Ternary scale theorems: Difference between revisions
m →Proposition 1 (Properties of SGA scales): Not sure if my proof really proves elimination-mos. |
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# If ''n'' is odd, ''S'' is abstractly SV3 (i.e. SV3 for almost all tunings). | # If ''n'' is odd, ''S'' is abstractly SV3 (i.e. SV3 for almost all tunings). | ||
# If ''n'' is odd, ''S'' = ''a''X ''b''Y ''b''Z is obtained from some mode of the (primitive) 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. | # If ''n'' is odd, ''S'' = ''a''X ''b''Y ''b''Z is obtained from some mode of the (primitive) 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. | ||
# If ''n'' is odd, ''S'' is pairwise-mos. That is, the following operations each result in a [[mos]]: setting L = M, setting L = s, and setting M = s. | # If ''n'' is odd, ''S'' is pairwise-mos. That is, the following operations each result in a [[mos]]: setting L = M, setting L = s, and setting M = s.<!-- | ||
# If ''n'' is odd, ''S'' is elimination-mos. That is, ''E''<sub>X</sub>(''S''), ''E''<sub>Y</sub>(''S''), ''E''<sub>Z</sub>(''S'') are all mosses. | # If ''n'' is odd, ''S'' is elimination-mos. That is, ''E''<sub>X</sub>(''S''), ''E''<sub>Y</sub>(''S''), ''E''<sub>Z</sub>(''S'') are all mosses.--> | ||
In particular, odd generator-offset scales always satisfy these properties (see Proposition 2 below). | In particular, odd generator-offset scales always satisfy these properties (see Proposition 2 below). | ||
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x x x ... x | x x x ... x | ||
x 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 rank-3 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). | and use the vectors (-1, 2) and (ceil(n/2), 1) as the Fokker block chromas. A rank-3 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) ==== | ==== Statement (6) ==== | ||
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# If chunk sizes of a binary scale form a mos, the scale itself must be a mos; see [[Recursive structure of MOS scales#Reflection of generators|Recursive structure of MOS scales, "Reflection of generators"]]. | # If chunk sizes of a binary scale form a mos, the scale itself must be a mos; see [[Recursive structure of MOS scales#Reflection of generators|Recursive structure of MOS scales, "Reflection of generators"]]. | ||
Lastly, ''E''<sub>X</sub>(''S'') is the mos ''b''Y ''b''Z; hence ''S'' is elimination-mos. | Lastly, ''E''<sub>X</sub>(''S'') is the mos ''b''Y ''b''Z; hence ''S'' is elimination-mos. | ||
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== Proposition 2 (Odd generator-offset scales are SGA) == | == Proposition 2 (Odd generator-offset scales are SGA) == | ||