Generator-offset property: Difference between revisions
m →Proof: important statement |
m →Proof: expand out expressions using g_3 to show more clearly that a_i are distinct |
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[Note: This is not true with SGA replaced with GO; [[blackdye]] is a counterexample that is MV4.] | [Note: This is not true with SGA replaced with GO; [[blackdye]] is a counterexample that is MV4.] | ||
==== Proof ==== | ==== Proof ==== | ||
Let e be the equave of ''S''. | |||
Assuming SGA, we have two chains of generator g<sub>0</sub> (going right). The two cases are: | Assuming SGA, we have two chains of generator g<sub>0</sub> (going right). The two cases are: | ||
CASE 1: EVEN LENGTH | CASE 1: EVEN LENGTH | ||
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Label the notes (1, ''j'') and (2, ''j''), 1 ≤ ''j'' ≤ (number of notes in the chain), for notes in the upper and lower chain respectively. | Label the notes (1, ''j'') and (2, ''j''), 1 ≤ ''j'' ≤ (number of notes in the chain), for notes in the upper and lower chain respectively. | ||
In case 1, let g<sub>1</sub> = (2, 1) − (1, 1) | In case 1, let g<sub>1</sub> = (2, 1) − (1, 1), g<sub>2</sub> = (1, 2) − (2, 1), and g<sub>3</sub> = (1, 1) − (''n''/2, 2) = (−''n''/2*g<sub>1</sub> − g<sub>1</sub> − ''n''/2*g<sub>2</sub>) mod e. We have the chain g<sub>1</sub> g<sub>2</sub> g<sub>1</sub> g<sub>2</sub> ... g<sub>1</sub> g<sub>3</sub> which visits every note in ''S''. | ||
Let ''r'' be odd and ''r'' ≥ 3. Consider the following abstract sizes for the interval class (''k''-steps) reached by stacking ''r'' generators: | Let ''r'' be odd and ''r'' ≥ 3. Consider the following abstract sizes for the interval class (''k''-steps) reached by stacking ''r'' generators: | ||
# from g<sub>1</sub> ... g<sub>1</sub>, we get a<sub>1</sub> = (''r'' − 1)/2*g<sub>0</sub> + g<sub>1</sub> = (''r'' + 1 | # from g<sub>1</sub> ... g<sub>1</sub>, we get a<sub>1</sub> = (''r'' − 1)/2*g<sub>0</sub> + g<sub>1</sub> = (''r'' + 1/2) g<sub>1</sub> + (''r'' − 1/2) g<sub>2</sub> | ||
# from g<sub>2</sub> ... g<sub>2</sub>, we get a<sub>2</sub> = (''r'' − 1)/2*g<sub>0</sub> + g<sub>2</sub> = (''r'' − 1 | # from g<sub>2</sub> ... g<sub>2</sub>, we get a<sub>2</sub> = (''r'' − 1)/2*g<sub>0</sub> + g<sub>2</sub> = (''r'' − 1/2) g<sub>1</sub> + (''r'' + 1/2) g<sub>2</sub> | ||
# from g<sub>2</sub> (...even # of gens...) g<sub>1</sub> g<sub>3</sub> g<sub>1</sub> (...even # of gens...) g<sub>2</sub>, we get a<sub>3</sub> = (''r'' − 1)/2 g<sub>1</sub> + (''r'' − 1)/2 g<sub>2</sub> + g<sub>3</sub> | # from g<sub>2</sub> (...even # of gens...) g<sub>1</sub> g<sub>3</sub> g<sub>1</sub> (...even # of gens...) g<sub>2</sub>, we get a<sub>3</sub> = (''r'' − 1)/2 g<sub>1</sub> + (''r'' − 1)/2 g<sub>2</sub> + g<sub>3</sub> ≡ (''r'' − ''n''/2 − 3/2)g<sub>1</sub> + (''r'' − ''n''/2 − 1/2)g<sub>2</sub> mod e. | ||
# 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'' − 3)/2 g<sub>2</sub> + g<sub>3</sub>. | # 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'' − 3)/2 g<sub>2</sub> + g<sub>3</sub> ≡ (''r'' − ''n''/2 − 1/2)g<sub>1</sub> + (''r'' − ''n''/2 − 3/2)g<sub>2</sub> mod e. | ||
Choose a tuning where g<sub>1</sub> and g<sub>2</sub> are both very close to but not exactly 1/2*g<sub>0</sub>, resulting in a scale very close to the mos generated by 1/2*g<sub>0</sub> | Choose a tuning where g<sub>1</sub> and g<sub>2</sub> are both very close to but not exactly 1/2*g<sub>0</sub>, resulting in a scale very close to the mos generated by 1/2*g<sub>0</sub> (i.e. g<sub>1</sub> and g<sub>2</sub> differ from 1/2*g<sub>0</sub> by ε, a quantity much smaller than the chroma of the ''n''/2-note mos generated by g<sub>0</sub>, which is |g<sub>3</sub> − g<sub>2</sub>|). Thus we have 4 distinct sizes for ''k''-steps: | ||
# a<sub>1</sub>, a<sub>2</sub> and a<sub>3</sub> are clearly distinct. | # a<sub>1</sub>, a<sub>2</sub> and a<sub>3</sub> are clearly distinct. | ||
# a<sub>4</sub> − a<sub>3</sub> = g<sub>1</sub> − g<sub>2</sub> ≠ 0, since the scale is a non-degenerate GO scale. | # a<sub>4</sub> − a<sub>3</sub> = g<sub>1</sub> − g<sub>2</sub> ≠ 0, since the scale is a non-degenerate GO scale. | ||
# a<sub>4</sub> − a<sub>1</sub> = g<sub>3</sub> − g<sub>2</sub> = (g<sub>3</sub> + g<sub>1</sub>) − (g<sub>2</sub> + g<sub>1</sub>) ≠ 0. This is exactly the chroma of the mos generated by g<sub>0</sub>. | # a<sub>4</sub> − a<sub>1</sub> = g<sub>3</sub> − g<sub>2</sub> = (g<sub>3</sub> + g<sub>1</sub>) − (g<sub>2</sub> + g<sub>1</sub>) ≠ 0. This is exactly the chroma of the mos generated by g<sub>0</sub>. | ||
# a<sub>4</sub> − a<sub>2</sub> = g<sub>1</sub> − | # a<sub>4</sub> − a<sub>2</sub> = g<sub>1</sub> − 2g<sub>2</sub> + g<sub>3</sub> = (g<sub>3</sub> − g<sub>2</sub>) + (g<sub>1</sub> − g<sub>2</sub>) = (chroma ± ε) ≠ 0 by choice of tuning. | ||
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, unconditionally SV3, GO scale must be of the form xy...xyxz. But this pattern is not abstractly SV3 if ''n'' ≥ 6, since 3-steps come in 4 sizes: xyx, yxy, yxz and xzx. Thus ''n'' = 4 and the scale is xyxz. (Note that xyxz is not SV3, since it has only two kinds of 2-steps, xy and xz.) This proves (1). | 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, unconditionally SV3, GO scale must be of the form xy...xyxz. But this pattern is not abstractly SV3 if ''n'' ≥ 6, since 3-steps come in 4 sizes: xyx, yxy, yxz and xzx. Thus ''n'' = 4 and the scale is xyxz. (Note that xyxz is not SV3, since it has only two kinds of 2-steps, xy and xz.) This proves (1). | ||