2L 3s
"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").