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A Framework for Topological Music Analysis (TMA)
Alberto Alcalá-Alvarez, Pablo Padilla-Longoria
In the present article we describe and discuss a framework for applying different topological data analysis (TDA) techniques to a music fragment given as a score in traditional Western notation. We first consider different sets of points in Euclidean spaces of different dimensions that correspond to musical events in the score, and obtain their persistent homology features. Then we introduce two families of simplicial complexes that can be associated to chord sequences, and calculate their main homological descriptors. These complexes lead us to the definition of dynamical systems modeling harmonic progressions. Finally, we show the results of applying the described methods to the analysis and stylistic comparison of fragments from three Brandenburg Concertos by J.S. Bach and two Graffiti by Mexican composer Armando Luna.