Morphisms of generalized interval systems and PR-groups

作者：Fiore Thomas M^*; Noll Thomas; Satyendra Ramon

来源：Journal of Mathematics and Music, 2013, 7(1): 3-27.

DOI：10.1080/17459737.2013.785724

• neo-Riemannian group
• PLR-group
• dual groups
• generalized interval systems
• morphism
• category
• cover
• Schoenberg
• 00A65
• 20B35
• 18B25

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 LawvereTierney 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. 103135].

• 出版日期2013-3-1