Generator-offset property: Difference between revisions
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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. | 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) notes and on some other note ''p' '' (not on the first chain) we'll have floor(''n''/2) notes | 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) notes and on some other note ''p' '' (not on the first chain) we'll have floor(''n''/2) notes. | ||
We must have gcd(''k'', ''n'') = 1. If not, since ''n'' is odd, gcd(''k'', ''n'') is an odd number at least 3, and the ''k''-steps must form more than 2 parallel chains. | We must have gcd(''k'', ''n'') = 1. If not, since ''n'' is odd, gcd(''k'', ''n'') is an odd number at least 3, and the ''k''-steps must form more than 2 parallel chains. | ||