Periodic scales and meters typically embody “organizational principles” – their pitches and onset times are not randomly distributed, but structured by rules or constraints. Identifying such principles is useful for understanding existing music and for generating novel music. In this article, we identify and discuss a novel organizational principle for scales and rhythms that we feel is of both theoretical interest and practical utility: perfect balance. When distributed around the circle, perfectly balanced rhythms and scales have their “centre of gravity” at the centre of the circle. The present article serves as a repository of the theorems and definitions crucial to perfect balance. It also further explores its mathematical ramifications by linking the existing theorems to algebraic number theory and computational optimizations. In the Online Supplement, http://www.dynamictonality.com/perfect_balance_files/, we provide audio samples of perfectly balanced rhythmic loops and microtonal scales, computational routines, and video demonstrations of some of the concepts.

周期刻度和仪表通常体现“组织原则”-它们的音调和开始时间不是随机分布的，而是由规则或约束构成的。确定这些原理对于理解现有音乐和产生新颖音乐很有用。在本文中，我们确定并讨论了一种新颖的音阶和节奏组织原理，我们认为这既具有理论意义又具有实用价值：完美的平衡。当围绕圆分布时，完美平衡的节奏和音阶的“重心”位于圆的中心。本文充当了对完美平衡至关重要的定理和定义的存储库。它还通过将现有定理与代数数论和计算优化联系起来，进一步探讨了其数学分支。在《在线增刊》（http://www.dynamictonality.com/perfect_balance_files/）中，我们提供了具有完美平衡的节奏循环和微音阶的音频样本，计算例程以及一些概念的视频演示。