Physics of Simple and Higher-Order Networks of Musical Spaces: From Laplace Operator to Dirac’s

Abstract

Music is inherently complex, with structures and interactions that unfold across multiple layers. The quantitative analysis of a diverse collection of digitized Western classical music scores and their corresponding networks has recently revealed patterns of harmonic complexity. Although notable works have used these approaches to study music, dyadic representations of interactions fall short in conveying the underlying complexity and depth. In this thesis, we present a multiscale approach to analyzing J. S. Bach’s Sonatas and Partitas for solo violin, and supplement the conventional topological analysis of these networks with music-theoretical interpretations. We begin with simple networks by constructing duration-weighted transition matrices to model melody and harmony networks and complement these with metrics from statistical physics, such as the partition function and communicability, originally employed in the context of socio-economic networks. We perform intra- and inter-network analyses to classify musical movements based on their structural connectivity and similarity metrics. Building upon these notions, we extend our analysis to capture higher-order interactions by introducing a higher-order network model, particularly a simplicial complex representation. Our analysis pertains to the study of the temporal evolution of these complexes and is centered around constructing the higher-order Hodge Laplacian and examining trends in the way the topological invariants and geometrical features evolve over time. We further extend our analysis to include musical pieces of similar genres by different composers in order to generalize our findings. Additionally, we incorporate AI-generated music to support and expand the scope of our analysis and assess the extent to which such music adheres to the structural principles of classical tonal composition from the perspective of topological and geometrical measures.