Ternary scale theorems: Difference between revisions

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

# A single-period MV3 is either (1) equivalent to XYZYX, (2) equivalent to XYXZXYX, (3) constructed from ''a''X ''b''Z with ''a'' even and gcd(''a'', ''b'') = 1 by replacing every other X with Y, (4) constructed from 2''a''X 2''b''Z with ''a'' odd and gcd(''a'', ''b'') = 1 by replacing every other X with Y, or (5) a "twisted" word constructed as follows:

## Start with ~~a power of~~ a multimos word ''w''(X, Z) = ''ka''X ''kb''Z such that ''ka'' 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 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 of the ~~borders~~ of these copies of the mos word ''w''.

## 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, and 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''.

# Single-period 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.

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 == Theorem 5 (Classification of MV3 scales) ==
 # A single-period MV3 is either (1) equivalent to XYZYX, (2) equivalent to XYXZXYX, (3) constructed from ''a''X ''b''Z with ''a'' even and gcd(''a'', ''b'') = 1 by replacing every other X with Y, (4) constructed from 2''a''X 2''b''Z with ''a'' odd and gcd(''a'', ''b'') = 1 by replacing every other X with Y, or (5) a "twisted" word constructed as follows:
-## Start with a power of a multimos word ''w''(X, Z) = ''ka''X ''kb''Z such that ''ka'' 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 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 of the borders of these copies of the mos word ''w''.
+## 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, and 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''.
 # Single-period 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.