Generalized Inversions and the Construction of Musical Group and Groupoid Actions

TL;DR

This paper extends transformational music theory by developing generalized inversion transformations and constructing musical group and groupoid actions, providing detailed methods for calculating these transformations in various contexts.

Contribution

It introduces a new concept of generalized inversion transformations and details the construction of musical group and groupoid actions, expanding the theoretical framework of transformational music theory.

Findings

Developed methods for calculating musical group actions.

Constructed groupoids of transformations between pitch-class sets.

Introduced generalized inversion transformations for partial transformations.

Abstract

Transformational music theory is a recent field in music theory which studies the possible transformations between musical objects, such as chords. In the framework of the theory initiated by David Lewin, the set of all transformations forms a group which acts simply transitively on the set of musical objects. In particular, neo-Riemannian theory focuses on the action of the T/I group or the PLR group, which are both isomorphic to the dihedral group D24, on the set of major and minor triads. It has been shown recently that generalized neo-Riemannian groups of transformations can be built as group extensions. By generalizing this construction, groupoids of transformations between different pitch-class sets can be obtained. The goal of this paper is to give details concerning the calculation of musical group and groupoid actions, both in the covariant and contravariant case. In doing so, we introduce a concept of generalized inversion transformations for the partial transformation between pitch-class sets.