Rank 3 scale: Difference between revisions

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== Rank-3 scales ==

=== MV3 scales ===

=== [[User:Inthar/MV3|MV3 scales]] ===

Maximum variety 3, or MV3 scales are a generalization of MOS scales (the scales of MV2) into rank-3.

For all MV3 scales apart from the ~~scale~~ abacaba, and ~~its~~ repetitions abacabaabacaba etc., at least two of the three steps must occur the same number of times, and ~~may be generated by an alternating generator sequence. Because of this~~, there always exists a generator for a MV3 scale ~~(other than the one exception)~~ such that the scale can be expressed as two parallel chains of this generator whose lengths are equal, or differ by 1.

For all MV3 scales apart from the scales abacaba, and it's repetitions abacabaabacaba etc., at least two of the three steps must occur the same number of times. Moreover, excluding the scales abacaba, abcba, and their repetitions, there always exists a generator for a MV3 scale such that the scale can be expressed as two parallel chains of this generator whose lengths are equal, or differ by 1.

'''Conjecture:''' The only symmetric MV3 scales are abacaba (and its repetitions) and the scales of the form a...ba...c. Therefore the only MV3 scales that are symmetric are the only MV3 scales that are also 3-[[SN scales]] (introduced below).

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When mappings are considered, PWF scales are rank-3 [[Gallery of wakalixes|''wakalixes'']] - Fokker blocks which are Fokker blocks in more than one way.

Not all trivalent scales are PWF. Only a single scale - ~~aabcb~~ - is trivalent and not PWF.

Not all trivalent scales are PWF. Only a single scale - abcba - is trivalent and not PWF.

Only a single PWF scale is symmetric - abacaba.

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 == Rank-3 scales ==
-=== MV3 scales ===
+=== [[User:Inthar/MV3|MV3 scales]] ===
 Maximum variety 3, or MV3 scales are a generalization of MOS scales (the scales of MV2) into rank-3.
-For all MV3 scales apart from the scale abacaba, and its repetitions abacabaabacaba etc., at least two of the three steps must occur the same number of times, and may be generated by an alternating generator sequence. Because of this, there always exists a generator for a MV3 scale (other than the one exception) such that the scale can be expressed as two parallel chains of this generator whose lengths are equal, or differ by 1.
+For all MV3 scales apart from the scales abacaba, and it's repetitions abacabaabacaba etc., at least two of the three steps must occur the same number of times. Moreover, excluding the scales abacaba, abcba, and their repetitions, there always exists a generator for a MV3 scale such that the scale can be expressed as two parallel chains of this generator whose lengths are equal, or differ by 1.
 '''Conjecture:''' The only symmetric MV3 scales are abacaba (and its repetitions) and the scales of the form a...ba...c. Therefore the only MV3 scales that are symmetric are the only MV3 scales that are also 3-[[SN scales]] (introduced below).
@@ Line 51: / Line 51: @@
 When mappings are considered, PWF scales are rank-3 [[Gallery of wakalixes|''wakalixes'']] - Fokker blocks which are Fokker blocks in more than one way.
-Not all trivalent scales are PWF. Only a single scale - aabcb - is trivalent and not PWF.
+Not all trivalent scales are PWF. Only a single scale - abcba - is trivalent and not PWF.
 Only a single PWF scale is symmetric - abacaba.