On the use of relational presheaves in transformational music theory

Traditional transformational music theory describes transformations between musical elements as functions between sets and studies their subsequent algebraic properties and their use for music analysis. This is formalized from a categorical point of view by the use of functors where is a category, often a group or a monoid. At the same time, binary relations have also been used in mathematical music theory to describe relations between musical elements, one of the most compelling examples being Douthett's and Steinbach's parsimonious relations on pitch-class sets. Such relations are often used in a geometrical setting, for example through the use of so-called parsimonious graphs to describe how musical elements relate to each other. This article examines a generalization of transformational approaches based on functors , called relational presheaves, which focuses on the algebraic properties of binary relations defined over sets of musical elements. While binary relations include the particular case of functions, they provide additional flexibility as they also describe partial functions and allow the definition of multiple images for a given musical element. Our motivation to expand the toolbox of transformational music theory is illustrated in this paper by practical examples of monoids and categories generated by parsimonious and common-tone cross-type relations. At the same time, we describe the interplay between the algebraic properties of such objects and the geometrical properties of graph-based approaches.