Generator-offset property: Difference between revisions

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Assume that the generator is a ''k''-step and ''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′'' = ''p'' + offset (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 by well-formedness with respect to the generator, the generators must form more than 2 parallel chains.

By modular arithmetic we have ''rk'' mod ''n'' = ''k''/2 iff ''r'' = ceil(''n''/2) mod ''n''. (Since gcd(2, ''n'') = 1, 2 is multiplicatively invertible mod ''n'', and we can multiply both sides by 2 to check this.) This proves that the offset, which must be reached after ceil(''n''/2) generator steps, is a ''k''/2-step, as desired. (If the offset wasn't reached in ceil(''n''/2) steps, the two generator chains either wouldn't be disjoint or wouldn't have the assumed lengths.)

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 Assume that the generator is a ''k''-step and ''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′'' = ''p'' + offset (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 by well-formedness with respect to the generator, the generators must form more than 2 parallel chains.
 By modular arithmetic we have ''rk'' mod ''n'' = ''k''/2 iff ''r'' = ceil(''n''/2) mod ''n''. (Since gcd(2, ''n'') = 1, 2 is multiplicatively invertible mod ''n'', and we can multiply both sides by 2 to check this.) This proves that the offset, which must be reached after ceil(''n''/2) generator steps, is a ''k''/2-step, as desired. (If the offset wasn't reached in ceil(''n''/2) steps, the two generator chains either wouldn't be disjoint or wouldn't have the assumed lengths.)