Recursive structure of MOS scales: Difference between revisions
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=== Reduction preserves the MOS property === | === Reduction preserves the MOS property === | ||
Suppose w(L, s) had three chunks L...s with r, r+1 and r+2 'L's. Then we have a length r+2 subword that's only 'L's, one that has one s at the end and one that has two 's's on either side, which means that the original scale was not MOS. Therefore the reduced word has two step sizes. The number of chunks is b, and gcd(a%b, (b-a%b)) = gcd(a%b, b) = gcd(a,b) by the Euclidean algorithm. | Suppose w(L, s) had three chunks L...s with r, r+1 and r+2 'L's. Then we have a length r+2 subword that's only 'L's, one that has one s at the end and one that has two 's's on either side, which means that the original scale was not MOS. Therefore the reduced word has two step sizes. The number of chunks is b, and gcd(a%b, (b-a%b)) = gcd(a%b, b) = gcd(a,b) by the Euclidean algorithm. Moreover, we have seen that the reduced word is generated. Thus the previous lemma shows that this reduced scale must be a MOS. | ||
=== Uniqueness and existence of the generator=== | === Uniqueness and existence of the generator=== | ||