Incorporating Voice Permutations into the Theory of Neo-Riemannian Groups and Lewinian Duality

TL;DR

This paper extends neo-Riemannian theory by incorporating permutations into T/I-PLR-duality, enhancing its applicability and resolving voice-leading issues, with mathematical proofs and musical examples.

Contribution

It constructs the dual group to permutations combined with affine maps, broadening the theoretical framework of neo-Riemannian groups and duality.

Findings

Dual group to permutations and affine maps is an internal direct product.

The Fiore--Noll dual group construction is validated in the finite case.

Clarifies the relationship between permutations and the RICH transformation.

Abstract

A familiar problem in neo-Riemannian theory is that the P, L, and R operations defined as contextual inversions on pitch-class segments do not produce parsimonious voice leading. We incorporate permutations into T/I-PLR-duality to resolve this issue and simultaneously broaden the applicability of this duality. More precisely, we construct the dual group to the permutation group acting on n-tuples with distinct entries, and prove that the dual group to permutations adjoined with a group G of invertible affine maps Z12 -> Z12 is the internal direct product of the dual to permutations and the dual to G. Musical examples include Liszt, R. W. Venezia, S. 201 and Schoenberg, String Quartet Number 1, Opus 7. We also prove that the Fiore--Noll construction of the dual group in the finite case works, and clarify the relationship of permutations with the RICH transformation.