Rank-3 scale theorems: Difference between revisions
→AG + unconditionally MV3 implies "ax by bz" and that the scale has odd size: blackdye is not ag Tags: Mobile edit Mobile web edit |
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# 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 g1 and g2 are both very close to but not exactly 1/2*g0 (i.e. they differ from 1/2*g0 by ε, a quantity much smaller than the chroma of the | Choose a tuning where g1 and g2 are both very close to but not exactly 1/2*g0 (i.e. they differ from 1/2*g0 by ε, a quantity much smaller than the chroma of the n/2-note mos generated by g0, which is |g3 - g2|). We have 4 distinct sizes for n/2-steps, a contradiction to unconditional-MV3: | ||
# a1, a2 and a3 are clearly distinct. | # a1, a2 and a3 are clearly distinct. | ||
# a4 - a3 = g1 - g2 != 0, since the scale is a non-trivial AG. | # a4 - a3 = g1 - g2 != 0, since the scale is a non-trivial AG. | ||