Perfect balance and circularly rich words

Abstract

A recent article [Milne, Andrew J., David Bulger, and Steffan A. Herff. 2017. “Exploring the Space of Perfectly Balanced Rhythms and Scales.” Journal of Mathematics and Music 11 (2):101--133] posits the notion of balance and contrasts it with evenness, which may be associated with the patterns of step intervals in either maximally even sets or well-formed scales. Scales defined as “perfectly balanced” are optimal in this regard. In this paper, we propose that perfectly balanced scales that display circular palindromic “richness” and also exhibit relatively few step “differences” may prove to be advantageous from a cognitive and musical perspective. For context, all circularly rich sets among 5-, 6-, and 7-note sets in 12-TET are identified. Finally, the minimal perfectly balanced sets in divisions of 30, 42, and 70 identified in Milne, Bulger, and Herff (2017) above are examined for the two constraints proposed here.

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