Ternary scale theorems: Difference between revisions

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## Interchange some of the Z's and X's at some of the boundaries of these copies of the mos word ''w''. Here, if ''w'' is a word and ''w'' = <i>u'uvv'</i>, the ''boundary'' between ''u'' and ''v'' is ''u''[len(''u'')]''v''[1] within w. (If w is a circular word and w = [w'] where w' is a linear word, replace w with w' in the previous sentence.) For example, let ''w'' be the multimos word 8X6Z, XXZXZXZXXZXZXZ. Then the border between the copies of the MOS subword XXZXZXZ are w[7]w[8] and w[14]w[1].

## Replace every other X with Y in ''w''.

# ~~primitive~~ MV3 scales not of type (4) are always SV3, and those of type (4) are SV3 with the exception of the ''n''/2-step (''n'' = scale length) which is variety 2.

# Primitive MV3 scales not of type (4) are always SV3, and those of type (4) are SV3 with the exception of the ''n''/2-step (''n'' = scale length) which is variety 2.

# ~~primitive~~ MV3 scales not of type (5) and not of the form XYZYX are ''balanced'': for any ''k'', any pair of k-steps has a difference that contains +1, -1, or 0 of each step size.

# Primitive MV3 scales not of type (5) and not of the form XYZYX are ''balanced'': for any ''k'', any pair of k-steps has a difference that contains +1, -1, or 0 of each step size.

=== Proof ===

Proven by Bulgakova, Buzhinsky and Goncharov (2023), "[https://www.sciencedirect.com/science/article/pii/S0304397522006417 On balanced and abelian properties of circular words over a ternary alphabet]" (and Theorem 4). Note that odd GO scales are type (3) in this classification.

@@ Line 233: / Line 233: @@
 ## Interchange some of the Z's and X's at some of the boundaries of these copies of the mos word ''w''. Here, if ''w'' is a word and ''w'' = <i>u'uvv'</i>, the ''boundary'' between ''u'' and ''v'' is ''u''[len(''u'')]''v''[1] within w. (If w is a circular word and w = [w'] where w' is a linear word, replace w with w' in the previous sentence.) For example, let ''w'' be the multimos word 8X6Z, XXZXZXZXXZXZXZ. Then the border between the copies of the MOS subword XXZXZXZ are w[7]w[8] and w[14]w[1].
 ## Replace every other X with Y in ''w''.
-# primitive MV3 scales not of type (4) are always SV3, and those of type (4) are SV3 with the exception of the ''n''/2-step (''n'' = scale length) which is variety 2.
+# Primitive MV3 scales not of type (4) are always SV3, and those of type (4) are SV3 with the exception of the ''n''/2-step (''n'' = scale length) which is variety 2.
-# primitive MV3 scales not of type (5) and not of the form XYZYX are ''balanced'': for any ''k'', any pair of k-steps has a difference that contains +1, -1, or 0 of each step size.
+# Primitive MV3 scales not of type (5) and not of the form XYZYX are ''balanced'': for any ''k'', any pair of k-steps has a difference that contains +1, -1, or 0 of each step size.
 === Proof ===
 Proven by Bulgakova, Buzhinsky and Goncharov (2023), "[https://www.sciencedirect.com/science/article/pii/S0304397522006417 On balanced and abelian properties of circular words over a ternary alphabet]" (and Theorem 4). Note that odd GO scales are type (3) in this classification.