A Musical Application of Persistent Homology

Musicians* use notes and chords as words of their personal dictionary to create musical phrases. Quite often, these phrases are shaped as tension patterns over time, drawing the attention of the listener to particular moments thanks to specific choices, frustrating its intuition through unexpected changes, or confirming its expectation with, for instance, a well-known cadence leading to resolution. The notion of tension has often been related to that of consonance, for which recent models developed in the XXth century have allowed one to provide explicit calculations and a quantitative method to compute it for n-notes chords. Traditionally, the geometric-oriented analysis of Music is based on isotropic spaces, we suggest a topological model belonging to the domain of symbolic-signal interaction aimed at the introduction of preferred directions in those spaces. In particular, the Tonnetz interpreted as a 2-dimensional simplicial complex is deformed via an height function computed as the consonance value of a fixed triad and the note associated to each element of its 0-skeleton. We observe how the geometric analysis of the different states of this variable geometry space leads to intuitive and interesting representations on either a melodic and harmonic superposition point of view. A direct way to achieve this information is given by the comparison, in terms of Wasserstein distance, among 0-dimensional persistent homology corner points diagrams associated to each shape generated by the action of the height function on the Tonnetz's vertices.