Ternary scale theorems: Difference between revisions

Line 58:

# from g1 (...odd # of gens...) g1 g3 g1 (...odd # of gens...) g1, we get a4 = (''r'' + 1)/2 g1 + (''r'' − 3)/2 g2 + g3 ≡ (''r'' − ''n''/2 − 1/2)g1 + (''r'' − ''n''/2 − 3/2)g2 mod e.

These are all distinct by '''Z'''-linear independence. By applying this argument to 1-steps, we see that there must be 4 step sizes in some tuning, a contradiction. Thus g1 and g2 must themselves be step sizes. Thus we see that an even-length, abstractly SV3, generator-offset scale must be of the form (xy)''r''xz. (Note that (xy)''r''xz is not SV3, since it has only two kinds of 2-steps, xy and xz.) This proves (1).

These are all distinct by '''Z'''-linear independence; hence there are at least 4 sizes for ''k''-steps. A 1-step must be reached by stacking an odd number of generators, thus by applying this argument to 1-steps, we see that there must be at least 4 step sizes in some tuning, a contradiction. Thus g1 and g2 must themselves be step sizes. Thus we see that an even-length, abstractly SV3, generator-offset scale must be of the form (xy)''r''xz. (Note that (xy)''r''xz is not SV3, since it has only two kinds of 2-steps, xy and xz.) This proves (1).

==== Statement (2) ====

@@ Line 58: / Line 58: @@
 # from g<sub>1</sub> (...odd # of gens...) g<sub>1</sub> g<sub>3</sub> g<sub>1</sub> (...odd # of gens...) g<sub>1</sub>, we get a<sub>4</sub> = (''r'' + 1)/2 g<sub>1</sub> + (''r'' &minus; 3)/2 g<sub>2</sub> + g<sub>3</sub> ≡ (''r'' &minus; ''n''/2 &minus; 1/2)g<sub>1</sub> + (''r'' &minus; ''n''/2 &minus; 3/2)g<sub>2</sub> mod e.
-These are all distinct by '''Z'''-linear independence. By applying this argument to 1-steps, we see that there must be 4 step sizes in some tuning, a contradiction. Thus g<sub>1</sub> and g<sub>2</sub> must themselves be step sizes. Thus we see that an even-length, abstractly SV3, generator-offset scale must be of the form (xy)<sup>''r''</sup>xz. (Note that (xy)<sup>''r''</sup>xz is not SV3, since it has only two kinds of 2-steps, xy and xz.) This proves (1).
+These are all distinct by '''Z'''-linear independence; hence there are at least 4 sizes for ''k''-steps. A 1-step must be reached by stacking an odd number of generators, thus by applying this argument to 1-steps, we see that there must be at least 4 step sizes in some tuning, a contradiction. Thus g<sub>1</sub> and g<sub>2</sub> must themselves be step sizes. Thus we see that an even-length, abstractly SV3, generator-offset scale must be of the form (xy)<sup>''r''</sup>xz. (Note that (xy)<sup>''r''</sup>xz is not SV3, since it has only two kinds of 2-steps, xy and xz.) This proves (1).
 ==== Statement (2) ====