User:Inthar/MV3: Difference between revisions
→Lemma 2: Sizes of chunks of any fixed letter form a MOS (except XXYZY): xyzzy is funnier |
→MV3 Theorem 1: use X, Y, Z for step sizes |
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== MV3 Theorem 1== | == MV3 Theorem 1== | ||
''With only one exception, at least two of the three steps must occur the same number of times. For example, it is possible to have a max-variety-3 scale with 3 small steps, 5 medium steps, and 3 large steps, because there are the same number of small steps as large steps. But a max-variety-3 scale with 3 small steps, 5 medium steps, and 4 large steps is impossible. (The | ''Suppose we have an MV3 scale word with steps X, Y and Z. With only one exception, at least two of the three steps must occur the same number of times. For example, it is possible to have a max-variety-3 scale with 3 small steps, 5 medium steps, and 3 large steps, because there are the same number of small steps as large steps. But a max-variety-3 scale with 3 small steps, 5 medium steps, and 4 large steps is impossible. (The two exception to this rule are "XYZYX", and "XYXZXYX", along with their repetitions "XYXZXYXXYXZXYX", etc.) Because of this, there always exists some "generator" interval for any max-variety-3 scale (other than the one exception) such that the scale can be expressed as two parallel chains of this generator which are almost equal in length (the lengths are either equal, or differ by 1).'' | ||
=== Lemma 1: The word made by any two of the step sizes is a MOS === | === Lemma 1: The word made by any two of the step sizes is a MOS === | ||
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If the occurrence of any class has an X inserted in the middle, then we can scoot it left or right until we have one of T1(possibly with inserted X's)+X, T2(possibly with inserted X's)+X, or T3 (possibly with inserted X's)+X. Scoot the string with the least X's to the left and you lose the X on the right, and gain another non-X letter on the left so you get a fourth variant of this interval class that contains T1 + X, a contradiction. (Check this again...) | If the occurrence of any class has an X inserted in the middle, then we can scoot it left or right until we have one of T1(possibly with inserted X's)+X, T2(possibly with inserted X's)+X, or T3 (possibly with inserted X's)+X. Scoot the string with the least X's to the left and you lose the X on the right, and gain another non-X letter on the left so you get a fourth variant of this interval class that contains T1 + X, a contradiction. (Check this again...) | ||
=== Lemma 2: Sizes of chunks of any fixed letter form a MOS (except in the case " | === Lemma 2: Sizes of chunks of any fixed letter form a MOS (except in the case "XYZYX") === | ||
WOLOG consider chunks of X. Use Q for both Y and Z. | WOLOG consider chunks of X. Use Q for both Y and Z. | ||