Perfect balance

a novel principle for the construction of musical scales and meters

Andrew J. Milne

David Bulger

Steffen A. Herff

William A. Sethares

pp. 97-108

in: Tom Collins, David Mérédith, Anja Volk (eds), Mathematics and computation in music, Berlin, Springer, 2015

Abstract

We identify a class of periodic patterns in musical scales or meters that are perfectly balanced. Such patterns have elements that are distributed around the periodic circle such that their "centre of gravity" is precisely at the circle's centre. Perfect balance is implied by the well established concept of perfect evenness (e.g., equal step scales or isochronous meters). However, we identify a less trivial class of perfectly balanced patterns that have no repetitions within the period. Such patterns can be distinctly uneven. We explore some heuristics for generating and parameterizing these patterns. We also introduce a theorem that any perfectly balanced pattern in a discrete universe can be expressed as a combination of regular polygons. We hope this framework may be useful for understanding our perception and production of aesthetically interesting and novel (microtonal) scales and meters, and help to disambiguate between balance and evenness; two properties that are easily confused.