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 pairwise well-formed (PWF), equivalent to abcba, or a "twisted" word constructed as follows:

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

## 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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# Non-twisted single-period MV3 scales 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 which are almost equal in length (the lengths are either equal, or differ by 1). This property is called the [[generator-offset property]] (GO).

~~Statements 1 through 3 are proven in Bulgakova, D. V., Buzhinsky, N., & Goncharov, Y. O. (2023). Statement 4 is proven in [[generator-offset property]].~~

=== Generating MV3 scales ===

@@ Line 10: / Line 10: @@
 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 pairwise well-formed (PWF), equivalent to abcba, or a "twisted" word constructed as follows:
 ## Start with a power of the mos word ''w''(X, Z) that begins with a X and ending with a Z and has an even number of X's.
 ## Interchange some of the Z's and X's at some of the borders of these copies of the mos word ''w''.
@@ Line 17: / Line 17: @@
 # Non-twisted single-period MV3 scales 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 which are almost equal in length (the lengths are either equal, or differ by 1). This property is called the [[generator-offset property]] (GO).
-Statements 1 through 3 are proven in Bulgakova, D. V., Buzhinsky, N., & Goncharov, Y. O. (2023). Statement 4 is proven in [[generator-offset property]].
 === Generating MV3 scales ===