Maximum variety: Difference between revisions

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Consider a max-variety-3 scale with 3 different step sizes. It is a mathematical fact that, 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.)

If in addition the scale has odd size, 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). This property is called the alternating-generator ([[AG]]) property. (Proof?)

If in addition the scale has odd size and does not have the same number of every step size, 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). This property is called the alternating-generator ([[AG]]) property. (Proof?)

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).

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 Consider a max-variety-3 scale with 3 different step sizes. It is a mathematical fact that, 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.)
-If in addition the scale has odd size, 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). This property is called the alternating-generator ([[AG]]) property. (Proof?)
+If in addition the scale has odd size and does not have the same number of every step size, 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). This property is called the alternating-generator ([[AG]]) property. (Proof?)
 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).