Recursive structure of MOS scales: Difference between revisions

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Proving that the chunk sizes form a mos is equivalent to proving that the places where the s's are in the mos form a mos in n-edo (which must be [[maximally even]] by above).

For now assume r ≥ 1. Let W' be the reduced word with step sizes σ = r\n and λ = (r + 1)\n, and assume that the s's does not form a mos. Then for some k, W' must have k-steps of the following sizes:

For now assume r ≥ 1. Let W'(λ, σ) be the reduced word with step sizes σ = r\n and λ = (r + 1)\n, and assume that the s's does not form 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 ~~the reduced word~~, 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', corresponding to an interval in the mos with (p₁(r + 1) + q₁r) L's and k s's

# p₂ λ's and q₂ σ's, represented by subword W₂(λ, σ) in ~~the reduced word~~, corresponding to an interval in the mos with (p₂(r + 1) + q₂r) L's and k s's. By slinking the word to the right if necessary we assume it begins in λ.

# 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 the word to the right if necessary we assume it begins in λ.

where pᵢ + qᵢ = k and p₂ - p₁ ≥ 2

@@ Line 346: / Line 346: @@
 Proving that the chunk sizes form a mos is equivalent to proving that the places where the s's are in the mos form a mos in n-edo (which must be [[maximally even]] by above).
-For now assume r ≥ 1. Let W' be the reduced word with step sizes σ = r\n and λ = (r + 1)\n, and assume that the s's does not form a mos. Then for some k, W' must have k-steps of the following sizes:
+For now assume r ≥ 1. Let W'(λ, σ) be the reduced word with step sizes σ = r\n and λ = (r + 1)\n, and assume that the s's does not form 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 the reduced word, 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', corresponding to an interval in the mos with (p₁(r + 1) + q₁r) L's and k s's
-# p₂ λ's and q₂ σ's, represented by subword W₂(λ, σ) in the reduced word, corresponding to an interval in the mos with (p₂(r + 1) + q₂r) L's and k s's. By slinking the word to the right if necessary we assume it begins in λ.
+# 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 the word to the right if necessary we assume it begins in λ.
 where pᵢ + qᵢ = k and p₂ - p₁ ≥ 2