Ternary scale theorems: 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 (1) equivalent to XYZYX, (2) constructed from aX bZ with a even by replacing every other X with Y, (3) constructed from | # A single-period MV3 is either (1) equivalent to XYZYX, (2) constructed from aX bZ with a even and gcd(a, b) = 1 by replacing every other X with Y, (3) constructed from 2aX 2bY with a odd and gcd(a, b) = 1 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. | ## 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''. | ||