Help Privacy Policy Disclaimer
  Advanced SearchBrowse




Journal Article

Voicing transformations of triads


Fiore,  Thomas M.
Max Planck Institute for Mathematics, Max Planck Society;


Noll,  Thomas
Max Planck Institute for Mathematics, Max Planck Society;

External Resource
Fulltext (restricted access)
There are currently no full texts shared for your IP range.
Fulltext (public)

(Preprint), 4MB

Supplementary Material (public)
There is no public supplementary material available

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
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.