Non-Gaussian distributions of melodic intervals in music: The Lévy-stable approximation

¹ Department of Engineering Sciences, Solid State Physics, The Ångström Laboratory, Uppsala University P.O. Box 534, SE-75121 Uppsala, Sweden
² Betel Music Institute, Campus Bromma - Åkeshovsvägen 29, SE-16839 Bromma, Sweden

^(a) gunnar.niklasson@angstrom.uu.se

Received: 29 September 2015
Accepted: 12 November 2015

Abstract

The analysis of structural patterns in music is of interest in order to increase our fundamental understanding of music, as well as for devising algorithms for computer-generated music, so called algorithmic composition. Musical melodies can be analyzed in terms of a “music walk” between the pitches of successive tones in a notescript, in analogy with the “random walk” model commonly used in physics. We find that the distribution of melodic intervals between tones can be approximated with a Lévy-stable distribution. Since music also exibits self-affine scaling, we propose that the “music walk” should be modelled as a Lévy motion. We find that the Lévy motion model captures basic structural patterns in classical as well as in folk music.