Forgetful Functors: Music, Monoids, Homotopy, and so on

John Rahn (University of Washington, USA)

Music analysis – and theory – depend on abstraction from the sensuous/cognitive experience of music, abstraction which clearly has to “forget” properties of the experience – music analysis is a kind of forgetful functor from category of music to the category of the model. For example, Oren Kolman has shown, in a brilliant paper, that any Lewin-Generalized Interval System (GIS) can be re-written as a group, so that all of group theory applies to any GIS, and draws far-reaching consequences from this. However, in group theory, isomorphic groups are treated as structurally identical, that is, equivalent. But in music, two group-isomorphic structures can be hugely different, as in a construal of the pitch structure according to circle-of-fourths-order vs chromatic-order syntaxes (the unit step in one is five steps in the other). So representing music using a GIS amounts to a forgetful functor from music to group.

David Lewin has, some twenty years ago, also modelled music as “transformational networks”, which are digraphs with arrows labelled consistently in a monoid; sometimes the vertices are labelled as well in a set of objects acted on by the arrows. These networks are clearly equivalent to the commutative diagrams of category theory. More recently, Guerino Mazzola has elaborated models of music in Grothendieck Topologies, using as his basic construct the category of contravariant set-valued hom-functors over modules, so as to obtain a category (set) that has the sub-object classifier (characteristic function) required for a Grothendieck Topology. In either case, Lewin networks or Mazzola compositions in Grothendieck Topologies, one promising avenue of exploration is to investigate the homotopy classes. But as always in music theory, the forgetful generalization has to be kept firmly in mind alongside what it forgets. (This could be construed as a special case of Hegelian dialectic!) For a model of music, both the class of all paths from one place to another, and each individual such path as colored by its membership in the homotopy class, are essential to understanding.

John Rahn is Professor of Music Composition and Music Theory at the
University of Washington. His compositions have been widely performed and
broadcast in North and South America and in Europe, from Argentina to
Romania. He finished a chamber opera called The New Mother in late 2000,
and Greek Bones for two trombones in 2005. During 2003-2004, Rahn taught composition and theory at ESMUC in Barcelona, where one of his students won the international Guinjoan Composition Prize. He was actively involved in the formation of the Society for Music Theory. Rahn served as Editor of Perspectives of New Music from 1983 to 1994, and again from 2000 to the present.His publications include Basic Atonal Theory (MacMillan), Perspectives on Musical Aesthetics (Norton), Music Inside Out (2001, and many articles in journals and proceedings in about nine countries.