A mathematical model for voice leading and its complexity

TL;DR

This paper introduces a mathematical framework for analyzing voice leading in music, quantifying its complexity through matrix representations and dynamic time warping, enabling classification of musical compositions.

Contribution

It formalizes voice leading using partial permutations and matrices, incorporating rhythm and rests, and develops a complexity measure for musical analysis.

Findings

Complexity vectors effectively differentiate musical pieces.

Dynamic Time Warping accurately measures similarity between compositions.

The approach visualizes complexity evolution over time.

Abstract

We give a formal definition of the musical concept of voice leading in mathematical terms, based on the idea of partial permutations of certain ordered multisets. Then we associate a partial permutation matrix with a voice leading in a unique way and write an algorithm to easily transform any musical composition into a sequence of such matrices; we then generalise it in order to include in the model also rhythmic independence and rests. From that we extract a vector whose components return information about the movements of the voices in the piece and hence about the complexity of the voice leading. We provide some examples by analysing three compositions, also visualising complexity as a point cloud for each piece. Finally, we interpret the sequence of complexity vectors associated with each composition, thus considering the position of each observation with respect to time. The Dynamic Time Warping allows us to compute the distance between two pieces and to show that our approach distinguishes well the examples that we took into account, exhibiting a strong indication that the notion of complexity we propose is a good tool to identify and classify musical pieces.