Towards a measure of harmonic complexity in western classical music

Abstract

We recently introduced the concept of dynamical score network to represent the harmonic progressions in any composition. Through a process of chord slicing, we obtain a representation of the score as a complex network, where every chord is a node and each progression (voice leading) links successive chords. In this paper, we use this representation to extract quantitative information about harmonic complexity from the analysis of the topology of these networks using state-of-the-art statistical mechanics techniques. Since complex networks support the communication of information by encoding the structure of allowed messages, we can quantify the information associated with locating specific addresses through the measure of the entropy of such network. In doing so, we then characterize properties of network topology, such as the degree distribution of a graph or the shortest paths between couples of nodes. Here, we report on two different evaluations of network entropy, diffusion entropy analysis (DEA) and the Kullback-Leibler divergence applied to the conditional degree matrix, and the measurements of complexity they provide, when applied to an extensive corpus of scores spanning 500 years of western classical music. Although the analysis is limited in scope, our results already provide quantitative evidence of an increase of such measures of harmonic complexity over the corpora we have analyzed.