Building Generalized Neo-Riemannian Groups of Musical Transformations as Extensions

TL;DR

This paper develops a mathematical framework for generalized musical transformations using group extensions, unifying traditional groups like dihedral groups and enabling new models for rhythm and time-span transformations.

Contribution

It introduces a method to construct transformation groups as extensions of base and shape groups, broadening the scope of musical transformation models.

Findings

Traditional groups like dihedral groups are special cases of the proposed framework.

Explicit construction of group actions on musical objects with internal symmetries.

Application of group extensions to model rhythms and time-spans.

Abstract

Chords in musical harmony can be viewed as objects having shapes (major/minor/etc.) attached to base sets (pitch class sets). The base set and the shape set are usually given the structure of a group, more particularly a cyclic group. In a more general setting, any object could be defined by its position on a base set and by its internal shape or state. The goal of this paper is to determine the structure of simply transitive groups of transformations acting on such sets of objects with internal symmetries. In the main proposition, we state that, under simple axioms, these groups can be built as group extensions of the group associated to the base set by the group associated to the shape set, or the other way. By doing so, interesting groups of transformations are obtained, including the traditional ones such as the dihedral groups. The knowledge of the group structure and product allows to explicitly build group actions on the objects. In particular we differentiate between left and right group actions and we show how they are related to non-contextual and contextual transformations. Finally we show how group extensions can be used to build transformational models of time-spans and rhythms.