Rank-3 scale: Difference between revisions
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[[Maximum variety]] 3 (MV3) scales are a generalization of MOS scales (the scales of MV2) into rank-3. | [[Maximum variety]] 3 (MV3) scales are a generalization of MOS scales (the scales of MV2) into rank-3. | ||
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. This is the [[generator-offset property]] (GO). | '''Conjecture:''' 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. This is the [[generator-offset property]] (GO). | ||
All GO scales of odd cardinality are MV3. The only GO scale of even cardinality is ''abac''. | All GO scales of odd cardinality are MV3. The only GO scale of even cardinality is ''abac''. | ||