User:Inthar/MV3: Difference between revisions
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== MV3 Theorem 1== | == MV3 Theorem 1== | ||
''With only one exception, at least two of the three steps must occur the same number of times. For example, it is possible to have a max-variety-3 scale with 3 small steps, 5 medium steps, and 3 large steps, because there are the same number of small steps as large steps. But a max-variety-3 scale with 3 small steps, 5 medium steps, and 4 large steps is impossible. (The one exception to this rule is "aabacab", along with its repetitions "aabacabaabacab", etc.) Because of this, there always exists some "generator" interval for any max-variety-3 scale (other than the one exception) such that the scale can be expressed as two parallel chains of this generator which are almost equal in length (the lengths are either equal, or differ by 1).'' | |||
=== Lemma 1: The word made by any two of the step sizes is a MOS === | === Lemma 1: The word made by any two of the step sizes is a MOS === | ||
Suppose we have three sizes of a class, T1, T2, T3 of Y's and Z's. | Suppose we have three sizes of a class, T1, T2, T3 of Y's and Z's. | ||
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''To be continued...'' | ''To be continued...'' | ||
== MV3 Theorem 2 == | == MV3 Theorem 2 == | ||
''Once you have chosen a rank-3 temperament and a specific generator interval, there is a mechanical procedure to generate all max-variety-3 scales of a certain size (of which there are, however, infinitely many).'' | ''Once you have chosen a rank-3 temperament and a specific generator interval, there is a mechanical procedure to generate all max-variety-3 scales of a certain size (of which there are, however, infinitely many).'' | ||