Introduction to algorithmic models of music cognition
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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

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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


[[[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#


[[[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


[[[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


[[[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


  • 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”


  • 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


  • 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)


  • 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


  • 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


  • 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.

  • 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.

  • 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.

  • 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.

  • Temperley, D. (2001). The Cognition of Basic Musical Structures. MIT Press, Cambridge, MA.

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