Maximum variety: Difference between revisions

Line 4:

Any scale with all equal steps (such as an [[EDO|EDO]]) has maximum variety 1. All [[MOSScales|MOS]] scales have maximum variety 2. An example of a scale with high max variety is the [[harmonic_series|harmonic series]], because the steps get gradually smaller as you go up the scale, and none of them are equal.

== Mathematical properties ==

* A [[balanced]] word or [[necklace]] in ''N'' letters has a maximum variety bound of <math> N \choose {\lceil N/2 \rceil}</math>.

==Max-variety-3 scales==

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]].

@@ Line 4: / Line 4: @@
 Any scale with all equal steps (such as an [[EDO|EDO]]) has maximum variety 1. All [[MOSScales|MOS]] scales have maximum variety 2. An example of a scale with high max variety is the [[harmonic_series|harmonic series]], because the steps get gradually smaller as you go up the scale, and none of them are equal.
+== Mathematical properties ==
+* A [[balanced]] word or [[necklace]] in ''N'' letters has a maximum variety bound of <math> N \choose {\lceil N/2 \rceil}</math>.
 ==Max-variety-3 scales==
 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]].