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.

Keywords:

• perfectly balanced scales
• well-formed scales
• Christoffel words
• combinatorics on words
• rich words

2010 Mathematics Subject Classification:

• 11J71
• 11K36
• 68R15 2012 Computing Classification Scheme: applied computing
• sound and music computing

Acknowledgments

I am very grateful for the careful work of the reviewers whose insights have provided important ideas for the development of this paper. Thanks as well to the other authors in this collection who have made important suggestions. Finally, thanks to Thomas Fiore for proposing and editing this special issue on this fascinating topic, and for comments on the galley proofs of this article.

Disclosure statement

No potential conflict of interest was reported by the author.

ORCID

Norman Carey http://orcid.org/0000-0002-6417-0952

Notes

1 See also Milne, Bulger, and Herff (2017, Section 7.2.1). Such studies in the context of the discrete Fourier transform are the focus of, among others, Quinn (2004), Callender (2007), and Amiot (2016).

2 I fell victim to conflating balance and evenness in a conversation with John Clough shortly before the publication of his and Douthett's landmark publication on maximally even sets (Clough and Douthett 1991). I suggested the playground roundabout with children trying to distribute their weight evenly as an analogy for maximal evenness. Clearly, it isn't.

3 In Clough, Engebretsen, and Kochavi (1999), objects such as the French sixth embedded in 12-TET are said to display a kind of “distributional evenness.”

4 See Milne, Bulger, and Herff (2017, Section 2).

5 See also Theorem 1.1 in Clough and Douthett (1991) and the proof of the first part of Theorem 1 in Carey and Clampitt (2012, 62–63).

6 Some authors, following Brlek et al. (2004), describe these words as “full” words instead of “rich.” The latter locution appears to have won out, and we will follow that practice here.

7 In any event, the specific limit of 1/2 is admittedly arbitrary, and thus it need not function as a strict binary cut-off.

8 All such scales can be found as segments of Lambda words and are therefore rich (Carey 2013). In that context it can be shown that such a scale is the product of two palindromes, and all of its conjugates appear in the Lambda word and so it is circularly rich.

9 A stronger statement that in fact characterizes rich words is that a word is rich if the first return of any palindrome is always a palindrome. Letters, as single-element palindromes, would be included in this broader characterization (Bucci et al. 2009b).

10 See Section 4.4.

11 This is the label that Milne, Bulger, and Herff (2017) employ. There are two distinct 7-gons in 30-TET, distinguished as 7a-in-30-gon and 7b-in-30-gon.

12 For example, the three words associated with 9-in-42-gons in Table 4, indexed a, b, and c, are laeealada > kaefeakaa > heaeffeae.