2L 3s: Difference between revisions

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== Scale tree ==
== Scale tree ==
Generator ranges:
{{Todo| expand |inline=1|comment=Add back entries from original scale tree.}}
* Chroma-positive generator: 480 cents (2\5) to 600 cents (1\2)
{{Scale tree}}
* Chroma-negative generator: 600 cents (1\2) to 720 cents (3\5)
 
{{Todo| cleanup |inline=1|comment=Rework this scale tree.}}
 
{| class="wikitable"
|-
! colspan="6" | Generator
! | Cents
! | s
! | L-s
! | |L-2s|
! | Scale steps
! | Trichord
! | Comments
|-
| | 2\5
| |
| |
| |
| |
| |
| | 480
| | 240
| | 0
| | 240
| | 1 1 1 1 1
| | 1 1
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| | 11\27
| | 488.89
| | 222.22
| | 44.44
| | 177.78
| | 6 5 5 6 5
| | 6 5
| style="text-align:center;" | Slendro (insofar as it resembles a MOS)
 
would be in this region
|-
| |
| |
| |
| |
| | 9\22
| |
| | 490.91
| | 218.18
| | 54.545
| | 163.64
| | 5 4 4 5 4
| | 5 4
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| | 16\39
| | 492.31
| | 215.38
| | 61.54
| | 153.85
| | 9 7 7 9 7
| | 9 7
| style="text-align:center;" | No-5's superpyth/dominant is around here
|-
| |
| |
| |
| | 7\17
| |
| |
| | 494.12
| | 211.76
| | 70.59
| | 141.18
| | 4 3 3 4 3
| | 4 3
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| | 19\46
| | 495.65
| | 208.7
| | 78.26
| | 130.435
| | 11 8 8 11 8
| | 11 8
| |
|-
| |
| |
| |
| |
| | 12\29
| |
| | 496.55
| | 206.9
| | 82.76
| | 124.14
| | 7 5 5 7 5
| | 7 5
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| | 17\41
| | 497.56
| | 204.88
| | 87.8
| | 117.07
| | 10 7 7 10 7
| | 10 7
| style="text-align:center;" | Pythagorean pentatonic is around here
|-
| |
| |
| | 5\12
| |
| |
| |
| | 500
| | 200
| | 100
| | 100
| | 3 2 2 3 2
| | 3 2
| style="text-align:center;" | Familiar 12-equal pentatonic
 
(also optimum rank range: L/s=3/2)
|-
| |
| |
| |
| |
| |
| |
| | 502.305
| | 195.39
| | 111.53
| | 83.86
| | pi 2 pi 2 2
| | pi 2
| |
|-
| |
| |
| |
| |
| |
| | 18\43
| | 502.33
| | 195.35
| | 111.63
| | 83.72
| | 11 7 7 11 7
| | 11 7
| |
|-
| |
| |
| |
| |
| | 13\31
| |
| | 503.23
| | 193.55
| | 116.13
| | 77.42
| | 8 5 5 8 5
| | 8 5
| style="text-align:center;" | Optimal meantone pentatonic
 
is around here
|-
| |
| |
| |
| |
| |
| |
| | 1200/(4-phi)
| | 192.43
| | 118.93
| | 73.50
| | phi 1 1 phi 1
| | phi 1
| style="text-align:center;" | Golden meantone
|-
| |
| |
| |
| |
| |
| | 21\50
| | 504
| | 192
| | 120
| | 72
| | 13 8 8 13 8
| | 13 8
| style="text-align:center;" |
|-
| |
| |
| |
| | 8\19
| |
| |
| | 505.26
| | 189.47
| | 126.32
| | 63.16
| | 5 3 3 5 3
| | 5 3
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| | 19\45
| | 506.67
| | 186.67
| | 133.33
| | 53.33
| | 12 7 7 12 7
| | 12 7
| |
|-
| |
| |
| |
| |
| |
| |
| | 507.18
| | 185.64
| | 135.9
| | 49.74
| | √3 1 √3 1 1
| | √3 1
| |
|-
| |
| |
| |
| |
| | 11\26
| |
| | 507.69
| | 184.615
| | 138.46
| | 46.15
| | 7 4 4 7 4
| | 7 4
| |
|-
| |
| |
| |
| |
| |
| | 14\33
| | 509.09
| | 181.82
| | 145.455
| | 36.36
| | 9 5 5 9 5
| | 9 5
| |
|-
| |
| | 3\7
| |
| |
| |
| |
| | 514.29
| | 171.43
| | 171.43
| | 0
| | 2 1 1 2 1
| | 2 1
| style="text-align:center;" | (Boundary of propriety: smaller
 
generators than this are strictly proper)
|-
| |
| |
| |
| |
| |
| | 13\30
| | 520
| | 160
| | 200
| | 40
| | 9 4 4 9 4
| | 9 4
| |
|-
| |
| |
| |
| |
| | 10\23
| |
| | 521.74
| | 156.52
| | 208.7
| | 52.17
| | 7 3 3 7 3
| | 7 3
| |
|-
| |
| |
| |
| |
| |
| | 17\39
| | 523.08
| | 153.84
| | 215.385
| | 61.54
| | 12 5 5 12 5
| | 12 5
| |
|-
| |
| |
| |
| | 7\16
| |
| |
| | 525
| | 150
| | 225
| | 75
| | 5 2 2 5 2
| | 5 2
| style="text-align:center;" | 5-note subset of pelog (insofar as it
 
resembles a MOS) would be in this region
|-
| |
| |
| |
| |
| |
| | 18\41
| | 526.83
| | 146.34
| | 234.15
| | 87.8
| | 13 5 5 13 5
| | 13 5
| |
|-
| |
| |
| |
| |
| |
| |
| | 600(25+√5)/31
| | 145.7
| | 235.75
| | 90.05
| | phi+1 1 1 phi+1 1
| | phi+1 1
| |
|-
| |
| |
| |
| |
| | 11\25
| |
| | 528
| | 144
| | 240
| | 96
| | 8 3 3 8 3
| | 8 3
| |
|-
| |
| |
| |
| |
| |
| |
| | 528.88
| | 142.24
| | 244.405
| | 102.17
| | e 1 e 1 1
| | e 1
| style="text-align:center;" | L/s = e
|-
| |
| |
| |
| |
| |
| | 15\34
| | 529.41
| | 141.18
| | 247.06
| | 105.88
| | 11 4 4 11 4
| | 11 4
| |
|-
| |
| |
| | 4\9
| |
| |
| |
| | 533.33
| | 133.33
| | 266.67
| | 133.33
| | 3 1 1 3 1
| | 3 1
| style="text-align:center;" | L/s = 3
|-
| |
| |
| |
| |
| |
| |
| | 535.36
| | 129.26
| | 276.835
| | 147.57
| | pi 1 pi 1 1
| | pi 1
| style="text-align:center;" | <span style="display: block; text-align: center;">L/s = pi</span>
|-
| |
| |
| |
| |
| |
| | 13\29
| | 537.93
| | 124.14
| | 289.655
| | 165.52
| | 10 3 3 10 3
| | 10 3
| |
|-
| |
| |
| |
| |
| | 9\20
| |
| | 540
| | 120
| | 240
| | 180
| | 7 2 2 7 2
| | 7 2
| |
|-
| |
| |
| |
| |
| |
| | 14\31
| | 541.935
| | 116.13
| | 309.68
| | 193.55
| | 11 3 3 11 3
| | 11 3
| |
|-
| |
| |
| |
| | 5\11
| |
| |
| | 545.45
| | 109.09
| | 327.27
| | 218.18
| | 4 1 1 4 1
| | 4 1
| style="text-align:center;" | L/s = 4
|-
| |
| |
| |
| |
| |
| | 11\24
| | 550
| | 100
| | 350
| | 250
| | 9 2 2 9 2
| | 9 2
| |
|-
| |
| |
| |
| |
| | 6\13
| |
| | 553.85
| | 92.31
| | 369.23
| | 276.92
| | 5 1 1 5 1
| | 5 1
| |
|-
| |
| |
| |
| |
| |
| | 7\15
| | 560
| | 80
| | 480
| | 400
| | 6 1 1 6 1
| | 6 1
| |
|-
| | 1\2
| |
| |
| |
| |
| |
| | 600
| | 0
| | 600
| | 600
| | 1 0 0 1 0
| | 1 0
| style="text-align:center;" | a degenerated pentatonic scale with only 2 different steps
|}


From a [[3-limit]] perspective, just make a chain of four 4/3's and octave-reduce, and you end up with pentatonic.
From a [[3-limit]] perspective, just make a chain of four 4/3's and octave-reduce, and you end up with pentatonic.

Revision as of 11:42, 3 February 2024

↖ 1L 2s ↑ 2L 2s 3L 2s ↗
← 1L 3s 2L 3s 3L 3s →
↙ 1L 4s ↓ 2L 4s 3L 4s ↘
Scale structure
Step pattern LsLss
ssLsL
Equave 2/1 (1200.0 ¢)
Period 2/1 (1200.0 ¢)
Generator size
Bright 2\5 to 1\2 (480.0 ¢ to 600.0 ¢)
Dark 1\2 to 3\5 (600.0 ¢ to 720.0 ¢)
Related MOS scales
Parent 2L 1s
Sister 3L 2s
Daughters 5L 2s, 2L 5s
Neutralized 4L 1s
2-Flought 7L 3s, 2L 8s
Equal tunings
Equalized (L:s = 1:1) 2\5 (480.0 ¢)
Supersoft (L:s = 4:3) 7\17 (494.1 ¢)
Soft (L:s = 3:2) 5\12 (500.0 ¢)
Semisoft (L:s = 5:3) 8\19 (505.3 ¢)
Basic (L:s = 2:1) 3\7 (514.3 ¢)
Semihard (L:s = 5:2) 7\16 (525.0 ¢)
Hard (L:s = 3:1) 4\9 (533.3 ¢)
Superhard (L:s = 4:1) 5\11 (545.5 ¢)
Collapsed (L:s = 1:0) 1\2 (600.0 ¢)
View • Talk • Edit
For the 3/2-equivalent 2L 3s pattern, see 2L 3s (3/2-equivalent).

2L 3s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 2 large steps and 3 small steps, repeating every octave. Generators that produce this scale range from 480 ¢ to 600 ¢, or from 600 ¢ to 720 ¢. This scale is the "Classic" pentatonic. Perhaps the most common scale in the world.

The meantone pentatonic scale, in which the generator approximates 4/3 but other intervals in the scale approximate 6/5 and 5/4, has by far the lowest harmonic entropy of all 5-note MOS scales, which explains the worldwide popularity of these scales and their very long history of use. It is also strictly proper.

Names

The TAMNAMS system suggests the name pentic, derived from an informal clipping of "pentatonic" that is sometimes used to refer to this scale.

Modes

  • 4|0 LsLss
  • 3|1 LssLs
  • 2|2 sLsLs
  • 1|3 sLssL
  • 0|4 ssLsL

Scales

Scale tree

Todo: expand

Add back entries from original scale tree.

Template: Scale tree is deprecated. Please use Template: MOS tuning spectrum instead. Details:
Use of a single Comments parameter has become unmaintainable. Existing scale trees should be migrated to the new template, where comments are entered using a step ratio p/q as a parameter: 
{{MOS tuning spectrum
| 3/2 = Example comment
| 4/3 = Another example comment
}}


The parameters tuning and depth have been replaced with Scale Signature and Depth, respectively.


Scale tree and tuning spectrum of 2L 3s
Generator(edo) Cents Step ratio Comments
Bright Dark L:s Hardness
2\5 480.000 720.000 1:1 1.000 Equalized 2L 3s
11\27 488.889 711.111 6:5 1.200
9\22 490.909 709.091 5:4 1.250
16\39 492.308 707.692 9:7 1.286
7\17 494.118 705.882 4:3 1.333 Supersoft 2L 3s
19\46 495.652 704.348 11:8 1.375
12\29 496.552 703.448 7:5 1.400
17\41 497.561 702.439 10:7 1.429
5\12 500.000 700.000 3:2 1.500 Soft 2L 3s
18\43 502.326 697.674 11:7 1.571
13\31 503.226 696.774 8:5 1.600
21\50 504.000 696.000 13:8 1.625
8\19 505.263 694.737 5:3 1.667 Semisoft 2L 3s
19\45 506.667 693.333 12:7 1.714
11\26 507.692 692.308 7:4 1.750
14\33 509.091 690.909 9:5 1.800
3\7 514.286 685.714 2:1 2.000 Basic 2L 3s
Scales with tunings softer than this are proper
13\30 520.000 680.000 9:4 2.250
10\23 521.739 678.261 7:3 2.333
17\39 523.077 676.923 12:5 2.400
7\16 525.000 675.000 5:2 2.500 Semihard 2L 3s
18\41 526.829 673.171 13:5 2.600
11\25 528.000 672.000 8:3 2.667
15\34 529.412 670.588 11:4 2.750
4\9 533.333 666.667 3:1 3.000 Hard 2L 3s
13\29 537.931 662.069 10:3 3.333
9\20 540.000 660.000 7:2 3.500
14\31 541.935 658.065 11:3 3.667
5\11 545.455 654.545 4:1 4.000 Superhard 2L 3s
11\24 550.000 650.000 9:2 4.500
6\13 553.846 646.154 5:1 5.000
7\15 560.000 640.000 6:1 6.000
1\2 600.000 600.000 1:0 → ∞ Collapsed 2L 3s

From a 3-limit perspective, just make a chain of four 4/3's and octave-reduce, and you end up with pentatonic.

From a 5-limit perspective, the most interesting temperaments with this kind of pentatonic scale are meantone and mavila.

There is also the interesting 2.3.7 temperament that tempers out 64/63 (archy, "no-fives dominant").

Retrieved from "https://en.xen.wiki/index.php?title=2L_3s&oldid=134199"