Rank 3 scale: Difference between revisions

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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~~. Equivalently, MV3 scales apart from the single exception can be generated by an alternating generator sequence of two generators, modulo the period~~.

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.

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

Only a single PWF scale is symmetric - abacaba.

Apart from abacaba, PWF scales can be generated by an alternating generator sequence of two generators, modulo the period.

PWF scales can only have odd numbers of notes.

~~Only a single PWF scale is symmetric - abacaba.~~

=== Pairwise DE/MOS scales ===

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 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. Equivalently, MV3 scales apart from the single exception can be generated by an alternating generator sequence of two generators, modulo the period.
+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.
 '''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 49: / Line 49: @@
 Not all trivalent scales are PWF. Only a single scale - aabcb - is trivalent and not PWF.
+Only a single PWF scale is symmetric - abacaba.
+Apart from abacaba, PWF scales can be generated by an alternating generator sequence of two generators, modulo the period.
 PWF scales can only have odd numbers of notes.
-Only a single PWF scale is symmetric - abacaba.
 === Pairwise DE/MOS scales ===