Generator-offset property: Difference between revisions
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==== 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. | ||
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. | ||