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
108edo tempers out the Pythagorean comma, 531441/524288, in the 3-limit and 1990656/1953125, the valensixthtone comma, in the 5-limit. In the 7-limit it tempers out 126/125 and 1029/1024, supporting valentine temperament, 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 the 108e val tempers out 121/120 and 176/175, 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 | -18 | +23 | -19 | +38 | +35 | -45 | +22 | +46 | +34 | -5 | |
| Steps (reduced) |
108 (0) |
171 (63) |
251 (35) |
303 (87) |
374 (50) |
400 (76) |
441 (9) |
459 (27) |
489 (57) |
525 (93) |
535 (103) | |
Intervals
| Steps | Cents | Ups and downs notation | Approximate ratios |
|---|---|---|---|
| 0 | 0 | D | 1/1 |
| 1 | 11.1111 | ^D, v8E♭ | |
| 2 | 22.2222 | ^^D, v7E♭ | 66/65, 78/77 |
| 3 | 33.3333 | ^3D, v6E♭ | 49/48 |
| 4 | 44.4444 | ^4D, v5E♭ | 36/35, 40/39, 77/75 |
| 5 | 55.5556 | ^5D, v4E♭ | 33/32 |
| 6 | 66.6667 | ^6D, v3E♭ | 26/25, 28/27, 80/77 |
| 7 | 77.7778 | ^7D, vvE♭ | |
| 8 | 88.8889 | ^8D, vE♭ | |
| 9 | 100 | D♯, E♭ | 35/33, 55/52 |
| 10 | 111.111 | ^D♯, v8E | 16/15 |
| 11 | 122.222 | ^^D♯, v7E | 15/14 |
| 12 | 133.333 | ^3D♯, v6E | |
| 13 | 144.444 | ^4D♯, v5E | 49/45 |
| 14 | 155.556 | ^5D♯, v4E | 35/32 |
| 15 | 166.667 | ^6D♯, v3E | 11/10, 54/49 |
| 16 | 177.778 | ^7D♯, vvE | |
| 17 | 188.889 | ^8D♯, vE | 39/35 |
| 18 | 200 | E | 9/8 |
| 19 | 211.111 | ^E, v8F | 44/39 |
| 20 | 222.222 | ^^E, v7F | 25/22 |
| 21 | 233.333 | ^3E, v6F | 8/7 |
| 22 | 244.444 | ^4E, v5F | 15/13 |
| 23 | 255.556 | ^5E, v4F | |
| 24 | 266.667 | ^6E, v3F | 7/6 |
| 25 | 277.778 | ^7E, vvF | 75/64 |
| 26 | 288.889 | ^8E, vF | 13/11, 33/28, 77/65 |
| 27 | 300 | F | |
| 28 | 311.111 | ^F, v8G♭ | |
| 29 | 322.222 | ^^F, v7G♭ | 77/64 |
| 30 | 333.333 | ^3F, v6G♭ | 40/33 |
| 31 | 344.444 | ^4F, v5G♭ | 39/32 |
| 32 | 355.556 | ^5F, v4G♭ | 16/13 |
| 33 | 366.667 | ^6F, v3G♭ | |
| 34 | 377.778 | ^7F, vvG♭ | 56/45 |
| 35 | 388.889 | ^8F, vG♭ | 5/4 |
| 36 | 400 | F♯, G♭ | 44/35 |
| 37 | 411.111 | ^F♯, v8G | 33/26 |
| 38 | 422.222 | ^^F♯, v7G | |
| 39 | 433.333 | ^3F♯, v6G | 9/7, 50/39, 77/60 |
| 40 | 444.444 | ^4F♯, v5G | |
| 41 | 455.556 | ^5F♯, v4G | 13/10 |
| 42 | 466.667 | ^6F♯, v3G | 21/16, 64/49 |
| 43 | 477.778 | ^7F♯, vvG | 33/25 |
| 44 | 488.889 | ^8F♯, vG | |
| 45 | 500 | G | 4/3 |
| 46 | 511.111 | ^G, v8A♭ | 35/26 |
| 47 | 522.222 | ^^G, v7A♭ | |
| 48 | 533.333 | ^3G, v6A♭ | 15/11, 49/36 |
| 49 | 544.444 | ^4G, v5A♭ | 48/35 |
| 50 | 555.556 | ^5G, v4A♭ | 11/8 |
| 51 | 566.667 | ^6G, v3A♭ | |
| 52 | 577.778 | ^7G, vvA♭ | 39/28 |
| 53 | 588.889 | ^8G, vA♭ | 45/32 |
| 54 | 600 | G♯, A♭ | |
| 55 | 611.111 | ^G♯, v8A | 64/45 |
| 56 | 622.222 | ^^G♯, v7A | 56/39 |
| 57 | 633.333 | ^3G♯, v6A | 75/52 |
| 58 | 644.444 | ^4G♯, v5A | 16/11 |
| 59 | 655.556 | ^5G♯, v4A | 35/24 |
| 60 | 666.667 | ^6G♯, v3A | 22/15, 72/49 |
| 61 | 677.778 | ^7G♯, vvA | 65/44, 77/52 |
| 62 | 688.889 | ^8G♯, vA | 52/35 |
| 63 | 700 | A | 3/2 |
| 64 | 711.111 | ^A, v8B♭ | |
| 65 | 722.222 | ^^A, v7B♭ | 50/33 |
| 66 | 733.333 | ^3A, v6B♭ | 32/21, 49/32 |
| 67 | 744.444 | ^4A, v5B♭ | 20/13, 77/50 |
| 68 | 755.556 | ^5A, v4B♭ | |
| 69 | 766.667 | ^6A, v3B♭ | 14/9, 39/25 |
| 70 | 777.778 | ^7A, vvB♭ | |
| 71 | 788.889 | ^8A, vB♭ | 52/33 |
| 72 | 800 | A♯, B♭ | 35/22 |
| 73 | 811.111 | ^A♯, v8B | 8/5 |
| 74 | 822.222 | ^^A♯, v7B | 45/28, 77/48 |
| 75 | 833.333 | ^3A♯, v6B | |
| 76 | 844.444 | ^4A♯, v5B | 13/8 |
| 77 | 855.556 | ^5A♯, v4B | 64/39 |
| 78 | 866.667 | ^6A♯, v3B | 33/20, 81/49 |
| 79 | 877.778 | ^7A♯, vvB | |
| 80 | 888.889 | ^8A♯, vB | |
| 81 | 900 | B | |
| 82 | 911.111 | ^B, v8C | 22/13, 56/33 |
| 83 | 922.222 | ^^B, v7C | 75/44 |
| 84 | 933.333 | ^3B, v6C | 12/7, 77/45 |
| 85 | 944.444 | ^4B, v5C | |
| 86 | 955.556 | ^5B, v4C | 26/15 |
| 87 | 966.667 | ^6B, v3C | 7/4 |
| 88 | 977.778 | ^7B, vvC | 44/25 |
| 89 | 988.889 | ^8B, vC | 39/22 |
| 90 | 1000 | C | 16/9 |
| 91 | 1011.11 | ^C, v8D♭ | 70/39 |
| 92 | 1022.22 | ^^C, v7D♭ | |
| 93 | 1033.33 | ^3C, v6D♭ | 20/11, 49/27 |
| 94 | 1044.44 | ^4C, v5D♭ | 64/35 |
| 95 | 1055.56 | ^5C, v4D♭ | |
| 96 | 1066.67 | ^6C, v3D♭ | |
| 97 | 1077.78 | ^7C, vvD♭ | 28/15 |
| 98 | 1088.89 | ^8C, vD♭ | 15/8 |
| 99 | 1100 | C♯, D♭ | 66/35 |
| 100 | 1111.11 | ^C♯, v8D | |
| 101 | 1122.22 | ^^C♯, v7D | |
| 102 | 1133.33 | ^3C♯, v6D | 25/13, 27/14, 77/40 |
| 103 | 1144.44 | ^4C♯, v5D | 64/33 |
| 104 | 1155.56 | ^5C♯, v4D | 35/18, 39/20 |
| 105 | 1166.67 | ^6C♯, v3D | |
| 106 | 1177.78 | ^7C♯, vvD | 65/33, 77/39 |
| 107 | 1188.89 | ^8C♯, vD | |
| 108 | 1200 | D | 2/1 |