Interval variety: Difference between revisions

Line 9:

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. 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 ==

For abstract scale words, the standard academic counterpart to the xen term ''variety'' is the ''abelian complexity function of a [[word]]'': a function ρab : '''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.

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.

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).

Line 28:

Line 29:

To illustrate this process, let us use the simplest and most familiar rank-3 system: [[5-limit|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 MOSes of Pythagorean, the rank-2 temperament with 3/2 as a generator.

Numbers of notes in Pythagorean MOSes follow the pattern 1+1=2, 2+1=3, 2+3=5, 5+2=7, 5+7=12... (related to the continued fraction for ~~log2~~(3/2)). The numbers of notes in the 5-limit max-variety-3 scales we're constructing will be related to these by having one of the numbers repeated. Therefore the possible sizes are:

Numbers of notes in Pythagorean MOSes 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 log2(3/2)). The numbers of notes in the 5-limit max-variety-3 scales we're constructing will be related to these by having one of the numbers repeated. Therefore the possible sizes are:

1+1+1=3, 2+1+1=4, 2+2+1=5, 2+2+3=7, 2+3+3=8, 5+2+2=9, 5+5+2=12, 5+5+7=17, 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.

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: ...{1/1,10/9,3/2,5/3}, {1/1,5/4,3/2,5/3}, {1/1,5/4,3/2,15/8}, {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 {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 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.

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

Line 57:

Line 58:

=== Examples testing for MV ===

==== MV2 ====

===== Positive =====

[[File:MV2 positive.png|200px|thumb|right|'''MV2''': For each generic interval class, we find a maximum of 2 different sizes.]]

Line 73:

Line 71:

===== Negative =====

[[File:MV2 negative.png|200px|thumb|right|'''Not MV2''': For the 3rd generic interval class, we find 3 different sizes.]]

Line 85:

Line 82:

==== MV3 ====

===== Positive =====

[[File:MV3 positive.png|200px|thumb|right|'''MV3''': For each generic interval class, we find a maximum of 3 sizes.]]

Line 99:

Line 94:

===== Negative =====

[[File:MV3 negative.png|200px|thumb|right|'''Not MV3''': For the 2nd generic interval class, we find 4 different sizes.]]

Line 110:

Line 104:

===== 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.]]

How about the 2L 3M 2s scale with pattern LMMsMLs.

How about the 2L 3M 2s scale with pattern LMMsMLs?

# L, M, s

@@ Line 9: / Line 9: @@
 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. 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 ==
-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.
+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''' &rarr; '''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).
@@ Line 28: / Line 29: @@
 To illustrate this process, let us use the simplest and most familiar rank-3 system: [[5-limit|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 MOSes of Pythagorean, the rank-2 temperament with 3/2 as a generator.
-Numbers of notes in Pythagorean MOSes follow the pattern 1+1=2, 2+1=3, 2+3=5, 5+2=7, 5+7=12... (related to the continued fraction for log2(3/2)). The numbers of notes in the 5-limit max-variety-3 scales we're constructing will be related to these by having one of the numbers repeated. Therefore the possible sizes are:
+Numbers of notes in Pythagorean MOSes 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're constructing will be related to these by having one of the numbers repeated. Therefore the possible sizes are:
-+1+1=3, 2+1+1=4, 2+2+1=5, 2+2+3=7, 2+3+3=8, 5+2+2=9, 5+5+2=12, 5+5+7=17, 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.
-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: ...{1/1,10/9,3/2,5/3}, {1/1,5/4,3/2,5/3}, {1/1,5/4,3/2,15/8}, {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 {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 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.
 /1 9/8 40/27 3/2 160/81 ... max variety 3
@@ Line 57: / Line 58: @@
 === Examples testing for MV ===
 ==== MV2 ====
 ===== Positive =====
 [[File:MV2 positive.png|200px|thumb|right|'''MV2''': For each generic interval class, we find a maximum of 2 different sizes.]]
@@ Line 73: / Line 71: @@
 ===== Negative =====
 [[File:MV2 negative.png|200px|thumb|right|'''Not MV2''': For the 3rd generic interval class, we find 3 different sizes.]]
@@ Line 85: / Line 82: @@
 ==== MV3 ====
 ===== Positive =====
 [[File:MV3 positive.png|200px|thumb|right|'''MV3''': For each generic interval class, we find a maximum of 3 sizes.]]
@@ Line 99: / Line 94: @@
 ===== Negative =====
 [[File:MV3 negative.png|200px|thumb|right|'''Not MV3''': For the 2nd generic interval class, we find 4 different sizes.]]
@@ Line 110: / Line 104: @@
 ===== 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.]]
-How about the 2L 3M 2s scale with pattern LMMsMLs.
+How about the 2L 3M 2s scale with pattern LMMsMLs?
 # L, M, s