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 | 11.111 | ^D, v8E♭ | |
| 2 | 22.222 | ^^D, v7E♭ | |
| 3 | 33.333 | ^3D, v6E♭ | |
| 4 | 44.444 | 38/37, 39/38, 40/39, 41/40 | ^4D, v5E♭ |
| 5 | 55.556 | 31/30, 32/31 | ^5D, v4E♭ |
| 6 | 66.667 | 26/25 | ^6D, v3E♭ |
| 7 | 77.778 | 23/22, 45/43 | ^7D, vvE♭ |
| 8 | 88.889 | 20/19, 39/37 | ^8D, vE♭ |
| 9 | 100 | 18/17, 35/33 | D♯, E♭ |
| 10 | 111.111 | 16/15 | ^D♯, v8E |
| 11 | 122.222 | 44/41 | ^^D♯, v7E |
| 12 | 133.333 | 40/37, 41/38 | ^3D♯, v6E |
| 13 | 144.444 | 25/23, 38/35 | ^4D♯, v5E |
| 14 | 155.556 | 35/32, 47/43 | ^5D♯, v4E |
| 15 | 166.667 | 11/10 | ^6D♯, v3E |
| 16 | 177.778 | 31/28, 41/37 | ^7D♯, vvE |
| 17 | 188.889 | 29/26, 39/35 | ^8D♯, vE |
| 18 | 200 | 37/33, 46/41 | E |
| 19 | 211.111 | 26/23, 35/31 | ^E, v8F |
| 20 | 222.222 | 25/22, 33/29 | ^^E, v7F |
| 21 | 233.333 | 8/7 | ^3E, v6F |
| 22 | 244.444 | 38/33 | ^4E, v5F |
| 23 | 255.556 | 22/19, 29/25 | ^5E, v4F |
| 24 | 266.667 | 7/6 | ^6E, v3F |
| 25 | 277.778 | 47/40 | ^7E, vvF |
| 26 | 288.889 | 13/11 | ^8E, vF |
| 27 | 300 | 44/37 | F |
| 28 | 311.111 | ^F, v8G♭ | |
| 29 | 322.222 | 47/39 | ^^F, v7G♭ |
| 30 | 333.333 | 40/33 | ^3F, v6G♭ |
| 31 | 344.444 | 39/32 | ^4F, v5G♭ |
| 32 | 355.556 | 43/35 | ^5F, v4G♭ |
| 33 | 366.667 | 21/17, 47/38 | ^6F, v3G♭ |
| 34 | 377.778 | 41/33, 46/37 | ^7F, vvG♭ |
| 35 | 388.889 | ^8F, vG♭ | |
| 36 | 400 | 29/23, 34/27 | F♯, G♭ |
| 37 | 411.111 | 19/15, 33/26 | ^F♯, v8G |
| 38 | 422.222 | 37/29 | ^^F♯, v7G |
| 39 | 433.333 | 9/7 | ^3F♯, v6G |
| 40 | 444.444 | 31/24 | ^4F♯, v5G |
| 41 | 455.556 | 13/10 | ^5F♯, v4G |
| 42 | 466.667 | 38/29 | ^6F♯, v3G |
| 43 | 477.778 | 29/22 | ^7F♯, vvG |
| 44 | 488.889 | ^8F♯, vG | |
| 45 | 500 | 4/3 | G |
| 46 | 511.111 | 39/29, 43/32, 47/35 | ^G, v8A♭ |
| 47 | 522.222 | ^^G, v7A♭ | |
| 48 | 533.333 | ^3G, v6A♭ | |
| 49 | 544.444 | 26/19 | ^4G, v5A♭ |
| 50 | 555.556 | 40/29 | ^5G, v4A♭ |
| 51 | 566.667 | 43/31 | ^6G, v3A♭ |
| 52 | 577.778 | ^7G, vvA♭ | |
| 53 | 588.889 | 45/32 | ^8G, vA♭ |
| 54 | 600 | 41/29 | G♯, A♭ |
| 55 | 611.111 | 37/26, 47/33 | ^G♯, v8A |
| 56 | 622.222 | 43/30 | ^^G♯, v7A |
| 57 | 633.333 | ^3G♯, v6A | |
| 58 | 644.444 | 29/20, 45/31 | ^4G♯, v5A |
| 59 | 655.556 | 19/13 | ^5G♯, v4A |
| 60 | 666.667 | 47/32 | ^6G♯, v3A |
| 61 | 677.778 | 37/25 | ^7G♯, vvA |
| 62 | 688.889 | ^8G♯, vA | |
| 63 | 700 | 3/2 | A |
| 64 | 711.111 | ^A, v8B♭ | |
| 65 | 722.222 | 44/29, 47/31 | ^^A, v7B♭ |
| 66 | 733.333 | 29/19 | ^3A, v6B♭ |
| 67 | 744.444 | 20/13, 43/28 | ^4A, v5B♭ |
| 68 | 755.556 | ^5A, v4B♭ | |
| 69 | 766.667 | 14/9 | ^6A, v3B♭ |
| 70 | 777.778 | 47/30 | ^7A, vvB♭ |
| 71 | 788.889 | 30/19, 41/26 | ^8A, vB♭ |
| 72 | 800 | 27/17, 46/29 | A♯, B♭ |
| 73 | 811.111 | ^A♯, v8B | |
| 74 | 822.222 | 37/23, 45/28 | ^^A♯, v7B |
| 75 | 833.333 | 34/21 | ^3A♯, v6B |
| 76 | 844.444 | ^4A♯, v5B | |
| 77 | 855.556 | 41/25 | ^5A♯, v4B |
| 78 | 866.667 | 33/20 | ^6A♯, v3B |
| 79 | 877.778 | ^7A♯, vvB | |
| 80 | 888.889 | ^8A♯, vB | |
| 81 | 900 | 37/22 | B |
| 82 | 911.111 | 22/13 | ^B, v8C |
| 83 | 922.222 | ^^B, v7C | |
| 84 | 933.333 | 12/7 | ^3B, v6C |
| 85 | 944.444 | 19/11 | ^4B, v5C |
| 86 | 955.556 | 33/19 | ^5B, v4C |
| 87 | 966.667 | 7/4 | ^6B, v3C |
| 88 | 977.778 | 44/25 | ^7B, vvC |
| 89 | 988.889 | 23/13 | ^8B, vC |
| 90 | 1000 | 41/23 | C |
| 91 | 1011.111 | 43/24 | ^C, v8D♭ |
| 92 | 1022.222 | ^^C, v7D♭ | |
| 93 | 1033.333 | 20/11 | ^3C, v6D♭ |
| 94 | 1044.444 | ^4C, v5D♭ | |
| 95 | 1055.556 | 35/19, 46/25 | ^5C, v4D♭ |
| 96 | 1066.667 | 37/20 | ^6C, v3D♭ |
| 97 | 1077.778 | 41/22 | ^7C, vvD♭ |
| 98 | 1088.889 | 15/8 | ^8C, vD♭ |
| 99 | 1100 | 17/9 | C♯, D♭ |
| 100 | 1111.111 | 19/10 | ^C♯, v8D |
| 101 | 1122.222 | 44/23 | ^^C♯, v7D |
| 102 | 1133.333 | 25/13 | ^3C♯, v6D |
| 103 | 1144.444 | 31/16 | ^4C♯, v5D |
| 104 | 1155.556 | 37/19, 39/20 | ^5C♯, v4D |
| 105 | 1166.667 | ^6C♯, v3D | |
| 106 | 1177.778 | ^7C♯, vvD | |
| 107 | 1188.889 | ^8C♯, vD | |
| 108 | 1200 | 2/1 | D |