User:Inthar/MV3: Difference between revisions
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Claim: A MV3 scale always has some subset (made by combining some of the two-step intervals into one) that is also MV3. | Claim: A MV3 scale always has some subset (made by combining some of the two-step intervals into one) that is also MV3. | ||
=== 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 === | ||
Suppose we have three sizes of a class, T1, T2, T3 of Y's and Z's. | Suppose we have three sizes of a class, T1, T2, T3 of Y's and Z's. | ||
If the occurrence of T1 has an X inserted in the middle, then we can scoot it left or right until there are no X's in the middle | If the occurrence of T1 has an X inserted in the middle, then we can scoot it left or right until there are no X's in the middle | ||
then we (can assume we) have one of T1+X, T2+X, or T3+X. Scoot this 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. | then we (can assume we) have one of T1+X, T2+X, or T3+X. Scoot this 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. | ||
=== Sizes of chunks of any fixed letter form a MOS === | === Sizes of chunks of any fixed letter form a MOS === | ||
WOLOG consider chunks of X. Use Q for both Y and Z. | WOLOG consider chunks of X. Use Q for both Y and Z. | ||