109edo
| ← 108edo | 109edo | 110edo → |
109 equal divisions of the octave (abbreviated 109edo or 109ed2), also called 109-tone equal temperament (109tet) or 109 equal temperament (109et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 109 equal parts of about 11 ¢ each. Each step represents a frequency ratio of 21/109, or the 109th root of 2.
Theory
109edo tempers out 20000/19683 in the 5-limit; 245/243, 2401/2400 and 65625/65536 in the 7-limit; 385/384, 1375/1372, and 4000/3993 in the 11-limit. It provides the optimal patent val for 7-limit octacot temperament, and 11 and 13 limit leapweek; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.
109edo has an excellent 7th harmonic, being a denominator of semiconvergent to log27, and it is overall a strong 2.5.7.11.19.23.31.41 subgroup tuning, with errors of less than 10% on all harmonics. Some commas it tempers out in this subgroup are 575/574, 1331/1330, 1375/1572, 2255/2244, 2300/2299, 6860/6859, 10241/10240.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.00 | +2.63 | -0.99 | -0.02 | -0.86 | -3.83 | +5.14 | -0.27 | -0.75 | +5.29 | -0.08 |
| relative (%) | +0 | +24 | -9 | -0 | -8 | -35 | +47 | -2 | -7 | +48 | -1 | |
| Steps (reduced) |
109 (0) |
173 (64) |
253 (35) |
306 (88) |
377 (50) |
403 (76) |
446 (10) |
463 (27) |
493 (57) |
530 (94) |
540 (104) | |
Subsets and supersets
109edo is the 29th prime EDO.
Nonoctave temperaments
Taking every 8 degree of 109edo produces a scale extremely close to 88cET.
Intervals
| Steps | Cents | Ups and downs notation | Approximate ratios |
|---|---|---|---|
| 0 | 0 | D | 1/1 |
| 1 | 11.0092 | ^D, v6E♭ | |
| 2 | 22.0183 | ^^D, v5E♭ | 78/77 |
| 3 | 33.0275 | ^3D, v4E♭ | 49/48, 50/49, 56/55 |
| 4 | 44.0367 | ^4D, v3E♭ | 40/39, 77/75 |
| 5 | 55.0459 | ^5D, vvE♭ | 33/32 |
| 6 | 66.055 | ^6D, vE♭ | 26/25, 80/77 |
| 7 | 77.0642 | ^7D, E♭ | 22/21 |
| 8 | 88.0734 | ^8D, v11E | 21/20 |
| 9 | 99.0826 | ^9D, v10E | 35/33, 52/49, 55/52 |
| 10 | 110.092 | ^10D, v9E | 16/15 |
| 11 | 121.101 | ^11D, v8E | 15/14 |
| 12 | 132.11 | D♯, v7E | 14/13 |
| 13 | 143.119 | ^D♯, v6E | 49/45 |
| 14 | 154.128 | ^^D♯, v5E | 12/11, 35/32 |
| 15 | 165.138 | ^3D♯, v4E | 11/10 |
| 16 | 176.147 | ^4D♯, v3E | |
| 17 | 187.156 | ^5D♯, vvE | 39/35, 49/44 |
| 18 | 198.165 | ^6D♯, vE | 28/25, 55/49 |
| 19 | 209.174 | E | 44/39 |
| 20 | 220.183 | ^E, v6F | 25/22 |
| 21 | 231.193 | ^^E, v5F | 8/7 |
| 22 | 242.202 | ^3E, v4F | |
| 23 | 253.211 | ^4E, v3F | |
| 24 | 264.22 | ^5E, vvF | 7/6, 64/55 |
| 25 | 275.229 | ^6E, vF | 75/64 |
| 26 | 286.239 | F | 13/11, 33/28 |
| 27 | 297.248 | ^F, v6G♭ | 77/65 |
| 28 | 308.257 | ^^F, v5G♭ | |
| 29 | 319.266 | ^3F, v4G♭ | 6/5, 77/64 |
| 30 | 330.275 | ^4F, v3G♭ | 40/33 |
| 31 | 341.284 | ^5F, vvG♭ | 39/32 |
| 32 | 352.294 | ^6F, vG♭ | 49/40, 60/49 |
| 33 | 363.303 | ^7F, G♭ | 16/13 |
| 34 | 374.312 | ^8F, v11G | 56/45 |
| 35 | 385.321 | ^9F, v10G | 5/4 |
| 36 | 396.33 | ^10F, v9G | 44/35, 49/39 |
| 37 | 407.339 | ^11F, v8G | |
| 38 | 418.349 | F♯, v7G | 14/11 |
| 39 | 429.358 | ^F♯, v6G | 32/25, 50/39, 77/60 |
| 40 | 440.367 | ^^F♯, v5G | |
| 41 | 451.376 | ^3F♯, v4G | 13/10 |
| 42 | 462.385 | ^4F♯, v3G | 64/49 |
| 43 | 473.394 | ^5F♯, vvG | 21/16 |
| 44 | 484.404 | ^6F♯, vG | 33/25 |
| 45 | 495.413 | G | 4/3 |
| 46 | 506.422 | ^G, v6A♭ | 75/56 |
| 47 | 517.431 | ^^G, v5A♭ | 35/26, 66/49 |
| 48 | 528.44 | ^3G, v4A♭ | |
| 49 | 539.45 | ^4G, v3A♭ | 15/11 |
| 50 | 550.459 | ^5G, vvA♭ | 11/8, 48/35 |
| 51 | 561.468 | ^6G, vA♭ | |
| 52 | 572.477 | ^7G, A♭ | 39/28 |
| 53 | 583.486 | ^8G, v11A | 7/5 |
| 54 | 594.495 | ^9G, v10A | 45/32, 55/39 |
| 55 | 605.505 | ^10G, v9A | 64/45, 78/55 |
| 56 | 616.514 | ^11G, v8A | 10/7 |
| 57 | 627.523 | G♯, v7A | 56/39 |
| 58 | 638.532 | ^G♯, v6A | |
| 59 | 649.541 | ^^G♯, v5A | 16/11, 35/24 |
| 60 | 660.55 | ^3G♯, v4A | 22/15 |
| 61 | 671.56 | ^4G♯, v3A | 65/44 |
| 62 | 682.569 | ^5G♯, vvA | 49/33, 52/35, 77/52 |
| 63 | 693.578 | ^6G♯, vA | |
| 64 | 704.587 | A | 3/2 |
| 65 | 715.596 | ^A, v6B♭ | 50/33 |
| 66 | 726.606 | ^^A, v5B♭ | 32/21 |
| 67 | 737.615 | ^3A, v4B♭ | 49/32, 75/49 |
| 68 | 748.624 | ^4A, v3B♭ | 20/13, 77/50 |
| 69 | 759.633 | ^5A, vvB♭ | |
| 70 | 770.642 | ^6A, vB♭ | 25/16, 39/25 |
| 71 | 781.651 | ^7A, B♭ | 11/7 |
| 72 | 792.661 | ^8A, v11B | |
| 73 | 803.67 | ^9A, v10B | 35/22, 78/49 |
| 74 | 814.679 | ^10A, v9B | 8/5, 77/48 |
| 75 | 825.688 | ^11A, v8B | 45/28 |
| 76 | 836.697 | A♯, v7B | 13/8 |
| 77 | 847.706 | ^A♯, v6B | 49/30, 80/49 |
| 78 | 858.716 | ^^A♯, v5B | 64/39 |
| 79 | 869.725 | ^3A♯, v4B | 33/20 |
| 80 | 880.734 | ^4A♯, v3B | 5/3 |
| 81 | 891.743 | ^5A♯, vvB | |
| 82 | 902.752 | ^6A♯, vB | |
| 83 | 913.761 | B | 22/13, 56/33 |
| 84 | 924.771 | ^B, v6C | 75/44 |
| 85 | 935.78 | ^^B, v5C | 12/7, 55/32 |
| 86 | 946.789 | ^3B, v4C | |
| 87 | 957.798 | ^4B, v3C | |
| 88 | 968.807 | ^5B, vvC | 7/4 |
| 89 | 979.817 | ^6B, vC | 44/25 |
| 90 | 990.826 | C | 39/22 |
| 91 | 1001.83 | ^C, v6D♭ | 25/14 |
| 92 | 1012.84 | ^^C, v5D♭ | 70/39 |
| 93 | 1023.85 | ^3C, v4D♭ | |
| 94 | 1034.86 | ^4C, v3D♭ | 20/11 |
| 95 | 1045.87 | ^5C, vvD♭ | 11/6, 64/35 |
| 96 | 1056.88 | ^6C, vD♭ | |
| 97 | 1067.89 | ^7C, D♭ | 13/7 |
| 98 | 1078.9 | ^8C, v11D | 28/15 |
| 99 | 1089.91 | ^9C, v10D | 15/8 |
| 100 | 1100.92 | ^10C, v9D | 49/26, 66/35 |
| 101 | 1111.93 | ^11C, v8D | 40/21 |
| 102 | 1122.94 | C♯, v7D | 21/11 |
| 103 | 1133.94 | ^C♯, v6D | 25/13, 77/40 |
| 104 | 1144.95 | ^^C♯, v5D | 64/33 |
| 105 | 1155.96 | ^3C♯, v4D | 39/20 |
| 106 | 1166.97 | ^4C♯, v3D | 49/25, 55/28 |
| 107 | 1177.98 | ^5C♯, vvD | 77/39 |
| 108 | 1188.99 | ^6C♯, vD | |
| 109 | 1200 | D | 2/1 |