Multifractal analyses of music sequences

Multifractal analysis is applied to study the fractal property of music. In this paper, a method is proposed to transform both the melody and rhythm of a music piece into individual sets of distributed points along a one-dimensional line. The structure of the musical composition is thus manifested and characterized by the local clustering pattern of these sequences of points. Specifically, the local Hölder exponent and the multifractal spectrum are calculated for the transformed music sequences according to the multifractal formalism. The observed fluctuations of the Hölder exponent along the music sequences confirm the non-uniformity feature in the structures of melodic and rhythmic motions of music. Our present result suggests that the shape and opening width of the multifractal spectrum plot can be used to distinguish different styles of music. In addition, a characteristic curve is constructed by mapping the point sequences converted from the melody and rhythm of a musical work into a two-dimensional graph. Each different pieces of music has its own unique characteristic curve. This characteristic curve, which also exhibits a fractal trait, unveils the intrinsic structure of music.