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 (tetracot comma) 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, 1375/1372, 2255/2244, 2300/2299, 6860/6859, 10241/10240.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +2.63 | -0.99 | -0.02 | -0.86 | -3.83 | +5.14 | -0.27 | -0.75 | +5.29 | -0.08 | +1.87 | +0.30 | -5.10 | -4.96 | -3.78 |
| Relative (%) | +0.0 | +23.9 | -9.0 | -0.2 | -7.8 | -34.8 | +46.7 | -2.4 | -6.8 | +48.0 | -0.7 | +17.0 | +2.7 | -46.3 | -45.0 | -34.3 | |
| Steps (reduced) |
109 (0) |
173 (64) |
253 (35) |
306 (88) |
377 (50) |
403 (76) |
446 (10) |
463 (27) |
493 (57) |
530 (94) |
540 (104) |
568 (23) |
584 (39) |
591 (46) |
605 (60) |
624 (79) | |
Subsets and supersets
109edo is the 29th prime edo, following 107edo and before 113edo. 436edo, which slices each step of 109edo in four, provides correction for the approximation to harmonic 3.
Nonoctave temperaments
Taking every 8 degree of 109edo produces a scale extremely close to 88cET.
Intervals
| Steps | Cents | Approximate ratios | Ups and downs notation |
|---|---|---|---|
| 0 | 0 | 1/1 | D |
| 1 | 11 | ^D, v6E♭ | |
| 2 | 22 | ^^D, v5E♭ | |
| 3 | 33 | ^3D, v4E♭ | |
| 4 | 44 | 38/37, 39/38, 40/39, 41/40 | ^4D, v3E♭ |
| 5 | 55 | 31/30, 32/31, 33/32 | ^5D, vvE♭ |
| 6 | 66.1 | 26/25 | ^6D, vE♭ |
| 7 | 77.1 | 23/22 | v5D♯, E♭ |
| 8 | 88.1 | 20/19, 41/39 | v4D♯, ^E♭ |
| 9 | 99.1 | 18/17 | v3D♯, ^^E♭ |
| 10 | 110.1 | 16/15, 33/31 | vvD♯, ^3E♭ |
| 11 | 121.1 | 15/14, 44/41 | vD♯, ^4E♭ |
| 12 | 132.1 | 41/38 | D♯, ^5E♭ |
| 13 | 143.1 | 25/23, 38/35 | ^D♯, v6E |
| 14 | 154.1 | 35/32, 47/43 | ^^D♯, v5E |
| 15 | 165.1 | 11/10 | ^3D♯, v4E |
| 16 | 176.1 | 31/28, 41/37 | ^4D♯, v3E |
| 17 | 187.2 | 39/35 | ^5D♯, vvE |
| 18 | 198.2 | 28/25, 37/33, 46/41 | ^6D♯, vE |
| 19 | 209.2 | 35/31, 44/39 | E |
| 20 | 220.2 | 25/22, 42/37 | ^E, v6F |
| 21 | 231.2 | 8/7 | ^^E, v5F |
| 22 | 242.2 | 23/20, 38/33 | ^3E, v4F |
| 23 | 253.2 | 22/19, 37/32 | ^4E, v3F |
| 24 | 264.2 | ^5E, vvF | |
| 25 | 275.2 | 34/29, 41/35 | ^6E, vF |
| 26 | 286.2 | 33/28, 46/39 | F |
| 27 | 297.2 | 19/16 | ^F, v6G♭ |
| 28 | 308.3 | 37/31 | ^^F, v5G♭ |
| 29 | 319.3 | ^3F, v4G♭ | |
| 30 | 330.3 | 23/19 | ^4F, v3G♭ |
| 31 | 341.3 | 28/23, 39/32 | ^5F, vvG♭ |
| 32 | 352.3 | 38/31 | ^6F, vG♭ |
| 33 | 363.3 | 37/30 | v5F♯, G♭ |
| 34 | 374.3 | 31/25, 36/29, 41/33 | v4F♯, ^G♭ |
| 35 | 385.3 | 5/4 | v3F♯, ^^G♭ |
| 36 | 396.3 | 39/31, 44/35 | vvF♯, ^3G♭ |
| 37 | 407.3 | 19/15 | vF♯, ^4G♭ |
| 38 | 418.3 | 14/11 | F♯, ^5G♭ |
| 39 | 429.4 | 32/25, 41/32 | ^F♯, v6G |
| 40 | 440.4 | 40/31 | ^^F♯, v5G |
| 41 | 451.4 | ^3F♯, v4G | |
| 42 | 462.4 | ^4F♯, v3G | |
| 43 | 473.4 | 25/19, 46/35 | ^5F♯, vvG |
| 44 | 484.4 | 37/28, 41/31, 45/34 | ^6F♯, vG |
| 45 | 495.4 | G | |
| 46 | 506.4 | ^G, v6A♭ | |
| 47 | 517.4 | 31/23 | ^^G, v5A♭ |
| 48 | 528.4 | 19/14 | ^3G, v4A♭ |
| 49 | 539.4 | 41/30 | ^4G, v3A♭ |
| 50 | 550.5 | 11/8 | ^5G, vvA♭ |
| 51 | 561.5 | ^6G, vA♭ | |
| 52 | 572.5 | 32/23, 39/28 | v5G♯, A♭ |
| 53 | 583.5 | 7/5 | v4G♯, ^A♭ |
| 54 | 594.5 | 31/22 | v3G♯, ^^A♭ |
| 55 | 605.5 | 44/31 | vvG♯, ^3A♭ |
| 56 | 616.5 | 10/7 | vG♯, ^4A♭ |
| 57 | 627.5 | 23/16 | G♯, ^5A♭ |
| 58 | 638.5 | ^G♯, v6A | |
| 59 | 649.5 | 16/11 | ^^G♯, v5A |
| 60 | 660.6 | 41/28 | ^3G♯, v4A |
| 61 | 671.6 | 28/19 | ^4G♯, v3A |
| 62 | 682.6 | 46/31 | ^5G♯, vvA |
| 63 | 693.6 | ^6G♯, vA | |
| 64 | 704.6 | A | |
| 65 | 715.6 | ^A, v6B♭ | |
| 66 | 726.6 | 35/23, 38/25 | ^^A, v5B♭ |
| 67 | 737.6 | ^3A, v4B♭ | |
| 68 | 748.6 | 37/24 | ^4A, v3B♭ |
| 69 | 759.6 | 31/20, 45/29 | ^5A, vvB♭ |
| 70 | 770.6 | 25/16, 39/25 | ^6A, vB♭ |
| 71 | 781.7 | 11/7 | v5A♯, B♭ |
| 72 | 792.7 | 30/19 | v4A♯, ^B♭ |
| 73 | 803.7 | 35/22 | v3A♯, ^^B♭ |
| 74 | 814.7 | 8/5 | vvA♯, ^3B♭ |
| 75 | 825.7 | 29/18 | vA♯, ^4B♭ |
| 76 | 836.7 | A♯, ^5B♭ | |
| 77 | 847.7 | 31/19 | ^A♯, v6B |
| 78 | 858.7 | 23/14 | ^^A♯, v5B |
| 79 | 869.7 | 38/23, 43/26 | ^3A♯, v4B |
| 80 | 880.7 | ^4A♯, v3B | |
| 81 | 891.7 | ^5A♯, vvB | |
| 82 | 902.8 | 32/19 | ^6A♯, vB |
| 83 | 913.8 | 39/23 | B |
| 84 | 924.8 | 29/17 | ^B, v6C |
| 85 | 935.8 | ^^B, v5C | |
| 86 | 946.8 | 19/11 | ^3B, v4C |
| 87 | 957.8 | 33/19, 40/23 | ^4B, v3C |
| 88 | 968.8 | 7/4 | ^5B, vvC |
| 89 | 979.8 | 37/21, 44/25 | ^6B, vC |
| 90 | 990.8 | 39/22 | C |
| 91 | 1001.8 | 25/14, 41/23 | ^C, v6D♭ |
| 92 | 1012.8 | ^^C, v5D♭ | |
| 93 | 1023.9 | 47/26 | ^3C, v4D♭ |
| 94 | 1034.9 | 20/11 | ^4C, v3D♭ |
| 95 | 1045.9 | ^5C, vvD♭ | |
| 96 | 1056.9 | 35/19, 46/25 | ^6C, vD♭ |
| 97 | 1067.9 | v5C♯, D♭ | |
| 98 | 1078.9 | 28/15, 41/22 | v4C♯, ^D♭ |
| 99 | 1089.9 | 15/8 | v3C♯, ^^D♭ |
| 100 | 1100.9 | 17/9 | vvC♯, ^3D♭ |
| 101 | 1111.9 | 19/10 | vC♯, ^4D♭ |
| 102 | 1122.9 | 44/23 | C♯, ^5D♭ |
| 103 | 1133.9 | 25/13 | ^C♯, v6D |
| 104 | 1145 | 31/16 | ^^C♯, v5D |
| 105 | 1156 | 37/19, 39/20 | ^3C♯, v4D |
| 106 | 1167 | ^4C♯, v3D | |
| 107 | 1178 | ^5C♯, vvD | |
| 108 | 1189 | ^6C♯, vD | |
| 109 | 1200 | 2/1 | D |
Music
- Teenagerges (2024) – tetracot[13] in 109edo tuning