Building generalized neo-Riemannian groups of musical transformations as extensions

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 with the base set by the group associated with the shape set, or the other way around. 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 one to explicitly build group actions on the objects. In particular, we differentiate between left and right group actions and 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.