User:Inthar/MV3: Difference between revisions
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=== 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 === | ||
Suppose | Suppose that some class C in the word of Ys and Zs has three sizes, T1, T2, T3. | ||
Assume T1, T2, T3 occur within a contiguous string of Y's and Z's. Then you get a 4th variant of this class (within the whole scale) by using a string of the same length including an X. | Assume T1, T2, T3 occur within a contiguous string of Y's and Z's. Then you get a 4th variant of this class (within the whole scale) by using a string of the same length including an X. | ||
So it's pretty obvious that you have to have MV2 within a contiguous string, but what about the whole string minus the X's? | So it's pretty obvious that you have to have MV2 within a contiguous string, but what about the whole string minus the X's? | ||
This part needs to be more careful, taking into account where X can appear: | |||
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...) | ||