Recursive structure of MOS scales: Difference between revisions
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(Assume that the MOS has length ''n''; the notation ''w''(X, Y) for a word ''w''(L, s) in L, s means w with step sizes X substituted for L and Y substituted for s.) | (Assume that the MOS has length ''n''; the notation ''w''(X, Y) for a word ''w''(L, s) in L, s means w with step sizes X substituted for L and Y substituted for s.) | ||
Suppose w(L, s) had three chunks L...s with r < r+a < r+b '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 that differ by 1. Repeating this argument for sequences of ''k'' chunks shows that any k-step in the reduced word must come in at most 2 sizes, showing that the reduced word is indeed MOS. | Suppose w(L, s) had three chunks L...s with r < r+a < r+b '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 that differ by 1. Repeating this argument for sequences of ''k'' > 1 chunks shows that any ''k''-step in the reduced word must come in at most 2 sizes, showing that the reduced word is indeed MOS. | ||
=== Preservation of generators === | === Preservation of generators === | ||