margulis 2005 - a model of melodic expectation

7/25/2019 Margulis 2005 - A Model of Melodic Expectation

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2005 BY THE REGENTS OF THE UNIVERSITY OF CALIFORNIAALL RIGHTS RESERVED .

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Address correspondence to Elizabeth Hellmuth Margulis, School of Music, NorthwesternUniversity, 711 Elgin Rd., Evanston, IL 60208. (e-mail: e-margulis@northw estern.edu)

ISSN: 0730-7829, electronic ISSN: 1533-8312. Please direct all requests for permissionto photocopy or reproduce article content to University of California Presss Rights andPermissions website, at w w w.ucpress.edu/journa ls/right s.htm.

M usic Perception

Summer 2005, Vol. 22, No. 4, 663714

A Model of M elodic Expectat ion

E L I Z A B E T H H E L L M U T H M A R G U L I S

N orthw estern University

A model of m elodic expectation is proposed. The model assigns ratingsto the expectedness of melodic events. The ratings depend on the hier-archic implementation o f three primary fa ctorssta bility, proximity, anddirectionand o ne seconda ry f acto rmobility. The model explicitly linksexpectancy ratings to aspects of listeners experiences of tension inmelody. An approach to temporal expectations is discussed but notquantified.

The model is situated within a framework for thinking about a typeof schematic melodic expectations. This article assesses the position of

these expectations w ithin the b roa der cognitive processes invoked in lis-tening to music. It suggests methods for investigating the expectationsempirically. Additionally, it outlines connections b etween the theorizedexpectatio ns and the dyna mic, aff ective contours of musical experience.

Received No vember 14, 2003, accepted O ctober 13, 2004

EXP ECTANCY has long been cited as a generator of musical affect.Leonard Meyer (1956) sparked contemporary theorists interest in thesubject. In Emot ion and M eaning in M usic, he appropriated MacCurdys(1925) modification of Deweys (1894) Conflict Theory of Emotions,

which suggested that affect, in general, arises from the inhibition of ten-dency. M eyer proposed tha t in music, more specifically, af fect . . . isaroused when an expectationa tendency to respondactivated by themusical stimulus situation, is temporarily inhibited or permanentlyblocked (M eyer, 1956, p. 31). This proposal has profound ra mifications.According to M eyer, it suggests that gra nted listeners w ho ha ve devel-oped reaction patterns appropriate to the work in question, the structureof the affective response to a piece of music can be studied by examiningthe music itself. . . . the study of the affective content of a particular w ork. . . can be made without continual and explicit reference to the respons-


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es of the listener or critic. That is, subjective content can be discussedobjectively (Meyer, 1956, p. 32).

Indeed, the most attractive aspect of the expectational approach tomusical affect is the potential it affords for directly connecting the struc-

ture of the musical line to the conto urs of listener experience. Yet t his pos-sible connection, which ostensibly motivated the extensive work onexpectancy t hat follow ed M eyer, ha s proven difficult to esta blish.

According to the psychologica l theories Meyer (1956, p. 20) cites, emo-tions are

themselves undifferentiated, affective experience is differentiated

because it involves awareness and cognition of a stimulus situation

which itself is necessarily differentiated. ... [This] explains and

account s for the existence and nat ure of the inta ngible, non-referential

affective states experienced in response to music. For in so far as the

stimulus situation, the music, is non-referential . . . there is no reason

to expect tha t our emotiona l experience of it should be referential. Theaffective experience made in response to music is specific and differen-

tiat ed but it is so in terms of the musical stimulus situation ra ther than

in terms of extra musical stimuli.

Expectancy violations, in other words, do not translate into affectiveexperiences in any obvious, transparent way. How might one articulatethe sensations tha t violations cause? Ho w might one systematize the rela-tionship between melodic line and listener experience via the conduit ofexpectancy?

The most expansive treatment of expectation in melody is Narmours

implicat ion-realization theory (1990, 1992). Although comprehensive andpenetrating as a theory of melody, the w ork do es not f ocus on connectionsbetween expectancy and affect. Many possible melodic structures aredescribed, but their experiential consequences are mentioned only occa-sionally. The intermittent references to affect usually entail the identifica-tion of moments that might surprise or shock a listener, rat her than theexposition of a system of dynamic expectancy-based affective fluctua-tions: a problem given that such moment-to-moment ebbing and flow ingseems to cha racterize musical experience better tha n isolat ed injections ofsurprise.

The present study builds on existing theories of melodic expectation,notably those of Narmour, Larson (1993, 2004), and Lerdahl (2001),reformulating and supplementing the principles in a way that enables thespecification of meaningful connections to listener experience. Ratherthan employ strings of symbols or d raw meta phoric relations, the presenttheory assigns expectedness rat ings (predictions about degree of expected-ness) to melodic events and associates these ratings with three distincttypes of tension (surprise-tension, denial-tension, and expectancy-ten-

Elizabeth Hellmuth Margulis664


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sion). This approach enables the graphical representation of moment-to-moment expectancy-based fluctuations in affect across a melodys course.

As a candidate for a mechanism of affect, expectation possesses sever-al advantages. The phenomenon is known to be a basic strategy of the

human mind; it underlies the ability to bring past experience to bear onthe future, the ability to prevent computational overload by constrainingthe number of outcomes worthy of consideration, and the ability to effi-ciently direct attention and mental resources (Masson, 1986; Posner &Snyder, 1975). Bharucha (1987) proposes tha t perceptual fa cilita tion (i.e.,expectation) arises naturally out of spreading activation in a networkwhere connectivity encodes musical relationships. The expectational the-ory o f musical a ffect gains plausibility by relying on a common cognitiveprocess.

The theory also benefits from its focus on the active, real-time experi-ence of listening. U nlike some music theories tha t seem to assume a listen-

er possesses a mental score of the entire movement before the second notehas sounded, a theory built around expectancy a bsorbs the reality of tem-poral experience into its foundation, preserving the distinction betweenpast events, which have become fact, and future events, which remainuncertain.1

Another significant advantage resides in the theorys appreciation of thecomplexity of musical experience. Rather than simplistically equating theminor mode w ith sadness, or a quick tempo w ith excitement, a n expecta-tional model examines a musical landscape as a dynamic succession ofevents, preserving the constant fluctuation in sensation that characterizesmusical experience.

Conceptual Framework

Meyer (1956, p. 25) observes that

all tendencies, even tho se w hich never reach the level of consciousness,

are expectations. For since a tendency is a kind of chain reaction in

which a present stimulus leads through a series of adjustments to a

more or less specified consequent, the consequent is alw ays implied in

the tendency, once the tendency ha s been bro ught into play.

Narmours implication-realization theory treats bottom-up, deeplyschematic expectations as well as top-down, stylistic-based ones. The pres-ent study restricts itself t o expectations o f t he former sort. It uses the term melodic expectations narrow ly to designate deeply schematic, auto mat-

A M odel of M elodic Expectation 665

1. For a discussion of the gap between final-state theories and real-time listening, aswell as an exposition of some potential bridges between the two, see Bigand (2003).


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ic primings not available for direct access, primings possibly instantiatedwithin a Fodorian module (Fodor, 1996; Justus & Bharucha, 2001).Although other senses of the term expectat ion a lmost certa inly apply to lis-teners experience of melody, the model treats o nly tho se expectations t hat

the listener is not aware of as such. For example, a listener hearingBeethovens Fifth Symphony for the eightieth time sustains very specificexpectations about which melodic events will ensue. Yet the information-ally encapsulated mo dule engaged in projective primings canno t access tha tbroa der know ledge and continues to pro duce the same uninformed predic-tions it would have on the first hearing (see Jackendoff, 1991).

The description deeply schematic relates to the distinction tha tBharucha (1987) and others make between schematic and veridicalexpectancies. According to Bharucha, schematic expectations are auto-matic predictions based on the implicit extrapolation of typical patternsfrom an extensive corpus of music, whereas veridical expectations repre-

sent specific knowledge about the way a particular piece goes. Bharuchauses the deceptive cadence (V-vi) to illustrate the difference; schematicexpectations predict continuation to the tonic, even when the listenerknows (via a veridical expectation, such as prior experience with thepiece) that vi will occur instead.

Although the distinction between schematic and veridical broadly cate-gorizes expectation types, the category of schematic itself encompassesmore than one variety of expectation. Specifically, schematic expectationsinhabit a continuum from relatively deep to relatively shallow, wheredepth relates to availability for direct access (from little to much availabil-ity), susceptibility to change through exposure (from little to much suscep-

tibility), and scope of application (from more universal to more limited).Examples of increasingly shallow schematic expectations might be: expec-tations for closure; expectations for cadential closure in tonal music;expectations for common cadence types in music from the classical peri-od; and expectations for common cadence figures in the music of M oza rt,where these expectations are increasingly available for access, increasing-ly susceptible to change through exposure to new pieces within the rele-vant repertoire, and increasingly limited in scope. The expectations dis-cussed in this article are thought to lie on the extreme deep side of theschematic spectrum, representing little conscious access availability, littlesusceptibility to change through exposure, and a wide scope of applica-

tion. To emphasize these chara cteristics, the expectat ions a re sometimesreferred to not simply as schematic, but as deeply schematic.

The friction between these deeply schematic expectations and the actu-ally occurring events, the model claims, registers in the listener as experi-ences of tension. The underlying expectancies are onlynaturally dis-cernible in these effects on tension and affective experience. Accordingly,the sort of melodic expectations discussed here are not equivalent to what



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composers usually do, to what sounds best, or to the continuations a lis-tener might sing. Composers more likely seek to create an optimal mix ofexpectedness and unexpectedness, a mix that listeners might try to emu-late w hen singing continuations. The notion of such an optimal mix is

not new to aesthetics. In a list of rules on the art of writing beautifulmelodies, eighteenth-century theorist Johann Mattheson repeatedly advis-es composers to make judicious use of expected structures steps andsmall intervals are preferred to la rge leaps as w ell as of unexpecteddeviat ions one should cleverly va ry such small steps (M at theson,1739, in Harriss, 1981, p. 312). In the most comprehensive applicat ion ofinformation theory to music, Abraham Moles (1966, p. 169) observes:

In order for the musical signal to be intelligible in the sense of inte-

grally perceived, the average transmitted information, integrated over

the max imum extent of presence, must b e on the ord er of the limiting

rate of a pperception. If it is much low er than t his rate, the signal seems

uninteresting; if it is much higher, the signal overwhelms the listenerand destroys his attention.

Some empirical evidence supports this idea. Simon and Wohlwill (1968),for example, showed that listeners preferred an original, complex musicalpassage over simplifications of it. Crandall (1967) presented people withsequences of nonsense words and asked them to rate individual words ona good-bad scale. Words that generated moderate uncertainty about thenext element in the sequence were rat ed closer to the good end o f the scalethan either w ords that generated a bsolute certainty or w ords that generat-ed no certainty. In a study of reactions to printed items, Maddi (1961)

showed that people experienced a more positive affect in relation to smalldeviations from expectedness than they did in relation to large deviationsor no deviations.2 These and o ther studies suggest tha t w hen novelty ismanipulated through degree of deviation fro m the familiar, . . . an interme-dia te degree of deviation is preferred (Berlyne, 1971, p. 194). A melodyfeaturing only the most expected continuations would not possess the richfluctuations in expectancy-based tension that (in part) make for good,interesting music. It can be broadly assumed that composers try to writeinteresting music. It can be similarly a ssumed tha t listeners try to sing con-tinuations that sound good to t hem. The problem w ith interpreting thesecontinuat ions as measures of deeply schemat ic expectedness is that good

does not straightforw ardly result from such expectedness. If good relatesto deeply schemat ic expectancies at all, it relates to the pat terns of t ensionthat a mix of expected and unexpected continuations creates.

Several experiments have investigated melodic expectations in thebroader (i.e., other than deeply schematic) sense by playing melodies and


2. For a discussion of these and similar studies, see Berlyne (1971).


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asking participants to sing continuations (Carlsen, 1981; Lake, 1987).Others have played listeners melodies with different probe tones append-ed. Listeners were asked to rate how well the different continuations fit(Cuddy & Lunney, 1995; Schellenberg, 1996; Schmuckler, 1989). These

experiments probably uncovered a broader type of expectation than thattargeted by the present model. First, the experiments required listeners toaccess their expectations by explicitly evaluating the expectedness of var-ious continuations or by singing expected continuations. The expectationsproposed here are theorized to be unavailable for such access. Second, lis-teners in these experiments were required to rate the goodness of fit ofcontinuations or to sing continuations: both t asks could be understood t oreveal (the first explicitly, the second implicitly) what par ticipants thoughtsounded good. As already observed, good most likely does not arisedirectly out of expectedness (in fact, constant capitulation to expectednessw ould most likely sound boringor ba d ); therefore, these tasks consti-

tute unsatisfactory measures of deeply schematic expectedness. Third, theexpectations exposed by the experiments were likely susceptible to theinfluence of familiarity. For example, had the subjects been played theopening measures of America the Beautiful to the A of America, t heymost likely would have sung (or most highly rated) a continuation to mer on a n a scending major sixth. Yet the present model suggests thateven for a memorized tune, schematic expectations will continue to favora different continuation than the major sixth; it is precisely this expecta-tion, the theory claims, tha t contributes to the particular af fective qualityof that leap. If expectations are responsible for the generation of affect,and if repeated exposure could nullify expectations, then familiar music

should lose its affective power. Experience suggests otherwise.Some expectations, and it is these that the present model explores, might

operate in an informationally encapsulated way, impervious to the influ-ence of knowledge or familiarity. These are not expectations in an every-da y sense, but ca n be thought o f more profitably a s schematic primings. Toisolat e their effects, experiments w ill have to follow listeners real-time rat -ings of tension and correlat e them with the expectancy models predictions(cf. Schubert, 20012002; Eerola, Toivia inen, & Krumhansl, 2002), or useevent-relat ed-potential (ERP) studies (cf. Besson & Fata, 1995; G rano t &D onchin, 2002; Tervaniemi, H uotilainen, Bra ttico, Ilmoniemi,Reinikainen, & Alho, 2003) or priming paradigms (cf. Bharucha &

Stoeckig, 1986) to reveal expectancy violations. Tillman and Bigand(2004) suggests tha t t he expectat ions uncovered b y priming pa rad igms areautomatic and unaffected by the prior repetition of expectancy-violatingpassages. For example, a listener will continue to expect the dominant toresolve to the tonic, even a fter hearing it repeat edly progress to the subme-diant. This finding supports the distinction proposed between the access



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availability of schematic and veridical expectations; although, if asked, thelistener may attest to expecting the submediant (a veridical expectation),the priming paradigm will reveal a (schematic) expectation for the tonic.

It is important to no te that deeply schemat ic does not necessarily

suggest innate. In its stability para meter, t he model outlined here incor-porates tonal context. Studies have suggested that principles of tonalorganization can be learned implicitly through simple exposure at an earlyage, resulting in their lat er automa tic and modular applicat ion (cf. Peretz& C oltheart, 2003). D eeply schematic implies the inability o f t heseexpectations to be directly accessed. Contrast this with shallowlyschematic, more easily accessed and described expectation types (e.g., theexpectation for a lyrical second theme in a sonata movement). Studieshave shown that priming in chord sequences depends on tonal relation-ships that are seemingly learned, rather than innate (Justus & Bharucha,2001), and that these harmonic primings resist repetition priming

(Tillman & Bigand, 2004). The robustness of the to na lity-ba sed pr imingsuncovered in these experiments suggests that tonal principles absorbed inrelatively early stages of development may play a role in informationallyencapsulated, schematic expectancies.

The model provides a ba seline associat ion betw een melodic structure andlistener experience. This association is an issue best explored empirically.The model outlined here contributes to that project by: (1) clearly specify-ing the nature of the expectations that might link structure and experience,(2) explicitly specifying the nature of the relationship between structure(expectancy and expectancy violation) and experience (tension types), and(3) providing a founda tion on w hich future inquiries can be built.

The Model

The model assigns expectedness ratings to melodic events.Expectedness ratings are predictions about the amount of deeply schemat-ic expectedness listeners will have for melodic events. Although the sym-bols in Narmours implication-realization (I-R) model clearly categorizemelodic segments on the ba sis of interval size and d irection, they only sec-ondarily denote the theorized expectedness of each segment. By directlyassigning a prediction of expectedness to each event, the present model

attempts to make its claims about expectancy more straightforward andaccessible. Because musical experience is dynamic and fluctua tes in quali-ty from moment to moment, it is advantageous that the model can assignrat ings from event to event a cross the course of a melody. The I-R modelsanalyses do not address the expectational content of all melodic events,but only of some (as discussed later in the comparative analysis of the



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opening measures of Mozart K. 282).3 The present model assigns ratingsby hierarchically applying three primary factorsstability, proximity, anddirectionand one secondary o ne, mobility. Temporality is discussed b utnot quantified. The expectancy ratings are correlated with three tension

types, facilitating the graphic depiction of fluctuations in tension acrossthe course of a melody. This article outlines each factor in turn before dis-cussing the fa ctors combination a nd joint operation, a s w ell as their the-orized impact on the experience of tension.4

Of the primary factors, twoproximity and directionstem fromparameters in Narmours I-R model. One, stability, stems from accountsof tonal pitch space and melodic attraction in Lerdahl (2001).Accordingly, a brief overview of these two extensive theories will permita better understand ing of the present models approa ch.

THE I-R MODEL

Na rmours implication-realization theory examines the way G esta ltprinciples extract implications from style shapes (configurations of amusical parameter such as interval size and directional contour). His basichypothesis is as follows:

if in a ny one para metric simplex t he elements o f a style-shape relat ion

of similarity, proximity, or common direction occur, then implication

of a further style-shape element of similarity, proximity, or common

direction takes place, except in cases where implication of intra- or

extraopus style-structural complexes are of such empirical confor-

mance so as to interfere. . . . [In other words] . . . If a parametric style

shape of elements a1 + a2 occurs, then a1 + a2 implies a3, unless a2 of arelevant conformant style structure of a

1+ a

2implying binterferes.

(Narmour, 1990, p. 70)


3. As an example, consider the three-note stepwise ascent C-D-E. Narmour wouldbracket these notes together as an instance of process, labeled [P]. The label signifies thatthe interval C-D created an implication for continued stepwise ascent that the E fulfilled.Thus, the lab el says something a bout Es expectednessnamely, tha t it w as highly expect-edbut it says nothing about Ds expectedness in light of the preceding C, or about Csexpectedness in light of w ha tever preceded it. The present mod el assesses the expectednessof every event, and avoids the gapped quality of the analyses in the I-R model, whereexpectedness is sometimes assessed only for every third note.

4. It should be noted that the model directly considers only pitch and time in the pro-duction of expectancy ratings. Other parameters most likely relateparticularly to theresultant melodic tensions. For example, a performer might choose to compound a tenseevent with an additionally unexpected dynamic accentuation, or, contrarily, to soften theeffect by withholding dynamic emphasis. Although expectancies regarding other parame-ters (timbre, dynamics, etc.) probably apply, the model is limited in scope and does notaddress them formally. For an account that may provide a starting point for an investiga-tion of timbre and expectation, see Tsang (2002).


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According to this hypothesis, similarity implies more similarity, and dif-ferentiation implies further d ifferentiation. Configurations of implicat ionfulfillment a nd denial in various para meters are categorized into ba sicstructures notat ed by strings of letters. [P], for example, symbolizes

process, a chain of small intervals all ascending or descending. It embod-ies fulfillment of both intervallic and registral direction implications. [R],on the other hand, symbolizes reversal, a large interval followed by asmall one in the opposite direction, a gain constituting realization of bot hintervallic and registral tendencies. Structures may be characterized by amix of fulfillment and denial. [IP], for example, designates a case wheresmall intervals follow one another in opposite directions, embodying ful-fillment of intervallic implications, but denial of registral ones. Narmoursmelodic ana lyses are mad e up of cha ins of these symbols, often a ccompa-nied by annotations, in smaller type, of metric, harmonic, dynamic, andother extramelodic fa ctors contributing to the segmentation a nd a nalysis.

The theorys conceptual background is problematic first in its founda-tiona l reliance on G esta lt principles (questioned for t heir dependence on apriori notions of good and best vague and definition-resistantterms), and second in its seemingly unjustified assertion that small inter-vals embody similarity and large intervals embody differentiation.Building on this categorization, Narmour postulates that small intervalsimply continuation to small intervals (a case of similarity implying simi-larity), but large intervals imply reversal to small intervals (a case of dif-ferentiation implying differentiation). Several questions arise: althoughpitches separated by a small interval are nearer to one another in frequen-cy, and could in this respect be viewed as more similar than distant ones,

do pitches a semitone apart really seem more similar than pitches a fifthor an octave apart (pitches more closely related in a tonal sense)? If largeintervals, as an instantiation of differentiation, imply further differentia-tion, should they not imply more large intervals, since large intervals bydefinition represent dissimilarity? Why do the principles apply only tointerval and registral direction? According to the theorys rules, shoulddynamic and rhythmic differentiation not suggest further differentiationas well?

Although the theoretical foundation seems open to question,Schellenberg (1996, 1997) showed that parts of the I-R theory could beexcised without damage to the models predictive power. For example,

Schellenberg observes tha t the notion o f small intervals implying small inter-vals and large intervals implying small intervals can be reduced to the ideathat pitches imply continuation to other nearby pitchesproximityaprinciple tha t has garnered empirical support (Schellenberg, 1996, 1997).

Narmours basic hypothesis, quoted above, embodies another of thetheorys core tensions. On the one hand, the I-R theory seeks to describe



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robust, innate, universal perceptual processes. On the other hand, it seeksto acknowledge the effect that learned schemata, stylistic idiosyncrasies,and other parameters (rhythm, harmony, etc.) can have on implicationformation. The result is a highly rigid and precise theory thatin analyt-

ic practicerelies almost entirely on except clauses.Presumably, expectancy interests theorists because of its potential role

in the generation of affective experience in listening. Yet the I-R theoryaddresses affect only occasionally and in passing. When the subject isbroached, musical experience sometimes seems to be depicted as animplausible ricochet between shock (thought to arise when expectationsare denied) and satisfaction (thought to arise when expectations are ful-filled).

The notational system of the I-R theory remains relatively opaque withregard to expectational content. The symbology consists primarily ofchains of letters that designate ba sic structures. The structures are cat -

egorized, however, not according to their degree of realization or denial,but according to their component parts (interval of this size in this direc-tion followed by one of that size in that direction). The method oftencauses raw taxonomy to supersede and obscure expectational content inanalyses and discussion. Although a reader steeped in Narmourian classi-fication might understand, at the sight of an [IP], that partial realizationhas occurred, a casual reader would probably have to flip to the glossaryof symbols in the back and reread the entry for intervallic process to find,couched among other points regarding the sequence of intervals and theirdirection, the summary of [IP]s realizational status.

Because the symbols do not speak directly to the implications and real-

izations they purport to describe, they necessarily diffuse some of the the-orys focus away from its chief aim: an examination of expectancies.Narmours inclusion, as Appendix 5 of his first book, of a catalog of the200-odd possible combinations of [IR]s, [D]s, and other structures thatmay occur in melodies reflects a turn away from expectancy and towardcategorizationa turn promoted by the notation. Indeed, at times,Narmour (1990, p. xiv) restates his aim in terms of categorization: Analytically, I chara cterize the w hole journey as an explicat ion of thegenetic code of melody, w ith the a im of discovering a consistent t ax ono-my of structural types. Na rmour ultimately succeeds more in thisendeavor tha n in the explicat ion of t he dynamic expecta tions that charac-

terize a listeners experience.Key connections between the implication-realization theory and the

present model are the commitment to an account of purely schematicexpectations, a nd t he centrality o f no tions of distance (related to interval-lic size in Narmours model and proximity in the present one) and direc-tion. Key distinctions include the incorporation of tonal relatedness



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(couched as stability within the present model, but excluded from the coreof Narmours account), the treatment of hierarchy (explicitly formalizedin the present model but not in Narmours) and of extramelodic factorssuch as meter and harmony (allowed to enter Narmours model in infor-

mal ways, but only permitted to enter the present model in formalizedones), the symbolic representation (category letters in the I-R model, andnumeric predictions of expectedness in the present one), the theorizedconnections to affect (relatively sparse and informal in the I-R model, butrelatively extensive and systematized in the present one), the centrality ofthe concept of closure (essential in the I-R model, but absent in the pres-ent one), the theoretical backgro und (relying on G esta lt notions vs. not ),and the analytic focus (categorization vs. the tracking of expectancies).5

MELODIC ATTRACTIONS

In contrast to Narmours approach of excluding tonal relationships tofocus on interval size and direction, Lerdahl (2001) presents a model oftension and expectat ion ba sed primarily on scale degree function . G iven acontext chord and key (determined by event governance rules), pitchclasses are assigned an anchoring strength, which represents the degree towhich that pitch can attract surrounding ones. For example, in a I of Cmajor context, the root C receives anchoring strength 4, the other mem-bers of the tonic triad (E and G ) anchoring strength 3, other diato nicpitches (D, F, A, and B) anchoring strength 2, and all chromatic pitchesanchoring strength 1. According to the theory, pitches tend to be at tract-ed to the nearest pitch with a higher anchoring strength (i.e., A, withanchoring strength 2, is attracted to neighbor G, with anchoring strength3). Tension, in this account , corresponds to instability.

Lerdahl quantifies the tendency of one pitch to resolve to another bydividing the anchoring strength of the potential attractor by the anchor-ing strength of the pitch under consideration, and multiplying the resultby the inverse square of the semitone distance between them. For exam-ple, A would be a ttracted to neighbor G by (3/2) (1/4) or .375.

Unlike the I-R theory, Lerda hls model explicitly ad dresses the degree towhich each melodic event is theorized to be expected (in his terminology,the degree to which the present event attracted the preceding one).Additionally, the predictions are presented in transparent notation (graphsof attractional fluctuations positioned across the top of scores) and with

explicit references to a proposed effect on experienced tension. However,the model does not incorporate direction, an omission that results insomewhat inflexible analyses. (In a C-major context, no directional con-


5. For a more comprehensive comparison between the implication-realization theoryand the present model, see Ch apter 3 o f M argulis (2003).


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text can make F more attracted to G than E). Additionally, its quantifica-tionparticularly the specification of proximity as the inverse square ofsemitone distancecauses problems. Because the semitone distancebetween a pitch and its repetition is 0, the model misleadingly predicts the

expectancy for pitch repetition to be infinite. Moreover, the formula priv-ileges movement by semitone to an inordinate degree, assigning hugeattractional ratings to continuations by semitone and comparativelyminuscule ratings to other possibilities. But experiments have suggested(Lake, 1987) that in many circumstances listeners expect a relativelybroad range of continuationsnot exclusively movement by semitone.

The present model takes the core of its event governance rules for thepara meter of stability from Lerdahls theory of melodic att ractions. It a lsoadopts the general approach of assigning a numeric prediction to eachmelodic eventin the case of Lerdahl, the prediction is of attraction, inthe case of the present model, the prediction is of expectedness. It uses

Lerdahls approach of linking predicted values to tension and graphing thetension fluctuations across the top of the score. However, tensions in thepresent model pertain exclusively to melody and are separated into threedifferent types; where Lerdahls model associates tension with instability,the present model proposes three tension types tha t a rise out of all a spectsof expectancy (not just stability). Also in contrast with Lerdahls theory ofmelodic attractions (a small part of a large study of pitch space), the pres-ent model attempts a thorough exposition of melodic expectations,including an extended account of direction and proximity, and a systemfor addressing hierarchic expectancies. As already discussed with respectto proximity, the quantified predictions of the present model differ sub-

stantially f rom Lerda hls.

MELODIC EXPECTATION

In comparison to these studies, a primary contribution of the currentpaper is the integration of the two frameworks into a single approach.One interpretation of Peretz and Coltheart (2003) suggests that tonalencoding might operate modularly, relying on principles of tonal organi-zation learned implicitly a nd a utoma tically in reaction t o early exposure.Stability, therefore, might be considered to act in conjunction with other,more Na rmourian a spects of melodic projection. In comparison t o t he I-

R t heory, the present mo del is more specific about the type of expectat ionsunder considerationdeeply schematic, automatic primings. It is alsomore consistent about pursuing its predictions on their own terms, with-out allusion to the contradicting effect of other expectancy types. Thepresent model was constructed around a theory of the way in whichschematic expectancy connects to musical affect, so expectancy predic-tions always connect explicitly to notions of melodic tension. Via the



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parameter of mobility, the present model successfully addresses the pitchrepetition issues that are problematic in Lerdahl (2001). Finally, in theform of the hierarchy parameter, the present model features a formalizedapproach to expectancies that are deeply schematic and automatic but

incorporate more context then just the two immediately preceding notes.The next sections describe the model parametersstability, proximity,

direction, mobility, and their hierarchic implementationone at a time.

Stability

Stability captures the intuition that, in general, listeners expect relative-ly stable melodic events. Tona l stab ility is a centra l concept in bot h musictheory (cf. Lerdahl & Jackendoff, 1983) and music cognition (cf.Krumhansl, 1990 and Bharucha, 1996). Its potential role in expectancy

has been articulated by Larson (1993) and Lerdahl (2001), but has notbeen treated w ithin the I-R mo del. One aim of t he present mo del is to inte-grate within a single framework the expectational approaches suggestedby existing theories.

Events are experienced differently in different tonal contexts.6 C, forexample, is maximally stable in a context ba sed o n I of C major, but min-imally stable in one based on I in F major. With slight adaptations, themodel adopts Lerdahls (2001) event governance rules to select an opera-tive chord and key in which stability may be evaluated. The bottom halfof Table 1 outlines the models event governance rules. Figure 1 illustratestheir operation in sample contexts. The boxes in the left column feature

nonchord tones, which shift the chord context, triggering an expectancyfor the dissonance to resolve to a chord member. The boxes in the bottomrow feature secondary chords, which shift the key context, triggering anexpecta ncy for movement to members of the tonicized key. O nly the upperright box, featuring a chord tone over a diatonic harmony, generates anexpectancy in the default context of I in C major.

The top half of Tab le 1 presents the models stab ility ra tings for eventswithin a given chord and key context. For major key contexts, it is clearwhat pitches count as diatonic. For minor key contexts, with their alter-nate versions of 6 and 7, some clarification is needed. Both versions of 6and 7 are considered diatonic until a particular version (e.g. raised 6)

occurs in the melody. An occurrence of the raised version of either 6 or 7


6. The stability parameter requires a governing chord and key context. Atonal or non-Western contexts may not allow for the extraction of such a context. For a listener notversed in t hese styles, t he other expectationa l par ameters (i.e., proximity, direction) mightsupersede. For a listener who had been immersed in these styles, the stability parametermight have developed with different specifications appropriately attuned to the relevantmusical vo cabula ries.


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TABLE 1Event Governance Rules Select the Tonal Context (a Chord and Key)

in Which an Event Is EvaluatedStability Rating Pitches in Context of a Chord and Key

6 C ho rd ro ot (a nd , a ft er a seven th in the melo dy, th e pit ch on e d ia to nic st epdown from it)

5 Chord third and fifth4 Other dia tonic pitches2 Chromatic pitches

Event Governance Rules

Context is I in current key, EXCEPT

When a secondary chord occurs, the context shifts to I in the tonicized key.When a melody note constitutes a nonchord tone with respect to the current harmony, the

context shifts to the current chord in the current key.When a melody note constitutes the seventh of the current chord, its lower diatonic neighbor

is promoted to the highest stability rating.When a strong predominant chord (such as an augmented sixth or Neapolitan) occurs, the

context shifts to V in the current key.

N OTEThe default context is I in the current key, except in the ca ses listed. Within th e chord a ndkey context established by the event governance rules, stability ratings are distributed as listed amongthe root of the context chord, its remaining members, pitches diatonic to the context key, and pitch-es foreign to it. Lerda hl (2001) also d istributes pitches accord ing to t he categories of roo t, th ird/fifth,diatonic, and chromatic, but uses a different quantification. The event governance rules are based onLerdahl (2001), w ith ad ded treatments for chord sevenths and strong predominants.

Fig. 1. The event go vernance rules esta blish the tona l context for expectancies abo ut thenext event. If a nonchord tone occurred over a dominant harmony in C major, the con-text key would remain C, but the context chord would shift to V. From the standpoint ofstability, the next event w ould be expected to resolve the preceding one to a chord to ne ofV. Three other scena rios a re depicted. Stab ility ra tings, depicted in Tab le 1, depend on theevent governance rules assignment of a tonal context.
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selects the raised version of both scale degrees as diatonic for theexpectancy targeting the immediately ensuing event (or, at higher levels,immediately ensuing span, as discussed later). The lowered versions ofeach are then consigned to chromatic status. Exactly the reverse occurs

follow ing the sounding of the low ered version of either: both low ered ver-sions are then considered diatonic and both raised versions, chromatic.

Each model factor assigns a rating to each melodic event; stability rat-ings are ultimately combined with ratings from the other factors to pro-duce overall expectancy ra tings for ea ch melodic event. G enera lly,expecta ncy ra tings represent the predicted expectedness of melodic events,with high ratings signifying high expectedness. The component parts ofthe overall ratingthe stability rating, the proximity rating, and so onrepresent the portion of expectedness attributable to the individual factor.A sta bility rating, in o ther w ords, captures the degree to w hich a melodicevent is predicted to be primed solely on the basis of its position within

the governing tona l context.Regarding the quantification of the model, it is important to note that

numeric values are invoked simply to mana ge the complexity of the predic-tions. It is not the precise value assigned to each event that is important,but rather the position in which that value situates the event with respectto surrounding events and other possible events. Quantification allows forthe relative ordering of expectancy predictions for the otherwise unman-ageably many possible melodic contexts. Additionally, it permits a speci-ficity in the models claims that makes empirical investigation easier.

The quantification of the model was established secondarily by inter-preting the results o f existing experiments (continua tion-sing-or-rat e par-

adigms) as reflections of an optimal mixture of expectedness and unex-pectedness, and primarily by consulting intuition to try to trace sensationsof tension to the originating expectancies. The details of the precise quan-tification of the expectancies are not themselves as important as theframework for future inquiry that the structure of the model provides.Experimental study can establish the best specification of each parameter.But such study could not o ccur in the absence of a conceptual fra me thatlinked structure to experience with explicit predictions. This model aimsto provide that conceptual frame.

Consider the stability ratings. In the default I of C-major context, Creceives a stability rating of 6, signifying strong expectedness. E and G

receive stability ratings of 5, D, F, A, and B receive stability ratings of 4,and the remaining chromatic pitches receive stab ility ra tings of 2. The par-ticularly low ratings for chromatic pitches reflect the experimental finding(Lake, 1987) that diatonic events are much more strongly expected thatnondiat onic ones. H ow ever, the primary motivat ion behind this and otherassignments of specific numerical ratings is a consideration of the experi-ences of tension predicted to arise from them.



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As described later in this article, unexpectedness is theorized to createa type of tension (termed surprise-tension) that registers in the listener asan experience of intensity or dynamism. Intuitive reflection reveals thatpitches foreign to the governing key (i.e., chromatic pitches) generate a

special intensity, more pronounced in comparison to that generated bydiatonic pitches than the intensity produced by the same diatonic pitchesseems in comparison to that produced by members of the governingchord. Accordingly, chromatic pitches receive a rating that is much lessthan that given to diatonic pitches, but diatonic pitches receive a ratingthat is only moderately smaller tha n tha t given to members of the govern-ing triad. On a ccount of the inverse relat ionship betw een expectancy a ndsurprise-tension discussed later, chromatic pitches particularly lowexpectancy ratings translate into particularly high amounts of surprise-tension.

These sorts of concerns determined the general relationship among the

different stab ility rat ings. The specific numeric va lues chosen reflect a con-sideration of the effect the numbers w ould have w hen combined w ith thevalues assigned by other parameters (proximity, etc.) to produce overallrat ings. Working ba ckw ard fro m experiences of melodic tension, the num-bers were set at a level that in combination with other factors producedthe desired tension predictions. As discussed later, the combined formulafor expectancy ratings makes it easy to list the core ratings for events afterany pitch in any key. Because these ratings are correlated with several ten-sion types, scrutiny of intuitive experience in multiple situations thatshared melodic successions allowed the predictions to be weighed, andsuggested, on occasion, reorderings or reassignments of parameter values.

In this wa y, the model wa s built do w nw ard from experience to prediction.It is reasonable to assume that empirical study across wider populationsw ill further modify the values.

Proximity

Proximity captures the intuition that listeners expect subsequent eventsto be relatively proxima te to preceding o nes. The I-R model theorizes thatlisteners expect small intervals to continue to additional small intervals inthe esta blished d irection a nd la rge intervals to reverse to small intervals in

the opposite direction. This amounts in part to theorizing that listenersexpect subsequent events to be proximate to preceding ones.

Schellenberg (1997) suggests that the principle of proximity underlies alarge pa rt of listener expecta tions. Tab le 2 presents the models prox imityratings, which are distributed differently than the proximity ratings inSchellenberg (1997) and in o ther models. Several f eatures make th is qua n-



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tification an improvement over those produced by more elegant formulas,such as the simple inverse and inverse square of semitone distance. First,the ratings for pitches less than an octave distant employ no decimals andare easily combined with other factors. Second, the ratings decrease less(4) from 1 to 2 semitones than they do from 2 to 3 semitones (7). Theplacement of the steepest drop between 2 and 3 semitones rather thanbetween 1 and 2 semitones reflects a preference for stepwise motionaffirmed in counterpoint textbooks (Kennan, 1998), statistical surveys ofWestern and world music (Vos & Troost, 1989), and psychological stud-

ies (Bregman, 1990; Schellenberg, 1996). In Western tonal music, as wellas in many other styles, steps are composed of 1 or 2 semitones.Composers use a preponderance of steps, and listeners almost a lways, onan expectancy scale, rate stepwise continuations higher than leaping ones(Schellenberg, 1996). Steps as a class are clearly more different f rom leapsas a class than semitone steps are from whole-tone steps. The proximityratings reflect that perceptual distinction. They avoid the undesirably highrat ings given to semitones by Lerdahls inverse-squa re rule (2001) and bet-ter follow empirical studies (Lake, 1987; Schmuckler, 1989), suggestingthat listeners expect a variety of possible continuations with comparablestrength and do not radically favor semitone neighbors.

After 3 semitones, proximity ratings drop increasingly gradually until,for very large intervals, expectancy ratings are equally low. The modelpredicts that a pitch 3 octaves away and a pitch 2 octaves away areexpected by about the same amount, but a pitch 1 octave away is muchless expected than a pitch 1 semitone away. At a smaller scale, the sameprinciple applies. The model predicts that a pitch 10 semitones away is


TABLE 2Proximity Ratings Increase as Semitone Distance Increases,Reflecting the Expectation That Pitches Will Proceed To

Relatively Nearby Continuations

Pitch Distance in Semitones Proximity Rating1 362 323 254 205 166 127 98 69 4

10 211 112 0.2513 0.02

>14 0.01


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expected slightly less than a pitch 7 semitones away, but a pitch 6 semi-tones aw ay is expected considerably less than a pitch 3 semitones aw ay.

The specific increments between the proximity ratings were selected,like the increments between the stability ratings, with end tension pre-

dictions in mind. In the case of proximity, the potential impact of direc-tion was especially attended to. As the formula (discussed later) forcombining individual f actor ratings reveals, sta bility a nd prox imity ra t-ings are multiplied and direction ratings are added. Thus after a givenpitch, say D in C major (see Tab le 4), the stab ility- and prox imity-ba sedrating for virtual (possible future) continuations can be listed (first C,than E, etc.). Ds directional context then shifts and reorders the ratings.For example, if D were leapt to from a much higher pitch, the direction-al impulse for reversal might add enough impetus for E to surpass C asthe most highly rated continuation. By considering the tension createdby multiple musical examples in which specific scale degrees were

leaptor steppedto, the desired interaction between proximity anddirection for each possible configuration was determined. Numeric val-ues were chosen so that direction reordered ratings (placing E before Cafter D) when intuitive experience warranted it, but preserved thedefault ranking (e.g., C before E after D) when appropriate (when, forexample, a progression f rom D to E instead of C creat ed more surprise-tension).

Direction

Support for Narmours (1990) idea that small intervals imply direction-al continuation but large intervals imply directional reversal has beenmixed. Schellenberg (1996, 1997) suggests tha t t he expecta tion for rever-sal af ter large intervals may be considerably stronger tha n the expectationfor continua tion a fter small ones. Figure 2 depicts the present mo dels dis-tribution of direction rat ings.

Within the category of small intervals, the more exaggeratedly small theinterval is, the stronger the expectation for continuation is proposed to be.Likewise, within the category of large intervals, the more exaggeratedlylarge the interval, the stronger the expectation for reversal is proposed tobe, until a cutoff point beyond which expectations are theorized to be

equivalently strong. (Extremely large intervals might suggest a melodythat is processed as a polyphonic combination of two componentmelodies). Between the categories of small and large, however, it can beseen that reversal expectations after even moderately large intervals aretheorized to grow much stronger than continuat ion expectations af ter thesmallest intervals.



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Tab le 3 show s the specific direction rat ings af ter differently sized inter-vals, ratings calculated in consideration of the proximity ratings to pro-duce tension predictions that matched experience. After a pitch repetition(interval size 0, for example the succession G -G ), direction ra tings w eak-ly favor continued lateral motion. Ratings after other intervals favor


Fig. 2. The xax is represents interval size, a nd the yax is represents strength o f d irection-al expectation. Dark shading represents an expectation for reversal, and light shadingindicates an expectation f or cont inuation. The influence of the direction para meter variesaccording t o the size of the interval betw een the tw o preceding not es. Very large intervalsare theorized to g enerat e a strong expectatio n for reversal, but very small intervals are the-orized to generate only a moderate expectation for continuation.

TABLE 3Ratings Given To Events That Lie in the Direction Implied by the

Interval Between the Preceding Two NotesInterval Size Direction Rating

0 6 for continuation1 20 for continuation2 12 for continuation3 6 for continuation4 05 6 for reversal6 12 for reversa l7 25 for reversa l

8 36 for reversa l9 52 for reversa l

!10 75 for reversa l

NOTEEvents lying in the opposite direction receive a rating of 0. For example, after an ascend-ing leap of a minor sixth (8 semitones), 36 is given to any event that constitutes a descent, 0 to any-thing that constitutes an ascent. (These values are different than those of later quantifications of theregistral direction parameter in Narmours implication-realization theory, cf. Schellenberg, 1996,1997.)
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either continuation or reversal. After an ascending interval, an event qual-ifies as a continuation if it lies above the last pitch and a reversal if it liesbelow it. After a d escending interval, a n event q ualifies as a continuationif it lies below the last pitch, and a reversal if it lies above it. Narmour

(1990) suggests that after a large interval pitch repetition should alsoqualify as reversal because of the differentiation between the initialascending or descending motion and the ensuing lateral motion.According to this notion, a repetition of A should count a s a partial rever-sal after the ascending leap C-A. In such cases, the absence of directionalcontinuation seems enough to qualify the repetition as a partial reversal.In the present model, pitch repetitions fulfill directional implications atonly 1/3 of their full value (rounded to t he nearest w hole number). Forexample, af ter the ascent C -A, a descent to neighbor ing G w ould receivea directional rating of 52, but repetition of the A would receive a direc-tional rating of 17, one third of 52.

Mobility

The secondary factor mobility captures the general intuition that amelody will move. Expectancies for self-repetition (the expectancy, forexample, that D will continue to D) have caused problems in previousmodels. Lerda hl (2001) excludes them fro m considerat ion, since his prox-imity rule (the inverse square of semitone distance) implies an infiniteexpecta tion for self-repetition (w here semitone d istance is 0). The diff icul-ties stem from repetitions in-between status; it counts neither as a contin-

uation of ascending or descending motion nor as a reversal of it, and itcounts neither as movement to a proximate or nonproximate pitch, butrather as something different. Before the inclusion of mobility, the presentmodel predicted dubiously strong and robust expectations for repetition.Mobility tempers these predictions by capturing the general expectationthat a melody will move. The principle operates within the model by low-ering expectation ratings for self-repetitions. Specifically, mobility worksby multiplying the stability and proximity rating for repetition by t he con-stant 2/3, to pena lize possible continua tions tha t do not move to a newpitch level. In all other cases, mobility is 1 and does not alter the overallexpectancy rating.

Factor Combination

Before hierarchic levels are considered, an events expectancy ratingmay be generated by using the following formula.



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EXPECTANCY FORMULA (BEFORE HIERARCHIC LEVELS ARE INCLUDED)

A pitch xis expected to follow pitch yby amount z:

z= (s p m) + d,

where s= the stability rat ing of x(see Table 1), p= the proximity ratingof x(see Table 2), d= the direction rating of x(see Table 3), and m= themobility rating of x(2/3 if xrepeats yand 1 in all other cases).

All factors that can apply, do, all that cannot, do not. For example,expectations concerning the second note of a piece use only sand pbecause no direction has been established. Consider the melody in Figure3. Expectancy rat ings could be generated for each event. Take the G inmeasure 1 as an example. In the default I of C major context, it receivesa stability rating o f 5. Tw o semitones aw ay from t he preceding A, itreceives a proximity rat ing of 32. G does not constitute a repetition of A,

and t herefore receives a va lue of 1 (rather than 2/3) for mobility.D irectiona lly the G sa tisfies the expecta tion generated by the preceding 9-semitone leap fro m C to A. Before hierarchic levels are considered, G stotal expectedness (212) is calculated by multiplying 5, 32, and 1, its sta-bility, proximity, and mobility ratings, and adding 52, the reversal ratinggenerated by the leap. As a contrasting example, take the F in measure 2.Its stability rating is 4, and its proximity rating (lying 5 semitones awayfrom the preceding C) is 16. Again, its mobility rating is 1. Its directionalrating is 0 because it does not lie in the direction predicted by the preced-ing descending step. It s total ra ting before hierarchic levels are considered,therefore, is (4 16 1) + 0, or 64.

A model overview can be provided by listing the core stability andproximity ratings after different events in a given tonal context. Such list-ings are valuable because they make general ratings trends apparent. Bystudying the distance between top-rated events, it can easily be seen howdifferent directiona l cont exts would a ffect the rat ings. Tab le 4 depictssome sample core sta bility/prox imity ra tings in an I of C major context.Although a pitchs theorized expectedness is calculated by combining allfa ctors, exa mining the stability/prox imity ra tings in isola tion gives an


Fig. 3. A melody, for every note of which an expectancy rating could be calculated. Theexpectancy ra tings signify the extent to w hich the continuation is predicted to be primed.Sample rating calculat ions are included fo r the G in measure 1 and the F in measure 2.
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excellent sense for the inner workings of the model; provided a tonal con-text, the ratings for possible continuations after any pitch can be concise-ly listed. Because direction depends not only on the identity of the previ-ously heard pitch, but also on the one before it (i.e., the interval createdby the two), if it were included on the table, the range of possible implica-tive situationseven in a single tonal contextwould become impracti-cably large; in C major, for example, ratings would have to be listed notjust after C, but after C when it followed B, after C when it followed D,and so on. Instead, a much more intuitive sense of the models predictions

can be gained by viewing a restricted list of sta bility/prox imity ra tings andimagining the shifts that different directional circumstances create.Contrast expectancy ratings after B, shown in row 2, with expectancy

ratings aft er G, show n in row 4. Leading-tone B is theorized to trigger amaximal expectation for the tonic, semitone-neighbor C. The next mostexpected continuation, A, receives a dramatically lower rating. Becausethe highest possible direction rating is 75, and As rating is 88 less thanCs, no directional context can cause A to surpass C as the most expectedcontinuation after B. Consider row 4, however. When only stability andproximity are considered, diatonic neighbors F and A tie as the mostexpected continuation after G. Following 5, directional context is always

the determining fa ctor in selecting

^

4 or

^

6 as the most expected continua-tion. Aside from the distinction in degree of susceptibility to directionalinfluence, expecta tions aft er B and G differ in another important w ay. Themost expected continuat ions after G receive a ra ting of 128, but the mostexpected continuation after B receives a rating of 216. The discrepancyindicates that expectations af ter B are strong, specific, and robust, but tha texpectations af ter G are w eak and diffuse. Situations in betw een these tw o


TABLE 4Sample Core Expectancy Ratings in an I of C Major Context

216 192 180 160 150 144 128 125 120 100 96 80

D C E B D G

FB C A D B E

G

F E G D C A

F

G F E G C B

A

NOTEThe leftmost column represents a pitch in C major. Subsequent columns show the com-bined (multiplied) stability and proximity ratings of pitches found within the span of one tritonebelow to o ne tritone above. For example, in the row fo r D , C refers to the C one step below, becauseit is the closest instance of C, and the only one within the tritone-to-tritone span. F refers to the F aminor third above D for the same reasons. As described in the text, the direction factor can reorderthe listings presented here.


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extremes can be seen in rows 1 and 3, which list expectancy ratings afterD and F in a default C-major context.

Hierarchy

Even deeply schematic expectancies depend not only on the last heardnote, or the last heard pair of notes, but also on a larger context.Hierarchic rules drawn from principles in Lerdahl and Jackendoff (1983)segment the music into time spans, choose a head, the most structural-ly significant event, for each span, and specify the heads influence onmelodic expectations. More important melodic events are predicted toplay a correspondingly larger role in expectancy format ion.

Time spans can be thought of a s apprehended rhyt hmic units in termsof w hich pitch structure is heard (Lerda hl & Ja ckendof f, 1983, p. 124).

Since many of the segmentation rules are couched in terms of preferencerules, some room for disagreement over the selection of a head exists.Such issues could be resolved by formalizing the preference rules morerigidly, but the task lies outside the scope of this essay. The present theo-ry does introduce one additional preference rule, to supplement Lerdahland Jackendoffs nine rules for time-span reduction. The additional rule ismeant to resolve ties in the case of more than one plausible head choice.Called Expecta ncy Fulfillment, it reads: O f the possib le choi ces fo rhead of a time-span T, w eakly prefer a choice that ful fi ll s expectationsgenerated by e, the head of the time-span immediately preceding T at thesame level. Since expectancies usually privilege stepwise motion, with the

highest rated continuations lying a whole or half step from the previouspitch, the expectancy fulfillment rule captures a listeners attempt to formstepwise progressions beneat h the musical surfa ce. O ther time-span reduc-tion rules specify, for exa mple, to prefer the selection o f a head t ha t is met-rically strong, intrinsically consonant, relatively closely related to thetonic, and, in the case of melody, relatively higher in terms of register(Lerdahl & Jackendoff, 1983).7

According to the theory proposed here, listeners treat the heads of thetime spans at a particular level as a sort of background melody aboutwhich expectations pertain. The principles of proximity, stability, mobili-ty, and d irection apply to the background melody just a s they apply to t he

notated one. Expectations about the level of the notated melody operatesimultaneously alongside expectations about background levels, forexample quarter-note, half-note, and whole-note levels. In other words, alistener is theorized to have a sense of not only what might happen next


7. For a full listing of the time-span reduction rules, see Lerdahl and Jackendoff(1983).


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locally, but also what might happen next slightly more globally. Thenotated melody is sometimes referred to in this article as the note-to-notelevel, to emphasize that it is composed of notes rather than heads.H ow ever, notes in this reading are entities within a tona l netw ork, and

event governance rules influence how stability is assessed within that net-work.

Figure 4 shows the time-span reduction for measures 14 of MozartsPiano Sonata in E major, w ith the score on the bo ttom line and reductionlevels on the lines above itfrom quarter-note to half-note, measure, andtwo measure. Although only the melodies are analyzed, the full scores ofmusical excerpts are included in this article. The harmonies influence theselection of chord and key contexts, w hich contribute to t he sta bility ra t-ings. Elements outside the melody can also influence the formation oftime-spans and the selection of heads, important in the hierarchic imple-mentation of the parameters of stability, proximity, and direction. At a

given level, notes depicted with the appropriate rhythmic value (i.e., quar-ter not e at the q uar ter-note level, ha lf no te at the ha lf-note level) representthe time-span heads: this notation is consistent with that used in Lerdahland Jackendoff (1983). The unstemmed noteheads (a new notational ele-ment) result from the implementa tion of a final-state theory w ithin a tem-porally situated context. They represent considered but ultimately dis-carded head candidates. At the start of a span, a listener cannot know theidentity of the spans best head. O nly the best cand idate in the part o f t hespan already heard can be known. The idea is that the listener assumes thebest cand idate from a mong the pitches already heard t o be head until fur-ther evidence accrues. Consider the start of measure 3. For lack of better

options, C is initially assumed to be head (observe the unstemmed note-head representing C at the quarter-note level). B soon supplants it, form-ing the actual head of that quarter-note span (observe the stemmed B onthe same level). Actual heads are notated differently than merely consid-ered ones because they ultimat ely play a role in the forma tion o f expectan-cies about future spans, as the next example explains.

O bserve tha t the C at the sta rt of measure 3, dissona nt w ith the under-lying harmony, is a highly unsat isfacto ry head candida te. Nevertheless, a tthe time point before any other notes have yet occurred within the span,it is the only candidate. The theory holds that listeners represent as headthe best ca ndidat e from a mong t hose already heard. Accordingly, it is pre-

dicted that listeners will often temporarily entertain the possibility of apoor head, especially at the start of a span, w here often the only notes thathave yet sounded are nonchord tones or similarly inferior options. Assoon as something more satisfactory occurs, the listener supplants his orher representation of the spans head with the better choice.

Note that in Lerdahl and Jackendoffs use of this notation, the final-state assessment of a pitchs structural importance is represented by the



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highest level at which that pitch is retained; the higher the level of reten-tion, the more structurally important the pitch. However, in the presentuse of the notation, no final-state assessment is implied. The inclusion asa stemless notehead of the C in measure 3 at not only the quarter-note,


Fig. 4. Time-span reduction f or m easures 14 of M oza rt, K . 282. Q uarter-note, ha lf-note,measure, and two-measure levels are depicted on separate staves. Unstemmed noteheadsrepresent considered but ultimately discarded head candidates. Expectancies project ateach level, targeting adjacent heads the way expectancies at the note-to-note level target

ad jacent pitches.


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but also the half-note, measure, and two-measure levels does not indicatethat the C has been retrospectively judged t o function structurally at all ofthese levels. Rather, it indicates that measure 3 initiates spans at not onlythe quarter-note level, but also all higher ones through the two-measure

level. Listeners are theorized to simultaneously pro ject cont inuat ions at allthese levels of the notated hierarchy, and for the first sixteenth note of allof the spans starting at the beginning of measure 3, they are theorized totemporarily sustain C as the only available head candidate. In otherwords, Cs notational presence on multiple levels at the start of measure 3says something not about a final-state a ssessment o f t he pitch, but ratherabout a theorized projectional and evaluative experience during one six-teenth not es duration at the start o f measure 3.

At the note-to-note level, expectations about the downbeat of measure3 (the C) are formed largely by the preceding two notes, F and D. At thequarter-note level, expectations about that same downbeat are formed

largely by the preceding two heads, C and F. At the half-note level, expec-tations about the same downbeat are most afected by the identity of theprevious two heads at that level, C and D. The C at the start of measure3 is more expected at some levels than at others. The way in which itsexpectancy ratings at different levels are combined to form an overallexpectancy rating is discussed shortly.

Lerdahl and Jackendoff use time-span reduction as a derivational steptoward prolongational reduction. For the purposes of the current model,time-span reduction suffices. It avoids the complications of prolongation-al reduction, which depends more substantially on final-state awarenessand resists a left-to-right temporal implementation. On its own, time-span

reduction effectively selects the most expectation-influencing pitch fromw ithin individual spans. In Lerdahl and Jackendoff, prolongationa l reduc-tion provides the connection to experience (position in the resultant treeis theorized to link to tension), but in the present model, the connectionto experience is made via expectancy ratings that evolve as the piece pro-gresses, obviating a final-state tree structure.

Expectations canno t a pply to levels with arbitra rily long t ime spans. AsJustin London (2002) remarks, the constraint on the scope of la rger tem-poral patterns is correlated with our sense of the psychological present(p. 536), a period whose upper limit researchers have variously placedbetween 5 and 10 seconds. London locates another important threshold

between 1.5 and 2 s, beyond which rhythmic synchronization becomesdifficult or impossible. These values suggest a scaling of expectancy rat-ings, according to which ratings for levels with spans longer than 2 sdiminish in importance.

Although at the shallowest level expectancy ratings measure the note-to-note fluctuation of tension, hierarchic ratings measure a sort of back-ground t ension. C onsider Figure 5, w hich presents a melody followed by



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a sequence of it an octave higher. If only note-to-note expectations wererepresented, the profiles for each statement, excepting the first two notes,would be identical. Yet the second statement (starting in m. 4) possessesan elevated degree of tension when compared with the first one, becauseof its registral distance and accordingly low background expectancy.H ierarchic expecta ncy ra tings register t he background proximity violationand assign a low value to the second statement, which, when combinedwith other levels ratings, produces slightly lower overall expectancy rat-ings for events within the second statement.

The method for computing and combining hierarchic expectancy rat-ings is presented here.

COMPLETE EXPECTANCY FORMULA8

1. At a given level, the sequence of heads of each of that levelstime spans forms a background melody.

2. The time-span segmenta tion, w ell-formedness, a nd preferencerules determine the head of each time-span at a given level.

a. At any point, the best candidat e for head is selected fromthe time spans already heard pitches. A listener may con-sider severa l different head possibilities over the course ofa span.

3. Expectancy ratings at the note-to-note level are calculated byapplying the previously presented formula, (s p m) + d.Expectancy ratings at other levels are calculated by applyingthe same formula to events (heads and head candidates) in theba ckground melody.

4. To determine a pitchs overall expecta ncy ra ting, produce aweighted average of the pitchs expectancy rating at each level.

a. A pitchs rating at the shallowest level is calculated byusing the expecta ncy formula, (s p m) + d.

b. A pitchs rating at a given hierarchic level is the rating forthe best yet-heard head candida te of t he time span w ithinw hich the pitch falls at t hat level.


Fig. 5. A melody consisting of a segment tha t is sequenced a n octa ve higher.

8. Peter Desain implemented the model in LISP. The program accepts representationsof pitches and associated tonal contexts, produces expectancy ratings, and graphicallydepicts the fluctuations in tension theorized to occur across the course of the melody. Ademo is planned for user access at w w w.nici.kun.nl/mmm under demos.


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c. Ea ch level receives a w eight reflecting its theorizedsalience.

i. Note-to-note ratings receive a weight of 15.ii. Ratings at levels beyond the note-to-note level, up

to and including levels w ith spans of 2 s duration,receive a weight of 5.

iii. Ratings at levels with time-span durations from 2 sup to and including 6 s receive a weight of 2.

iv. No levels with time-span durations longer than 6 sform.

d. Do not include hierarchic levels whose time spans occupya duration equivalent to or less than the durationbetween the current and the last heard note. This ruleprevents a span from receiving a w eight both at the note-to-note level and a hierarchic level.

5. The formula for overall expectedness can be expressed as

" wi[(s

i p

i m

i) + d

i]

" wi

where i= the level under consideration, wi= the weight of the

level under consideration (15 for the note-to-note level, 5 forlevels with spans up to 2 s, 2 for levels with spans up to 6 s),si

= the stability rating for the pitch or head candidate at thatlevel, p

i= the proximity rating for the pitch or head candidate

at that level, mi= the mobility rating for the pitch or head

candidate at that level, and di= the direction rating for the

pitch or head candidate at that level.

Using the weighted average of different levels ratings ensures thatexpectations derived from adjacent events play a greater role than moredistant hierarchic ones in the determination of overall expectancy ratings.To produce a w eighted a verage, an events expectancy ra ting a t each levelis multiplied by that levels weight (specified in the preceding formula).The resultant products, composed of a rating multiplied by a weight, aresummed. Finally, the resultant sum is divided by the sum of the weights toproduce the overall expectancy rating. The values of the weights reflectthe diminished salience spans a re theorized to possess over longer periods,

as suggested by London (2002). They were calculated to produce ratingsthat matched intuitive tension profiles, but are quite susceptible to futuremodification. Certain factors, for example, likely cause listeners to shifttheir attending to deeper levels, whereas other factors most likely steertheir at tending tow ard shallow er ones. These factors are not t reated w ith-in this model and w ould most likely change the quantification o f t he levelweights.



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The formulation of the parameters interrelationships affords themodel certain conceptual and procedural advantages. Consider the facttha t ma ny melodies proceed prima rily by step. The impact of direction istheorized t o b e minimal a fter small intervals. Accordingly, t here are ma ny

readily available musical situations in which expectancy-based tensionresults primarily from factors other than direction (e.g., stability andproximity). H ow ever, there are nota bly few er musical situat ions in whichexpectancy-based tension results primarily fro m fa ctors other tha n stab il-ity, or fa ctors other tha n proximity; in tona l music, there is usually a hier-archy of stability relations at play, and in most music, there is usuallysome distance from the preceding pitch to the current one. In otherwords, proximity and stability do not drop out in numerous situationsthe way that direction does. The configuration of the primary factorswithin the formula captures this dissimilarity: stability and proximityapply multiplicatively but direction is added, sometimes dramatically

reordering the ratings (e.g., af ter a leap in a certa in direction), other timesleaving them fundamentally intact (e.g., after a small interval in a certaindirection).

This conceptualization of factor combination made it easier to tracethe tensions to their sources in the different parameters of expectancy.By looking at the many situations w here direction applied w eakly or notat all, predictions about the other, relatively isolated factors could bemade more easily. For example, after introspecting about sensations oftension in various circumstances, it seemed clear tha t a fter the pitch F inC major, continuation t o G is normally more surprising than continua-tion to E. It w as then possible to a ssess the impact o f d irection by exa m-

ining expectancies after F in situations where F was itself preceded bydifferent notes, to determine the point at which E became a more sur-prising continuation tha n G (this might occur, fo r example, w hen F w asleapt to from further above). The additivity of the parameter of direc-tion contributed strongly to the logic and organization of this proce-dure. Similarly, the formulation of the model is intended to contributeto the ease of und ersta nding its claims. For exa mple, Tab le 4 lists sta bil-ity- and proximity-based expectancy ratings after pitches in C major.This kind of table makes the models predictions compact and easier tounderstand; if ratings based on stability, proximity, and direction werelisted, t he table w ould explode in complexity. R at ings wo uld have to be

included not just for D when it followed C, but also for D when it fol-lowed the two-note sequence B-C, the two-note sequence A-C, and thetw o-note seq uence G -C , a nd so on, ma king it much harder to gra sp thegist o f t he model. G iven the a dditivity of direction, readers can examinea t ab le based solely on stab ility and proximity and imagine the effect ofdirectional context by adding the appropriate increment to events lyingin a specified direction.



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Effects of Deeply Schematic Expectations on Listener Experience

In broa d t erms, the model proposes three forms of association betw eenexpectancy and affect. One of the persistent problems afflicting expecta-tional t heories is the difficulty of correlating expectancy w ith a nything buta unidimensional account of the clearly multidimensional phenomenon ofmusical affect. Although expectancy is thought to be interesting becauseof its impact on the generation of musical affect (Meyer, 1956), the natureof this connection remains obscure. Existing accounts tend to be impov-erished, postulating a simple alternation between shock (in the case ofunexpected events) and contentment (in the case of expected ones). Yet, a sliterary theorist Charles Altieri (2003, p. 10) observes, the arts should becapable of providing a fertile ground for exploring the subtleties of affec-tive experience in a ll doma ins:

M any af fects have pow er in our lives because they emerge as immedi-ate aspects of the kind of attention we pay to the world and to our-

selves. And how we feel is often shaped less by belief per se than by

how we experience the fit of various elements. Here works of art are

instructive because so much depends on their internal dynamics, that

is, on matters of structure and pacing and angle of perspective.

Aesthetics, in other words, is a domain that encourages a deeperengagement with affect, rather than a more superficial one. But a deeperengagement requires w illingness to conf ront t he elusive: ra ther than dis-miss w hat seems inchoa te or indefinite, w e may ha ve to treat these qua li-ties as fundamental features our affective lives are constantly negotiating(Altieri, 2003, p. 12). For this to occur, the first step . . . has to be devel-oping a vocabulary to a rticulate and d istinguish amo ng musics manynear-ineffable affective aspects (Altieri, 2003, p. 46).

The model presented here attempts to take a step in this direction byoffering a vocabulary for different aspects of affective engagement withmelody. This vocabulary respects the notion t hat, in the arts, differentversions of intentionality come into play, especially modes of intentional-ity connected to values like intensity and connectedness rather than to dis-cursive propositions tha t evalua te possible actions (Altieri, 2003, p. 3).It includes terms related to the experience of intensity (surprise-tension),to the highlighting of a melodys apparent intentionality (denial-tension),

and to the impression of desire or forward-directedness in melody(expectancy-tension). This taxonomy was inspired by Lerdahls (2001)separa tion of the phenomena of melodic at traction and implicat ive denial.If it is true that af fects are w ays of being moved that supplement sensa-tion w ith a t least a minimal degree of imaginative projection (Altieri,2003, p. 47), then it makes sense that musical affects would be fleeting



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and projective in the way described by these tension types. However, aspeople are typically not accustomed to introspecting to this degree abouttheir ephemeral and dynamic melodic impressions, it makes sense that itmight take some time, and some practice, to come to match the tension

types with the experiences they describe.The first expectancy-related t ension type the model proposes is termed

surprise-tension, and it correlates inversely with expectancy ratings.Highly predictable events (those with high expectancy ratings) generatelittle surprise-tension, but ext remely unpredicta ble events (those w ith lowexpectancy ratings) generate considerable amounts. Although named forsurprise, the tension deriving from unexpectedness registers not as a con-scious experience of shock, but rather as a subtle experience of intensityand dynamism. It motivates closer attention from the listener.

The sforzando leap to D in the second full measure of the Mendelssohnexcerpt (Figure 6) marks a local peak in surprise-tension. Because D

denies a strong expectat ion for G generated by the preceding F, it alsocreates high denial-tension. This second tension type correlates directlywith implicative denial. High denial-tension creates a sense of will, inten-tion, or determinedness.

Implicat ive denial is calculated w ith the follow ing formula, t aken fromLerdahl (2001).

IMPLICATIVE DENIAL FORMULA

Em

Er,

where Em

= the amount by which the maximally expected pitch was

expected and Er = the amount by w hich the actual realization w as expected.Whereas the expectancy rating measures the expectedness of a given

continuation in relation to all possible continuations for all possibleevents, t he implicat ive denial ra ting measures the continua tions expected-ness in relat ion only to those continuations possible follow ing the actua l-ly occurring event. Figure 7 illustrates the difference between expectancy


Fig. 6. M endelssohn, Song Without Words, O p. 53, N o. 4, measures 58.


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and implicative denial. Although one event, a descending minor sixth leapto B in a C-major context, for example, might permit a continuation, inthis case to C, with an expectancy rating of 252, another, for example aleap from C to E in C major, might permit a maximally rated continua-tion o f only 144, in this case to F. H ad the B from t he first scenario insteadproceeded to A, w ith an expecta ncy rating of 128, it w ould have generat-ed enormous implicative denial (124), but had the E from the second sce-nario proceeded to D, also with an expectancy rating of 128, it wouldhave generated little implicative denial (16). The distinction stems fromthe fact that certain circumstances (a large leap to the leading tone) canproduce stronger, more dramatic expectations than others (a skip to themediant).

A third tension type, expectancy-tension, pertains not to the degree towhich an event satisfies or denies expectations created by precedingevents, but to the strength of expectation generated by an event aboutfuture ones. It correlates with the maximum expectancy an event gener-


Fig. 7. The unstemmed noteheads represent two possible continuations of the melody inhalf notes. In both examples, one continuation (A in the first excerpt, D in the second)receives an expectancy rating of 128. Because surprise-tension is inversely proportional toexpectancy rating, both continuations are predicted to generate the same amount of sur-prise-tension. In t he first exa mple, how ever, the A thw art s a

margulis 2005 - a model of melodic expectation

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