Generator-offset property: Difference between revisions

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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 two of the temperings X = Y, Y = Z and X = Z. The third tempering follows by symmetry (by taking the other chirality).

===== Statement (6) =====

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 ''E''<sub>X</sub>(''S'') 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:

# The chunk sizes of X form a mos, and taking every ''k''th note of an ''n''-note mos yields a mos.

# The sum of sizes of consecutive chunks of X (1st chunk with 2nd chunk, 3rd with 4th, ...) must form a mos.

# If chunk sizes of a 2-step-size scale form a mos, the scale itself must be a mos; see [[Recursive structure of MOS scales]].

Since ''E''<sub>X</sub>(''S'') is the mos ''b''Y ''b''Z, ''S'' is elimination-mos.

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

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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 two of the temperings X = Y, Y = Z and X = Z. The third tempering follows by symmetry (by taking the other chirality).
 ===== Statement (6) =====
-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 ''E''<sub>X</sub>(''S'') 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:
+# The chunk sizes of X form a mos, and taking every ''k''th note of an ''n''-note mos yields a mos.
+# The sum of sizes of consecutive chunks of X (1st chunk with 2nd chunk, 3rd with 4th, ...) must form a mos.
+# If chunk sizes of a 2-step-size scale form a mos, the scale itself must be a mos; see [[Recursive structure of MOS scales]].
+Since ''E''<sub>X</sub>(''S'') is the mos ''b''Y ''b''Z, ''S'' is elimination-mos.
 === Proposition 2 (Odd GO scales are SGA) ===