Generator-offset property: Difference between revisions
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Then the scale is SGA. | Then the scale is SGA. | ||
Note: The third condition is required. Without it, this is false for the 5-note scale 0 250 300 700 750 1200 with alternants 700 and 50: the 1-steps form the chain 250 50 400 50 450 and the 2-steps form the chain 300 450 700 250 500. | |||
==== Proof ==== | ==== Proof ==== | ||
Assume that ''k'' is even. (If ''k'' is not even, invert the generator.) On some tonic p we have a chain of ceil(''n''/2) generators and on some other note ''p' '' (not on the first chain) we'll have floor(''n''/2) generators. We have ''k''*ceil(''n''/2) notes on ''p'' and ''k''*floor(''n''/2) notes on ''p' '' = ''p'' + offset. | Assume that ''k'' is even. (If ''k'' is not even, invert the generator.) On some tonic p we have a chain of ceil(''n''/2) generators and on some other note ''p' '' (not on the first chain) we'll have floor(''n''/2) generators. We have ''k''*ceil(''n''/2) notes on ''p'' and ''k''*floor(''n''/2) notes on ''p' '' = ''p'' + offset. | ||