Maximum variety: Difference between revisions

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The commonly discussed [[MOS]] property can be characterized as follows, [[MOS#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]].

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.

When discussing scale patterns with three abstract step sizes a, b and c, unlike in the "rank-2" case one must distinguish between ''abstractly MV3'' scale patterns, 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.

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

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 The commonly discussed [[MOS]] property can be characterized as follows, [[MOS#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]].
-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.
+When discussing scale patterns with three abstract step sizes a, b and c, unlike in the "rank-2" case one must distinguish between ''abstractly MV3'' scale patterns, 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.
 There is a theorem classifying all possible MV3 scales; see [[Ternary scale theorems]].