A framework for topological music analysis (TMA)

Abstract

In the present article we describe and discuss a framework for applying different topological data analysis (TDA) techniques to a music fragment given as a score in traditional Western notation. We first consider different sets of points in Euclidean spaces of different dimensions that correspond to musical events in the score, and obtain their persistent homology features. Then we introduce two families of simplicial complexes that can be associated with chord sequences, and leverage homology to compute their salient features. Finally, we show the results of applying the described methods to the analysis and stylistic comparison of fragments from three Brandenburg Concertos by J.S. Bach and two pieces from the Graffiti series by Mexican composer Armando Luna.

Keywords:

• Topological data analysis (TDA)
• simplicial complexes
• persistent homology
• music analysis

2020 Mathematics Subject Classifications:

• 00A65
• 55N31

Disclosure statement

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

Notes

1 Here we mean smooth in the sense Boulez talks about smooth time in music, which he opposes to striated (“discrete”) time (see Boulez 1986), developed later by Deleuze and Guattari: “Boulez says that in a smooth space-time one occupies without counting, whereas in a striated space-time one counts in order to occupy” (see Guattari and Deleuze 1987, 477).

2 In general, notes representing unpitched sounds could be included in the analysis, for instance assigning a numerical “pitch” value sufficiently distant from the ones representing actual pitches.

3 Armando Luna Ponce (1964–2015). Mexican composer born in the city of Chihuahua. He studied at the National Conservatory of Music of Mexico (where he later taught composition and music analysis) and the Carnegie Mellon University in the U.S.A. He mainly produced chamber and symphonic music for acoustic instruments, in what he came to name a ludic-eclectic-neorampageous style. Many of his pieces take the suite structure of brief movements as a model, incorporating Renaissance and Baroque dances from the European traidition as well as many other genres from both academic and folk origins. Also, a considerable amount of his works are hommages dedicated to different composers of the Western academic music canon, whose language and style are synthesized and reinterpreted.

4 The equivalences of letters A,B,…,H correspond to the usual German letter system for the notes of the diatonic scale. The equivalence S = mi  also comes from the German “Es,” which stands for E and is pronounced like the letter “s.” It was used for example by Shostakovich to musically encode his name as the motif re-mi-do-si (D-S-C-H). See, for example DSCH motif in Wikipedia.

5 It has been suggested by one of the reviewers of the present paper that perhaps the Cgal C++ library might be computationally more efficient for persistent homology calculations. This computational geometry project is at the time of submission unknown to the authors, who shall definitely try it out for future work.

6 For some tests including higher dimensions in homology, ran over 32-bar samples ($\sim 500$ event-points), there were overflow problems in the execution of the script. From the documentation of the Ripser library: “[In] It [sic] practice, anything above $H_1$ is very slow.”

7 So far, we have run the present methods on over 100 fragments, from 13 classical, baroque, renaissance composers, as well as from traditional Indian and Mexican music.

8 $H_0$-diagrams are necessarily non-empty, as a finite data set is a bounded set in some ${\mathbb{R}}^n$, and so in the successive construction of Vietoris-Rips complexes eventually at least one connected component is always persistent.