Thomas M. Fiore

Musical Actions of Dihedral Groups

The sequence of pitches which form a musical melody can be transposed or inverted. Since the 1970s, music theorists have modeled musical transposition and inversion in terms of an action of the dihedral group of order 24. More recently music theorists have found an intriguing second way that the dihedral group of order 24 acts on the set of major and minor chords. We illustrate both geometrically and algebraically how these two actions are {\it dual}. Both actions and their duality have been used to analyze works of music as diverse as Hindemith and the Beatles.

Commuting groups and the topos of triads

The goal of this article is to clarify the relationship between the topos of triads and the neo-Riemannian PLR-group. To do this, we first develop some theory of generalized interval systems: 1) we prove the well known fact that every pair of dual groups is isomorphic to the left and right regular representations of some group (Cayleys Theorem), 2) given a simply transitive group action, we show how to construct the dual group, and 3) given two dual groups, we show how to easily construct sub dual groups. Examples of this construction of sub dual groups include Cohns hexatonic systems, as well as the octatonic systems. We then enumerate all Z12-subsets which are invariant under the triadic monoid and admit a simply transitive PLR-subgroup action on their maximal triadic covers. As a corollary, we realize all four hexatonic systems and all three octatonic systems as Lawvere-Tierney upgrades of consonant triads.

Model structures on the category of small double categories

In this paper we obtain several model structures on DblCat, the category of small double categories. Our model structures have three sources. We first transfer across a categorification-nerve adjunction. Secondly, we view double categories as internal categories in Cat and take as our weak equivalences various internal equivalences defined via Grothendieck topologies. Thirdly, DblCat inherits a model structure as a category of algebras over a 2‐monad. Some of these model structures coincide and the different points of view give us further results about cofibrant replacements and cofibrant objects. As part of this program we give explicit descriptions for and discuss properties of free double categories, quotient double categories, colimits of double categories, horizontal nerve and horizontal categorification. 18D05, 18G55; 55P99, 55U10

A Thomason model structure on the category of small n-fold categories

We construct a cofibrantly generated Quillen model structure on the category of small n-fold categories and prove that it is Quillen equivalent to the standard model structure on the category of simplicial sets. An n-fold functor is a weak equivalence if and only if the diagonal of its n-fold nerve is a weak equivalence of simplicial sets. This is an n-fold analogue to Thomasons Quillen model structure on Cat. We introduce an n-fold Grothendieck construction for multisimplicial sets, and prove that it is a homotopy inverse to the n-fold nerve. As a consequence, we completely prove that the unit and counit of the adjunction between simplicial sets and n-fold categories are natural weak equivalences.

Morphisms of generalized interval systems and PR-groups

We begin the development of a categorical perspective on the theory of generalized interval systems (GISs). Morphisms of GISs allow the analyst to move between multiple interval systems and connect transformational networks. We expand the analytical reach of the Sub Dual Group Theorem of Fiore and Noll [Commuting groups and the topos of triads. In: Agon C, Amiot E, Andreatta M, Assayag G, Bresson J, Mandereau J, editors. Proceedings of the 3rd International Conference Mathematics and Computation in Music - MCM 2011. Lecture Notes in Computer Science, Springer; 2011] and the generalized contextual group of Fiore and Satyendra [Generalized contextual groups. Music Theory Online. 2005;11] by combining them with a theory of GIS morphisms. Concrete examples include an analysis of Schoenberg, String Quartet in D minor, op. 7, and simply transitive covers of the octatonic set. This work also lays the foundation for a transformational study of Lawvere–Tierney upgrades in the topos of triads of Noll [The topos of triads. In: Colloquium on mathematical music theory. Graz: Karl-Franzens-Univ. Graz; 2005. p. 103–135].

Waldhausen Additivity: classical and quasicategorical

We use a simplicial product version of Quillen’s Theorem A to prove classical Waldhausen Additivity of

A thomason model structure on the category of small n-fold categories

wS_\bullet

Hexatonic Systems and Dual Groups in Mathematical Music Theory

wS∙, which says that the “subobject” and “quotient” functors of cofiber sequences induce a weak equivalence