Rank-3 scale theorems: Difference between revisions
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# a4 - a1 = g3 - g2 = (g3 + g1) - (g2 + g1) != 0. This is exactly the chroma of the mos generated by g0. | # a4 - a1 = g3 - g2 = (g3 + g1) - (g2 + g1) != 0. This is exactly the chroma of the mos generated by g0. | ||
# a4 - a2 = g1 - 2 g2 + g3 = (g3 - g2) + (g1 - g2) = (chroma ± ε) != 0 by choice of tuning. | # a4 - a2 = g1 - 2 g2 + g3 = (g3 - g2) + (g1 - g2) = (chroma ± ε) != 0 by choice of tuning. | ||
(For n = 4, the above argument doesn't work because a3 = a4, and xyxz is a counterexample.) | |||
In case 2, let (2,1)-(1,1) = g1, (1,2)-(2,1) = g2 be the two alternating generators. Let g3 be the leftover generator after stacking alternating g1 and g2. Then the generator circle looks like g1 g2 g1 g2 ... g1 g2 g3. Then the generators corresponding to a step are: | In case 2, let (2,1)-(1,1) = g1, (1,2)-(2,1) = g2 be the two alternating generators. Let g3 be the leftover generator after stacking alternating g1 and g2. Then the generator circle looks like g1 g2 g1 g2 ... g1 g2 g3. Then the generators corresponding to a step are: | ||