Commuting groups and the topos of triads. (English) Zbl 1335.00121

Fiore, Thomas M.; Noll, Thomas

Agon, Carlos (ed.) et al., Mathematics and computation in music. Third international conference, MCM 2011, Paris, France, June 15–17, 2011. Proceedings. Berlin: Springer (ISBN 978-3-642-21589-6/pbk). Lecture Notes in Computer Science 6726. Lecture Notes in Artificial Intelligence, 69-83 (2011).

Summary: 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 (Cayley’s 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 Cohn’s hexatonic systems, as well as the octatonic systems. We then enumerate all \(\mathbb Z_{12}\)-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.
For the entire collection see [Zbl 1216.00007].

MSC:

00A65	Mathematics and music
18B25	Topoi

Keywords:

PLR-group; duality; sub dual groups; hexatonic systems; octatonic; topos; topos of triads; triadic monoid