Morphisms of Generalized Interval Systems and PR-Groups

TL;DR

This paper develops a categorical framework for generalized interval systems (GIS), introducing morphisms to connect different systems and extend existing theorems, with applications in music analysis and theoretical foundations.

Contribution

It introduces a new categorical perspective on GIS by defining morphisms, expanding the applicability of existing theorems, and providing concrete musical and theoretical examples.

Findings

Extended the Sub Dual Group Theorem using GIS morphisms

Analyzed Schoenberg's String Quartet with the new framework

Laid groundwork for transformational study of Lawvere--Tierney upgrades

Abstract

We begin the development of a categorical perspective on the theory of generalized interval systems (GIS's). Morphisms of GIS's 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--Noll (2011) and the generalized contextual group of Fiore--Satyendra (2005) 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 (2005).