Generator-offset property: Difference between revisions
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=== Theorem 5 (Classification of MV3 scales) === | === Theorem 5 (Classification of MV3 scales) === | ||
# A single-period MV3 is either PWF, equivalent to XYZYX, or a "twisted" word constructed as follows: | # A single-period MV3 is either (1) PWF, (2) equivalent to XYZYX, or (3) 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. | ## 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 (possibly none) of the borders of these copies of the mos word ''w''. | ||
## Replace every other X with Y in ''w''. | ## Replace every other X with Y in ''w''. | ||
# | # Single-period MV3 scales not of type (3) are always SV3. | ||
==== Proof ==== | ==== Proof ==== | ||
Proven in 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]". | Proven in 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]". | ||