Persistent Homology and Harmonic Analysis
摘要
In this paper, we apply persistent homology, one of the main tools in Topological Data Analysis, to the study of harmonic complexity of musical pieces. A directed weighted graph of chords provides a flexible structure that allows to adapt the distances between chords to a training corpus and to retain the asymmetry of human perception of chord progressions. Each musical piece of the corpus is modelled as a sequence of chords, which is used to compute the persistent homology of the simplicial complexes obtained by the Dowker filtration. The values associated to persistence barcodes are then studied to link the geometric properties of the simplicial complexes to the harmonic characteristics of the musical pieces.