Symmetrical Hierarchies in Musical 12-Tone Rows - CANCELED

Anil Venkatesh, Adelphi University

A musical 12-tone row is defined as any of the 12! possible orderings of notes in the Western chromatic scale. The musical notes of a 12-tone composition must always arise in the same order, cycling repeatedly through a predetermined “row” of twelve notes.  The repetitive structure of 12-tone music lends itself to mathematical study. In 2003, Hunter and von Hippel investigated symmetry in 12-tone rows, using group theory to enumerate equivalence classes of rows under a group of music-theoretic symmetries. They found that highly symmetric rows constitute just 0.13% of the 12! possibilities, and yet these rows arise in 10% of the actual compositions of Schoenberg and Webern. Their findings provided strong evidence that these composers favored symmetric rows, yet offered no account for 90% of the corpus. In this paper, we introduce a system to measure symmetries on multiple time scales, ranging from the full 12-note symmetries of Hunter and von Hippel down to much shorter repetitions and symmetries that were not detected in their analysis. Our analysis gives rise to a much richer set of symmetry classes for 12-tone rows, organized in a directed acyclic graph from least to most symmetric on all time scales. By examining the graph distributions of actual compositions versus the set of all 12-tone rows, we show that Schoenberg and Webern favored symmetry on many different time scales, choosing compositions with symmetric substructures far more frequently than dictated by the uniform distribution on 12-tone rows.