Hidden Categories: A New Perspective on Lewin’s Generalized Interval Systems and Klumpenhouwer Networks

In this work we provide a categorical formalization of several constructions found in transformational music theory. We first revisit David Lewin’s construction of a Generalized Interval System (GIS) to show that even a subset of the GIS conditions already implies a sequence of functors between categories. When all the conditions in Lewin’s definition are fullfilled, this sequence involves the category of elements \(\int _\textbf{G} S\) for the group action \(S :\textbf{G} \rightarrow \textbf{Sets}\) 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 \(\textbf{Cat}\) -based diagrammatic perspective, and present a new definition of categorical transformational networks, or CT-Nets, in more general musical categories. We show 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 \(\textbf{Rel}\) -based transformational networks which occur in well-known music-theoretical constructions such as Douthett’s and Steinbach’s Cube Dance.