Transformations of Triads and Seventh Chords: Group Extensions and Duality

TL;DR

This paper develops a mathematical framework using group actions and extensions to analyze transformations of triads and seventh chords in music, extending Lewin's transformational theory with new algebraic structures.

Contribution

It introduces a method to construct simply transitive group actions on combined sets of chords using equivariant bijections and extends Lewin dual pairs through short exact sequences and central extensions.

Findings

Constructed a simply transitive group action on combined chord sets.

Extended Lewin dual pairs via algebraic structures like short exact sequences.

Applied the framework to jazz and classical music examples.

Abstract

Transformational music theory, pioneered by David Lewin, uses simply transitive group actions to analyze music. In this paper, we construct a simply transitive group action on a disjoint union of two sets, built from a simply transitive action on each set and an equivariant bijection connecting them. Motivational examples are the omnibus progression and the reflected omnibus progression, which involve the consonant triads and the dominant/half-diminished seventh chords, connected by the inclusion bijection. We provide other examples from Jazz tunes. More generally, we combine multiple simply transitive group actions via a "meta-rotation"; examples include a simply transitive group acting on consonant triads and a variety of seventh chords, as well as a meta-rotation that realizes the root position seventh chord sequence of the flattening transformation (described by Clough-Douthett's J-function). The constructions in our theorems extend Lewin dual pairs to Lewin dual pairs. We formulate the constructions in terms of short exact sequences and central extensions as well. Contextual groups are also elucidated: the interval content of the generating pitch-class segment determines whether or not a generalized contextual group is generated by contextual inversions.