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 $\mathbf{\text{C}}\to \mathbf{\text{Sets}}$ where $\mathbf{\text{C}}$ 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–Steinbach and Cohn, transformational music theory has also relied on the use of relations between sets of the musical elements. Thus, K-net formalizations should be extended further to take this aspect into account. The present article 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 $\mathbf{\text{Rel}}$ rather than $\mathbf{\text{Sets}}$. We illustrate the potential of relational PK-nets with selected examples, by analyzing pop music and revisiting the work of Douthett–Steinbach and Cohn.

在强调音乐对象之间可能转换的变换音乐理论领域，Klumpenhouwer网络（K-net）构成了一个有用的框架，在群体理论和图论中都有联系。最近尝试以最一般的形式对K-net进行形式化证明了与类别理论的更深层次的联系。这些形式化使用集合中的图，即函子$\mathbf{\text{C}}\to \mathbf{\text{套装}}$ 哪里 $\mathbf{\text{C}}$通常是一个很小的类别，为音乐对象上的已知组或类半体动作提供了一个通用框架。但是，继Douthett–Steinbach和Cohn的工作之后，变换音乐理论也依赖于音乐元素集之间关系的使用。因此，应该进一步扩展K-net形式化以考虑到这一方面。本文提出了一个称为关系PK- net的新框架，该框架是我们先前关于多Klumpenhouwer网络（PK-nets）的工作的扩展，其中我们考虑了图$\mathbf{\text{相对}}$ 而不是 $\mathbf{\text{套装}}$。我们通过分析流行音乐并重新审视Douthett-Steinbach和Cohn的著作，来举例说明相关PK网络的潜力。