - E - F sh a r p . I t i s e a sy t oi m a g i n e a n y n u m b e r o f m e l o d i c e m b e l l i s h m e n t s i n t e r n a l to t h e s e q u e n c e . T h i st y p i c a ll y h a p p e n s w h e n c h i l d r e n s in g a n d p e r f o r m e r s i m p r o v i s e . L i k e t h e s y n t a xo f a s e n t e n c e ( C h o m s k y 1 95 7) , m u s i c a l e l a b o r a t i o n s c a n c o n t i n u e r e c u r s i v e l y t oa n i n d e f i n it e l e v e l o f c o m p l e x i t y ; fo r e x a m p l e , s o n a t a f o r m i s a n e x p a n s i o n o f t h eC l a s si c al p h r a se ( S c h e n k e r 1 93 5; R o se n 1 97 2; L e r d a h l a n d J a c k e n d o f f 1 98 3) . T h i sf e a t u r e e n a b l e s p i t c h r e l a t i o n s t o b e d e sc r i b e d h i e r a r c h i c a l l y b y a t r e e n o t a t i o n .

S e r i a li sm in s t e a d d e p e n d s o n sp e c i f i c o r d e r i n g s o f t h e e l e m e n t s o f a s e t.D i s t i n ct i o n s a ri s e f r o m p e r m u t a t i o n s o f t h e e l e m e n t s ; f o r e x a m p l e , t h e i n v e r s i o no f a 1 2 - t o n e r o w h a s t h e s a m e 1 2 p i t c h c l a s se s b u t i n a d i f f e r e n t o r d e r . T h e o r d e rp o s i t i o n o f t h e e l e m e n t s i s t h e r e f o r e e s se n t i a l t o t h e i d e n t i t y o f i n d i v i d u a l s e tf o r m s . F r o m t h i s i t f o l l o w s t h a t i n t e r n a l e l a b o r a t i o n o f a n y e l e m e n t b y o t h e re l e m e n t s w i ll u n d e r m i n e a se t f o r m . C o n s i d e r t h e r o w o f S c h o e n b e r g ' s V i ol inC o n c e r t o ( 1 9 3 6 ) , s h o w n i n F i g u r e 8 w i t h o r d e r n u m b e r s g i v e n a b o v e . T h es e q u e n c e B - E- F s h a r p a p p e a r s a t o r d e r n u m b e r s 3 - 5. I f o n e w a n t e d t o e l a b o r at eE , s a y , w i t h G o r w i t h B f l at - A , c r e a t i n g t h e s e q u e n c e s B - E - G- E- F sh a r p o r B- E- Bf l at -A- E - F sh a r p , t h e i n t e g r i t y o f t h e r o w wo u l d b e d e s t r o y e d , s i n c e G , B f la t,~- It must be said that, if only from the evidence in Schoenberg's 12-tone sketches (Hyde 1980),Schoenberg's (1941) account of the 12-tone system is partly misleading. Plainly he u sed the systemnot just to generate rows bu t to create certain systematic but non-serial relationships among subsets.This, however, does not affect my argument, which concerns the cognitive opacity of serialstructures (tone rows) as such.


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6 redLerdahl

and A have order positions elsewhere in the row. A tone row is not anelaborational structure. (The same point could be made with reference not toelements but to the intervals between elements.)The little research tha t has been directed towards serialism (as in Franc6s 1958;Dowling 1972; Deutsch 1982; Bruner 1984) suppo rts the contention that permu-tational structures are hard to learn and remember. Since other hu ma n activitiesare not organized in such a fashion, it is hardly surprising that the issue has ingeneral been ignored by psychologists. The sit.uation reminds me, though, ofChomsky's (1965) observation that many logically possible grammatical con-structions neve r occur, such as for ming interrogatives from declaratives by wordreversal or by exchange of odd and even words. There is unfort unately no oneon whom to test these linguistic counter-examples. Music offers a uniqueopport unity in this regard. Surely children can be fou nd who have been raisedon a st eady diet of serial music. Do they identify tone rows? A negative answerwill provide strong counter-ev idence concerning the structure of musical cogni-tion, and may suggest inherent limitations on cognitive organization in otherdomains as well.Ironically, Schoenberg was much preoccupied with the issue of comprehensi-bility. I suspect this is one reason w hy in his 12-tone phase he ad opt ed Classicalmotivic, phrasal, and formal structures. As a result, his serial music satisfies therhythmic Constraints 3-6. But the permutational basis of his pitch organizationassures a gap betwe en the compositional and listening grammars.It is of course possible to organize the combinations and sequences of indi-vidual rows on a hierarchical rather than permutational basis. Boulez, Babbitt,and others have done just that. But these higher-level hierarchies are extremelydifficult to cognize in a hierarchical fashion because their un derl ying basis, therow, remains non-tree-like. There are fur ther reasons as well, which brings us tothe other causes of serialism's cognitive opacity.The second cause is the avoidance of sensory consonance and dissonance asan input to the system (Constraint 12). This avoidance in turn stems from thehistorical collapse of the stability conditions underlying Classical tonal pieces.The pre-serial atonal period had established an aesthetic in which hierarchicalpitch relations gave way to purely contextual, associative ones. The 12-tonesystem reinforced this trend thro ugh constant reshuffling of the total chromaticand thro ugh co nstant repetition of the intervallic patterns of the row. Concernedabout the natural basis of his new aesthet ic, Schoenberg (1911, 1941) invoked thedoctrine of the emancipation of the dissonance : consonance and dissonanceare not opposites but are on a continuum determined by the overtone series;dissonances are harder to compre hend than consonances; but acquaintance withthe more remote overtones has ma de dissonan t intervals just as comprehensibleas consonan t intervals; so now intervals may be treated as equally consonan t (ordissonant). Babbitt (1965) goes a step further, debunking the overtone series as abasis for pitch systems and claiming that consonance-dissonance distinctionsamo ng intervals are entirely contextual in origin.Neither Schoenberg nor Babbitt distinguishes between sensory and musicalconsonance and dissonance. Though the details of pitch perception are not fullyresolved, there is no doub t abou t the objective existence of sensory consonanceand dissonance. For intervals between pitches with harmonic partials, thedegree of sensory dissonance bears an important relationship to the overtone


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ognitive onstraints 7Ord er numb er: 0 1 2 3 4 5 6 7 8 9 10 1112 -t on es et : A B~ E~ B E F~ C C G Aw D F

igure 8ser ies, by v i r tue of the de gree of in ter ference am ong par t ia l s wi th in the cr it ica lbandwidth . Thus far Schoenberg , a t l eas t , i s r igh t . What one does wi th th i smus ical ly i s another mat ter . C lass ica l tonal i ty founds mus ical consonance andd i s sonance l a rge l y on s enso ry consonance and d i s sonance (pa r t i a l excep t i onmust be made for the perfect four th , the minor t r i ad , and so on) . Ser ia l i sm, onthe o the r hand , t rea t s in tervall ic com binat ions in terms of the row and in ef fectdef ines mus ical conso nanc e an d d i ssona nce out of ex i s tence (wi th the g lar ingexcept ion of the octave: p itch c lasses are need ed) . But th is s t ra tegy , whi leperfect ly log ical , doe s not neut ra l ize sens ory conso nanc e and d i ssonance. Thesens ory d i sso nanc e of a seve nth re mains greater than that of a s ixth , regard lessof the mus ical purp ose s to which these in terval s are put.There i s an impor tan t psychological conseque nce: by ignor ing sen sory d i s t inc-t ions , ser ia li sm creates mus ical contexts that are not app reh en de d h ierarch ical ly .On e p i t ch o r hexachord ma y be as soc i a ted wi t h ano t he r p i t ch o r hexachord , bu tthese re la t ionships are not easi ly hear d in a dom inat ing -subo rdina t ing mann er .The requis i t e in tu i t ions of s tab il i ty and ins tab i l ity are miss ing . An d beca usehierarch ies are not in fer red , to ne row s and thei r combinat ions are d i f f icu l t tocomprehend (C ons t ra i n t 2 ) .The th i rd cause i s that ser ia l i sm does not induce a p i tch space where spat ia ld i s tance corre la tes wi th cogni t ive d i s tance (Cons t ra in t 15) . Assume for s im-pl ic ity that ser ia l p i t ch space looks l ike the combina t ional space sho wn inFigure 9 for Schoenb erg ' s Vio lin Concer to . He re the f i rst s ix p i tch c lasses of theinver ted se t , t ransposed up f ive semi tones , are ident ica l to the second s ix p i tchclasses of the pr im e se t, tho ug h in a d i f feren t order . For a var ie ty of reason s sucha space was usefu l to Schoenberg (see Schoenberg 1941) . A complete 12- tonepi tch space wo uld include t ransp os i t ions of th i s and o ther re la t ionships , jus t ast he t ona l space o f Fi gure 7 wo u l d have t o be en l a rged fo r o t he r cho rds andregions .

P0: A Bw E~ B E Fs C C= G ~ O F15: D C~ A~ C G F B B~ E ~ A F -

igure9On e might supp ose that a s tep in such h ighly chromat ic mus ic i s a semi-tone. But this is not reflected at all in the pi tc h space of Figure 9. Or, mo re in thesp i ri t o f the row, one might say that a s tep occurs f rom any p i tch c lass to anyhor izonta l ly ad jacent p i tch c lass . But in th is case s tep d i s tance var ies wi ld ly f rom

one ad jacency to the next , and there i s no corre la t ion wi th psychoacous t ic ( logfrequen cy) d i s tance . In shor t , ser ia l ism does not incorporate a ny psychological lycohe rent no t ion of s tep and sk ip . The l i s tener cons equ ent ly has d i f fi cu l tylocat ing a pi tch as close to or far from another pi tch.


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8 redLerdahI

Further causes of serialism's cognitive opacity could be adduced, such as itsfailure to provide consistent exposure from piece to piece to a limited numbe r ofalphabets (Constraint 16). But the main points have been addressed. We turnnow to some aesthetic issues that have been lurking behind this entirediscussion.

omprehensibil i ty and valueThere is no obvious relationship betw een the comprehensib ility of a piece and itsvalue. Many masterpieces are esoteric, while most ephemeral music is all toocomprehensible. On the other hand, if a piece cannot be under stood , h ow can itbe good? Most wo uld agree that comprehensibility is a necessary if not sufficientcondition for value.Care must be taken with this formulation in three respects. First, comprehen-sion presupposes listening competence for the music in question. This com-petence varies with ability and especially with exposure, but is not less real forthat. Second, comprehension pertains to the listening grammar rather than tothe compositional grammar. A serial piece may be understood in non-serialways. Third, we are talking about intuitive rather than analytic comprehension.Along the lines of Fodor (1983), the mind's music module must be able spon-taneously to form mental representations of musical structure from musicalsurfaces. This is quite different from using the all-purpose reasoning faculty tofigure out the structure of a piece.With these provisos in mind, I think the above formulation stands.Appreciation depe nds on cognition. I now want to go an aesthetic step further.

Aesthe tic Claim : The best music utilizes the full potential of ourcognitive resources.This seemingly innocuous statement carries weight because a great deal isbecoming kno wn about ho w musical cognition works. I have outlin ed aspects ofthis under sta ndi ng in the constraints presented above. Following the m will not

guarantee quality. I maintain only that following them will lead to cognitivelytransparent musical surfaces, and that this is in itself a positive value; and,conversely, that not following them will lead in varying degrees to cognitivelyopaque surfaces, and that this is in itself a negative value.This stance can be refined thr oug h the notio n of musical complexity , whichis to be constrasted with musical complicatedness (Lerdahl 1985). A musicalsurface is complicated if it has numerous non-redundant events per unit time.Complexity refers not to musical surfaces but to the richness of the structuresinferred from surfaces and to the richness of their (unconscious) derivation bythe listener. For example, a grouping structure is complex if it is deeply embed-ded and reveals structural patterns within and across levels. The derivation ofsuch a structure is complex if all the grouping preference rules come into playand if they conflict with one an othe r to a certain degree. (Total reinforcement ofthe rules wou ld pr oduce a stereotypical groupin g structure; total conflict wou ldcreate intolerable ambiguity.) I take complicatedness to be a neutral value and


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Cognitive Constraints 119complexity to be a positive one. Many musical surfaces meet the various con-straints, but only those that lead to complexity empl oy the full potential of ourcognitive resources .All sorts of music satisfy these criteria - for example, Indian raga, Japanesekoto, jazz, and most Western art music. Balinese gamelan falls short withrespect to its primitive pitch space. Rock music fails on g rou nds of insufficientcomplexity. Much contemporary music pursues complicatedness as compen-sation for a lack of complexity. In short, these criteria allow for infinite variety,but only along certain lines.I find this conclusion both exciting and - initially, at least - alarming. It isexciting because psychology really does have something substantive to sayabout how music might be; here is the foundation I was seeking. It is alarmingbecause the constraints are tighter than I had bargained for. Like the old avant-gardists, I dream of the breath of other planets. Yet my argument has led frompitch hierarchies (Constraints 7-8) to an approximation of pure intervals (Con-stra int 12), to diatonic scales and the circle of fifths (Const rain t 14), and to a pitchspace that prominently includes triads (Constraint 17).However, the constraints do not prescribe outworn styles. Rather they pro-vide a prototype (Rosch 1975). Let me first give an uncontroversial rhythmicexample. A musical surface in which the note values are multiples of 2 isintrinsically more stable and easier to cognize than one in whic h the note valuesare multiples of 7 and 11. This does not mea n tha t the latter surface is somehowimpermissible. It instead amounts to the observation that, because of the result-ant ease in forming a metrical structure, note values that are multiples of smallerprime number s are easier to process and remember, and therefore that multiplesof 2 (or 3) inevitably remain a cognitive reference point for more complicatedrhy thms. I claim that a similar situation holds for pitch: the structure in Figure 7(with its hierarchically organized octave, fifth, triad, and diatonic scale) remainsa reference point for other kinds of pitch organization, not because of its culturalubiqu ity but because it incorporates all of the cons traints de veloped above. I takethis as a given, with or against which a composer can play creatively. Of courseone may opt for a less constrained pitch space. But if a composer chooses thespace represented by Figure 7, I am sure there are innumerable and radicallynew ways to use and extend it. The future is open.

My second aesthetic step was discussed above; I list it here only forcompleteness.A e s t h e t i c C l a i m : The best music arises from a n alliance of a compos-itional grammar with the listening grammar.

This claim does not exclude the artifice hidden in a Bach fugue or a Brahmsintermezzo. Such artifice is rooted in the bedrock of a natur al compositionalgrammar. At our present musical juncture, however, composers would do wellto heed the claim.This claim carries with it a historical implication. The avant-gardists fromWagner to Boulez thou ght of music in terms of a progressivist philosophy ofhistory: a new work achieved value by its supposed role en rou te to a better (or atleast more sophisticated) future. My second aesthetic claim in effect rejects thisattitude in favour of the older view that music-making should be based on


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120 Fred Lerdahlna ture . For the anc ients , na tu re ma y have res id ed in the musi c of the spheres ,

but for us i t l ie s in the musi ca l min d. I th in k the m usic of the futu re wi l l emergele ss f r om twen t i e th - cen tu r y p r ogr e ssiv i s t a es the ti c s t han f r om n ewly acqu i r edknowl edge o f t he s t r uc tu r e o f mus ica l pe r cep t ion and cogn i t i on .ReferencesBabbitt, M. (1965) The structure and function of musical theory. College Music Symposium, 5, 10-21.(Reprinted in 1972 in Perspectives on contemporary music theory (ed. B. Boretz and E.T. Cone),Norton, New York.)Balzano, G.J. (1980) The group-theoretic description of 12-fold and microtonal pitch systems.Computer Music Journal, 4, (4), 66-84.Balzano, G.J. (1982) The pitch set as a level of description for studying musical pitch perception. InMusic, mind and brain (ed. M. Clynes). Plenum, New York.Bharucha, J.J. (1984a) Anchoring effects in music: the resolution of dissonance. Cognitive Psychology,16, 485-518.Bharucha, J.J. (1984b) Event hierarchies, tonal hierarchies, and assimilation: a reply to Deutsch andDowling. Journal of Experimental Psychology: General, 113 (3), 421-5.Boulez, P. (1954)Le marteau sans maftre. Universal Edition, London.Boulez, P. (1963) Penser la musique aujourd hui. Schott, Mainz. (Tr. 1971 by S. Bradshaw and R.R.Bennett. Harvard University Press, Cambridge, Mass.)Boulez, P. (1964) Alea. Perspectives of new music, 3 (1), 42-53.Bruner, C.L. (1984) The perception of contemporary pitch structures. Music Perception, 2 (1), 25-40.Burns, E.M. and Ward. W.D. (1982) Intervals, scales, and tuning. In The psychology of music (ed. D.Deutsch). Academic Press, New York.Cherry, E.C. (1953) Some experiments on the recognition of speech, with one and two ears. Journal ofthe Acoustical Society of America, 25, 975-9.Chomsky, N. (1957)Syntactic structures. Mouton, The Hague.Chomsky, N. (1965)Aspects of the theory of syntax. MIT Press, Cambridge, Mass.Deli~ge, I.S. (1985) Les r6gles preferentielles de groupement dans la perception musicale.Unpublished Ph.D. thesis. Universit6 de Li6ge.Deutsch, D. (1982) The processing of pitch combinations. In The psychology of music (ed. D. Deutsch).Academic Press, New York.Deutsch, D. and Feroe, J. (1981) The internal representation of pitch sequences in tonal music.Psychological Review, 88, 503-22.Dowling, W.J. (1972) Recognition of melodic transformations: inversion, retrograde, and retrograde-inversion. Perception and Psychophysics, 12, 417-21.Dowling, W.J. and Harwood, D.L. (1986) Music cognition. Academic Press, New York.Fodor, J.A. (1983) The modularity of mind. MIT Press, Cambridge, Mass.Franc6s, R. (1958) Laperception de la musique. Vrin, Paris.Fux, J.J. (1725)Gradus ad parnassum. Joannis Petri van Gehlen, Vienna. (Tr. and ed. 1965 by A. Mann,

Norton, New York).Hindemith, P. (1937) Unterweisung im tonsatz, Vol. 1. Schott, Mainz. (Tr . 1942 by A. Mendel.Associated Music Publishers, New York.)Hyde, M. (1980) The roots of form in Schoenberg's sketches. Journal of Music Theory, 24 (1), 1-36.Koblyakov, L. (1977) P. Boulez, Le Marteau sans Maltre, analysis of pitch structure. Zeitschrift furMusiktheorie, 8, (1), 24-39.Krumhansl, C.

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