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Introduction to algorithmic models of music cognition. David Meredith Aalborg University. Algorithmic models of music cognition. Most recent theories of music cognition have been rule systems, algorithms or computer programs
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Introduction to algorithmic models of music cognition David Meredith Aalborg University
Algorithmic models of music cognition • Most recent theories of music cognition have been rule systems, algorithms or computer programs • Take representation of musical passage as input and output a structural description • Structural description should correctly describe aspects of how a listener interprets the passage
Algorithmic models of music cognition • Models take different types of input • audio signals representing sound • representations of notated scores • piano-roll representations • Type of input depends on purpose of model
Algorithmic models of music cognition • A structural description represents a listener’s interpretation – so cannot be tested directly • Need to hypothesise how the listener’s interpretation will influence his or her behaviour
Longuet-Higgins’ model (1976) • Computer program that takes a performance of a melody as input and predicts key, pitch names, metre, notated note durations and onsets, phrasing and articulation OUTPUT: [[[24 C STC] [[-5 G STC] [0 G STC]]] [[1 AB] [-1 G TEN]]] [[[REST] [4 B STC]] [1 C TEN]]
Longuet-Higgins’ model (1976) • Uses score as a ground truth • Assumes pitch names, metre, phrasing, key, etc. should be as notated in an authoritative score of the passage performed • Note fourth note here spelt as an Ab not a G# OUTPUT: [[[24 C STC] [[-5 G STC] [0 G STC]]] [[1 AB] [-1 G TEN]]] [[[REST] [4 B STC]] [1 C TEN]]
Longuet-Higgins’ model (1976) • Even calculating notated duration and onset of each note is not trivial because performed durations and onsets will not correspond exactly to those in the score • e.g., need to decide whether timing difference is due to tempo change or change in notated value OUTPUT: [[[24 C STC] [[-5 G STC] [0 G STC]]] [[1 AB] [-1 G TEN]]] [[[REST] [4 B STC]] [1 C TEN]]
Longuet-Higgins’ model (1976) • Program assumes that perception of rhythm is independent of perception of tonality • So rhythm perceived not affected by pitch • actually not strictly true (cf. compound melody) • Assumes metre independent of dynamics • can perceive metre on harpsichord and organ where dynamics not controlled • Only considers metres in which beats within a single level are equally-spaced • One or two equally-spaced beats between consecutive beats at the next higher level OUTPUT: [[[24 C STC] [[-5 G STC] [0 G STC]]] [[1 AB] [-1 G TEN]]] [[[REST] [4 B STC]] [1 C TEN]]
Longuet-Higgins’ model of rhythm • To start, listener assumes binary metre • Changes interpretation if given enough evidence • current metre implies a syncopation • current metre implies excessive change in tempo • If enough evidence, then changes to a metre where no syncopation and/or smaller change in tempo implied
Longuet-Higgins’ model of tonality • Estimates value of sharpness of each note • i.e., position on line of fifths • Theory has six rules • First rule says that notes should be spelt so they are as close as possible to the tonic on the line of fifths • Other rules control how algorithm deals with chromatic intervals and modulations • e.g., second rule says that if current key implies two consecutive chromatic intervals, then change key so that both become diatonic
Longuet-Higgins’ model: Output • Section of coranglais solo from Act III of Wagner’s Tristan und Isolde • Triplets in first beat of fifth bar • Grace note in seventh bar • Output agrees with original score here • In a larger study (Meredith 2006, 2007) LH’s model correctly predicts 98.21% of pitch names in a 195972 note corpus • cf. 99.44% spelt correctly by Meredith’s PS13s1 algorithm
Lerdahl and Jackendoff’s (1983) Generative Theory of Tonal Music (GTTM) • Probably the most influential and frequently-cited theory in music cognition • Takes a musical surface as input and produces a structural description that predicts aspects of an expert listener’s interpretation • not entirely clear what information assumed in input • predicts “final state” of listener’s interpretation – not “real-time” experience of listening
GTTM • Four interacting modules • Grouping structure: motives, themes, phrases, sections • Metrical structure: “hierarchical pattern of beats” • Time-span reduction: how some events elaborate or depend on other events • Prolongational reduction: the “ebb-and-flow of tension”
GTTM • Each module contains two types of rule • Well-formedness rules: define a class of possible analyses • Preference rules: isolate best well-formed analyses • Modules depend on each other (sometimes circularly!) • Metre requires grouping • Grouping requires time-span reduction • Time-span reduction requires metre • Therefore not trivial to implement the theory computationally • Though some have tried (e.g., Temperley (2001), Hamanakaet al. (2005, 2007))
Temperley and Sleator’sMelisma system • Temperley (2001) presents a computational theory of music cognition, deeply influenced by GTTM • see Meredith (2002) for a detailed review • Uses well-formedness rules and preference rules like GTTM • Models six aspects of musical structure • metre • phrasing • contrapuntal structure • pitch-spelling • harmonic structure • key-structure
Melisma • Consists of five programs that should be piped as shown at left • Evaluated output by comparison with scores • 46 excerpts from a harmony text book (Kostka and Payne, 1995, 1995b)
Melisma • Input in the form of a note-list or piano-roll giving onset time, duration and MIDI note number of each note • Must first infer metre using meter program • But harmony can influence metre and vice-versa, so should use a “two-pass” method as shown • The notelist and beatlist are then given as input to the other programs
Using Temperley’s model to explain listening, composition, performance and style • Melisma programs scan music from left to right, keeping note of the analyses that best satisfy the preference rules at each point • Ambiguity: Two or more best analyses at a given point • Revision: The best analysis at a given point is not part of the best analysis at a later point • Expectation: We most expect events that lead to an analysis that doesn’t conflict with the preference rules • Style: A piece is in the style of the preference rules if it satisfies them not too well (boring) and does not conflict with them too much (incomprehensible) • Composition: Compose a piece that optimally satisfies the preference rules • Performance: Temporal and dynamic expression aimed at conveying structure that best satisfies the preference rules
Summary • Can model music cognition using algorithms that generate structural descriptions from musical surfaces • We can evaluate such algorithms by comparing their output with expert analyses and authoritative scores • Some well-developed theories of music cognition take the form of preference-rule systems containing • Well-formedness rules that define a class of legal analyses • Preference rules that identify the well-formed analyses that best describe the listener’s experience
References • Hamanaka, M., Hirata, K. & Tojo, S. (2005). ATTA: Automatic time-span tree analyzer based on extended GTTM. Proceedings of the Sixth International Conference on Music Information Retrieval (ISMIR 2005), London. pp. 358—365. http://ismir2005.ismir.net/proceedings/1015.pdf • Hamanaka, M., Hirata, K. & Tojo, S. (2007). ATTA: Implementing GTTM on a computer. Proceedings of the Eighth International Conference on Music Information Retrieval (ISMIR 2007), Vienna. pp. 285-286. http://ismir2007.ismir.net/proceedings/ISMIR2007_p285_hamanaka.pdf • Kostka, S. & Payne, D. (1995a). Tonal Harmony. New York: McGraw-Hill. • Kostka, S. & Payne, D. (1995b). Workbook for Tonal Harmony. New York: McGraw-Hill. • Lerdahl, F. and Jackendoff, R. (1983). A Generative Theory of Tonal Music. MIT Press, Cambridge, MA. • Longuet-Higgins, H. C. (1976). The perception of melodies. Nature, 263(5579), 646-653. • Longuet-Higgins, H. C. (1987). The perception of melodies. In H. C. Longuet-Higgins (ed.), Mental Processes: Studies in Cognitive Science, pp. 105-129. British Psychological Society/MIT Press, London/Cambridge, MA. • Meredith, D. (2002). Review of David Temperley’sThe Cognition of Basic Musical Structures (Cambridge, MA: MIT Press, 2001). Musicae Scientiae, 6(2), pp. 287-302. • Meredith, D. (2006). The ps13 pitch spelling algorithm. Journal of New Music Research, 35(2), pp. 121-159. http://taylorandfrancis.metapress.com/link.asp?id=q679l61r31m18460 • Meredith, D. (2007). Computing Pitch Names in Tonal Music: A Comparative Analysis of Pitch Spelling Algorithms. D. Phil. dissertation. Faculty of Music, University of Oxford. http://www.titanmusic.com/papers/public/meredith-dphil-final.pdf • Temperley, D. (2001). The Cognition of Basic Musical Structures. MIT Press, Cambridge, MA.