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 ~~===~~

There is a theorem classifying MV3 scales; see [[Ternary scale theorems]].

~~{{main|~~Ternary scale theorems}}

~~# 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''.~~

~~## 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.~~

~~# Single-period MV3 scales not of type (1) or (5) are ''balanced'': for any ''k'', any pair of k-steps has a difference that contains +1, -1, or 0 of each step size~~.

=== Generating MV3 scales ===

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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 ===
+There is a theorem classifying MV3 scales; see [[Ternary scale theorems]].
-{{main|Ternary scale theorems}}
-# 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''.
-## 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.
-# Single-period MV3 scales not of type (1) or (5) are ''balanced'': for any ''k'', any pair of k-steps has a difference that contains +1, -1, or 0 of each step size.
 === Generating MV3 scales ===