Well-formedness in two dimensions: a generalization of Carey and Clampitt's theorem

Marek Žabka

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Musicology and Musical Analysis Diverse Music Education Insights Music Technology and Sound Studies

Abstract

The famous results of Carey and Clampitt [N. Carey and D. Clampitt, Aspects of well-formed scales, Music Theory Spectr. 11 (1989), pp. 187–206] focus on scales generated by one interval and explain why some of these scales are preferable to others. Those preferable are called well-formed (WF). Their explanation is based on a theorem showing equivalence between ‘symmetry’ and ‘closure’. In this paper, we propose and prove a generalization of this theorem. Instead of scales with a single generator, tone systems generated by two intervals are considered. In addition, various examples are given to illustrate the developed theoretical framework. Among them, the ancient Indian 22-śruti system is interpreted as a WF two-dimensional tone system generated by the fifth and the śruti. Finally, we draft open problems pertaining to the presented theory.

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References

1988Generalized Musical Intervals and Transformations 1997Neo-Riemannian Operations, Parsimonious Trichords, and Their "Tonnetz" Representations 1998Introduction to Neo-Riemannian Theory: A Survey and a Historical Perspective 1989Aspects of Well-Formed Scales 1985Variety and Multiplicity in Diatonic Systems 2007Generalized Musical Intervals and Transformations 2002The Topos of Music 2003The topos of music: geometric logic of concepts, theory, and performance

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