Relational PK-Nets for Transformational Music Analysis

Abstract

In the field of transformational music theory, which emphasizes the possible transformations between musical objects, Klumpenhouwer networks (K-Nets) constitute a useful framework with connections in both group theory and graph theory. Recent attempts at formalizing K-Nets in their most general form have evidenced a deeper connection with category theory. These formalizations use diagrams in sets, i.e. functors $\mathbf{C} \to \mathbf{Sets}$ where $\mathbf{C}$ is often a small category, providing a general framework for the known group or monoid actions on musical objects. However, following the work of Douthett and Cohn, transformational music theory has also relied on the use of relations between sets of the musical elements. Thus, K-Net formalizations have to be further extended to take this aspect into account. This work proposes a new framework called relational PK-Nets, an extension of our previous work on Poly-Klumpenhouwer networks (PK-Nets), in which we consider diagrams in $\mathbf{Rel}$ rather than $\mathbf{Sets}$. We illustrate the potential of relational PK-Nets with selected examples, by analyzing pop music and revisiting the work of Douthett and Cohn.