Axiomatic scale theory. Issue 3 (1st September 2020)

• Record Type: Journal Article
• Authors: Harasim, Daniel; Schmidt, Stefan E.; Rohrmeier, Martin
• Is Part Of: Journal of mathematics and music. Volume 14:Issue 3(2020)
• Page Start: 223
• Page End: 244
• Publication Date: 2020-09-01
• Subjects: scales -- mathematical scale theory -- axiomatic modeling -- cyclic ordered sets -- generalized interval systems -- maximally even sets -- well-formed scales; 08C10 -- 06F99; Music theory -- Periodicals; Music -- Mathematical models -- Periodicals; Music -- Mathematics -- Periodicals; 781.2
• DOI: 10.1080/17459737.2019.1696899
• Languages: English
• ISSNs: 1745-9737

Abstract

Abstract : Scales are a fundamental concept of musical practice around the world. They commonly exhibit symmetry properties that are formally studied using cyclic groups in the field of mathematical scale theory. This paper proposes an axiomatic framework for mathematical scale theory, embeds previous research, and presents the theory of maximally even scales and well-formed scales in a uniform and compact manner. All theorems and lemmata are completely proven in a modern and consistent notation. In particular, new simplified proofs of existing theorems such as the equivalence of non-degenerate well-formedness and Myhill's property are presented. This model of musical scales explicitly formalizes and utilizes the cyclic order relation of pitch classes.