Groupoids and Wreath Products of Musical Transformations: a Categorical Approach from poly-Klumpenhouwer Networks

TL;DR

This paper introduces a categorical approach to transformational music theory using groupoids and wreath products, extending Klumpenhouwer networks to a more flexible and algebraically rich framework for analyzing post-tonal music.

Contribution

It develops a novel groupoid-based framework for transformational music analysis, connecting poly-Klumpenhouwer networks with wreath products via groupoid bisections.

Findings

Constructed groupoids of musical transformations.

Applied the framework to analyze Berg's Four pieces for clarinet and piano.

Linked groupoids to wreath products through bisections.

Abstract

Transformational music theory, pioneered by the work of Lewin, shifts the music-theoretical and analytical focus from the "object-oriented" musical content to an operational musical process, in which transformations between musical elements are emphasized. In the original framework of Lewin, the set of transformations often form a group, with a corresponding group action on a given set of musical objects. Klumpenhouwer networks have been introduced based on this framework: they are informally labelled graphs, the labels of the vertices being pitch classes, and the labels of the arrows being transformations that maps the corresponding pitch classes. Klumpenhouwer networks have been recently formalized and generalized in a categorical setting, called poly-Klumpenhouwer networks. This work proposes a new groupoid-based approach to transformational music theory, in which transformations of PK-nets are considered rather than ordinary sets of musical objects. We show how groupoids of musical transformations can be constructed, and an application of their use in post-tonal music analysis with Berg's Four pieces for clarinet and piano, Op. 5/2. In a second part, we show how groupoids are linked to wreath products (which feature prominently in transformational music analysis) through the notion of groupoid bisections