Recursive structure of MOS scales: Difference between revisions
| Line 402: | Line 402: | ||
(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 | 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. | ||
Without loss of generality assume r ≥ 1 (otherwise flip the roles of L and s). Let W'(λ, σ) be the reduced word with step sizes λ (for the larger chunk) and σ (for the smaller chunk), and assume that W' is not a mos. Then for some k, W' must have k-steps of the following sizes: | Without loss of generality assume r ≥ 1 (otherwise flip the roles of L and s). Let W'(λ, σ) be the reduced word with step sizes λ (for the larger chunk) and σ (for the smaller chunk), and assume that W' is not a mos. Then for some k, W' must have k-steps of the following sizes: | ||