2L 3s

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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.

Scale Tree

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").