2L 3s

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"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.

Generator Cents s L-s |L-2s| Scale steps Trichord Comments
2\5 480 240 0 240 1 1 1 1 1 1 1
11\27 488.89 222.22 44.44 177.78 6 5 5 6 5 6 5 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
16\39 492.31 215.38 61.54 153.85 9 7 7 9 7 9 7 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
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
17\41 497.56 204.88 87.8 117.07 10 7 7 10 7 10 7 Pythagorean pentatonic is around here
5\12 500 200 100 100 3 2 2 3 2 3 2 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 Optimal meantone pentatonic

is around here

1200/(4-phi) 192.43 118.93 73.50 phi 1 1 phi 1 phi 1 Golden meantone
21\50 504 192 120 72 13 8 8 13 8 13 8
8\19 505.26 189.47 126.32 63.16 5 3 3 5 3 5 3
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 (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 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 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 L/s = 3
535.36 129.26 276.835 147.57 pi 1 pi 1 1 pi 1 L/s = pi
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 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 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 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 ("no-fives dominant").