Ternary scale theorems: Difference between revisions

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==== Statement (4) ====

~~For (4), assume~~ ''S'' is ''a''X ''b''Y ''b''Z, ''a'' odd. If ''b'' = 1, there's nothing to prove, so assume ''b'' > 1.

Assume ''S'' is ''a''X ''b''Y ''b''Z, ''a'' odd. If ''b'' = 1, there's nothing to prove, so assume ''b'' > 1.

Assume, possibly after inverting the generator, that the imperfect generator of ''T'' has ''j'' + 1 W's and the perfect generator has ''j'' W's. Consider the two alternants, detemperings of the generator ''i''X + ''j''W of the ''a''X 2''b''W mos ''T''(X, W) = ''S''(X, W, W) where gcd(''j'', 2''k'') = 1.

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==== Statement (5) ====

~~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:

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

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==== 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 (corresponding to either Y or Z) turns it into a mos:

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 (corresponding to either Y or Z) 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.

@@ Line 73: / Line 73: @@
 ==== Statement (4) ====
-For (4), assume ''S'' is ''a''X ''b''Y ''b''Z, ''a'' odd. If ''b'' = 1, there's nothing to prove, so assume ''b'' > 1.
+Assume ''S'' is ''a''X ''b''Y ''b''Z, ''a'' odd. If ''b'' = 1, there's nothing to prove, so assume ''b'' > 1.
 Assume, possibly after inverting the generator, that the imperfect generator of ''T'' has ''j'' + 1 W's and the perfect generator has ''j'' W's. Consider the two alternants, detemperings of the generator ''i''X + ''j''W of the ''a''X 2''b''W mos ''T''(X, W) = ''S''(X, W, W) where gcd(''j'', 2''k'') = 1.
@@ Line 88: / Line 88: @@
 ==== Statement (5) ====
-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:
+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
@@ Line 94: / Line 94: @@
 ==== 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 (corresponding to either Y or Z) turns it into a mos:
+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 (corresponding to either Y or Z) 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.