Generator-offset property: Difference between revisions

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# ''S'' is of the form ''a''x ''b''y ''b''z for some permutation (x, y, z) of (L, M, s).

# The length of ''S'' is odd, or ''S'' is equivalent to xyxz.

# ''S'' = ''a''X ''b''Y ''b''Z is obtained from some mode of the (single-period) mos ''a''X 2''b''W by replacing all the W's successively with alternating Y's and Z's (or alternating Z's and Y's for the other chirality, fixing the mode of ''a''X 2''b''W).

# If ''n'' is odd, ''S'' = ''a''X ''b''Y ''b''Z is obtained from some mode of the (single-period) mos ''a''X 2''b''W by replacing all the W's successively with alternating Y's and Z's (or alternating Z's and Y's for the other chirality, fixing the mode of ''a''X 2''b''W). The two alternants differ by replacing one Y with a Z.

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

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

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

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Since ''a'' + 2''b'' ≥ 5, there are at least 4 perfect generators, so there must be at least one of each of (a), (b), and (c), giving a contradiction to SV3. [Whenever the root of the (''i'' + ''j'')-step are moved within ''S'', the numbers of Y's and Z's change one at a time and reach a maximum at some choice of the root and a minimum with another choice, guaranteeing that intermediate values are reached. We can use this "intermediate value theorem" argument because (d) occurs at only one note.]

Any generator of ''a''X 2''b''W must have an odd number of W steps (Otherwise, intervals with an odd number of W steps can't be generated.). ~~This with~~ (4~~) immediately gives (5~~).

Any generator of ''a''X 2''b''W must have an odd number of W steps (Otherwise, intervals with an odd number of W steps can't be generated.). We have finished proving (4).

For (6), odd-numbered SGA scales are [[Fokker block]]s (in the 2-dimensional lattice generated by the generator and the offset). To see this, consider the following lattice depiction of such a scale:

For (5), odd-numbered SGA scales are [[Fokker block]]s (in the 2-dimensional lattice generated by the generator and the offset). To see this, consider the following lattice depiction of such a scale:

x x x ... x

x x x ... x x

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).

For (7), 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) ===

@@ Line 33: / Line 33: @@
 # ''S'' is of the form ''a''x ''b''y ''b''z for some permutation (x, y, z) of (L, M, s).
 # The length of ''S'' is odd, or ''S'' is equivalent to xyxz.
-# ''S'' = ''a''X ''b''Y ''b''Z is obtained from some mode of the (single-period) mos ''a''X 2''b''W by replacing all the W's successively with alternating Y's and Z's (or alternating Z's and Y's for the other chirality, fixing the mode of ''a''X 2''b''W).
+# If ''n'' is odd, ''S'' = ''a''X ''b''Y ''b''Z is obtained from some mode of the (single-period) mos ''a''X 2''b''W by replacing all the W's successively with alternating Y's and Z's (or alternating Z's and Y's for the other chirality, fixing the mode of ''a''X 2''b''W). The two alternants differ by replacing one Y with a Z.
-# The two alternants differ by replacing one Y with a Z.
 # ''S'' is pairwise-mos. That is, the following operations each result in a mos: setting L = M, setting L = s, and setting M = s.
 # ''S'' is elimination-mos. That is, "tempering out" any one step size results in a mos.
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 Since ''a'' + 2''b'' ≥ 5, there are at least 4 perfect generators, so there must be at least one of each of (a), (b), and (c), giving a contradiction to SV3. [Whenever the root of the (''i'' + ''j'')-step are moved within ''S'', the numbers of Y's and Z's change one at a time and reach a maximum at some choice of the root and a minimum with another choice, guaranteeing that intermediate values are reached. We can use this "intermediate value theorem" argument because (d) occurs at only one note.]
-Any generator of ''a''X 2''b''W must have an odd number of W steps (Otherwise, intervals with an odd number of W steps can't be generated.). This with (4) immediately gives (5).
+Any generator of ''a''X 2''b''W must have an odd number of W steps (Otherwise, intervals with an odd number of W steps can't be generated.). We have finished proving (4).
-For (6), odd-numbered SGA scales are [[Fokker block]]s (in the 2-dimensional lattice generated by the generator and the offset). To see this, consider the following lattice depiction of such a scale:
+For (5), odd-numbered SGA scales are [[Fokker block]]s (in the 2-dimensional lattice generated by the generator and the offset). To see this, consider the following lattice depiction of such a scale:
   x x x ... x
   x x x ... x x
 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).
-For (7), 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) ===