2L 3s

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↖ 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 ¢)
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 scale, which is 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.

Scale properties

This article uses TAMNAMS conventions for the names of this scale's intervals and scale degrees. The use of 1-indexed ordinal names is reserved for interval regions.

Intervals

Intervals of 2L 3s
Intervals Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-pentstep Perfect 0-pentstep P0ms 0 0.0 ¢
1-pentstep Minor 1-pentstep m1ms s 0.0 ¢ to 240.0 ¢
Major 1-pentstep M1ms L 240.0 ¢ to 600.0 ¢
2-pentstep Diminished 2-pentstep d2ms 2s 0.0 ¢ to 480.0 ¢
Perfect 2-pentstep P2ms L + s 480.0 ¢ to 600.0 ¢
3-pentstep Perfect 3-pentstep P3ms L + 2s 600.0 ¢ to 720.0 ¢
Augmented 3-pentstep A3ms 2L + s 720.0 ¢ to 1200.0 ¢
4-pentstep Minor 4-pentstep m4ms L + 3s 600.0 ¢ to 960.0 ¢
Major 4-pentstep M4ms 2L + 2s 960.0 ¢ to 1200.0 ¢
5-pentstep Perfect 5-pentstep P5ms 2L + 3s 1200.0 ¢

Generator chain

Generator chain of 2L 3s
Bright gens Scale degree Abbrev.
6 Augmented 2-pentdegree A2md
5 Augmented 0-pentdegree A0md
4 Augmented 3-pentdegree A3md
3 Major 1-pentdegree M1md
2 Major 4-pentdegree M4md
1 Perfect 2-pentdegree P2md
0 Perfect 0-pentdegree
Perfect 5-pentdegree		 P0md
P5md
−1 Perfect 3-pentdegree P3md
−2 Minor 1-pentdegree m1md
−3 Minor 4-pentdegree m4md
−4 Diminished 2-pentdegree d2md
−5 Diminished 5-pentdegree d5md
−6 Diminished 3-pentdegree d3md

Modes

Scale degrees of the modes of 2L 3s
UDP Cyclic
order 		Step
pattern 		Scale degree (pentdegree)
0 1 2 3 4 5
4|0 1 LsLss Perf. Maj. Perf. Aug. Maj. Perf.
3|1 3 LssLs Perf. Maj. Perf. Perf. Maj. Perf.
2|2 5 sLsLs Perf. Min. Perf. Perf. Maj. Perf.
1|3 2 sLssL Perf. Min. Perf. Perf. Min. Perf.
0|4 4 ssLsL Perf. Min. Dim. Perf. Min. Perf.

Mode names

There are three sets of mode names: descriptive, modal (5 of the 7 heptatonic modes), and traditional Chinese.

Modes of 2L 3s
UDP Cyclic
order		 Step
pattern		 Descriptive Modal Chinese
4|0 1 LsLss Fifthless Phrygian Jue
3|1 3 LssLs Minor Aeolian Yu
2|2 5 sLsLs Thirdless Minor* Dorian Shang
1|3 2 sLssL Thirdless Major* Mixolydian Zhi
0|4 4 ssLsL Major Ionian Gong

* Thirdless Minor/Major is also known as Suspended Minor/Major

Scales

Scale list

Scale tree

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
13\32 487.500 712.500 7:6 1.167
11\27 488.889 711.111 6:5 1.200 Slendro (insofar as it resembles a MOS) would
be in this region
20\49 489.796 710.204 11:9 1.222
9\22 490.909 709.091 5:4 1.250
25\61 491.803 708.197 14:11 1.273
16\39 492.308 707.692 9:7 1.286 No-5s superpyth/dominant is around here
23\56 492.857 707.143 13:10 1.300
7\17 494.118 705.882 4:3 1.333 Supersoft 2L 3s
26\63 495.238 704.762 15:11 1.364
19\46 495.652 704.348 11:8 1.375
31\75 496.000 704.000 18:13 1.385
12\29 496.552 703.448 7:5 1.400
29\70 497.143 702.857 17:12 1.417
17\41 497.561 702.439 10:7 1.429
22\53 498.113 701.887 13:9 1.444 Pythagorean pentatonic is around here
5\12 500.000 700.000 3:2 1.500 Soft 2L 3s
Familiar 12-equal pentatonic
23\55 501.818 698.182 14:9 1.556
18\43 502.326 697.674 11:7 1.571
31\74 502.703 697.297 19:12 1.583
13\31 503.226 696.774 8:5 1.600 Optimal meantone pentatonic is around here
34\81 503.704 696.296 21:13 1.615
21\50 504.000 696.000 13:8 1.625
29\69 504.348 695.652 18:11 1.636
8\19 505.263 694.737 5:3 1.667 Semisoft 2L 3s
27\64 506.250 693.750 17:10 1.700
19\45 506.667 693.333 12:7 1.714
30\71 507.042 692.958 19:11 1.727
11\26 507.692 692.308 7:4 1.750
25\59 508.475 691.525 16:9 1.778
14\33 509.091 690.909 9:5 1.800
17\40 510.000 690.000 11:6 1.833
3\7 514.286 685.714 2:1 2.000 Basic 2L 3s
Scales with tunings softer than this are proper
16\37 518.919 681.081 11:5 2.200
13\30 520.000 680.000 9:4 2.250
23\53 520.755 679.245 16:7 2.286
10\23 521.739 678.261 7:3 2.333
27\62 522.581 677.419 19:8 2.375
17\39 523.077 676.923 12:5 2.400
24\55 523.636 676.364 17:7 2.429
7\16 525.000 675.000 5:2 2.500 Semihard 2L 3s
Five-note subset of pelog (insofar as it
resembles a MOS) would be in this region
25\57 526.316 673.684 18:7 2.571
18\41 526.829 673.171 13:5 2.600
29\66 527.273 672.727 21:8 2.625
11\25 528.000 672.000 8:3 2.667
26\59 528.814 671.186 19:7 2.714
15\34 529.412 670.588 11:4 2.750
19\43 530.233 669.767 14:5 2.800
4\9 533.333 666.667 3:1 3.000 Hard 2L 3s
17\38 536.842 663.158 13:4 3.250
13\29 537.931 662.069 10:3 3.333
22\49 538.776 661.224 17:5 3.400
9\20 540.000 660.000 7:2 3.500
23\51 541.176 658.824 18:5 3.600
14\31 541.935 658.065 11:3 3.667
19\42 542.857 657.143 15:4 3.750
5\11 545.455 654.545 4:1 4.000 Superhard 2L 3s
16\35 548.571 651.429 13:3 4.333
11\24 550.000 650.000 9:2 4.500
17\37 551.351 648.649 14:3 4.667
6\13 553.846 646.154 5:1 5.000
13\28 557.143 642.857 11:2 5.500
7\15 560.000 640.000 6:1 6.000
8\17 564.706 635.294 7:1 7.000
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 2.3.7 temperament that tempers out 64/63 (archy, "no-fives dominant").

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