From K-Nets to PK-Nets: A Categorical Approach

@article{Popoff2016FromKT,
  title={From K-Nets to PK-Nets: A Categorical Approach},
  author={Alexandre Popoff and Carlos Ag{\'o}n and Moreno Andreatta and Andr{\'e}e C. Ehresmann},
  journal={Perspectives of New Music},
  year={2016},
  volume={54},
  pages={5-63}
}
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9 Citations
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9 Citations

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Relational poly-Klumpenhouwer networks for transformational and voice-leading analysis
TLDR
The present article proposes a new framework called relational PK-nets, an extension of the previous work on poly-Klumpenhouwer networks (PK-nets), in which diagrams in sets are considered rather than functors in sets.
From Dürer's Magic Square to Klumpenhouwer Tesseracts: On Melencolia (2013) by Philippe Manoury
Meter networks: a categorical framework for metrical analysis
TLDR
A framework based on category theory which unifies the simultaneous consideration of timepoints, metrical relations, and meter inclusion founded on the category Rel of sets and binary relations is developed.
On the use of relational presheaves in transformational music theory
  • A. Popoff
  • Mathematics
    Journal of Mathematics and Music
  • 2020
Traditional transformational music theory describes transformations between musical elements as functions between sets and studies their subsequent algebraic properties and their use for music…
Groupoids and Wreath Products of Musical Transformations: A Categorical Approach from poly-Klumpenhouwer Networks
TLDR
This work proposes a new groupoid-based approach to transformational music theory, in which transformations of PK-nets are considered rather than ordinary sets of musical objects.
Tropical Generalized Interval Systems
TLDR
The impossibility to build tropical GIS in the finite case is finally proven and discussed and a new framework allows to broaden the GIS model introducing a new operation and consequently new musical and conceptual insights and applications.
From Music to Mathematics and Backwards: Introducing Algebra, Topology and Category Theory into Computational Musicology
Despite a long historical relationship between mathematics and music, the interest of mathematicians is a recent phenomenon. In contrast to statistical methods and signal-based approaches currently…
Opycleid: A Python package for transformational music theory
TLDR
Transformational music theory has progressively shifted the music-theoretical and analytical process from an “object-oriented” point of view to one where the transformations between musical elements are emphasized.
Relational PK-Nets for Transformational Music Analysis
TLDR
This work proposes a new framework called relational PK-Nets, an extension of their previous work on Poly-Klumpenhouwer networks, in which they consider diagrams in $\mathbf{Rel}$ rather than $\Mathbf{Sets}$.

References

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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…
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.
Towards a categorical theory of creativity
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