Generator-offset property: Difference between revisions
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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: | 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- | # ''a''<sub>4</sub> − ''a''<sub>3</sub> = ''g''<sub>1</sub> − ''g''<sub>2</sub> != 0, since the scale is a non-degenerate alt-gen 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> − 2 ''g''<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. | # ''a''<sub>4</sub> − ''a''<sub>2</sub> = ''g''<sub>1</sub> − 2 ''g''<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. | ||