Recursive structure of MOS scales: Difference between revisions

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# 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 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.

Here, pᵢ + qᵢ = k and p₂ - p₁ ≥ 2.

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

Let K = p₁(r + 1) + q₁r + k. Consider the following sizes for (K+1)-steps:

# w₁(L, s) = the word sW₁(Lr+1s, Lrs) [W₁ interpreted as a subword of the original mos], with (k + 1) s's

# w₂(L, s) = the first K+1 letters of W₂(Lr+1s, Lrs)

# w₂(L, s) = the first K+1 letters of W₂(Lr+1s, Lrs) (which must be longer than w₁(L, s), since W₂ had more λ's than W₁)

# w₃(L, s) = the last K+1 letters of W₂(Lr+1s, Lrs)

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Case 1

If w₂ and w₃ (which have the same length) contain the same number of complete chunks~~, one case is~~ (X denotes a chunk boundary, < > are chunk boundaries that are also the boundary of the word, [] are non-chunk-boundary word boundaries.)

Suppose w₂ and w₃ (which have the same length) contain the same number of complete chunks. (Below X denotes a chunk boundary, < > are chunk boundaries that are also the boundary of the word, [] are non-chunk-boundary word boundaries.)

Case 1.1:

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 # 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 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.
-Here, pᵢ + qᵢ = k and p₂ - p₁ ≥ 2.
+Here, pᵢ + qᵢ = k and we assume p₂ - p₁ ≥ 2.
 Let K = p₁(r + 1) + q₁r + k. Consider the following sizes for (K+1)-steps:
 # w₁(L, s) = the word sW₁(L<sup>r+1</sup>s, L<sup>r</sup>s) [W₁ interpreted as a subword of the original mos], with (k + 1) s's
-# w₂(L, s) = the first K+1 letters of W₂(L<sup>r+1</sup>s, L<sup>r</sup>s)
+# w₂(L, s) = the first K+1 letters of W₂(L<sup>r+1</sup>s, L<sup>r</sup>s) (which must be longer than w₁(L, s), since W₂ had more λ's than W₁)
 # w₃(L, s) = the last K+1 letters of W₂(L<sup>r+1</sup>s, L<sup>r</sup>s)
@@ Line 418: / Line 418: @@
 Case 1
-If w₂ and w₃ (which have the same length) contain the same number of complete chunks, one case is (X denotes a chunk boundary, < > are chunk boundaries that are also the boundary of the word, [] are non-chunk-boundary word boundaries.)
+Suppose w₂ and w₃ (which have the same length) contain the same number of complete chunks. (Below X denotes a chunk boundary, < > are chunk boundaries that are also the boundary of the word, [] are non-chunk-boundary word boundaries.)
 Case 1.1: