Beyond the Geometry of Music

People have been modeling music geometrically for more than 300 years. Early in the twenty first century it became clear that many existing models were orbifolds, quotients of R^n (known to musicians as "n-note ordered pitch space") by various symmetry groups (e.g. the "OPTI" symmetries containing octave shifts, permutations, translations and reflections). Recently, however, it has become clear that some closely related musical models are not orbifolds, either because the parent space is inherently discrete or because the symmetry group does not act properly discontinuously. Music theorists thus need to be able to work with a wider range of quotients. My talk will survey some of the fascinating interactions between geometry and music, and will describe a new and more comprehensive framework using groupoids.