Generator-offset property: Difference between revisions

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

# The length of ''S'' is odd, or ''S'' is equivalent to (xy)<sup>''r''</sup>xz for some integer ''r'' ≥ 1.

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

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

# 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 (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.

# 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.

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

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

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

@@ Line 32: / Line 32: @@
 Let ''S'' be a 3-step-size scale word in L, M, and s of length ''n'', and suppose ''S'' is SGA. Then:
 # The length of ''S'' is odd, or ''S'' is equivalent to (xy)<sup>''r''</sup>xz for some integer ''r'' ≥ 1.
-# ''S'' is of the form ''a''x ''b''y ''b''z for some permutation (x, y, z) of (L, M, s).
+# If ''n'' is odd, ''S'' is of the form ''a''x ''b''y ''b''z for some permutation (x, y, z) of (L, M, s).
 # 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 (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.
+# 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.
-# ''S'' is elimination-mos. That is, "tempering out" any one step size results in a mos.
+# If ''n'' is odd, ''S'' is elimination-mos. That is, "tempering out" any one step size results in a mos.
 In particular, odd GO scales always satisfy these properties (see Proposition 2 below).