Topology of musical data

Abstract

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 bar-codes 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 bar codes show many interesting features. Examples show that individual pieces may span the complete space (in which case the classical and the datadriven analyses agree), or they may span only part of the space.

Mathematics and Computation in Music

“…We accomplish this by first partitioning the measures into lines using the standard Knuth dynamic programming approach [29]. As bar lines are treated like other stand-alone symbols, this simply amounts to fixing the horizontal location of each right bar line while optimizing over the remaining parameters.…”

“…In the paper he explored the degree of interrelatedness of the seemingly different notions of musical distance and discussed the appropriate conditions under which each could be used. Sethares and Budney discuss metrics on musical data in relation to topology where they recover the circularity of octave-reduced musical scales and the circle of fifths [29]. In his book, Tymoczko showed several types of chord spaces and discussed their utility [34] (also see [32]).…”

Bach Is the Father of Harmony: Revealed by a 1/f Fluctuation Analysis across Musical Genres

“…For example, the musical chords and voice leadings were modelled by geometric space [ 6 , 7 ]. The topology analysis [ 8 ] and compositional data analysis [ 9 ] were both used to investigate musical structures. Johannes Kepler’s “ The Harmony of the World ” was inspired by music [ 10 ].…”

Demonstration of topological data analysis on a quantum processor

“…There are many interesting and useful applications of topological data analysis. For instance, in the field of image recognition, Carlsson et al found that high-contrast 3×3 pixel patches from grayscale digital images concentrate near the surface of a Klein bottle in a higher-dimensional space [4]; in the field of signal processing, Perea and Harer found that persistent homology can detect periodicity in time-series data preventing noise [5], which is very stable and accurate especially in the presence of damping; in unsupervised machine learning, persistent homology also provides a powerful tool for the analysis of musical data, exploring common features of classical scores [56].…”