Recursive structure of MOS scales: Difference between revisions
| Line 409: | Line 409: | ||
Here, pᵢ + qᵢ = k and we assume 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: | Let K = p₁(r + 1) + q₁r + k. Consider the following sizes for (K+1)-steps in w: | ||
# 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. w₁ is K+1 letters long. | # 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. w₁ is K+1 letters long. | ||
# 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) | # 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) | ||