Rank-3 scale: Difference between revisions
→MV3 scales: I don't know a proof |
→MV3 scales: This part is falsified by the 14-note SV3 LLLMLLLsLLMLLs: "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." |
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== MV3 scales == | == MV3 scales == | ||
[[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. We often speak of strict-variety 3 (SV3) instead, meaning that every interval class has ''exactly'' three sizes. | ||
'''Conjecture:''' For all odd-cardinality | '''Conjecture:''' For all odd-cardinality SV3 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. | ||
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''. | ||