Generator-offset property: Difference between revisions

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* is generator-offset (chain lengths off by 1)

* has odd length ''n''

* the generator is ~~always~~ subtended ~~by a ''k''-step for a fixed ''k''~~.

* the generator is constantly subtended.

Then the scale is SGA.

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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.

==== 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' = p'' + offset (not on the first chain) we'll have floor(''n''/2) notes.

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.

@@ Line 65: / Line 65: @@
 * is generator-offset (chain lengths off by 1)
 * has odd length ''n''
-* the generator is always subtended by a ''k''-step for a fixed ''k''.
+* the generator is constantly subtended.
 Then the scale is SGA.
@@ Line 71: / Line 71: @@
 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 ====
-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' = p'' + offset (not on the first chain) we'll have floor(''n''/2) notes.
+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.