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, or (3) a "twisted" word constructed as follows:

# 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:

## 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''.

## Interchange some of the Z's and X's at some (not none) of the borders of these copies of the mos word ''w''.

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

# Single-period MV3 scales not of type (3), or ~~of type~~ (3) ~~with ''a'' even and with no twisting at step 2,~~ are always SV3.

# Single-period MV3 scales not of type (3) or (4) are always SV3.

=== 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).

@@ Line 227: / Line 227: @@
 == Theorem 5 (Classification of MV3 scales) ==
-# A single-period MV3 is either (1) pairwise well-formed, (2) equivalent to XYZYX, or (3) a "twisted" word constructed as follows:
+# 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:
 ## 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''.
+## Interchange some of the Z's and X's at some (not none) of the borders of these copies of the mos word ''w''.
 ## Replace every other X with Y in ''w''.
-# Single-period MV3 scales not of type (3), or of type (3) with ''a'' even and with no twisting at step 2, are always SV3.
+# Single-period MV3 scales not of type (3) or (4) are always SV3.
 === 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).