Interval variety: Difference between revisions
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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. 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 == | == 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''' | 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 == | ||
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An [[interval class]] is the set of all interval sizes of a scale which span a given number of steps. For example, in the [[5L_2s|diatonic scale]], the 2-step class consists of the minor third and the major third, because those are the only intervals that are divided into 2 steps of the scale. The "variety" of an interval class is the number of different intervals in it. The maximum variety of a scale is simply the maximum variety of any interval class. | An [[interval class]] is the set of all interval sizes of a scale which span a given number of steps. For example, in the [[5L_2s|diatonic scale]], the 2-step class consists of the minor third and the major third, because those are the only intervals that are divided into 2 steps of the scale. The "variety" of an interval class is the number of different intervals in it. The maximum variety of a scale is simply the maximum variety of any interval class. | ||
Any scale with all equal steps (such as an [[ | Any scale with all equal steps (such as an [[edo]]) has maximum variety 1. All [[mos scale]]s have maximum variety 2. An example of a scale with high maximum variety is the [[harmonic series]], because the steps get gradually smaller as you go up the scale, and none of them are equal. | ||
== Maximum variety 3 scales == | == Maximum-variety-3 scales == | ||
The commonly discussed [[ | 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]]. | ||
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]]. | ||
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Once you have chosen a rank-3 temperament and a specific generator interval, there is a mechanical procedure to generate all max-variety-3 scales of a certain size (of which there are, however, infinitely many). | Once you have chosen a rank-3 temperament and a specific generator interval, there is a mechanical procedure to generate all max-variety-3 scales of a certain size (of which there are, however, infinitely many). | ||
To illustrate this process, let us use the simplest and most familiar rank-3 system: [[ | To illustrate this process, let us use the simplest and most familiar rank-3 system: [[5-limit]] JI, and let us use 3/2 as a generator. Because 3/2 is the generator, these max-variety-3 scales will be related (albeit not in a simple 1-to-1 way) with mos scales of Pythagorean tuning, the rank-2 temperament with 3/2 as a generator. | ||
Numbers of notes in Pythagorean | Numbers of notes in Pythagorean mos scales follow the pattern {{nowrap| 1 + 1 {{=}} 2 }}, {{nowrap| 2 + 1 {{=}} 3 }}, {{nowrap| 2 + 3 {{=}} 5 }}, {{nowrap| 5 + 2 {{=}} 7 }}, {{nowrap| 5 + 7 {{=}} 12 }}… (related to the continued fraction for log<sub>2</sub>(3/2)). The numbers of notes in the 5-limit max-variety-3 scales we are constructing will be related to these by having one of the numbers repeated. Therefore the possible sizes are: | ||
: {{nowrap|1 + 1 + 1 {{=}} 3}}, {{nowrap|2 + 1 + 1 {{=}} 4}}, {{nowrap|2 + 2 + 1 {{=}} 5}}, {{nowrap|2 + 2 + 3 {{=}} 7}}, {{nowrap|2 + 3 + 3 {{=}} 8}}, {{nowrap|5 + 2 + 2 {{=}} 9}}, {{nowrap|5 + 5 + 2 {{=}} 12}}, {{nowrap|5 + 5 + 7 {{=}} 17}}, {{nowrap|5 + 7 + 7 {{=}} 19}} | : {{nowrap| 1 + 1 + 1 {{=}} 3 }}, {{nowrap| 2 + 1 + 1 {{=}} 4 }}, {{nowrap| 2 + 2 + 1 {{=}} 5 }}, {{nowrap| 2 + 2 + 3 {{=}} 7 }}, {{nowrap| 2 + 3 + 3 {{=}} 8 }}, {{nowrap| 5 + 2 + 2 {{=}} 9 }}, {{nowrap| 5 + 5 + 2 {{=}} 12 }}, {{nowrap| 5 + 5 + 7 {{=}} 17 }}, {{nowrap| 5 + 7 + 7 {{=}} 19 }}, … | ||
If the number of notes is even, the max-variety-3 scale consists of two chains of 3/2 of equal length, each of which contains half the notes. If the number of notes is odd, the two 3/2-chains differ by 1 in length. | If the number of notes is even, the max-variety-3 scale consists of two chains of 3/2 of equal length, each of which contains half the notes. If the number of notes is odd, the two 3/2-chains differ by 1 in length. | ||
Any scale at all with only 3 notes has max variety 3, so let | Any scale at all with only 3 notes has max variety 3, so let us 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 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 | 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 ({{nowrap| 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 us 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 | 1/1 9/8 40/27 3/2 160/81 … max variety 3 | ||
1/1 10/9 9/8 40/27 3/2 | 1/1 10/9 9/8 40/27 3/2 … fails, max variety 4 | ||
1/1 10/9 9/8 3/2 5/3 | 1/1 10/9 9/8 3/2 5/3 … fails, max variety 4 | ||
1/1 9/8 5/4 3/2 5/3 | 1/1 9/8 5/4 3/2 5/3 … max variety 3 | ||
1/1 9/8 5/4 3/2 15/8 | 1/1 9/8 5/4 3/2 15/8 … fails, max variety 5 | ||
1/1 9/8 45/32 3/2 15/8 | 1/1 9/8 45/32 3/2 15/8 … max variety 3 | ||
1/1 135/128 9/8 45/32 3/2 | 1/1 135/128 9/8 45/32 3/2 … fails, max variety 4 | ||
1/1 135/128 9/8 3/2 405/256 | 1/1 135/128 9/8 3/2 405/256 … fails, max variety 4 | ||
1/1 9/8 1215/1024 3/2 405/256 | 1/1 9/8 1215/1024 3/2 405/256 … max variety 3 | ||
=== Examples testing for MV === | === Examples testing for MV === | ||
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# LLL, LLs | # LLL, LLs | ||
Actually we only have to check up to halfway, because all of the generic interval classes beyond this are period complements of the ones we already checked. And so it | Actually we only have to check up to halfway, because all of the generic interval classes beyond this are period complements of the ones we already checked. And so it is confirmed: this is an MV2. | ||
===== Negative ===== | ===== Negative ===== | ||
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# LLL, LLs, sLs — stop! | # LLL, LLs, sLs — stop! | ||
We | We have found that for the generic interval class 3, this scale has three different specific intervals, so it is not MV2. | ||
==== MV3 ==== | ==== MV3 ==== | ||
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===== Conditional ===== | ===== Conditional ===== | ||
[[File:MV3 conditional.png|200px|thumb|right|'''Conditionally MV3''': For the 2nd and 3rd generic interval classes, if the condition that MM=Ls is met, then this scale has a maximum of 3 different sizes; otherwise it has 4 and is therefore not MV3.]] | [[File:MV3 conditional.png|200px|thumb|right|'''Conditionally MV3''': For the 2nd and 3rd generic interval classes, if the condition that {{nowrap| MM = Ls }} is met, then this scale has a maximum of 3 different sizes; otherwise it has 4 and is therefore not MV3.]] | ||
How about the 2L 3M 2s scale with pattern LMMsMLs? | How about the 2L 3M 2s scale with pattern LMMsMLs? | ||
# L, M, s | # L, M, s | ||
# LM, MM, Ms, Ls — stop! | # LM, MM, Ms, Ls — stop! … but wait. What if MM=Ls? Then actually this would still be only 3 specific intervals. So let's go with that, and continue. | ||
# LMM=LLs, MMs, LMs | # LMM=LLs, MMs, LMs | ||
So this scale is Conditionally MV3 (MM=Ls). | So this scale is Conditionally MV3 ({{nowrap| MM = Ls }}). | ||
== Facts about variety properties of scale words == | == Facts about variety properties of scale words == | ||
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'''…({{nowrap| '''''n'' − 1''' }}) is SV-''n''.}} | ||
{{Proof|contents=All ''k''-letter subwords of '''0123''' | {{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'''…({{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''' | 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 ({{nowrap|'''r | Now consider the ''k''-letter noncircular subword ({{nowrap| '''r − 1''' }})({{nowrap| '''r − 2''' }})…'''210012'''…({{nowrap| '''s − 2''' }})({{nowrap| '''s − 1''' }}), where {{nowrap| ''r'', ''s'' ≥ 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'' ≥ ''k'' }}. | ||
A similar thing goes with ({{nowrap|'''''n'' | 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 ''k''/2 subwords, each corresponding to a solution of {{nowrap| ''r'' + ''s'' − 1 ≥ ''k'' }}. | ||
Thus there are | Thus there are | ||
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* 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''' | ** 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 (''k'' | ** 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. | ||
[[Category:Combinatorics on words]] | [[Category:Combinatorics on words]] | ||