Maximum variety: Difference between revisions

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==Max-variety-3 scales==

The commonly discussed [[MOS]] property can be defined as follows, among other equivalent ways: Every set of (non-unison reduced) generic intervals has size at most 2. We can rephrase this as saying that the maximum variety of the scale is 2. '''~~Max-~~variety-3''' scales ~~are~~ a generalization of 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.

The commonly discussed [[MOS]] property can be defined as follows, among other equivalent ways: Every set of (non-unison reduced) generic intervals has size at most 2. We can rephrase this as saying that the maximum variety of the scale is 2, or that the scale is maximum variety 2 (MV2). '''Maximum variety 3''' (MV3) scales is a generalization of the MV2 property to [[ternary scale]]s. 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.

When discussing scale patterns with three abstract step sizes a, b and c, unlike in the "rank-2" case one must distinguish between ''unconditionally MV3'' scale patterns or ''abstractly MV3'' ones, patterns that are MV3 regardless of what concrete sizes a, b, and c have, and ''conditionally MV3'' patterns, which have tunings that are not MV3. For example, MMLs is conditionally MV3 because it is only MV3 when L, M and s are chosen such that MM = Ls. When we say that an abstract scale pattern is MV3, the former meaning is usually intended.

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 ==Max-variety-3 scales==
-The commonly discussed [[MOS]] property can be defined as follows, among other equivalent ways: Every set of (non-unison reduced) generic intervals has size at most 2. We can rephrase this as saying that the maximum variety of the scale is 2. '''Max-variety-3''' scales are a generalization of 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.
+The commonly discussed [[MOS]] property can be defined as follows, among other equivalent ways: Every set of (non-unison reduced) generic intervals has size at most 2. We can rephrase this as saying that the maximum variety of the scale is 2, or that the scale is maximum variety 2 (MV2). '''Maximum variety 3''' (MV3) scales is a generalization of the MV2 property to [[ternary scale]]s. 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.
 When discussing scale patterns with three abstract step sizes a, b and c, unlike in the "rank-2" case one must distinguish between ''unconditionally MV3'' scale patterns or ''abstractly MV3'' ones, patterns that are MV3 regardless of what concrete sizes a, b, and c have, and ''conditionally MV3'' patterns, which have tunings that are not MV3. For example, MMLs is conditionally MV3 because it is only MV3 when L, M and s are chosen such that MM = Ls. When we say that an abstract scale pattern is MV3, the former meaning is usually intended.