Using Monoidal Categories in the Transformational Study of Musical Time-Spans and Rhythms

TL;DR

This paper introduces a categorical framework using monoidal categories to analyze transformations of musical time-spans and rhythms, extending previous group-based approaches to multi-timeline and complex rhythmic structures.

Contribution

It demonstrates that generalizations of time-span transformations have a categorical foundation in free monoidal categories generated by groups, enabling analysis of intertimeline exchanges and complex rhythms.

Findings

Categorical framework models time-span transformations.

Symmetric monoidal categories handle intertimeline exchanges.

Complex rhythms are represented using nested time-span brackets.

Abstract

Transformational musical theory has so far mainly focused on the study of groups acting on musical chords, one of the most famous example being the action of the dihedral group D24 on the set of major and minor chords. Comparatively less work has been devoted to the study of transformations of time-spans and rhythms. D. Lewin was the first to study group actions on time-spans by using a subgroup of the affine group in one dimension. In our previous work, the work of Lewin has been included in the more general framework of group extensions, and generalizations to time-spans on multiple timelines have been proposed. The goal of this paper is to show that such generalizations have a categorical background in free monoidal categories generated by a group-as-category. In particular, symmetric monoidal categories allow to deal with the possible interexchanges between timelines. We also show that more general time-spans can be considered, in which single time-spans are encapsulated in a "bracket" of time-spans, which allows for the description of complex rhythms.