Abstract scales are formalized as a cyclic group of classes of projection functions related to iterations of the scale generator. Their representatives in the frequency domain are used to built cyclic sequences of tone iterates satisfying the closure condition. The refinement of cyclic sequences with regard to the best closure provides a constructive algorithm that allows to determine cyclic scales avoiding continued fractions. New proofs of the main properties are obtained as a consequence of the generating procedure. When the scale tones are generated from the two elementary factors associated with the generic widths of the step intervals we get the partition of the octave leading to the fundamental Bézout's identity relating several characteristic scale indices. This relationship is generalized to prove a new relationship expressing the partition that the frequency ratios associated with the two sizes composing the different step-intervals induce to a specific set of octaves.

摘要比例尺被形式化为与比例尺生成器迭代相关的投影函数类的循环组。它们在频域中的代表用于建立满足闭合条件的单调迭代的循环序列。关于最佳闭合的循环序列的细化提供了一种建设性的算法，该算法可以确定循环比例，避免连续分数。作为生成过程的结果，获得了主要特性的新证明。当根据与步长间隔的通用宽度相关的两个基本因子生成音阶音调时，我们得到八度音阶的划分，从而导致了与多个特征音阶索引相关的基本贝索特身份。