Quantification of Determinism in Music using Iterated Function Systems

Brian Meloon and Julien C. Sprott
Department of Physics
University of Wisconsin at Madison

ABSTRACT

This article proposes a novel technique for exhibiting and quantifying the determinism in music. A written score of music is modeled as a dynamical system employing an iterated function system to generate a picture from the music. This picture is then analyzed using methods of chaos theory and time-series analysis to quantify the determinism. Comparisons with random and chaotic control data and with some algorithmic compositions are made. The method might be useful for cataloging different musical styles or perhaps even testing authenticity of musical compositions.

Ref: B. Meloon and J. C. Sprott, Empirical Studies of the Arts 15, 3-13 (1997)

The manuscript of the complete paper is available in pdf format.

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Fig. 1. A written score of music cannot be modeled as a dynamical system without certain simplifications.
[Figure 1]

Fig. 2. Two examples of Iterated Function Systems: (a) with 4 possible input values; (b) using pitch classes as input values.
[Figure 2a] (a)

[Figure 2b] (b)

Fig. 3. Examples of the output of the IFS process: (a) with uniformly random notes; (b) Mozart's "Sonata in C."
[Figure 3a] (a)

[Figure 3b] (b)

Fig. 4. The dimension curve for Mozart's "Sonata in C," typical of the dimension curves that are generated from real music.
[Figure 4]

Fig. 5. The dimension curves for Mozart's "Sonata in C" and three types of control music derived from it.
[Figure 5]

Fig. 6. The dimension curves for Mozart's "Sonata in C" and a piece of music created algorithmically from L-Systems.
[Figure 6]