Persistent Homology and Discrete Fourier Transform

This book proposes contributions to various problems in the field of topological analysis of musical data: the objects studied are scores represented symbolically by MIDI files, and the tools used are the discrete Fourier transform and persistent homology. The manuscript is divided into three parts: the first two are devoted to the study of the aforementioned mathematical objects and the implementation of the model. More precisely, the notion of DFT introduced by Lewin is generalized to the case of dimension two, by making explicit the passage of a musical bar from a piece to a subset of Z/tZ×Z/pZ, which leads naturally to a notion of metric on the set of musical bars by their Fourier coefficients. This construction gives rise to a point cloud, to which the filtered Vietoris-Rips complex is associated, and consequently a family of barcodes given by persistent homology. This approach also makes it possible to generalize classical results such as Lewin's lemma and Babitt's Hexachord theorem. The last part of this book is devoted to musical applications of the model: the first experiment consists in extracting barcodes from artificially constructed scores, such as scales or chords. This study leads naturally to song harmonization process, which reduces a song to its melody and chord grid, thus defining the notions of graph and complexity of a piece. Persistent homology also lends itself to the problem of automatic classification of musical style, which will be treated here under the prism of symbolic descriptors given by statistics calculated directly on barcodes. Finally, the last application proposes a encoding of musical bars based on the Hausdorff distance, which leads to the study of musical textures.