arXiv · 1009.3065
We describe what might be called the "Hall-fusion" bialgebra constructed from a promonoidal double, and mention the corresponding face version for probicategories.
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Brian Day. 2010-09-15. A Hall-Fusion Bialgebra. https://arxiv.org/abs/1009.3065
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We extend Mazzola's counterpoint model using category theory, generalizing from the category $\mathbf{Set}$ to an arbitrary topos other than $\mathbf{Set}$. This generalization suggests that counterpoint's essential structure depends on specific categorical conditions rather than classical set-theoretic reasoning. A key contribution is identifying sufficient requirements for a well-behaved counterpoint theory in a topos: some version of Zorn's Lemma (GJZL), and two-valuedness and split supports (NS). Within a topos, we introduce (weak) quasidichotomies alongside the classical notion of dichotomy. These structures capture varying degrees of oppositional structure between consonance and dissonance, with weak quasidichotomies preserving the non-Boolean flexibility essential to musical practice while quasidichotomies represent maximal opposition short of complete partition. We prove a generalized counterpoint theorem giving sufficient conditions for the existence of admitted successors. When the ambient topos turns non-zero successor objects into points, admitted succession can be iterated to form counterpoint paths, which may terminate at consonances with no admitted successor. The framework naturally accommodates counterpoint with sets instead of pure pitches, relaxing the ``yes/no'' character of classical consonance definitions and emphasizing context-dependence. Mazzola's model allows a Kuratowski closure operator induced by a polarity, which defines an internal topology enabling algebraic-topological analysis of counterpoint structure. We conclude by showing this construction generalizes to involutive morphisms. This categorical approach provides foundations for understanding both the historical evolution of contrapuntal practice and cross-cultural divergences in interval organization.
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