115edo
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Prime factorization
5 × 23
Step size
10.4348¢
Fifth
67\115 (699.13¢)
Semitones (A1:m2)
9:10 (93.91¢ : 104.3¢)
Consistency limit
7
Distinct consistency limit
7
| ← 114edo | 115edo | 116edo → |
115 equal divisions of the octave (abbreviated 115edo or 115ed2), also called 115-tone equal temperament (115tet) or 115 equal temperament (115et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 115 equal parts of about 10.4 ¢ each. Each step represents a frequency ratio of 21/115, or the 115th root of 2.
Theory
The equal temperament tempers out 1594323/1562500 (unicorn comma) and 2109375/2097152 (semicomma) in the 5-limit, as well as 43046721/41943040 (python comma) and [19 10 -15⟩ (trisedodge comma); 225/224, 1728/1715, and 9920232/9765625 in the 7-limit. Using the patent val, it supports the newspeak temperament, tempering out 441/440, 1375/1372, and 1944/1925 in the 11-limit. The wizardharry comma, 4000/3993 is also tempered out.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -2.82 | -0.23 | +1.61 | +4.79 | +1.73 | +4.69 | -3.05 | -0.61 | +5.10 | -1.22 | -2.19 |
| Relative (%) | -27.1 | -2.2 | +15.4 | +45.9 | +16.5 | +44.9 | -29.2 | -5.8 | +48.8 | -11.7 | -21.0 | |
| Steps (reduced) |
182 (67) |
267 (37) |
323 (93) |
365 (20) |
398 (53) |
426 (81) |
449 (104) |
470 (10) |
489 (29) |
505 (45) |
520 (60) | |
Intervals
| Steps | Cents | Approximate ratios | Ups and downs notation |
|---|---|---|---|
| 0 | 0 | 1/1 | D |
| 1 | 10.4 | ^D, E♭♭ | |
| 2 | 20.9 | ^^D, ^E♭♭ | |
| 3 | 31.3 | ^3D, ^^E♭♭ | |
| 4 | 41.7 | 40/39, 41/40, 42/41, 43/42, 44/43 | ^4D, ^3E♭♭ |
| 5 | 52.2 | 33/32, 34/33, 35/34 | v4D♯, ^4E♭♭ |
| 6 | 62.6 | 29/28 | v3D♯, v4E♭ |
| 7 | 73 | 24/23 | vvD♯, v3E♭ |
| 8 | 83.5 | 21/20, 43/41 | vD♯, vvE♭ |
| 9 | 93.9 | D♯, vE♭ | |
| 10 | 104.3 | 17/16 | ^D♯, E♭ |
| 11 | 114.8 | 31/29, 46/43, 47/44 | ^^D♯, ^E♭ |
| 12 | 125.2 | 43/40 | ^3D♯, ^^E♭ |
| 13 | 135.7 | 40/37 | ^4D♯, ^3E♭ |
| 14 | 146.1 | 25/23, 37/34 | v4D𝄪, ^4E♭ |
| 15 | 156.5 | 23/21, 35/32 | v3D𝄪, v4E |
| 16 | 167 | 11/10, 43/39 | vvD𝄪, v3E |
| 17 | 177.4 | 31/28, 41/37 | vD𝄪, vvE |
| 18 | 187.8 | 29/26, 39/35 | D𝄪, vE |
| 19 | 198.3 | 28/25, 37/33, 46/41 | E |
| 20 | 208.7 | 35/31, 44/39 | ^E, F♭ |
| 21 | 219.1 | 42/37 | ^^E, ^F♭ |
| 22 | 229.6 | 8/7 | ^3E, ^^F♭ |
| 23 | 240 | 23/20 | ^4E, ^3F♭ |
| 24 | 250.4 | 37/32 | v4E♯, ^4F♭ |
| 25 | 260.9 | 43/37 | v3E♯, v4F |
| 26 | 271.3 | 48/41 | vvE♯, v3F |
| 27 | 281.7 | 20/17 | vE♯, vvF |
| 28 | 292.2 | E♯, vF | |
| 29 | 302.6 | 25/21, 31/26 | F |
| 30 | 313 | ^F, G♭♭ | |
| 31 | 323.5 | 35/29, 41/34, 47/39 | ^^F, ^G♭♭ |
| 32 | 333.9 | 40/33 | ^3F, ^^G♭♭ |
| 33 | 344.3 | 39/32 | ^4F, ^3G♭♭ |
| 34 | 354.8 | 43/35 | v4F♯, ^4G♭♭ |
| 35 | 365.2 | 21/17 | v3F♯, v4G♭ |
| 36 | 375.7 | 41/33, 46/37 | vvF♯, v3G♭ |
| 37 | 386.1 | 5/4 | vF♯, vvG♭ |
| 38 | 396.5 | 39/31, 44/35 | F♯, vG♭ |
| 39 | 407 | 43/34 | ^F♯, G♭ |
| 40 | 417.4 | 14/11 | ^^F♯, ^G♭ |
| 41 | 427.8 | 32/25, 41/32 | ^3F♯, ^^G♭ |
| 42 | 438.3 | ^4F♯, ^3G♭ | |
| 43 | 448.7 | 48/37 | v4F𝄪, ^4G♭ |
| 44 | 459.1 | 30/23, 43/33 | v3F𝄪, v4G |
| 45 | 469.6 | 21/16, 38/29 | vvF𝄪, v3G |
| 46 | 480 | 29/22, 33/25 | vF𝄪, vvG |
| 47 | 490.4 | F𝄪, vG | |
| 48 | 500.9 | G | |
| 49 | 511.3 | 39/29, 43/32, 47/35 | ^G, A♭♭ |
| 50 | 521.7 | 23/17 | ^^G, ^A♭♭ |
| 51 | 532.2 | 34/25 | ^3G, ^^A♭♭ |
| 52 | 542.6 | 26/19, 41/30 | ^4G, ^3A♭♭ |
| 53 | 553 | 11/8 | v4G♯, ^4A♭♭ |
| 54 | 563.5 | v3G♯, v4A♭ | |
| 55 | 573.9 | 39/28, 46/33 | vvG♯, v3A♭ |
| 56 | 584.3 | 7/5 | vG♯, vvA♭ |
| 57 | 594.8 | 31/22 | G♯, vA♭ |
| 58 | 605.2 | 44/31 | ^G♯, A♭ |
| 59 | 615.7 | 10/7 | ^^G♯, ^A♭ |
| 60 | 626.1 | 33/23 | ^3G♯, ^^A♭ |
| 61 | 636.5 | ^4G♯, ^3A♭ | |
| 62 | 647 | 16/11 | v4G𝄪, ^4A♭ |
| 63 | 657.4 | 19/13 | v3G𝄪, v4A |
| 64 | 667.8 | 25/17 | vvG𝄪, v3A |
| 65 | 678.3 | 34/23, 37/25 | vG𝄪, vvA |
| 66 | 688.7 | G𝄪, vA | |
| 67 | 699.1 | A | |
| 68 | 709.6 | ^A, B♭♭ | |
| 69 | 720 | 44/29, 47/31 | ^^A, ^B♭♭ |
| 70 | 730.4 | 29/19, 32/21 | ^3A, ^^B♭♭ |
| 71 | 740.9 | 23/15, 43/28 | ^4A, ^3B♭♭ |
| 72 | 751.3 | 37/24 | v4A♯, ^4B♭♭ |
| 73 | 761.7 | v3A♯, v4B♭ | |
| 74 | 772.2 | 25/16 | vvA♯, v3B♭ |
| 75 | 782.6 | 11/7 | vA♯, vvB♭ |
| 76 | 793 | A♯, vB♭ | |
| 77 | 803.5 | 35/22 | ^A♯, B♭ |
| 78 | 813.9 | 8/5 | ^^A♯, ^B♭ |
| 79 | 824.3 | 37/23 | ^3A♯, ^^B♭ |
| 80 | 834.8 | 34/21, 47/29 | ^4A♯, ^3B♭ |
| 81 | 845.2 | v4A𝄪, ^4B♭ | |
| 82 | 855.7 | 41/25 | v3A𝄪, v4B |
| 83 | 866.1 | 33/20 | vvA𝄪, v3B |
| 84 | 876.5 | vA𝄪, vvB | |
| 85 | 887 | A𝄪, vB | |
| 86 | 897.4 | 42/25, 47/28 | B |
| 87 | 907.8 | ^B, C♭ | |
| 88 | 918.3 | 17/10 | ^^B, ^C♭ |
| 89 | 928.7 | 41/24 | ^3B, ^^C♭ |
| 90 | 939.1 | 43/25 | ^4B, ^3C♭ |
| 91 | 949.6 | v4B♯, ^4C♭ | |
| 92 | 960 | 40/23 | v3B♯, v4C |
| 93 | 970.4 | 7/4 | vvB♯, v3C |
| 94 | 980.9 | 37/21 | vB♯, vvC |
| 95 | 991.3 | 39/22 | B♯, vC |
| 96 | 1001.7 | 25/14, 41/23 | C |
| 97 | 1012.2 | ^C, D♭♭ | |
| 98 | 1022.6 | ^^C, ^D♭♭ | |
| 99 | 1033 | 20/11 | ^3C, ^^D♭♭ |
| 100 | 1043.5 | 42/23 | ^4C, ^3D♭♭ |
| 101 | 1053.9 | 46/25 | v4C♯, ^4D♭♭ |
| 102 | 1064.3 | 37/20 | v3C♯, v4D♭ |
| 103 | 1074.8 | vvC♯, v3D♭ | |
| 104 | 1085.2 | 43/23 | vC♯, vvD♭ |
| 105 | 1095.7 | 32/17 | C♯, vD♭ |
| 106 | 1106.1 | ^C♯, D♭ | |
| 107 | 1116.5 | 40/21 | ^^C♯, ^D♭ |
| 108 | 1127 | 23/12 | ^3C♯, ^^D♭ |
| 109 | 1137.4 | ^4C♯, ^3D♭ | |
| 110 | 1147.8 | 33/17 | v4C𝄪, ^4D♭ |
| 111 | 1158.3 | 39/20, 41/21, 43/22 | v3C𝄪, v4D |
| 112 | 1168.7 | vvC𝄪, v3D | |
| 113 | 1179.1 | vC𝄪, vvD | |
| 114 | 1189.6 | C𝄪, vD | |
| 115 | 1200 | 2/1 | D |