一个音阶的发生器数量
About the number of generators of a musical scale
摘要：
几个音阶，如大调音阶，可以被描述为有限的等差数列模八度音阶，即循环群中的等差数列的块。因此，在一个循环群中有多少不同的等差数列将给出相同的支撑集的问题。我们证明了这个数总是一个totient数，并描述了不同的可能情况。特别是，存在具有任意大量不同生成元的比例尺，但没有具有14个生成元的比例尺。给出了一些相关的结果和推广，例如关于离散傅里叶变换的特征刻画，以及关于环R/Z上的有限或无限等差数列的结果。
英文摘要：
Several musical scales, like the major scale, can be described as finite arithmetic sequences modulo octave, i.e. chunks of an arithmetic sequence in a cyclic group. Hence the question of how many different arithmetic sequences in a cyclic group will give the same support set. We prove that this number is always a totient number and characterize the different possible cases. In particular, there exists scales with an arbitrarily large number of different generators, but none with 14 generators. Some connex results and extensions are also given, for instance on characterization via a Discrete Fourier Transform, and about finite or infinite arithmetic sequences in the torus R/Z.
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