Homological persistence in time series: an application to music classification

Abstract

Meaningful low-dimensional representations of dynamical processes are essential to better understand the mechanisms underlying complex systems, from music composition to learning in both biological and artificial intelligence. We suggest to describe time-varying systems by considering the evolution of their geometrical and topological properties in time, by using a method based on persistent homology. In the static case, persistent homology allows one to provide a representation of a manifold paired with a continuous function as a collection of multisets of points and lines called persistence diagrams. The idea is to fingerprint the change of a variable-geometry space as a time series of persistence diagrams, and afterwards compare such time series by using dynamic time warping. As an application, we express some music features and their time dependency by updating the values of a function defined on a polyhedral surface, called the Tonnetz. Thereafter, we use this time-based representation to automatically classify three collections of compositions according to their style, and discuss the optimal time-granularity for the analysis of different musical genres.

Keywords:

• Tonnetz
• topology
• time-series analysis
• persistent homology
• dynamic time warping
• classification
• style

2010 Mathematics Subject Classification:

• 00A65
• 20B35
• 18B25

2012 Computing Classification Scheme:

• Applied computing

Acknowledgments

The authors acknowledge the precious comments of Massimo Ferri on an early version of this paper. We thank the reviewers for their careful reading of the manuscript, and their valuable and insightful comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 https://mgbergomi.github.io/html/def_ton_and_sounds/examples/deformed_tonnetz_int_sound_pers.html