Maximum variety: Difference between revisions

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==Max-variety-3 scales==

'''Max-variety-3''' scales are an attempt to generalize distributional evenness (closely related to the MOS property) to scales with three different step sizes rather than two (for example, those related to rank-3 [[Regular_Temperaments|regular temperaments]]). The construction of max-variety-3 scales is significantly more complicated than that of MOSes, but not much more difficult to understand if the right approach is used.

=== MV3 Structure Theorem ===

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 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 [[generator-offset property]] (GO). (Proof?)

=== Generating MV3 scales ===

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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 ==Max-variety-3 scales==
 '''Max-variety-3''' scales are an attempt to generalize distributional evenness (closely related to the MOS property) to scales with three different step sizes rather than two (for example, those related to rank-3 [[Regular_Temperaments|regular temperaments]]). The construction of max-variety-3 scales is significantly more complicated than that of MOSes, but not much more difficult to understand if the right approach is used.
+=== MV3 Structure Theorem ===
 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 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 [[generator-offset property]] (GO). (Proof?)
+=== Generating MV3 scales ===
 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).