Dynamical and topological tools for (modern) music analysis

Abstract : In this work, we suggest a collection of novel models for the representation of music. These models are endowed with two main features. First, they originate from a topological and geometrical inspiration; second, their low dimensionality allows to build simple and informative visualisations. We tackle the problem of music representation following three non-orthogonal directions. First, we propose an interpretation of counterpoint as a multivariate time series of partial permutation matrices, whose observations are characterised by a degree of complexity. After providing both a static and a dynamic representation of counterpoint, voice leadings are reinterpreted as a special class of partial singular braids, and their main features are visualised. Thereafter, we give a topological interpretation of the Tonnetz (a graph commonly used in computational musicology), whose vertices are deformed by both a harmonic and a consonance-oriented function. The shapes derived from these deformations are classified using the formalism of persistent homology. Thus, this novel representation of music is evaluated on a collection of heterogenous musical datasets. Finally, a combination of the two approaches is proposed. A model at the crossroad between the signal and symbolic analysis of music uses multiple sequences alignment to provide an encompassing, novel viewpoint on the musical inspiration transfer among compositions belonging to different artists, genres and time. Then, music is represented as a time series of topological fingerprints, allowing the comparison of pairs of time-varying shapes in both topological and musical terms.
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Mattia Giuseppe Bergomi. Dynamical and topological tools for (modern) music analysis. General Topology [math.GN]. Université Pierre et Marie Curie - Paris VI, 2015. English. ⟨NNT : 2015PA066465⟩. ⟨tel-01293602⟩

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