Fractal Patterns in Music

TL;DR

This paper introduces a novel, local method for detecting fractal patterns in music based on self-similarity across temporal scales, offering a new way to analyze musical complexity.

Contribution

It presents a new approach to identify fractal patterns in music that is based on self-similarity and does not rely on averaging, differing from previous statistical methods.

Findings

Fractal patterns are detectable in musical compositions using the proposed method.

The approach successfully identifies fractal structures in musical renditions of the Cantor Set and Koch Curve.

Application to masterful compositions shows the presence of fractal patterns across five centuries.

Abstract

If our aesthetic preferences are affected by fractal geometry of nature, scaling regularities would be expected to appear in all art forms, including music. While a variety of statistical tools have been proposed to analyze time series in sound, no consensus has as yet emerged regarding the most meaningful measure of complexity in music, or how to discern fractal patterns in compositions in the first place. Here we offer a new approach based on self-similarity of the melodic lines recurring at various temporal scales. In contrast to the statistical analyses advanced in recent literature, the proposed method does not depend on averaging within time-windows and is distinctively local. The corresponding definition of the fractal dimension is based on the temporal scaling hierarchy and depends on the tonal contours of the musical motifs. The new concepts are tested on musical 'renditions' of the Cantor Set and Koch Curve, and then applied to a number of carefully selected masterful compositions spanning five centuries of music making.