2013/07/04 by Ryan Budney, William A. Sethares, William Sethares · 1 voice · 1 citation
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #cs.SD #math.AT #math.ST
paper · pdf · doi:10.1080/17459737.2013.850597
openalex publication_date 2014/01/02 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
The musical realm is a promising area in which to expect to find nontrivial topological structures. This paper describes several kinds of metrics on musical data, and explores the implications of these metrics in two ways: via techniques of classical topology where the metric space of all-possible musical data can be described explicitly, and via modern data-driven ideas of persistent homology which calculates the Betti-number barcodes of individual musical works. Both analyses are able to recover three well-known topological structures in music: the circularity of octave-reduced musical scales, the circle of fifths, and the rhythmic repetition of timelines. Applications to a variety of musical works (for example, folk music in the form of standard MIDI files) are presented, and the barcodes show many interesting features. Examples show that individual pieces may span the complete space (in which case the classical and the data-driven analyses agree), or they may span only part of the space.