Interval variety: Difference between revisions

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== Maximum-variety-3 scales ==

The commonly discussed [[mos]] property can be characterized as follows, [[MOS scale #~~Definition~~|~~as well as in~~ 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) is the generalization of the MV2 characterization of the ~~MOS~~ property to [[ternary scale]]s. Other characterizations of the mos property, such as [[distributional evenness]] and [[generator|having a generator]], ~~do not~~ generalize to ~~properties that are equivalent to MV3 in higher [[arity|arities]]~~.

The commonly discussed [[mos]] property can be characterized as follows, as well as in [[MOS scale #Equivalent definitions and generalizations|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) is the generalization of the MV2 characterization of the mos property to [[ternary scale]]s. Other characterizations of the mos property, such as [[distributional evenness]] and [[generator|having a generator]], generalize to ternary scales differently.

There is a theorem classifying all possible MV3 scales; see [[Ternary scale theorems]].

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 == Maximum-variety-3 scales ==
-The commonly discussed [[mos]] property can be characterized as follows, [[MOS scale #Definition|as well as in 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) is the generalization of the MV2 characterization of the MOS property to [[ternary scale]]s. Other characterizations of the mos property, such as [[distributional evenness]] and [[generator|having a generator]], do not generalize to properties that are equivalent to MV3 in higher [[arity|arities]].
+The commonly discussed [[mos]] property can be characterized as follows, as well as in [[MOS scale #Equivalent definitions and generalizations|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) is the generalization of the MV2 characterization of the mos property to [[ternary scale]]s. Other characterizations of the mos property, such as [[distributional evenness]] and [[generator|having a generator]], generalize to ternary scales differently.
 There is a theorem classifying all possible MV3 scales; see [[Ternary scale theorems]].