Generator-offset property: Difference between revisions
m →Proposition 3 (Properties of even GO scales): The proof in Theorem 1 did not apply to all even GO scales, just the ones that were SGA. |
m →Proof |
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For (6), consider the mos ''a''X 2''b''W as chunks of X separated by W (tempering Y and Z together into W). Eliminating every other W turns it into a mos, because the sum of sizes of consecutive chunks of X (1st chunk with 2nd chunk, 3rd with 4th, ...) must form a mos. This is because the chunk sizes of X form a mos, and taking every ''k''th note of an ''n''-note mos where ''k'' divides ''n'' yields a mos. Since the result of setting X = 0 is the mos ''b''Y ''b''Z, ''S'' is elimination-mos. | For (6), consider the mos ''a''X 2''b''W as chunks of X separated by W (tempering Y and Z together into W). Eliminating every other W turns it into a mos, because the sum of sizes of consecutive chunks of X (1st chunk with 2nd chunk, 3rd with 4th, ...) must form a mos. This is because the chunk sizes of X form a mos, and taking every ''k''th note of an ''n''-note mos where ''k'' divides ''n'' yields a mos. Since the result of setting X = 0 is the mos ''b''Y ''b''Z, ''S'' is elimination-mos. | ||
=== Proposition 2 (Odd GO scales are SGA) === | === Proposition 2 (Odd GO scales are SGA) === | ||
Suppose that a periodic scale satisfies the following: | Suppose that a periodic scale satisfies the following: | ||