Generator-offset property: Difference between revisions

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# The length of ''S'' is either odd, or 4 (and ''S'' is of the form ''xyxz'').

# S = aX bY bZ is obtained from the (single-period) mos aX 2bW 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).

# 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'' (PMOS~~)~~. That~~ is, ~~the result of identifying any two step sizes of~~ ''S'' ~~is always a mos.~~

# If len(''S'') is odd, ''S'' has the property that the following operations result in a mos: setting L = M, and setting M = s.

~~# ''S'' is ''monotone-mos'' (MMOS). That is, each of~~ the following operations ~~results~~ in a mos: setting L = M, setting M = s~~, and setting s = 0.~~

~~# If len(''S'') is odd, ''S'' is a [[billiard scale]]~~.~~-->~~

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

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The generator of aX 2bW must have an odd number of W steps; if it had an even number of W steps, it would be generated by stacking the generator of the mos aX bW' with W' = 2W, a contradiction. This with (4) immediately gives (5).

For (6) and (7), odd-numbered SGA scales are [[Fokker block]]s* (in the 2-dimensional lattice generated by the generator and the offset), hence are rank-3 billiard scales. Rank-3 billiard scales project to mosses when one removes all instances of one step size, in particular s. Thus it suffices to prove that the result of equating any two step sizes is a mos. Equating Y and Z equates the swung alternants, resulting in the mos aX 2bW by (5). It remains to prove that equating X and Y equates the generator with the difference between the start of the 2nd chain with the end of the 1st one or vice versa; similarly for equating X and Z. Consider the swung-generator-alternant chain g1 g2 .... g1 g2 g3. Assume that the offset g1 = δ has one more Y (and one fewer Z) than g2 = g - δ. Then g3 (which becomes the imperfect generator of aX 2bW) either has one more X or one fewer X than both g1 and g2. If g3 has one more X, then g3 + g1 is equated to g1 + g2 = g, as desired. If g3 has one fewer X, then simply use the inverted generator class. ~~Assume that the offset g1 = δ has one fewer Y (and one more Z) than g2. Then we use the inverted generator class iff g3 has one ''more'' X than both g1 and g2. <math>\square</math>~~

For (6) and (7), 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:

~~<nowiki>*</nowiki>~~ To see this, consider the following lattice:

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.

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. [Proof to be continued...]

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

@@ Line 33: / Line 33: @@
 # The length of ''S'' is either odd, or 4 (and ''S'' is of the form ''xyxz'').
 # S = aX bY bZ is obtained from the (single-period) mos aX 2bW 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).
-# 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'' (PMOS). That is, the result of identifying any two step sizes of ''S'' is always a mos.
+#  If len(''S'') is odd, ''S'' has the property that the following operations result in a mos: setting L = M, and setting M = s.
-# ''S'' is ''monotone-mos'' (MMOS). That is, each of the following operations results in a mos: setting L = M, setting M = s, and setting s = 0.
-# If len(''S'') is odd, ''S'' is a [[billiard scale]].-->
 In particular, odd GO scales always satisfy these properties (see Proposition 2 below).
@@ Line 85: / Line 83: @@
 The generator of aX 2bW must have an odd number of W steps; if it had an even number of W steps, it would be generated by stacking the generator of the mos aX bW' with W' = 2W, a contradiction. This with (4) immediately gives (5).
-For (6) and (7), odd-numbered SGA scales are [[Fokker block]]s* (in the 2-dimensional lattice generated by the generator and the offset), hence are rank-3 billiard scales. Rank-3 billiard scales project to mosses when one removes all instances of one step size, in particular s. Thus it suffices to prove that the result of equating any two step sizes is a mos. Equating Y and Z equates the swung alternants, resulting in the mos aX 2bW by (5). It remains to prove that equating X and Y equates the generator with the difference between the start of the 2nd chain with the end of the 1st one or vice versa; similarly for equating X and Z. Consider the swung-generator-alternant chain g1 g2 .... g1 g2 g3. Assume that the offset g1 = δ has one more Y (and one fewer Z) than g2 = g - δ. Then g3 (which becomes the imperfect generator of aX 2bW) either has one more X or one fewer X than both g1 and g2. If g3 has one more X, then g3 + g1 is equated to g1 + g2 = g, as desired. If g3 has one fewer X, then simply use the inverted generator class. Assume that the offset g1 = δ has one fewer Y (and one more Z) than g2. Then we use the inverted generator class iff g3 has one ''more'' X than both g1 and g2. <math>\square</math>
+For (6) and (7), 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:
-<nowiki>*</nowiki> To see this, consider the following lattice:
   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.
+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. [Proof to be continued...]
 === Proposition 2 (Odd GO scales are SGA) ===