Interval variety: Difference between revisions

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The '''interval variety''' of an [[interval class]] in a [[scale]] is the number of different [[interval quality|interval qualities]] available for that interval class. For example, the interval class "fifth" in the [[5L 2s|diatonic]] scale has interval variety 2, because there are two sizes of fifths in that scale: 6 perfect fifths and 1 diminished fifth.

The concept of interval variety can be applied to all interval classes of a scale at once. The most signficant of such properties is the highest interval variety, or [[#Maximum variety|maximum variety]]. Other properties might include the mean interval variety, median interval variety, and lowest interval variety. 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).

The concept of interval variety can be applied to all interval classes of a scale at once. The most signficant of such properties is the highest interval variety, or [[#Maximum variety|maximum variety]]. Other properties might include the mean interval variety, median interval variety, and lowest interval variety. In addition, '''strict variety''' ('''SV''') 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 {{nowrap|'''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~~.

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 {{nowrap| '''0''' + '''3''' {{=}} '''1''' + '''2''' }} will be minimum variety 3.

== Terminology ==

For abstract scale words, the standard academic counterpart to the xen term ''variety'' is the ''abelian complexity function of a [[word]]'': a function ρab: {{nowrap|'''N''' → '''N'''}} where ~~ρ~~ab(''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.

For abstract scale words, the standard academic counterpart to the xen term ''variety'' is the ''abelian complexity function of a [[word]]'': a function ρab: {{nowrap| '''N''' → '''N''' }} where ρab(''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.

== Maximum variety ==

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* 9 notes: none

* 10 notes: '''0010020302''' and '''0102103012'''

* 11 to 14 notes: none

* 11 to 15 notes: none

(Note that abstract SV4-ness implies that a scale pattern is [[primitive]], or single-period.)

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== Open questions ==

* Why are (abstractly) SV4 scale patterns seemingly so rare?

** Conjecture: There are only finitely many SV4/MV4 [[circular word]]s.

** Conjecture: There are only finitely many SV4/MV4 quaternary [[circular word]]s.

** Conjecture: For all ''n'' greater than a sufficiently large ''m'', the longest abstractly SV-''n'' word is '''0123'''…({{nowrap| '''''n'' − 2''' }})({{nowrap| '''''n'' − 1''' }})({{nowrap| '''''n'' − 2''' }})…'''3210''', with length {{nowrap| 2''n'' − 1 }}.

** The following conjecture may be key to proving the ones above: For a sufficiently long [[arity|ternary]] noncircular word, there exists {{nowrap| ''k'' > 1 }} such that the interval class of ''k''-steps has at least 3 sizes and the interval class of {{nowrap|(''k'' − 1)}}-steps also has at least 3 sizes.

@@ Line 5: / Line 5: @@
 The '''interval variety''' of an [[interval class]] in a [[scale]] is the number of different [[interval quality|interval qualities]] available for that interval class. For example, the interval class "fifth" in the [[5L 2s|diatonic]] scale has interval variety 2, because there are two sizes of fifths in that scale: 6 perfect fifths and 1 diminished fifth.
-The concept of interval variety can be applied to all interval classes of a scale at once. The most signficant of such properties is the highest interval variety, or [[#Maximum variety|maximum variety]]. Other properties might include the mean interval variety, median interval variety, and lowest interval variety. 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).
+The concept of interval variety can be applied to all interval classes of a scale at once. The most signficant of such properties is the highest interval variety, or [[#Maximum variety|maximum variety]]. Other properties might include the mean interval variety, median interval variety, and lowest interval variety. In addition, '''strict variety''' ('''SV''') 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 {{nowrap|'''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.
+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 {{nowrap| '''0''' + '''3''' {{=}} '''1''' + '''2''' }} will be minimum variety 3.
 == Terminology ==
-For abstract scale words, the standard academic counterpart to the xen term ''variety'' is the ''abelian complexity function of a [[word]]'': a function &rho;<sup>ab</sup>: {{nowrap|'''N''' → '''N'''}} where &rho;<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.
+For abstract scale words, the standard academic counterpart to the xen term ''variety'' is the ''abelian complexity function of a [[word]]'': a function &rho;<sup>ab</sup>: {{nowrap| '''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.
 == Maximum variety ==
@@ Line 148: / Line 148: @@
 * 9 notes: none
 * 10 notes: '''0010020302''' and '''0102103012'''
-* 11 to 14 notes: none
+* 11 to 15 notes: none
 (Note that abstract SV4-ness implies that a scale pattern is [[primitive]], or single-period.)
@@ Line 159: / Line 159: @@
 == Open questions ==
 * Why are (abstractly) SV4 scale patterns seemingly so rare?
-** Conjecture: There are only finitely many SV4/MV4 [[circular word]]s.
+** Conjecture: There are only finitely many SV4/MV4 quaternary [[circular word]]s.
 ** Conjecture: For all ''n'' greater than a sufficiently large ''m'', the longest abstractly SV-''n'' word is '''0123'''…({{nowrap| '''''n'' − 2''' }})({{nowrap| '''''n'' − 1''' }})({{nowrap| '''''n'' − 2''' }})…'''3210''', with length {{nowrap| 2''n'' − 1 }}.
 ** The following conjecture may be key to proving the ones above: For a sufficiently long [[arity|ternary]] noncircular word, there exists {{nowrap| ''k'' > 1 }} such that the interval class of ''k''-steps has at least 3 sizes and the interval class of {{nowrap|(''k'' − 1)}}-steps also has at least 3 sizes.