Rank-3 scale theorems: Difference between revisions
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# from g2 (even) g1 g3 g1 (even) g2, get a3 = (n/2-1) g1 + (n/2-1) g2 + g3 | # from g2 (even) g1 g3 g1 (even) g2, get a3 = (n/2-1) g1 + (n/2-1) g2 + g3 | ||
# from g1 (odd) g1 g3 g1 (odd) g1, get a4 = n/2 g1 + (n/2-2) g2 + g3. | # from g1 (odd) g1 g3 g1 (odd) g1, get a4 = n/2 g1 + (n/2-2) g2 + g3. | ||
Choose a tuning where g0 is different enough from g3 + g1 (the imperfect gen of the mos generated by g0), and where g1 and g2 are both very close to but not exactly 1/2*g0. We have 4 distinct sizes for n/2-steps, a contradiction to MV3: | Choose a tuning where g0 is different enough from g3 + g1 (the imperfect gen of the mos generated by g0), and where g1 and g2 are both very close to but not exactly 1/2*g0. We have 4 distinct sizes for n/2-steps, a contradiction to unconditional-MV3: | ||
(1) a1, a2 and a3 are clearly distinct. | (1) a1, a2 and a3 are clearly distinct. | ||
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if a step is an odd number of generators (since the scale size is odd, we can always ensure this by taking octave complements of all the generators). The first two sizes must occur the same number of times. | if a step is an odd number of generators (since the scale size is odd, we can always ensure this by taking octave complements of all the generators). The first two sizes must occur the same number of times. | ||
This proof shows that AG and MV3 scales must have odd size. | This proof shows that AG and unconditionally-MV3 scales must have odd size. | ||
==== 3-DE implies MV3 (WIP) ==== | ==== 3-DE implies MV3 (WIP) ==== | ||