Interval variety - Revision history

Inthar: /* Open questions */

2026-03-19T00:08:54Z

Open questions

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	== Open questions ==		== Open questions ==
	* Why are (abstractly) SV4 scale patterns seemingly so rare?		* 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 }}.		** 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.		** 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.

Inthar: /* Abstractly SV4 scale patterns */

2026-01-06T15:55:15Z

Abstractly SV4 scale patterns

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	* 9 notes: none		* 9 notes: none
	* 10 notes: '''0010020302''' and '''0102103012'''		* 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.)		(Note that abstract SV4-ness implies that a scale pattern is [[primitive]], or single-period.)

FloraC: Note the abbreviation SV

2025-07-09T11:01:23Z

Note the abbreviation SV

← Older revision		Revision as of 11:01, 9 July 2025
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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 '''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.		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.

FloraC: - explanations that only make sense to programmers

2025-07-07T14:37:20Z

- explanations that only make sense to programmers

← Older revision		Revision as of 14:37, 7 July 2025
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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 '''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''' 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 ==		== 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>: {{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.		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>: {{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 ==		== Maximum variety ==

FloraC: /* Maximum-variety-3 scales */ Improve wording a bit

2025-07-07T14:19:51Z

Maximum-variety-3 scales: Improve wording a bit

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	== Maximum-variety-3 scales ==		== Maximum-variety-3 scales ==
	The commonly discussed [[mos]] property can be characterized as follows, [[MOS scale #~~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]]~~.		The commonly discussed [[mos]] property can be characterized as follows, as well as in [[MOS scale #Equivalent definitions and generalizations\|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]], generalize to ternary scales differently.

	There is a theorem classifying all possible MV3 scales; see [[Ternary scale theorems]].		There is a theorem classifying all possible MV3 scales; see [[Ternary scale theorems]].

FloraC: Style & fixes

2025-07-07T13:46:32Z

Style & fixes

Show changes

FloraC: Maximum variety deserves its own section

2025-07-07T13:30:35Z

Maximum variety deserves its own section

← Older revision		Revision as of 13:30, 7 July 2025
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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 '''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. ~~Here are some~~ such properties:		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).
	* Highest interval variety ~~(see also~~ [[maximum variety]])
	* Mean interval variety
	* Median interval variety
	* 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).

	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. In other words, variety terms ''are overloaded and have different "types"'', namely <code>ConcreteScale -> Boolean</code> and <code>AbstractScaleWord -> Boolean</code>, depending on context.
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	== Terminology ==		== 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>: {{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.		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>: {{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 ==
	The '''maximum variety''' ('''MV''') of a scale is a way of quantifying how many different "flavors" of intervals there are in it. Scales with high maximum variety have many different intervals of similar size occuring at different places in the scale. Scales with low maximum variety may be easier for composers and listeners to understand, because there is more uniformity and consistence between different parts of the scale. In a low maximum variety scale, simply knowing how many scale steps an interval spans gives you a lot of information about the interval (by narrowing it down to a small set of choices).		The '''maximum variety''' ('''MV''') of a scale is a way of quantifying how many different "flavors" of intervals there are in it. Scales with high maximum variety have many different intervals of similar size occuring at different places in the scale. Scales with low maximum variety may be easier for composers and listeners to understand, because there is more uniformity and consistence between different parts of the scale. In a low maximum variety scale, simply knowing how many scale steps an interval spans gives you a lot of information about the interval (by narrowing it down to a small set of choices).

Recentlymaterialized: 8:9:10:12:15:16 is MV5

2025-01-14T19:54:47Z

8:9:10:12:15:16 is MV5

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	Any scale at all with only 3 notes has max variety 3, so let's begin with the 4-note scales. A 4-note scale consists of two parallel 3/2 intervals separated by some other 5-limit interval. Although strictly speaking this could be any other 5-limit interval at all, if the two chains are separated from each other there will be no potential for 5-limit harmony. For these 4-note scales it turns out that all the configurations work, so we can easily list all the possible scales: ...{{nowrap\|{{(}}1/1, 10/9, 3/2, 5/3{{)}}}}, {{nowrap\|{{(}}1/1, 5/4, 3/2, 5/3{{)}}}}, {{nowrap\|{{(}}1/1, 5/4, 3/2, 15/8{{)}}}}, {{nowrap\|{{(}}1/1, 45/32, 3/2, 125/64{{)}}}}... Only the ones where the two chains are lined up closely have good potential for harmony.		Any scale at all with only 3 notes has max variety 3, so let's begin with the 4-note scales. A 4-note scale consists of two parallel 3/2 intervals separated by some other 5-limit interval. Although strictly speaking this could be any other 5-limit interval at all, if the two chains are separated from each other there will be no potential for 5-limit harmony. For these 4-note scales it turns out that all the configurations work, so we can easily list all the possible scales: ...{{nowrap\|{{(}}1/1, 10/9, 3/2, 5/3{{)}}}}, {{nowrap\|{{(}}1/1, 5/4, 3/2, 5/3{{)}}}}, {{nowrap\|{{(}}1/1, 5/4, 3/2, 15/8{{)}}}}, {{nowrap\|{{(}}1/1, 45/32, 3/2, 125/64{{)}}}}... Only the ones where the two chains are lined up closely have good potential for harmony.

	The 5-note scales will consist of a chain of 3 notes and a parallel chain of 2 notes. The 3-note chain has the ~~same~~ pattern as {{nowrap\|{{(}}1/1, 9/8, 3/2{{)}}}}, and ~~in order for it~~ to make ~~an actual~~ max-variety-3 scale, the other 2 notes must ~~fall in~~ between 9/8 and 3/2, and in between 3/2 and 2/1. If either of the notes falls between 1/1 and 9/8, the scale will not be max-variety-3. Let's look at all the scales we get as we move the 2-note chain past the 3-note chain in the 5-limit lattice.		The 5-note scales will consist of a chain of 3 notes and a parallel chain of 2 notes. The 3-note chain has the pattern {{nowrap\|{{(}}1/1, 9/8, 3/2{{)}}}}, and to make a max-variety-3 scale, the other chain of 2 notes must be separated by 4/3, falling between the large (L = 4/3) steps of the 3-note chain (between 9/8 and 3/2, and between 3/2 and 2/1). If either of the notes falls between 1/1 and 9/8, the scale will not be max-variety-3. Let's look at all the scales we get as we move the 2-note chain past the 3-note chain in the 5-limit lattice.

	1/1 9/8 40/27 3/2 160/81 ... max variety 3		1/1 9/8 40/27 3/2 160/81 ... max variety 3
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	1/1 9/8 5/4 3/2 5/3 ... max variety 3		1/1 9/8 5/4 3/2 5/3 ... max variety 3

	1/1 9/8 5/4 3/2 15/8 ... max variety 3		1/1 9/8 5/4 3/2 15/8 ... fails, max variety 5

	1/1 9/8 45/32 3/2 15/8 ... max variety 3		1/1 9/8 45/32 3/2 15/8 ... max variety 3

FloraC: +interwiki

2024-12-18T08:30:05Z

+interwiki

← Older revision		Revision as of 08:30, 18 December 2024
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			{{Interwiki
			\| en = Interval variety
			\| ro = Game N-intervalare
			}}
	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 '''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.

ArrowHead294 at 11:02, 11 August 2024

2024-08-11T11:02:31Z

← Older revision		Revision as of 11:02, 11 August 2024
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	== Terminology ==		== 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>: {{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.		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>: {{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.
	The '''maximum variety''' ('''MV''') of a scale is a way of quantifying how many different "flavors" of intervals there are in it. Scales with high maximum variety have many different intervals of similar size occuring at different places in the scale. Scales with low maximum variety may be easier for composers and listeners to understand, because there is more uniformity and consistence between different parts of the scale. In a low maximum variety scale, simply knowing how many scale steps an interval spans gives you a lot of information about the interval (by narrowing it down to a small set of choices).		The '''maximum variety''' ('''MV''') of a scale is a way of quantifying how many different "flavors" of intervals there are in it. Scales with high maximum variety have many different intervals of similar size occuring at different places in the scale. Scales with low maximum variety may be easier for composers and listeners to understand, because there is more uniformity and consistence between different parts of the scale. In a low maximum variety scale, simply knowing how many scale steps an interval spans gives you a lot of information about the interval (by narrowing it down to a small set of choices).

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	In the following, two letters are to be considered the same if their numerical values are congruent modulo ''n''.		In the following, two letters are to be considered the same if their numerical values are congruent modulo ''n''.

	{{theorem\|contents=For all ''n'' ≥ 1, the word '''0123'''...('''''n''-1''') is SV''n''.}}		{{theorem\|contents=For all ''n'' ≥ 1, the word '''0123'''...({{nowrap\|'''''n'' − 1'''}}) is SV''n''.}}

	{{Proof\|contents=All ''k''-letter subwords of '''0123'''...('''''n''-1''') is of the form ('''i''')('''i+1''')...('''i+k-1'''), and there are exactly ''n'' of them.}}		{{Proof\|contents=All ''k''-letter subwords of '''0123'''...({{nowrap\|'''''n'' − 1'''}}) is of the form ('''i''')({{nowrap\|'''i + 1'''}})...({{nowrap\|'''i + k − 1'''}}), and there are exactly ''n'' of them.}}

	{{theorem\|contents=For all ''n'' ≥ 1, the word '''0123'''...('''''n''-2''')('''''n''-1''')('''''n''-2''')...'''3210''' is SV''n''.}}		{{theorem\|contents=For all ''n'' ≥ 1, the word '''0123'''...({{nowrap\|'''''n'' − 2'''}})({{nowrap\|'''''n'' − 1'''}})({{nowrap\|'''''n'' − 2'''}})...'''3210''' is SV''n''.}}

	{{Proof\|contents=We prove this by dividing this word into four overlapping noncircular subwords which cover all cases.		{{Proof\|contents=We prove this by dividing this word into four overlapping noncircular subwords which cover all cases.

	Consider the subwords '''0123'''...('''''n''-2''')('''''n''-1''') and ('''''n''-1''')('''''n''-2''')...'''3210'''. If we treat these two words as noncircular, then there are ''n''-''k'' distinct ''k''-letter subwords.		Consider the subwords '''0123'''...({{nowrap\|'''''n'' − 2'''}})({{nowrap\|'''''n'' − 1'''}}) and ({{nowrap\|'''''n'' − 1'''}})({{nowrap\|'''''n'' − 2'''}})...'''3210'''. If we treat these two words as noncircular, then there are {{nowrap\|''n'' − ''k''}} distinct ''k''-letter subwords.

	Now consider the ''k''-letter noncircular subword ('''r-1''')('''r-2''')...'''210012'''...('''s-2''')('''s-1'''), where ''r'', ''s'' ≥ 1. Note that interchanging ''r'' and ''s'' yields equivalent subwords. If ''k'' is odd, there are (''k''-1)/2 distinct subwords of this kind; if ''k'' is even, there are ''k''/2 subwords, each corresponding to a solution of ''r'' + ''s'' ≥ ''k''.		Now consider the ''k''-letter noncircular subword ({{nowrap\|'''r − 1'''}})({{nowrap\|'''r − 2'''}})...'''210012'''...({{nowrap\|'''s − 2'''}})({{nowrap\|'''s − 1'''}}), where {{nowrap\|''r'', ''s'' &geqslant; 1}}. Note that interchanging ''r'' and ''s'' yields equivalent subwords. If ''k'' is odd, there are ({{nowrap\|''k'' − 1}})/2 distinct subwords of this kind; if ''k'' is even, there are ''k''/2 subwords, each corresponding to a solution of {{nowrap\|''r'' + ''s'' &geqslant; ''k''}}.

	A similar thing goes with ('''''n''-''r''-1''')...('''''n''-2''')('''''n''-1''')('''''n''-2''')..('''''n''-''s''-1'''). If ''k'' is odd, there are (''k''+1)/2 distinct subwords of this kind; if ''k'' is even, there are ''k''/2 subwords, each corresponding to a solution of ''r'' + ''s'' - 1 ≥ ''k''.		A similar thing goes with ({{nowrap\|'''''n'' − ''r'' − 1'''}})...({{nowrap\|'''''n'' − 2'''}})({{nowrap\|'''''n'' − 1'''}})({{nowrap\|'''''n'' − 2'''}})..({{nowrap\|'''''n'' − ''s'' − 1'''}}). If ''k'' is odd, there are ({{nowrap\|''k'' + 1}})/2 distinct subwords of this kind; if ''k'' is even, there are {{nowrap\|''k'' / 2}} subwords, each corresponding to a solution of {{nowrap\|''r'' + ''s'' − 1 ≥ ''k''}}.

	Thus there are		Thus there are

	~~(''~~n'' - ''k~~'')~~ + ~~(''~~k''-1)/2 + ~~(''~~k''+1)/2 = ''n'' (''k'' odd)		<math>n - k + \frac{k - 1}{2} + \frac{k + 1}{2} = n \qquad \left(k\,\text{odd}\right)</math>

	(''n'' - ''k'') + 2(''k~~''/~~2) = ''n'' (''k'' even)		<math>(n - k) + 2\left(\frac{k}{2}\right) = n \qquad \left(k\,\text{even}\right)</math>

	distinct ''k''-letter subwords, so the word is SV''n''.}}		distinct ''k''-letter subwords, so the word is SV''n''.}}
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	Conjecture: there are no SV4 scale patterns with more than 10 notes.		Conjecture: there are no SV4 scale patterns with more than 10 notes.

	=== Maximum variety of balanced words ===		=== Maximum variety of balanced words ===
	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 {\left\lceil N/2 \right\rceil}</math>.

	== Open questions ==		== Open questions ==
	* Why are (abstractly) SV4 scale patterns seemingly so rare?		* 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 [[circular word]]s.
	** Conjecture: For all ''n'' greater than a sufficiently large ''m'', the longest abstractly SV''n'' word is '''0123'''...('''''n''−2''')('''''n''−1''')('''''n''−2''')...'''3210''', with length 2''n'' - 1.		** 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 ''k'' > 1 such that the interval class of ''k''-steps has at least 3 sizes and the interval class of (''k'' − 1)-steps also has at least 3 sizes.		** 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 (''k'' − 1)-steps also has at least 3 sizes.

	[[Category:Combinatorics on words]]		[[Category:Combinatorics on words]]
	[[Category:Scale]]		[[Category:Scale]]

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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 '''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.		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.