Maximum variety: Difference between revisions
→Mathematical overview: abelian complexity is a function rather than a maximum of that function |
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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. | 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 overview == | == Mathematical overview == | ||
A standard academic counterpart to | A standard academic counterpart to the xen term ''variety'' is the ''abelian complexity function of a [[word]]'': a function ρ<sup>ab</sup> : '''N''' -> '''N''' where ρ<sup>ab</sup>(''n'') is the number of distinct "sizes" that length-''n'' subwords can have in a word | ||
* A [[balanced]] word or [[necklace]] in ''N'' letters has a maximum variety bound of <math> N \choose {\lceil N/2 \rceil}</math>. | * A [[balanced]] word or [[necklace]] in ''N'' letters has a maximum variety bound of <math> N \choose {\lceil N/2 \rceil}</math>. | ||