A Compression-Based Model of
Musical Learning
David Meredith
DMRN+7, Queen Mary University of London, 18 December 2012
The weather in Denmark
The goal of music analysis
• The goal of music analysis is to
find the best possible
explanations for musical works
• The “best possible”
explanation for a musical work
is one that allows you to
– remember it most easily
– identify errors most accurately
– predict best what will come
next
– ...
• The best possible explanation
– is as simple as possible
– accounts for as much detail as
possible
Lerdahl and Jackendoff (1983, p.205)
• These two criteria often
conflict
sequence of such collect ions. Corresponds t
" prime" operat or.
t r ansl at e( N, V) Translat es t he not e or not e set ,
vect ors or vect or sums, V.
A musical analysis as a program
A n Example M EL Encoding
For egr ound
M iddl egr ound
• A musical analysis can be
represented as a computer
program or algorithm
MEL25;
n1 = not e( 0, 90) ;
/ / Fi r st not e
p = coor ds( 1, - 1) ;
/ / Cor r esponds t o p ( " pr evi ous" ) oper at or i n Deut sch and Fer oe
n = coor ds( 1, 1) ;
/ / Cor r esponds t o n ( " next " ) oper at or i n Deut sch and Fer oe
ms1 = maskSt r uct ur e( 2, 2, 3) ;
/ / Tr i ad mask st r uct ur e
s1 = mask( 6, 2, 2, 3, 2, 3) ;
/ / Backgr ound scal e: Gb pent at oni c
s2 = mask( 6, 2, 2, 1, 2, 2, 2, 1) ;
/ / Gb maj or scal e
T1 = maskSequence( mask( 0, 4, 2, 6) ) ;
/ / Backgr ound r hyt hm
T2 = maskSequence( mask( 0, 1) ) ;
/ / Tat um t i me mask sequence
T3 = maskSequence( mask( 0, 2) ) ;
/ / Ti me mask sequence f or al t er nat e semi quaver s
P1 = maskSequence( s2, mask( 3, ms1) ) ;
/ / Subdomi nant t r i ad i n Gb maj or
P2 = maskSequence( s1) ;
/ / Pi t ch mask sequence f or backgr ound ( Gb pent at oni c)
P3 = maskSequence( s2, mask( 0, ms1) ) ;
/ / Toni c t r i ad i n Gb maj or
S1 = space( T1, P2) ;
/ / Backgr ound space
S2 = space( T2, P3) ;
/ / Space f or f i r st f our semi quaver s
S3 = space( T2, P1) ;
/ / Space f or vect or v4
S4 = space( T3, P3) ;
/ / Space f or vect or v2
v1 = vect or ( p, S1) ;
// \
v2 = vect or ( p, S4) ;
/ / | Vect or s - see f i gur e - >
v3 = vect or ( n, S2) ;
// |
v4 = vect or ( n, S3) ;
// /
Q1 = r epeat ( 2, v1) ;
/ / Sequence of 2 v1 vect or s i n backgr ound space
Q2 = r epeat ( 2, v2) ;
/ / Sequence of 2 v2 vect or s i n mi ddl egr ound space
R1 = pr oduct ( v2, v3) ;
/ / Car t esi an pr oduct of v2 and v3
R2 = pr oduct ( Q2, v3) ;
/ / Car t esi an pr oduct of Q2 = <v2, v2> and v3
add( t r ansl at e( n1,
pr oduct ( Q1,
/ / <v1, v1>
sequence( R1,
/ / v2 x v3
vect or SumSet ( v4) , / / v4
R2) ) ) ) ;
/ / <v2, v2> x v3
pr i nt ( ) ;
/ / Pr i nt s t o t he consol e
(2012)
drMeredith
aw( ) ;
/ / Dr aws a gr aph i n a wi ndow
pl ay( 100) ;
/ / Pl ays r esul t i ng not e set , wi t h t at um = 100ms
– The program must generate a
representation of the music to
be explained as its only output
• The program is usually a
compact or compressed
encoding of its output
• The program is a description of
its output
• If this description is short
enough, it becomes an
explanation of its output
B ack gr o
Program length as a measure of
complexity
P(p(0,0),p(0,1),p(1,0),p(1,1),p(2,0),p(2,1),p(2,2),p(2,3),p(3,0),p(3,1),p(3,2),p(3,3))
t(P(p(0,0),p(0,1),p(1,0),p(1,1)),V(v(2,0),v(2,2)))
• From Kolmogorov
(1965) complexity
theory:
– can use the length of
a program to measure
the complexity of its
corresponding
explanation
– The shorter the
program, the simpler
and better the
explanation
Music analysis aims to compress music
P(p(1,27),p(2,26),p(3,27),p(4,28),p(5,26),p(6,25),p(7,26),
p(8,27),p(9,25),p(10,24),p(11,25),p(12,26))
t(P(p(1,27),p(2,26),p(3,27), p(4,28)),V(v(4,-1),v(8,-2)))
• Since the best explanations
are the shortest descriptions,
the aim of music analysis is to
compress music as much as
possible
Meredith, Lemström and Wiggins (2002)
Perceptual organisations of surfaces
Analysis of Chopin Op.10, No.5
Schenker (1925, p.92)
• Music analysis aims to find
the most satisfying
perceptual organisations
that are consistent with a
musical surface
– could be a score or a
performance
Analysis of first bar of Chopin Op.10, No.5
Meredith (2012)
Chater (1996, p.571)
Likelihood vs. Simplicity
• Theories of perceptual
organisation mostly based
on either
Likelihood
Simplicity
– Likelihood principle: Prefer
the most probable
interpretation (Helmholtz,
1910)
– Simplicity principle: Prefer
the simplest interpretation
(Koffka, 1935)
• Chater (1996) showed that
the two principles are
mathematically identical
Musical objects are interpreted
in the context of larger containing objects
• A musical object (phrase,
section, movement,
work, corpus, ...) is
usually interpreted
within the context of
some larger object that
contains it
M
I
T
WS P
C
F
– e.g., a work is often
interpreted in the context
of its composer’s other
works, or other works in
the same genre or form
or other works for the
same instrument(s)
Analysts look for short programs that
compute collections of musical objects
P
• The analyst tries to find
the shortest program
that computes a set of in
extenso descriptions of
– the object to be
explained (the
explanandum)
– other objects, related to
the explanandum,
defining a context within
which the explanandum
is to be interpreted
Listener interprets new music in the
context of previously heard music
P
P'
• When the (expert)
listener interprets a
new piece, the
existing explanation
(program), P, for all
music previously
heard is modified (as
little as possible), to
produce a new
program, P', to
account for the new
music in addition
Perceived structure represented by
process by which object is generated
P
P'
• Perceived structure of
new musical object
represented by
specific way in which
P' computes that
object
• On this view, both
music analysis and
music perception are
the compression of
collections of musical
objects
Perception and analysis are nonoptimal compression
• Both analyst and (expert) listener aim to find
shortest encodings
• Neither analyst nor listener achieve this aim in
general
• Hampered by limitations of perceptual system
• e.g., require recognizable patterns to be fairly compact
in pitch-time space (Collins et al., 2011)
Individual differences depend on order
of presentation of context
• Prefer to re-use previous encodings wherever
possible
– “greedy algorithm”: means that way in which a
new object is understood depends on the order of
presentation of previous objects
• Implies that each individual will have a
different interpretation of the same musical
object that depends, not only on what
previous music has been heard, but also the
order in which it was encountered
Simple example using SIATECLearn
Simple example using SIATECLearn
WTC I – Top-ranked patterns
Cover
BWV 846a
BWV 846b
BWV 847a
BWV 847b
Learn
WTC I - Top-ranked patterns
Cover
BWV 848a
BWV 848b
BWV 849a
BWV 849b
Learn
Summary
• Main claim is that both music analysis and music
perception can be thought of as having the goal of
compressing music
• A non-optimal, greedy compression strategy which
maximises reuse of existing encodings provides an
explanation for individual differences in interpretation
• A computational model based on the geometric,
SIATEC pattern discovery algorithm has been adapted
to implement a very simple version of this general idea
and applied to Bach’s WTC I
– Results are promising, but output needs to be studied in
more depth to determine its significance
Links
• Slides can be downloaded from
– http://www.titanmusic.com/papers.php
• SIATECCover source code
– http://tinyurl.com/cbrorn7
• SIATECLearn source code
– http://tinyurl.com/d78huwo
References
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Chater, N. (1996). Reconciling simplicity and likelihood principles in perceptual
organization. Psychological Review, 103(3):566-581.
Collins, T., Laney, R., Willis, A. and Garthwaite, P. H. (2011). Modeling pattern
importance in Chopin’s Mazurkas. Music Perception, 28(4):387-414.
Koffka, K. (1935). Principles of Gestalt Psychology. Harcourt Brace, New York.
Kolmogorov, A.N. (1965). Three approaches to the quantitative definition of
information. Problems of Information Transmission, 1(1):1-7.
Lerdahl, F. and Jackendoff, R. (1983). A Generative Theory of Tonal Music. MIT
Press, Cambridge, MA.
Meredith, D. (2012). A geometric language for representing structure in
polyphonic music. Proceedings of the 13th International Society for Music
Information Retrieval Conference, Porto, Portugal.
Meredith, D., Lemström, K. and Wiggins. G. A. (2002). Algorithms for discovering
repeated patterns in multidimensional representations of polyphonic music.
Journal of New Music Research, 31(4):321-345.
Schenker. H. (1925). Das Meisterwerk in der Musik (Vol. 1). Drei Masken Verlag,
Munich.
von Helmholtz. H. L. F. (1910/1962). Treatise on Physiological Optics. Dover, New
York.