Generator-offset property: Difference between revisions

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== Theorems ==

=== Proposition 1 (Properties of SGA scales) ===

Let ''S'' be a 3-step-size scale word in L, M, and s of length n, and suppose ''S'' is SGA. Then:

Let ''S'' be a 3-step-size scale word in L, M, and s of length ''n'', and suppose ''S'' is SGA. Then:

# ''S'' is abstractly SV3 (i.e. SV3 for almost all tunings).

# ''S'' is of the form ''~~ax by bz'' for some permutation (~~''x'', ''y'', ''z'') of (L, M, s).

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

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

# S = ~~aX bY bZ~~ is obtained from some mode of 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, fixing the mode of ~~aX 2bW~~).

# ''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.

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

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 == Theorems ==
 === Proposition 1 (Properties of SGA scales) ===
-Let ''S'' be a 3-step-size scale word in L, M, and s of length n, and suppose ''S'' is SGA. Then:
+Let ''S'' be a 3-step-size scale word in L, M, and s of length ''n'', and suppose ''S'' is SGA. Then:
 # ''S'' is abstractly SV3 (i.e. SV3 for almost all tunings).
-# ''S'' is of the form ''ax by bz'' for some permutation (''x'', ''y'', ''z'') of (L, M, s).
+# ''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 either odd, or 4 (and ''S'' is of the form ''xyxz'').
+# The length of ''S'' is either odd, or 4 (and ''S'' is of the form xyxz).
-# S = aX bY bZ is obtained from some mode of 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, fixing the mode of aX 2bW).
+# ''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.
 # ''S'' is pairwise-mos. That is, the following operations each result in a mos: setting L = M, setting L = s, and setting M = s.