[Paper Review] Incorporating Voice Permutations into the Theory of Neo-Riemannian Groups and Lewinian Duality

Thomas M. Fiore, Thomas Noll+1 more|arXiv (Cornell University)|Jan 17, 2013

Musicology and Musical Analysis4 references5 citations

TL;DR

This paper resolves a foundational issue in neo-Riemannian theory by incorporating voice permutations into the $T/I$-$PLR$ duality framework, showing that the dual group to the permutation group acting on $n$-tuples with distinct pitch classes is isomorphic to the internal direct product of the duals of permutations and affine maps $\mathbb{Z}_{12}\to\mathbb{Z}_{12}$. The key result is that the $\mathrm{RICH}$ transformation, previously problematic due to fixed points, becomes part of a simply transitive group action when restricted to octatonic subsets, enabling consistent parsimonious voice leading in musical examples like Schoenberg's Op. 7 and Liszt's S. 201.

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.

Motivation & Objective

• To resolve the inconsistency between voice-leading parsimony and the $PLR$ transformations in neo-Riemannian theory, which fail to produce parsimonious voice leading due to fixed points in the $\mathrm{RICH}$ transformation.
• To extend the classical $T/I$-$PLR$ duality to include permutations of voices in pitch-class segments, thereby broadening the scope of transformational music theory.
• To prove that the dual group to the semidirect product of the permutation group and the $T/I$ group is the internal direct product of the duals of each component.
• To clarify the role of the $\mathrm{RICH}$ transformation in the context of voice-leading parsimony and its compatibility with simply transitive group actions on restricted pitch-class sets.

Proposed method

• Construct the dual group to the permutation group acting on $n$-tuples with distinct entries in $\mathbb{Z}_{12}$, using the Fiore–Noll duality framework.
• Prove that the dual group to the semidirect product of the permutation group $\Sigma_n$ and the $T/I$ group is the internal direct product of the dual of $\Sigma_n$ and the dual of $T/I$.
• Define the $\mathrm{RICH}$ transformation as the composition of a voice exchange $(1\;3)$ and a contextual inversion $J^{2,3}$, modeling retrograde inversion enchaining.
• Analyze the cycle decomposition of $\mathrm{RICH}$ on all 144 permutations of major and minor triads to identify fixed points and orbit structures.
• Restrict the action of $\mathrm{RICH}$ to octatonic subsets (e.g., $\{0,2,3,4,6,7,9,10\}$) to eliminate fixed points and achieve simply transitive group actions.
• Use explicit cycle notation and musical examples (Schoenberg Op. 7, Liszt S. 201) to demonstrate that $\mathrm{RICH}$ acts simply transitively on selected subsets, enabling parsimonious voice leading.

Experimental results

Research questions

• RQ1 How can voice permutations be formally incorporated into the $T/I$-$PLR$ duality framework to restore voice-leading parsimony?
• RQ2 Why does the $\mathrm{RICH}$ transformation fail to act simply transitively on the full set of 144 ordered triads, and can this failure be remedied?
• RQ3 What is the group-theoretic structure of the dual group when permutations are adjoined with the $T/I$ group in neo-Riemannian theory?
• RQ4 Can the $\mathrm{RICH}$ transformation be embedded in a simply transitive group action on a subset of pitch-class segments, and if so, under what conditions?
• RQ5 How do the cycle decompositions of $\mathrm{RICH}$ on different triadic subsets (e.g., octatonic scales) relate to musical voice leading and transformational networks?

Key findings

• The dual group to the semidirect product of the permutation group $\Sigma_n$ and the $T/I$ group is isomorphic to the internal direct product of the dual of $\Sigma_n$ and the dual of $T/I$, resolving a foundational issue in duality theory.
• The $\mathrm{RICH}$ transformation, defined as $(1\;3) \circ J^{2,3}$, has fixed points when acting on all 144 permutations of major and minor triads, preventing it from being part of a simply transitive group action.
• When restricted to octatonic subsets such as $\{0,2,3,4,6,7,9,10\}$, the $\mathrm{RICH}$ transformation acts simply transitively, as evidenced by cycle decompositions with no fixed points.
• The $\mathrm{RICH}$ transformation contains cycles of length 24 (like $RL$), 8 (like $PR$), and 6 (like $PL$), and its 6th and 8th powers have fixed points, confirming its non-simply transitive action on the full set.
• Musical examples from Schoenberg’s String Quartet No. 1, Op. 7, and Liszt’s *Venezia e destruzione*, S. 201, demonstrate that $\mathrm{RICH}$ acts simply transitively on octatonic subsets, enabling parsimonious voice leading.
• The $\mathrm{RICH}$ transformation's action on the octatonic scale $\{2,3,5,6,8,9,11,0\}$ includes a cycle of length 8 that matches the cello motive in measures 8–10, confirming its musical relevance.