Modelling the perception and cognition of musical structure

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Modelling the perception and cognition of
musical structure

• David Meredith
• ltdave_at_titanmusic.comgt
• Centre for Cognition, Computation and Culture
• Goldsmiths College, University of London

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Algorithmic models of music cognition
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Longuet-Higgins modelLonguet-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.

• OUTPUT
• 24 C STC -5 G STC 0 G STC 1 AB -1
  G TEN REST 4 B STC 1 C TEN

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Longuet-Higgins model of rhythm

• Assumes listener initially assumes pure binary
  metre
• But willing to change mind at any metrical level
• Evidence for change in metre
• Current metre implies syncopation
• No note onset at beginning of next higher
  metrical unit
• Current metre implies excessively large change in
  tempo
• Metre changed if
• evidence for change and
• other division does not imply syncopation or
  excessive tempo change

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Longuet-Higgins model of tonality

• For each note, estimates value of sharpness
  position of pitch name on line of fifths
• Theory of tonality consists of six rules
• First ensures each note spelt so name is as close
  as possible to local tonic on line of fifths
• Other rules control how algorithm deals with
  chromatic intervals and modulations
• e.g., second rule states that if current key
  implies two consecutive chromatic intervals, then
  key should be changed so that both become diatonic

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Longuet-Higgins model Output

• Section of cor anglais solo from Act III of
  Wagners Tristan und Isolde
• Change from binary to ternary in first beat of
  fifth bar (triplets)
• Grace note correctly identified in seventh bar
• Agrees fully with original score in tonal and
  rhythmic indications
• Wagner marked all triplets as staccato fault
  with performance, not program!
• 98.21 notes spelt correctly (3508 errors) in a
  195972 note corpus of classical and baroque music
• Cf. 99.44 spelt correctly (1100 errors) by
  Merediths PS13s1 algorithm
• Meredith, D. (2006). The ps13 pitch spelling
  algorithm. Journal of New Music Research, 35(2),
  pp. 121-159.

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Lerdahl and Jackendoffs Generative Theory of
Tonal Music (GTTM)Lerdahl, F. and Jackendoff, R.
(1983). A Generative Theory of Tonal Music. MIT
Press, Cambridge, MA.

• WELL-FORMEDNESS RULES define CLASS of POSSIBLE
  structural descriptions
• PREFERENCE RULES used to find BEST
  structuraldescriptions

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Lerdahl and Jackendoffs theory of grouping
structure

• Listener automatically segments music into
  structural units of various sizes called groups
• Grouping structure of a passage is way that it is
  perceived to be segmented into groups
• Grouping can be viewed as the most basic
  component of musical understanding (Lerdahl and
  Jackendoff, 1983, p.13)

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Lerdahl and Jackendoffsgrouping well-formedness
rules

• GWFR 1 Any contiguous sequence of pitch-events,
  drum beats, or the like can constitute a group,
  and only contiguous sequences can constitute a
  group.
• GWFR 2 A piece constitutes a group.
• GWFR 3 A group may contain smaller groups.
• GWFR 4 If a group G1 contains part of a group G2,
  then it must contain all of G2.
• GWFR 5 If a group G1 contains a smaller group G2
  then G1 must be exhaustively partitioned into
  smaller groups.

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The Gestalt principles of proximity and
similarity in vision and in music
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Lerdahl and Jackendoffssecond grouping
preference rule

• GPR 2 (Proximity) Consider a sequence of four
  notes n1, n2, n3, n4. All else being equal, the
  transition n2n3 may be heard as a group boundary
  if
• a. (Slur/Rest) the interval of time from the end
  of n2 to the beginning of n3 is greater than that
  from the end of n1 to the beginning of n2 and
  that from the end of n3 to the beginning of n4,
  or if
• b. (Attack-Point) the interval of time between
  the attack points of n2 and n3 is greater than
  that between the attack points of n1 and n2 and
  that between the attack points of n3 and n4.

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Lerdahl and Jackendoffsthird preference rule

• GPR 3 (Change) Consider a sequence of four notes
  n1, n2, n3, n4. All else being equal, the
  transition n2n3 may be heard as a group boundary
  if
• a. (Register) the transition n2n3 involves a
  greater intervallic distance than both n1n2 and
  n3n4, or if
• b. (Dynamics) the transition n2n3 involves a
  change in dynamics and n1n2 and n3n4 do not, or
  if
• c. (Articulation) the transition n2n3 involves
  a change in articulation and n1n2 and n3n4 do
  not, or if
• d. (Length) n2 and n3 are of different lengths
  and both pairs n1, n2 and n3, n4 do not differ in
  length.
• (One might add further cases to deal with such
  things as change in timbre or instrumentation.)

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Temperley and Sleators Melisma music
analyserTemperley, D. (2001). The Cognition of
Basic Musical Structures. MIT Press, Cambridge,
MA.Meredith, D. (2002). Review of David
Temperleys The Cognition of Basic Musical
Structures (Cambridge, MA MIT Press, 2001).
Musicae Scientiae, 6(2), pp. 287-302.
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Temperleys theory of contrapuntal structure
Input representation
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Temperleys contrapuntal well-formedness rules
(CWFRs)

• CWFR 1 A stream must consist of a set of
  temporally contiguous squares on the plane.
• CWFR 2 A stream may be only one square wide in
  the pitch dimension.
• CWFR 3 Streams may not cross in pitch.
• CWFR 4 Each note must be entirely included in a
  single stream.

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Temperleys contrapuntal preference rules (CPRs)

• CPR 1 (Pitch Proximity Rule) Prefer to avoid
  large leaps within streams.
• CPR 2 (New Stream Rule) Prefer to minimize the
  number of streams.
• CPR 3 (White Square Rule) Prefer to minimize the
  number of white squares in streams.
• CPR 4 (Collision Rule) Prefer to avoid cases
  where a single square is included in more than
  one stream.

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Using Temperleys theory to model listening,
composition, performance and style

• Temperley and Sleators programs scan the music
  from left to right, keeping note of the analyses
  that best satisfy the preference rules so far at
  each point.
• Ambiguity Two or more best analyses at a given
  point in the music.
• Revision The best analysis at some point in the
  music does not form part of the best analysis at
  some later point.
• Expectation The most expected events are those
  that will lead to an analysis that best satisfies
  the preference rules.
• Style A passage is in the style defined by a set
  of preference rules if the analysis that best
  satisfies the preference rules achieves a score
  that is not too high (boring) and not too low
  (incomprehensible).
• Composition Choices guided by goal to produce
  piece that satisfies preference rules to just the
  right extent.
• Performance Temporal and dynamic expression
  geared towards conveying structure in accordance
  with analysis that best satisfies the preference
  rules.

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Summing up

• We can attempt to model the perception and
  cognition of musical structure by constructing
  algorithms that take representations of musical
  passages as input and generate structural
  descriptions of those passages as output
• We can evaluate such algorithms by comparing
  their output with expert human analyses and
  authoritative scores
• Can express a theory of musical structure as a
  preference rule system consisting of
• Well-formedness rules that define the class of
  legal structural descriptions
• Preference rules the legal structural
  descriptions that best satisfy the preference
  rules are predicted to be the ones that listeners
  are most likely to hear