Hidden Categories: a New Perspective on Lewin's Generalized Interval Systems and Klumpenhouwer Networks

Abstract

In this work we provide a categorical formalization of several constructions found in transformational music theory. We first revisit David Lewin's original theoretical construction of Generalized Interval Systems (GIS) to show that it implicitly defines categories. When all the conditions in Lewin's definition are fullfilled, such categories coincide with the category of elements $\int_\mathbf{G} S$ for the group action $S \colon \mathbf{G} \to \mathbf{Sets}$ implied by the GIS structure. By focusing on the role played by categories of elements in such a context, we reformulate previous definitions of transformational networks in a $\mathbf{Cat}$-based diagrammatical perspective, and present a new definition of transformational networks (called CT-Nets) in general musical categories. We show incidently how such an approach provides a bridge between algebraic, geometrical and graph-theoretical approaches in transformational music analysis. We end with a discussion on the new perspectives opened by such a formalization of transformational theory, in particular with respect to $\mathbf{Rel}$-based transformational networks which occur in well-known music-theoretical constructions such as Douthett's and Steinbach's Cube Dance.