Relational poly-Klumpenhouwer networks for transformational and voice-leading analysis

@article{Popoff2018RelationalPN,
  title={Relational poly-Klumpenhouwer networks for transformational and voice-leading analysis},
  author={Alexandre Popoff and Moreno Andreatta and Andr{\'e}e C. Ehresmann},
  journal={Journal of Mathematics and Music},
  year={2018},
  volume={12},
  pages={35 - 55}
}
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 where is often a small category, providing a general framework for the known group… 
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References

SHOWING 1-10 OF 19 REFERENCES
Thoughts on Klumpenhouwer Networks and Mathematical Models: The Synergy of Sets and Graphs
TLDR
This essay responds to one of the four central issues in Buchler's thoughtful article: K-nets and their association with dual transformation.
A Categorical Generalization of Klumpenhouwer Networks
TLDR
This article proposes a functorial framework for generalizing some constructions of transformational theory via the concept of set-valued poly-K-nets (henceforth PK-nets), focusing on Klumpenhouwer Networks.
Klumpenhouwer Networks and Some Isographies that Involve Them
Networks involving T and I operations are useful for interpreting pcsets, and for other purposes. Certain groups of isographies among such networks, being isomorphic to the T/I group itself, are
Voice-Leading Parsimony in the Music of Alexander Scriabin
Discussions of pitch structure in Scriabin's later oeuvre typically take as their point of departure a description of the mystic chord, a member of set class 6-34. Dahlhaus (1987) describes this
The Inner and Outer Automorphisms of Pitch-Class Inversion and Transposition: Some Implications for Analysis with Klumpenhouwer Networks
All Klumpenhouwer network analysis is carried out using the outer automorphisms defined by Lewin to relate the pitch-class transformations that label corresponding arrows on isographic networks.1 It
The Topos of Triads
The article studies the topos Sets of actions of an 8-element monoid T on sets. It is called the triadic topos as T is isomorphic to the monoid of affine transformations of the twelve tone system
Commuting Groups and the Topos of Triads
TLDR
The goal of this article is to clarify the relationship between the topos of triads and the neo-Riemannian PLR-group by developing some theory of generalized interval systems and enumerating all Z12-subsets which are invariant under the triadic monoid.
Parsimonious Graphs: A Study in Parsimony, Contextual Transformations, and Modes of Limited Transposition
Connections between parsimonious structures and modes of limited transposition from three set classes are explored. A graph-theoretic approach proves useful in illustrating the symmetries inherent in
Audacious Euphony: Chromaticism and the Triad's Second Nature
Table of Contents 1. Mapping the Triadic Universe Three Methods for Calculating Triadic Distance Triads in Chromatic Space Remarks on Syntax and Maps 2. Hexatonic Cycles: A First Preliminary Model of
From a Categorical Point of View: K-nets as Limit Denotators
TLDR
The interpretation of Klumpenhouwer Networks is presented as a special limit construction in the framework of the topos-theoretic denotator architecture developed in [19], offering a vast generalization of the network concept and the formally closed recursive construction of iterated networks.
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