Generator-offset property: Difference between revisions

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# from ''g''1 (...odd # of gens...) ''g''1 ''g''3 ''g''1 (...odd # of gens...) ''g''1, we get ''a''4 = (''r'' + 1)/2 ''g''1 + (''r'' − 3)/2 ''g''2 + ''g''3.

Choose a tuning where ''g''1 and ''g''2 are both very close to but not exactly 1/2*''g''0, resulting in a scale very close to the mos generated by 1/2 ''g''0. (i.e. ''g''1 and ''g''2 differ from 1/2*''g''0 by ε, a quantity much smaller than the chroma of the ''n''/2-note mos generated by ''g''0, which is |''g''3 − ''g''2|). ~~Assuming ''n'' > 4,~~ we have 4 distinct sizes for k-steps~~, a contradiction to unconditional-MV3~~:

Choose a tuning where ''g''1 and ''g''2 are both very close to but not exactly 1/2*''g''0, resulting in a scale very close to the mos generated by 1/2 ''g''0. (i.e. ''g''1 and ''g''2 differ from 1/2*''g''0 by ε, a quantity much smaller than the chroma of the ''n''/2-note mos generated by ''g''0, which is |''g''3 − ''g''2|). Thus we have 4 distinct sizes for k-steps:

# ''a''1, ''a''2 and ''a''3 are clearly distinct.

# ''a''4 − ''a''3 = ''g''1 − ''g''2 != 0, since the scale is a non-trivial AG.

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 # 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>.
-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> &minus; ''g''<sub>2</sub>|). Assuming ''n'' > 4, we have 4 distinct sizes for k-steps, a contradiction to unconditional-MV3:
+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> &minus; ''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>4</sub> &minus; ''a''<sub>3</sub> = ''g''<sub>1</sub> &minus; ''g''<sub>2</sub> != 0, since the scale is a non-trivial AG.