Maximum variety: Difference between revisions

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When discussing scale patterns with three abstract step sizes a, b and c, unlike in the "rank-2" case one must distinguish between ''unconditionally MV3'' scale patterns or ''abstractly MV3'' ones, patterns that are MV3 regardless of what concrete sizes a, b, and c have, and ''conditionally MV3'' patterns, which have tunings that are not MV3. For example, MMLs is conditionally MV3 because it is only MV3 when L, M and s are chosen such that MM = Ls. When we say that an abstract scale pattern is MV3, the former meaning is usually intended.

=== Classification of MV3 scales ===

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

# A single-period MV3 is either (1) equivalent to XYZYX, (2) equivalent to XYXZXYX, (3) [[generator-offset]] with odd length, (4) constructed from 2aX 2bY 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 ''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''.

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 When discussing scale patterns with three abstract step sizes a, b and c, unlike in the "rank-2" case one must distinguish between ''unconditionally MV3'' scale patterns or ''abstractly MV3'' ones, patterns that are MV3 regardless of what concrete sizes a, b, and c have, and ''conditionally MV3'' patterns, which have tunings that are not MV3. For example, MMLs is conditionally MV3 because it is only MV3 when L, M and s are chosen such that MM = Ls. When we say that an abstract scale pattern is MV3, the former meaning is usually intended.
 === Classification of MV3 scales ===
-# A single-period MV3 is either (1) equivalent to XYZYX, (2) equivalent to XYXZXYX, (3) [[generator-offset]] with odd length, or (4) constructed from 2aX 2bY with a odd and gcd(a, b) = 1 by replacing every other X with Y, or (5) a "twisted" word constructed as follows:
+# A single-period MV3 is either (1) equivalent to XYZYX, (2) equivalent to XYXZXYX, (3) [[generator-offset]] with odd length, (4) constructed from 2aX 2bY 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 ''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''.