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 $\mathbf{C}\to \mathbf{S}\mathbf{e}\mathbf{t}\mathbf{s}$ where $\mathbf{C}$ 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 $\mathbf{C}\to \mathbf{R}\mathbf{e}\mathbf{l}$, 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.

传统的转换音乐理论将音乐元素之间的转换描述为集合之间的函数，并研究它们随后的代数性质及其在音乐分析中的用途。这是通过使用函子从分类的角度形式化的$\mathbf{C}\to \mathbf{小号}\mathbf{e}\mathbf{吨}\mathbf{s}$在哪里$\mathbf{C}$是一个类别，通常是一个组或一个幺半群。同时，二元关系也被用于数学音乐理论来描述音乐元素之间的关系，最引人注目的例子之一是 Douthett 和 Steinbach 在音高类集合上的简约关系。这种关系通常用于几何设置，例如通过使用所谓的简约图来描述音乐元素如何相互关联。本文研究了基于函子的转换方法的概括$\mathbf{C}\to \mathbf{R}\mathbf{e}\mathbf{l}$，称为关系预滑层，它侧重于在音乐元素集上定义的二元关系的代数性质。虽然二元关系包括函数的特殊情况，但它们提供了额外的灵活性，因为它们还描述了部分函数并允许为给定的音乐元素定义多个图像。本文通过简约和同音交叉类型关系生成的类群和类别的实际示例说明了我们扩展转换音乐理论工具箱的动机。同时，我们描述了此类对象的代数属性与基于图的方法的几何属性之间的相互作用。