  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.
