109edo
| ← 108edo | 109edo | 110edo → |
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 (%) | +0.0 | +23.9 | -9.0 | -0.2 | -7.8 | -34.8 | +46.7 | -2.4 | -6.8 | +48.0 | -0.7 | |
| 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 | 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 |