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.