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 ~~pairwise well-formed~~ (~~PWF~~), equivalent to ~~abcba~~, or 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, 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:

## Start with a power of a ~~mos~~ word ''w''(X, Z) that ~~begins with~~ a X and ~~ends with~~ a Z ~~and has an even number~~ of X's.

## 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''.

# ~~Non-twisted single~~-period MV3 scales are always SV3 ~~(strict variety 3)~~.

# Single-period MV3 scales not of type (3) or (4) are always SV3.

# If the ~~scale is PWF~~, ~~with one exception abacaba, there always exists some "generator" interval such~~ that ~~the scale can be expressed as two parallel chains of this generator whose lengths differ by~~ 1~~. This property is called the [[generator~~-~~offset property]] (GO)~~.

# Single-period MV3 scales not of type (4) and not of the form XYZYX 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 ===
-# A single-period MV3 is either pairwise well-formed (PWF), equivalent to abcba, or 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, 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:
-## Start with a power of a mos word ''w''(X, Z) that begins with a X and ends with a Z and has an even number of X's.
+## 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''.
-# Non-twisted single-period MV3 scales are always SV3 (strict variety 3).
+# Single-period MV3 scales not of type (3) or (4) are always SV3.
-# If the scale is PWF, with one exception abacaba, there always exists some "generator" interval such that the scale can be expressed as two parallel chains of this generator whose lengths differ by 1. This property is called the [[generator-offset property]] (GO).
+# Single-period MV3 scales not of type (4) and not of the form XYZYX 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 ===