Interval variety: Difference between revisions

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In addition, '''strict variety''' scales, such as single-period [[MOS scale]]s and [[trivalent scale]]s, have the same interval variety for all interval classes (except the unison, which always trivially has interval variety 1).

For scales of [[arity]] at least 3, there is a critical distinction between '''abstract interval variety''' and '''conditional interval variety'''. ''~~Abstract~~'' means that the interval variety property holds for any tuning of the scale steps, and ''conditional'' means that the interval variety property only holds for a particular tuning of the steps.

For scales of [[arity]] at least 3, there is a critical distinction between '''free''' or '''abstract interval variety''' and '''conditional interval variety'''. ''Free'' or ''abstract'' means that the interval variety property holds for any tuning of the scale steps, and ''conditional'' means that the interval variety property only holds for a particular tuning of the steps.

== Terminology ==

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 (abelianizations, living in a free abelian group over the step sizes) that length-''n'' subwords can have in a word.

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 In addition, '''strict variety''' scales, such as single-period [[MOS scale]]s and [[trivalent scale]]s, have the same interval variety for all interval classes (except the unison, which always trivially has interval variety 1).
-For scales of [[arity]] at least 3, there is a critical distinction between '''abstract interval variety''' and '''conditional interval variety'''. ''Abstract'' means that the interval variety property holds for any tuning of the scale steps, and ''conditional'' means that the interval variety property only holds for a particular tuning of the steps.
+For scales of [[arity]] at least 3, there is a critical distinction between '''free''' or '''abstract interval variety''' and '''conditional interval variety'''. ''Free'' or ''abstract'' means that the interval variety property holds for any tuning of the scale steps, and ''conditional'' means that the interval variety property only holds for a particular tuning of the steps.
 == Terminology ==
 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 (abelianizations, living in a free abelian group over the step sizes) that length-''n'' subwords can have in a word.