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

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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 -\textgreater 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.

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### BibTeX entry

@article{Fiore2013, 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 -\textgreater 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.}, author = {Fiore, Thomas M. and Noll, Thomas and Satyendra, Ramon}, month = {jan}, pages = 15, title = {Incorporating Voice Permutations into the Theory of Neo-Riemannian Groups and Lewinian Duality}, url = {http://arxiv.org/abs/1301.4136 http://arxiv.org/pdf/1301.4136v1}, year = 2013, archivePrefix = {arXiv}, eprint = {1301.4136}, primaryClass = {math.GR}, urldate = {2013-01-21}, collections = {Music,The groups group} }