108edo
| ← 107edo | 108edo | 109edo → |
108 equal divisions of the octave (abbreviated 108edo or 108ed2), also called 108-tone equal temperament (108tet) or 108 equal temperament (108et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 108 equal parts of about 11.1 ¢ each. Each step represents a frequency ratio of 21/108, or the 108th root of 2.
Theory
108et tempers out the Pythagorean comma in the 3-limit and 1990656/1953125 (valentine comma) in the 5-limit. In the 7-limit it tempers out 126/125 and 1029/1024, supporting valentine, and making for a good tuning for it and for starling temperament, the planar temperament tempering out 126/125. In the 11-limit the patent val tempers out 540/539 and 1344/1331, and the 108e val tempers out 121/120, 176/175, 385/384, and 441/440, supporting 11-limit valentine for which it is again a good tuning.
108edo is the smallest 12n-edo which offers a decent alternative fifth to 12edo's fifth, that is 27edo's superpyth fifth.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -1.96 | +2.58 | -2.16 | +4.24 | +3.92 | -4.96 | +2.49 | +5.06 | +3.76 | -0.59 |
| Relative (%) | +0.0 | -17.6 | +23.2 | -19.4 | +38.1 | +35.3 | -44.6 | +22.4 | +45.5 | +33.8 | -5.3 | |
| Steps (reduced) |
108 (0) |
171 (63) |
251 (35) |
303 (87) |
374 (50) |
400 (76) |
441 (9) |
459 (27) |
489 (57) |
525 (93) |
535 (103) | |
Subsets and supersets
Since 108 factors into 22 × 33, 108edo has subset edos 2, 3, 4, 6, 9, 12, 18, 27, 36, and 54.
Intervals
| Steps | Cents | Approximate ratios | Ups and downs notation |
|---|---|---|---|
| 0 | 0 | 1/1 | D |
| 1 | 11.1 | ^D, ^E♭♭ | |
| 2 | 22.2 | ^^D, ^^E♭♭ | |
| 3 | 33.3 | ^3D, ^3E♭♭ | |
| 4 | 44.4 | 38/37, 39/38, 40/39, 41/40 | ^4D, ^4E♭♭ |
| 5 | 55.6 | 31/30, 32/31 | v4D♯, v4E♭ |
| 6 | 66.7 | 26/25 | v3D♯, v3E♭ |
| 7 | 77.8 | 23/22, 45/43 | vvD♯, vvE♭ |
| 8 | 88.9 | 20/19, 39/37 | vD♯, vE♭ |
| 9 | 100 | 18/17, 35/33 | D♯, E♭ |
| 10 | 111.1 | 16/15 | ^D♯, ^E♭ |
| 11 | 122.2 | 44/41 | ^^D♯, ^^E♭ |
| 12 | 133.3 | 40/37, 41/38 | ^3D♯, ^3E♭ |
| 13 | 144.4 | 25/23, 38/35 | ^4D♯, ^4E♭ |
| 14 | 155.6 | 35/32, 47/43 | v4D𝄪, v4E |
| 15 | 166.7 | 11/10 | v3D𝄪, v3E |
| 16 | 177.8 | 31/28, 41/37 | vvD𝄪, vvE |
| 17 | 188.9 | 29/26, 39/35 | vD𝄪, vE |
| 18 | 200 | 37/33, 46/41 | E |
| 19 | 211.1 | 26/23, 35/31 | ^E, ^F♭ |
| 20 | 222.2 | 25/22, 33/29 | ^^E, ^^F♭ |
| 21 | 233.3 | 8/7 | ^3E, ^3F♭ |
| 22 | 244.4 | 38/33 | ^4E, ^4F♭ |
| 23 | 255.6 | 22/19, 29/25 | v4E♯, v4F |
| 24 | 266.7 | 7/6 | v3E♯, v3F |
| 25 | 277.8 | 47/40 | vvE♯, vvF |
| 26 | 288.9 | 13/11 | vE♯, vF |
| 27 | 300 | 44/37 | F |
| 28 | 311.1 | ^F, ^G♭♭ | |
| 29 | 322.2 | 47/39 | ^^F, ^^G♭♭ |
| 30 | 333.3 | 40/33 | ^3F, ^3G♭♭ |
| 31 | 344.4 | 39/32 | ^4F, ^4G♭♭ |
| 32 | 355.6 | 43/35 | v4F♯, v4G♭ |
| 33 | 366.7 | 21/17, 47/38 | v3F♯, v3G♭ |
| 34 | 377.8 | 41/33, 46/37 | vvF♯, vvG♭ |
| 35 | 388.9 | vF♯, vG♭ | |
| 36 | 400 | 29/23, 34/27 | F♯, G♭ |
| 37 | 411.1 | 19/15, 33/26 | ^F♯, ^G♭ |
| 38 | 422.2 | 37/29 | ^^F♯, ^^G♭ |
| 39 | 433.3 | 9/7 | ^3F♯, ^3G♭ |
| 40 | 444.4 | 31/24 | ^4F♯, ^4G♭ |
| 41 | 455.6 | 13/10 | v4F𝄪, v4G |
| 42 | 466.7 | 38/29 | v3F𝄪, v3G |
| 43 | 477.8 | 29/22 | vvF𝄪, vvG |
| 44 | 488.9 | vF𝄪, vG | |
| 45 | 500 | 4/3 | G |
| 46 | 511.1 | 39/29, 43/32, 47/35 | ^G, ^A♭♭ |
| 47 | 522.2 | ^^G, ^^A♭♭ | |
| 48 | 533.3 | ^3G, ^3A♭♭ | |
| 49 | 544.4 | 26/19 | ^4G, ^4A♭♭ |
| 50 | 555.6 | 40/29 | v4G♯, v4A♭ |
| 51 | 566.7 | 43/31 | v3G♯, v3A♭ |
| 52 | 577.8 | vvG♯, vvA♭ | |
| 53 | 588.9 | 45/32 | vG♯, vA♭ |
| 54 | 600 | 41/29 | G♯, A♭ |
| 55 | 611.1 | 37/26, 47/33 | ^G♯, ^A♭ |
| 56 | 622.2 | 43/30 | ^^G♯, ^^A♭ |
| 57 | 633.3 | ^3G♯, ^3A♭ | |
| 58 | 644.4 | 29/20, 45/31 | ^4G♯, ^4A♭ |
| 59 | 655.6 | 19/13 | v4G𝄪, v4A |
| 60 | 666.7 | 47/32 | v3G𝄪, v3A |
| 61 | 677.8 | 37/25 | vvG𝄪, vvA |
| 62 | 688.9 | vG𝄪, vA | |
| 63 | 700 | 3/2 | A |
| 64 | 711.1 | ^A, ^B♭♭ | |
| 65 | 722.2 | 44/29, 47/31 | ^^A, ^^B♭♭ |
| 66 | 733.3 | 29/19 | ^3A, ^3B♭♭ |
| 67 | 744.4 | 20/13, 43/28 | ^4A, ^4B♭♭ |
| 68 | 755.6 | v4A♯, v4B♭ | |
| 69 | 766.7 | 14/9 | v3A♯, v3B♭ |
| 70 | 777.8 | 47/30 | vvA♯, vvB♭ |
| 71 | 788.9 | 30/19, 41/26 | vA♯, vB♭ |
| 72 | 800 | 27/17, 46/29 | A♯, B♭ |
| 73 | 811.1 | ^A♯, ^B♭ | |
| 74 | 822.2 | 37/23, 45/28 | ^^A♯, ^^B♭ |
| 75 | 833.3 | 34/21 | ^3A♯, ^3B♭ |
| 76 | 844.4 | ^4A♯, ^4B♭ | |
| 77 | 855.6 | 41/25 | v4A𝄪, v4B |
| 78 | 866.7 | 33/20 | v3A𝄪, v3B |
| 79 | 877.8 | vvA𝄪, vvB | |
| 80 | 888.9 | vA𝄪, vB | |
| 81 | 900 | 37/22 | B |
| 82 | 911.1 | 22/13 | ^B, ^C♭ |
| 83 | 922.2 | ^^B, ^^C♭ | |
| 84 | 933.3 | 12/7 | ^3B, ^3C♭ |
| 85 | 944.4 | 19/11 | ^4B, ^4C♭ |
| 86 | 955.6 | 33/19 | v4B♯, v4C |
| 87 | 966.7 | 7/4 | v3B♯, v3C |
| 88 | 977.8 | 44/25 | vvB♯, vvC |
| 89 | 988.9 | 23/13 | vB♯, vC |
| 90 | 1000 | 41/23 | C |
| 91 | 1011.1 | 43/24 | ^C, ^D♭♭ |
| 92 | 1022.2 | ^^C, ^^D♭♭ | |
| 93 | 1033.3 | 20/11 | ^3C, ^3D♭♭ |
| 94 | 1044.4 | ^4C, ^4D♭♭ | |
| 95 | 1055.6 | 35/19, 46/25 | v4C♯, v4D♭ |
| 96 | 1066.7 | 37/20 | v3C♯, v3D♭ |
| 97 | 1077.8 | 41/22 | vvC♯, vvD♭ |
| 98 | 1088.9 | 15/8 | vC♯, vD♭ |
| 99 | 1100 | 17/9 | C♯, D♭ |
| 100 | 1111.1 | 19/10 | ^C♯, ^D♭ |
| 101 | 1122.2 | 44/23 | ^^C♯, ^^D♭ |
| 102 | 1133.3 | 25/13 | ^3C♯, ^3D♭ |
| 103 | 1144.4 | 31/16 | ^4C♯, ^4D♭ |
| 104 | 1155.6 | 37/19, 39/20 | v4C𝄪, v4D |
| 105 | 1166.7 | v3C𝄪, v3D | |
| 106 | 1177.8 | vvC𝄪, vvD | |
| 107 | 1188.9 | vC𝄪, vD | |
| 108 | 1200 | 2/1 | D |