Wreaths for Rahn, and Valuable Exchanges

WREATHS FOR RAHN, AND VALUABLE EXCHANGES JASON YUST HE YEARS WHEN I STUDIED with John Rahn, 2001–06, were, in retrospect, an inflection point in mathematical music theory. There was a sort of Cambrian explosion with the breaking down of a geographic barrier (the Atlantic Ocean), leading in short order to the founding of an international society and journal, the Society for Mathematics and Computation in Music and the Journal of Mathematics and Music, whose flagstone was the mission of intercontinental dialogue. At around the same time we lost, far too soon, two of the most important American pioneers in mathematical music theory when both John Clough and David Lewin died in 2003. The daunting task of honoring their legacy was, in its own way, an important impetus that accelerated and guided subsequent developments in the field. I would be putting myself in good company to say that discovering Lewin’s work was a defining moment in my own intellectual development , and it was Rahn’s guidance that led me into this universe of ideas. Early in my studies at University of Washington, he introduced me to the whole idea of groups in music theory; not only Lewin, but other authors, including many gems of mathematical music theory published in the pages of Perspectives. When I studied serialism with T 404 Perspectives of New Music him, he pointed me towards Mead’s (1988) excellent exegesis of the twelve-tone system and an intriguing paper by Stanfield (1984) about the exchange operation (which exchanges the pitch-class numbers and order numbers of a row). Around the same time he brought Michael Leyton to the UW to give a lecture, and was investigating the application of Leyton’s mathematical theories of shape to music theory. Leyton (2001) showed how symmetry might be an essential part of the description of the form of an object, even though the object itself might not be literally symmetrical. His memorable analogy is a smashed soda can on the floor of the subway station: one conceives this shape by imagining some ideal symmetrical shape, a cylinder, and applying some deformations to it. The asymmetry of the smashed can encodes a process by which its shape came into being. Leyton’s basic mathematical tool was a group-theoretic construction called the wreath product. For Rahn’s take on Leyton, see Rahn (2003), 18–25. Rahn introduced me to Leyton as I was learning about another wreath-product group, Julian Hook’s UTT (Uniform Triadic Transformation ) group. I first encountered Hook’s work at a special session of the American Mathematical Society Spring Sectional in Baton Rouge that John generously brought me to in 2003, but most will know it from Hook (2002). The application of this to twelve-tone music, a natural extension, is explored in an outstanding paper by Hook and Douthett (2008). Since then, some excellent work extending these has been published by the Society of Mathematics and Computation in Music (Fiore, Noll, Satyendra 2013a) and the Journal of Mathematics and Music (Fiore, Noll, Satyendra 2013) both of which Rahn was instrumental in helping to get off the ground. (See also Fiore and Noll [2016].) One of my fond memories of graduate school was the “aha!” moment I had in this serialism seminar. While inventing symmetrical tone rows at the little upright piano in the dungeonous theory TA office in the School of Music (according to the lettering on the door it was actually the “Sprinkler Supply Valve Room”), I realized that rotationally symmetrical rows, those that map onto themselves by some combination of rotation of order positions and transposition, could actually be described with a wreath product group! (N.B.: the exclamation point is not for you, dear reader, but for my 25-year-old self.) Here’s how it works: Assume your row has some symmetry such as T4r4, where “r4” means to rotate the order positions ahead by four places. This is satisfied precisely if the four augmented triads, the orbits of T4, are assigned to the orbits of r4, order positions {0, 4, 8}, {1, 5, 9}, etc. The augmented triads can be assigned in any permutation, and they can start from...