User:Inthar/MV3: Difference between revisions

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== MV3 Theorem 1==

''Suppose we have an MV3 scale word with steps X, Y and Z. 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 two exception to this rule are "XYZYX", and "XYXZXYX", along with their repetitions "XYXZXYXXYXZXYX", 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).''

''Suppose we have an MV3 scale word with steps X, Y and Z. With only one exception ("XYXZXYX"), 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 two exception to this rule are "XYZYX", and "XYXZXYX", along with their repetitions "XYXZXYXXYXZXYX", 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 (except in the case "XYZYX") ===

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 == MV3 Theorem 1==
-''Suppose we have an MV3 scale word with steps X, Y and Z. 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 two exception to this rule are "XYZYX", and "XYXZXYX", along with their repetitions "XYXZXYXXYXZXYX", 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).''
+''Suppose we have an MV3 scale word with steps X, Y and Z. With only one exception ("XYXZXYX"), 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 two exception to this rule are "XYZYX", and "XYXZXYX", along with their repetitions "XYXZXYXXYXZXYX", 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 (except in the case "XYZYX") ===