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).

It is crucial to remember that variety properties of a concrete scale and variety properties of an abstract scale word ''mean different things''. Namely, if certain linear relations hold between step sizes, the abstract scale word may satisfy different variety properties than the concrete tuning of it. The modifier ''abstractly'' is used to emphasize that the variety property holds for the scale pattern represented by the scale, not merely or necessarily for the concrete scale. For example, the scale pattern '''0102103012''' is abstractly minimum variety 4, but (almost all) tunings of the pattern that satisfy '''0''' + '''3''' = '''1''' + '''2''' will be minimum variety 3.

It is crucial to remember that variety properties of a concrete scale and variety properties of an abstract scale word ''mean different things''. Namely, if certain linear relations hold between step sizes, the abstract scale word may satisfy different variety properties than the concrete tuning of it. The modifier ''abstractly'' is used to emphasize that the variety property holds for the scale pattern represented by the scale, not merely or necessarily for the concrete scale. For example, the scale pattern '''0102103012''' is abstractly minimum variety 4, but (almost all) tunings of the pattern that satisfy '''0''' + '''3''' = '''1''' + '''2''' will be minimum variety 3. In other words, variety terms ''are overloaded and have different "types"'', namely <code>ConcreteScale -> Boolean</code> and <code>AbstractScaleWord -> Boolean</code>, depending on context.

== Terminology ==

For abstract scale words, the 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).
-It is crucial to remember that variety properties of a concrete scale and variety properties of an abstract scale word ''mean different things''. Namely, if certain linear relations hold between step sizes, the abstract scale word may satisfy different variety properties than the concrete tuning of it. The modifier ''abstractly'' is used to emphasize that the variety property holds for the scale pattern represented by the scale, not merely or necessarily for the concrete scale. For example, the scale pattern '''0102103012''' is abstractly minimum variety 4, but (almost all) tunings of the pattern that satisfy '''0''' + '''3''' = '''1''' + '''2''' will be minimum variety 3.
+It is crucial to remember that variety properties of a concrete scale and variety properties of an abstract scale word ''mean different things''. Namely, if certain linear relations hold between step sizes, the abstract scale word may satisfy different variety properties than the concrete tuning of it. The modifier ''abstractly'' is used to emphasize that the variety property holds for the scale pattern represented by the scale, not merely or necessarily for the concrete scale. For example, the scale pattern '''0102103012''' is abstractly minimum variety 4, but (almost all) tunings of the pattern that satisfy '''0''' + '''3''' = '''1''' + '''2''' will be minimum variety 3. In other words, variety terms ''are overloaded and have different "types"'', namely <code>ConcreteScale -> Boolean</code> and <code>AbstractScaleWord -> Boolean</code>, depending on context.
 == Terminology ==
 For abstract scale words, the 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.