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.

=== MV3 ~~Structure Theorem (conjectured)~~ ===

=== Classification of MV3 scales ===

~~(TODO~~: ~~Investigate Bulgakova, D~~. V., ~~Buzhinsky~~, N., ~~& Goncharov, Y. O~~. (~~2023) which seems to have something similar to this~~)

# A single-period MV3 is either PWF, equivalent to abcba, or a "twisted" word constructed as follows:

## Start with a power of the mos word ''w'' 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''.

## Replace every other X with Y in ''w''.The PWF scales are exactly the single-period rank-3 [[billiard scale]]s.

# Non-twisted 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).

~~Consider a(n unconditionally~~, ~~so independently of tuning) max-variety-3 scale with 3 different step sizes~~. ~~It is a mathematical fact that~~, ~~with only one exception~~, ~~at least two of the three steps must occur '''the same number of times'''~~. ~~For example, it is possible to have a max-variety-3 scale with 3 small steps~~, ~~5 medium steps, and 3 large steps~~, ~~because there are the same number of small steps as large steps~~. ~~But a max-variety-3 scale with 3 small steps, 5 medium steps, and 4 large steps is impossible~~. (~~The one exception to this rule is "aabacab", along with its repetitions "aabacabaabacab", etc.~~)

Statements 1 and 2 are proven in Bulgakova, D. V., Buzhinsky, N., & Goncharov, Y. O. (2023), statement 3 is proven in [[generator-offset property]].

~~If in addition the scale has odd size and does not have the same number of every step size~~, ~~there always exists some "generator" interval for any max-variety-~~3 ~~scale (other than the one exception) 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)~~. ~~(Proof?)~~

=== 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.
-=== MV3 Structure Theorem (conjectured) ===
+=== Classification of MV3 scales ===
-(TODO: Investigate Bulgakova, D. V., Buzhinsky, N., & Goncharov, Y. O. (2023) which seems to have something similar to this)
+# A single-period MV3 is either PWF, equivalent to abcba, or a "twisted" word constructed as follows:
+## Start with a power of the mos word ''w''  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''.
+## Replace every other X with Y in ''w''.The PWF scales are exactly the single-period rank-3 [[billiard scale]]s.
+# Non-twisted 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).
-Consider a(n unconditionally, so independently of tuning) max-variety-3 scale with 3 different step sizes. It is a mathematical fact that, with only one exception, at least two of the three steps must occur '''the same number of times'''. For example, it is possible to have a max-variety-3 scale with 3 small steps, 5 medium steps, and 3 large steps, because there are the same number of small steps as large steps. But a max-variety-3 scale with 3 small steps, 5 medium steps, and 4 large steps is impossible. (The one exception to this rule is "aabacab", along with its repetitions "aabacabaabacab", etc.)
+Statements 1 and 2 are proven in Bulgakova, D. V., Buzhinsky, N., & Goncharov, Y. O. (2023), statement 3 is proven in [[generator-offset property]].
-If in addition the scale has odd size and does not have the same number of every step size, there always exists some "generator" interval for any max-variety-3 scale (other than the one exception) 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). (Proof?)
 === Generating MV3 scales ===