Generator-offset property: Difference between revisions

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# The two alternants differ by replacing one Y with a Z.

# ''S'' is pairwise-mos. That is, the following operations result in a mos: setting L = M, setting L = s, and setting M = s.

# ~~If S = aX bY bZ, then setting either Y = 0 or Z = 0 results in a mos. Since the result of setting X = 0 is the mos bY bZ,~~ ''S'' is elimination-mos.

# ''S'' is elimination-mos. That is, "tempering out" any one step size results in a mos.

In particular, odd GO scales always satisfy these properties (see Proposition 2 below).

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and use the vectors (-1, 2) and (ceil(n/2), 1) as the Fokker block chromas. A Fokker block has the property that tempering out by each of the chromas gives two mosses. These correspond to the L = M and M = s temperings. The L = s case follows by symmetry.

For (7), consider the mos aX 2bW 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 kth note of an n-note mos where k divides n yields a mos.

For (7), consider the mos aX 2bW 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 kth note of an n-note mos where k divides n yields a mos. Since the result of setting X = 0 is the mos bY bZ, ''S'' is elimination-mos.

=== Proposition 2 (Odd GO scales are SGA) ===

@@ Line 35: / Line 35: @@
 # The two alternants differ by replacing one Y with a Z.
 # ''S'' is pairwise-mos. That is, the following operations result in a mos: setting L = M, setting L = s, and setting M = s.
-# If S = aX bY bZ, then setting either Y = 0 or Z = 0 results in a mos. Since the result of setting X = 0 is the mos bY bZ, ''S'' is elimination-mos.
+# ''S'' is elimination-mos. That is, "tempering out" any one step size results in a mos.
 In particular, odd GO scales always satisfy these properties (see Proposition 2 below).
@@ Line 89: / Line 89: @@
 and use the vectors (-1, 2) and (ceil(n/2), 1) as the Fokker block chromas. A Fokker block has the property that tempering out by each of the chromas gives two mosses. These correspond to the L = M and M = s temperings. The L = s case follows by symmetry.
-For (7), consider the mos aX 2bW 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 kth note of an n-note mos where k divides n yields a mos.
+For (7), consider the mos aX 2bW 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 kth note of an n-note mos where k divides n yields a mos. Since the result of setting X = 0 is the mos bY bZ, ''S'' is elimination-mos.
 === Proposition 2 (Odd GO scales are SGA) ===