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Simulating Melodic and Harmonic Expectations for Tonal Cadences Using Probabilistic Models JOURNAL OF NEW MUSIC RESEARCH, 2017 https://doi.org/10.1080/09298215.2017.1367010 Simulating melodic and harmonic expectations for tonal cadences using probabilistic models David R. W. Searsa,MarcusT.Pearceb,WilliamE.Caplina and Stephen McAdamsa aMcGill University, Canada; bQueen Mary University of London, UK ABSTRACT ARTICLE HISTORY This study examines how the mind’s predictive mechanisms contribute to the perception of Received 5 November 2016 cadential closure during music listening. Using the Information Dynamics of Music model (or Accepted 14 July 2017 IDyOM) to simulate the formation of schematic expectations—a finite-context (or n-gram) model KEYWORDS that predicts the next event in a musical stimulus by acquiring knowledge through unsupervised Cadence; expectation; statistical learning of sequential structure—we predict the terminal melodic and harmonic events statistical learning; from 245 exemplars of the five most common cadence categories from the classical style. Our segmental grouping; findings demonstrate that (1) terminal events from cadential contexts are more predictable than n-gram models those from non-cadential contexts; (2) models of cadential strength advanced in contemporary cadence typologies reflect the formation of schematic expectations; and (3) a significant decrease in predictability follows the terminal note and chord events of the cadential formula. 1. Introduction regularities that listeners will learn and remember. The tonal cadence is a case in point. As a recurrent temporal In the intellectual climate now prevalent, many scholars formula appearing at the ends of phrases, themes and view the brain as a ‘statistical sponge’ whose purpose is larger sections in music of the common-practice period, to predict the future (Clark, 2013). While descending a the cadence provides perhaps the clearest instance of staircase, for example, even slightly misjudging the height phrase-level schematic organisation in the tonal system. or depth of each step could be fatal, so the brain predicts To be sure, cadential formulæ flourished in eighteenth- future steps by building a mental representation of the century compositional practice by serving to ‘mark the staircase, using incoming auditory, visual, haptic and breathing places in the music, establish the tonality, and proprioceptive cues to minimise potential prediction render coherent the formal structure’, thereby cementing errors and update the representation in memory. Re- their position ‘throughout the entire period of common searchers sometimes call these representations harmonic practice’ (Piston, 1962, p. 108). As a conse- schemata—‘active, developing patterns’ whose units are quence, Sears (2015, 2016)hasarguedthatcadencesare serially organised, not simply as individual members learned and remembered as closing schemata, whereby coming one after the other, but as a unitary mass (Bartlett, the initial events of the cadence activate the correspond- 1932, p. 201). Over the course of exposure, these ing schematic representation in memory, allowing schematic representations obtain greater specificity, listeners to form expectations for the most probable con- Downloaded by [Queen Mary University of London] at 02:52 22 November 2017 thereby increasing our ability to navigate complex sen- tinuations in prospect. The subsequent realisation of those sory environments and predict future outcomes. expectations then serves to close offboth the cadence Among music scholars, this view was first crystallised itself, and perhaps more importantly, the longer phrase- by Meyer (1956, 1967), with the resurgence of associa- structural process that subsumes it. tionist theories in the cognitive sciences—which placed There is a good deal of support for the role played by the brain’s predictive mechanisms at the forefront of contemporary research in music psychology—following expectation and prediction in the perception of soon thereafter. Krumhansl (1990) has suggested, for ex- closure (Huron, 2006; Margulis, 2003; Meyer, 1956; ample, that composers often exploit the brain’s Narmour, 1990), with scholars also sometimes potential for prediction by organising events on the suggesting that listeners possess schematic representa- musical surface to reflect the kinds of statistical tions for cadences and other recurrent closing patterns CONTACT David R. W. Sears [email protected] David R. W. Sears, Texas Tech University, J.T. & Margaret Talkington College of Visual & Performing Arts, Holden Hall, 1011 Boston Ave., Lubbock, TX 79409, USA The underlying research materials for this article can be accessed at https://doi.org/10.1080/09298215.2017.1367010. © 2017 Informa UK Limited, trading as Taylor & Francis Group 2 D. R. W. SEARS ET AL. (Eberlein, 1997; Eberlein & Fricke, 1992; Gjerdingen, demonstrated the degree to which IDyOM can simu- 1988; Meyer, 1967; Rosner & Narmour, 1992; Temperley, late the responses of listeners in tasks involving melodic 2004). Yet currently very little experimental evidence segmentation (Pearce, Müllensiefen, & Wiggins, 2010), justifies the links between expectancy, prediction, and subjective ratings of predictive uncertainty (Hansen & the variety of cadences in tonal music or indeed, more Pearce, 2014), subjective and psychophysiological emo- specifically, in music of the classical style (Haydn, Mozart, tional responses to expectancy violations (Egermann, and Beethoven), where the compositional significance of Pearce, Wiggins, & McAdams, 2013), and behavioural cadential closure is paramount (Caplin, 2004; Hepokoski (Omigie, Pearce, & Stewart, 2012; Pearce & Wiggins, & Darcy, 2006; Ratner, 1980; Rosen, 1972). This point is 2006; Pearce, Ruiz, Kapasi, Wiggins, & Bhattacharya, somewhat surprising given that the tonal cadence is the 2010), electrophysiological (Omigie, Pearce, Williamson, quintessential compositional device for suppressing ex- &Stewart, 2013), and neural measures of melodic pitch pectations for further continuation (Margulis, 2003). The expectations (Pearce, Ruiz, et al., 2010). And yet, the harmonic progression and melodic contrapuntal motion majority of these studies were limited to the simulation of within the cadential formula elicit very definite expecta- melodic pitch expectations, so this investigation develops tions concerning the harmony, the melodic scale degree new representation schemes that also permit the prob- and the metric position of the goal event. As Huron abilistic modelling of harmonic sequences in complex puts it, ‘it is not simply the final note of the cadence polyphonic textures. that is predictable; the final note is often approached To consider how IDyOM might simulate schematic in a characteristic or formulaic manner. If cadences are expectations in cadential contexts, this study adopts a truly stereotypic, then this fact should be reflected in corpus-analytic approach, using the many methods of measures of predictability’ (2006, p. 154). If Huron is statistical inference developed in the experimental right, applying a probabilistic approach to the cadences sciences to examine a few hypotheses about cadential from a representative corpus should allow us to examine expectancies. To that end, Section 2 provides a brief sum- these claims empirically. mary and discussion of the cadence concept, as well as This study applies and extends a probabilistic account the typology on which this study is based (Caplin, 1998, of expectancy formation called the Information Dynam- 2004), and then offers three hypotheses designed to ex- ics of Music model (or IDyOM)—a finite-context (or n-gram) model that predicts the next event in a mu- amine the link between prediction and cadential closure. sical stimulus by acquiring knowledge through unsu- Next, Section 3 introduces the multiple viewpoints frame- pervised statistical learning of sequential structure—to work employed by IDyOM, and Section 4 describes the examine how the formation, fulfilment, and violation methods for estimating the conditional probability func- of schematic expectations may contribute to the percep- tion for individual melodic or harmonic viewpoints using tion of cadential closure during music listening (Pearce, maximum likelihood (ML) estimation and the prediction- 2005). IDyOM is based on a class of Markov models commonly used in statistical language modelling by-partial-match (PPM) algorithm. We then present in (Manning & Schütze, 1999), the goal of which is to sim- Section 5 the corpus of expositions and the annotated ca- ulate the learning mechanisms underlying human cogni- dence collection from Haydn’s string quartets and tion. Pearce explains, describe Pearce’s procedure for improving model per- It should be possible to design a statistical learning algo- formance by combining viewpoint models into a single rithm ... with no initial knowledge of sequential depen- compositepredictionforeachmelodicorharmonicevent dencies between melodic events which, given exposure in the sequence. Finally, Section 6 presents the results of Downloaded by [Queen Mary University of London] at 02:52 22 November 2017 to a reasonable corpus of music, would exhibit similar the computational experiments, and Section 7 concludes patterns of melodic expectation to those observed in experiments with human subjects. (Pearce, 2005, p. 152) by discussing limitations of the modelling approach and considering avenues for future research. Unlike language models, which typically deal with unidimensional inputs, IDyOM generates predictions for 2. The classical cadence multidimensional melodic sequences
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