Detecting Symmetries of All Cardinalities With Application to Musical 12-Tone Rows

Popularized by Arnold Schoenberg in the mid-20th century, the method of twelve-tone composition produces musical compositions based on one or more orderings of the equal-tempered chromatic scale. The work of twelve-tone composers is famously challenging to traditional Western tonal and structural sensibilities; even so, group theoretic approaches have determined that 10% of certain composers' works contain a highly unusual classical symmetry of music. We extend this result by revealing many symmetries that were previously undetected in the works of Schoenberg, Webern, and Berg. Our approach is computational rather than group theoretic, scanning each composition for symmetries of many different cardinalities. Thus, we capture partial symmetries that would be overlooked by more formal means. Moreover, our methods are applicable beyond the narrow scope of twelve-tone composition. We achieve our results by first extending the group-theoretic notion of symmetry to encompass shorter motives that may be repeated and reprised in a given composition, and then comparing the incidence of these symmetries between the work of composers and the space of all possible 12-tone rows. We present four candidate hierarchies of symmetry and show that in each model, between 75% and 95% of actual compositions contained high levels of internal symmetry.