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.

最近的文章[Milne，Andrew J.，David Bulger和Steffan A. Herff。2017年。“探索完美平衡的节奏和音阶空间。” 数学与音乐学报11（2）：101--133]提出了平衡的概念，并将其与均匀性进行了对比，均匀性可能与步长间隔的模式相关，即最大程度的偶数集或格式良好的标度。在这方面，定义为“完美平衡”的量表是最佳的。在本文中，我们提出，从认知和音乐的角度来看，显示圆形回文“丰富度”并且还表现出相对少的阶跃“差异”的完美平衡音阶可能被证明是有利的。对于上下文，在12-TET中识别出5、6和7音符集中的所有圆形富集。最后，检查了上面Milne，Bulger和Herff（2017）中确定的30、42和70的最小最小平衡集，以了解此处提出的两个约束。