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Journal Article

#### Voicing transformations of triads

##### External Resource

https://doi.org/10.1137/16M1110054

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##### Fulltext (public)

arXiv:1603.09636.pdf

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##### Supplementary Material (public)

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##### Citation

Fiore, T. M., & Noll, T. (2018). Voicing transformations of triads.*
SIAM Journal on Applied Algebra and Geometry,* *2*(2), 281-313. doi:10.1137/16M1110054.

Cite as: https://hdl.handle.net/21.11116/0000-0004-470C-0

##### Abstract

Motivated by analytical methods in mathematical music theory, we determine the structure of the subgroup $\mathcal{J}$ of $GL(3,\mathbb{Z}_{12})$ generated by the three voicing reflections. We determine the centralizer of $\mathcal{J}$ in both $GL(3,\mathbb{Z}_{12})$ and the monoid ${Aff}(3,\mathbb{Z}_{12})$ of affine transformations, and recover a Lewinian duality for trichords containing a generator of $\mathbb{Z}_{12}$. We present a variety of musical examples, including 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 setwise stabilizer $\mathcal{H}$ in $\Sigma_3 \ltimes \mathcal{J}$ of root position triads. 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. Some of the proofs are located in the Supplementary Material file, so that this main article can focus on the applications.