  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.
