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

The analysis of structural patterns in music is of interest in order to increase ourfundamental understanding of music, as well as for devising algorithms for computer-generatedmusic, 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 L´evy-stable distribution. Since music also exibits self-affine scaling,we propose that the “music walk” should be modelled as a L´evy motion. We find that the L´evy motion model captures basic structural patterns in classical as well as in folk music.