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 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.

For all odd-cardinality MV3 scales apart from the scales abacaba, and its 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, and na nb nc scales, 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.

All [[AG]] scales of odd cardinality are MV3~~, and it is conjectured that all odd MV3 scales satisfying certain mild restrictions are AG~~. The only AG scale of even cardinality is abac.

All [[AG]] scales of odd cardinality are MV3. The only AG scale of even cardinality is abac.

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

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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 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.
+For all odd-cardinality MV3 scales apart from the scales abacaba, and its 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, and na nb nc scales, 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.
-All [[AG]] scales of odd cardinality are MV3, and it is conjectured that all odd MV3 scales satisfying certain mild restrictions are AG. The only AG scale of even cardinality is abac.
+All [[AG]] scales of odd cardinality are MV3. The only AG scale of even cardinality is abac.
 '''Conjecture:''' The only mirror-symmetric MV3 scales are abacaba (and its repetitions) and the scales of the form a...ba...c. Therefore the only MV3 scales that are mirror-symmetric are the only MV3 scales that are also 3-[[SN scales]] (introduced below).