Voicing Transformations and a Linear Representation of Uniform Triadic Transformations

TL;DR

This paper explores the mathematical structure of voicing transformations in music, revealing connections to uniform triadic transformations and providing new insights into musical symmetries and dualities.

Contribution

It determines the structure of the subgroup generated by voicing reflections and links it to uniform triadic transformations, offering a novel mathematical framework for musical analysis.

Findings

The subgroup J is characterized within GL(3,Z12).

The stabilizer H is shown to represent Hook's U group.

The retrograde inversion RICH belongs to the stabilizer H.

Abstract

Motivated by analytical methods in mathematical music theory, we determine the structure of the subgroup J of GL(3,Z12) generated by the three voicing reflections. As applications of our Structure Theorem, we determine the structure of the stabilizer H in Sigma3 semi-direct product J of root position triads, and show that H is a representation of Hook's uniform triadic transformations group U. We also determine the centralizer of J in both GL(3,Z12) and the monoid Aff(3,Z12) of affine transformations, and recover a Lewinian duality for trichords containing a generator of Z12}. We present a variety of musical examples, including the Wagner's hexatonic Grail motive and the diatonic falling fifths as cyclic orbits, an elaboration of our earlier work with Satyendra on Schoenberg, String Quartet in D minor, op. 7, and an affine musical map of Joseph Schillinger. Finally, we observe, perhaps unexpectedly, that the retrograde inversion enchaining operation RICH (for arbitrary 3-tuples) belongs to the representation H. This allows a more economical description of a passage in Webern, Concerto for Nine Instruments, op. 24 in terms of a morphism of group actions.