Recursive structure of MOS scales: Difference between revisions

Line 404:

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 (corresponding to the chunk of L's of size r+1) and σ (corresponding the chunk of size r), and assume that W' is not a mos. Then for some k, W' must have k-steps of the following sizes:

# p₁ λ's and q₁ σ's, represented by subword W₁(λ, σ) in W'~~, corresponding to an interval in the mos with (p₁(r + 1) + q₁r) L's and k s's~~

# p₁ λ's and q₁ σ's, represented by subword W₁(λ, σ) in W'.

# p₂ λ's and q₂ σ's, represented by subword W₂(λ, σ) in W'~~, corresponding to an interval in the mos with (p₂(r + 1) + q₂r) L's and k s's~~. By slinking W₂ to the right until it begins in λ, which will never decrease the number of λ's, we can assume W₂ begins in λ.

# p₂ λ's and q₂ σ's, represented by subword W₂(λ, σ) in W'. By slinking W₂ to the right until it begins in λ, which will never decrease the number of λ's, we can assume W₂ begins in λ.

Here, pᵢ + qᵢ = k and we assume p₂ - p₁ ≥ 2.

@@ Line 404: / Line 404: @@
 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 (corresponding to the chunk of L's of size r+1) and σ (corresponding the chunk of size r), and assume that W' is not a mos. Then for some k, W' must have k-steps of the following sizes:
-# p₁ λ's and q₁ σ's, represented by subword W₁(λ, σ) in W', corresponding to an interval in the mos with (p₁(r + 1) + q₁r) L's and k s's
+# p₁ λ's and q₁ σ's, represented by subword W₁(λ, σ) in W'.
-# p₂ λ's and q₂ σ's, represented by subword W₂(λ, σ) in W', corresponding to an interval in the mos with (p₂(r + 1) + q₂r) L's and k s's. By slinking W₂ to the right until it begins in λ, which will never decrease the number of λ's, we can assume W₂ begins in λ.
+# p₂ λ's and q₂ σ's, represented by subword W₂(λ, σ) in W'. By slinking W₂ to the right until it begins in λ, which will never decrease the number of λ's, we can assume W₂ begins in λ.
 Here, pᵢ + qᵢ = k and we assume p₂ - p₁ ≥ 2.