Ternary scale theorems: Difference between revisions

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== Theorem 5 (Classification of MV3 scales) ==

# A single-period MV3 is either (1) ~~pairwise well-formed~~, (2) ~~equivalent to XYZYX~~, (3) ~~constructed from~~ ''w''(X, Z)^2 (power of a primitive mos ''w''(X, Z) with #X odd) by replacing every other X with Y, or (4) a "twisted" word constructed as follows:

# A single-period MV3 is either (1) equivalent to XYZYX, (2) constructed from aX bZ with a even by replacing every other X with Y, (3) ''w''(X, Z)^2 (power of a primitive mos ''w''(X, Z) with #X odd) by replacing every other X with Y, or (4) a "twisted" word constructed as follows:

## Start with a power of a multimos word ''w''(X, Z) = ''ka''X ''kb''Z such that ''a'' is even and each ''a''X ''b''Z subword of ''w'' is of the form X''P''(X, Z)Z where ''P''(X, Z) is a palindrome.

## Interchange some of the Z's and X's at some (possibly none) of the borders of these copies of the mos word ''w''.

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# Single-period MV3 scales not of type (4) 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).

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 PWF scales are S<sub>3</sub>-action images of φ(the mos aX bY), where a is even, b is odd, and φ is the operation of replacing every other X with Z.

[[Category:Math]]

[[Category:Ternary scale]]

[[Category:Scale]]

@@ Line 227: / Line 227: @@
 == Theorem 5 (Classification of MV3 scales) ==
-# A single-period MV3 is either (1) pairwise well-formed, (2) equivalent to XYZYX, (3) constructed from ''w''(X, Z)^2 (power of a primitive mos ''w''(X, Z) with #X odd) by replacing every other X with Y, or (4) a "twisted" word constructed as follows:
+# A single-period MV3 is either (1) equivalent to XYZYX, (2) constructed from aX bZ with a even by replacing every other X with Y, (3) ''w''(X, Z)^2 (power of a primitive mos ''w''(X, Z) with #X odd) by replacing every other X with Y, or (4) a "twisted" word constructed as follows:
 ## Start with a power of a multimos word ''w''(X, Z) = ''ka''X ''kb''Z such that ''a'' is even and each ''a''X ''b''Z subword of ''w'' is of the form X''P''(X, Z)Z where ''P''(X, Z) is a palindrome.
 ## Interchange some of the Z's and X's at some (possibly none) of the borders of these copies of the mos word ''w''.
@@ Line 234: / Line 234: @@
 # Single-period MV3 scales not of type (4) 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).
+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 PWF scales are S<sub>3</sub>-action images of φ(the mos aX bY), where a is even, b is odd, and φ is the operation of replacing every other X with Z.
 [[Category:Math]]
 [[Category:Ternary scale]]
 [[Category:Scale]]