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 | 10.435 | ^D, E♭♭ | |
| 2 | 20.87 | 78/77 | ^^D, v8E♭ |
| 3 | 31.304 | 50/49, 56/55, 64/63 | ^3D, v7E♭ |
| 4 | 41.739 | 40/39 | ^4D, v6E♭ |
| 5 | 52.174 | 33/32 | ^5D, v5E♭ |
| 6 | 62.609 | 80/77 | ^6D, v4E♭ |
| 7 | 73.043 | 25/24 | ^7D, v3E♭ |
| 8 | 83.478 | 21/20, 22/21 | ^8D, vvE♭ |
| 9 | 93.913 | 55/52 | D♯, vE♭ |
| 10 | 104.348 | 35/33, 52/49 | ^D♯, E♭ |
| 11 | 114.783 | 16/15 | ^^D♯, v8E |
| 12 | 125.217 | 14/13 | ^3D♯, v7E |
| 13 | 135.652 | ^4D♯, v6E | |
| 14 | 146.087 | ^5D♯, v5E | |
| 15 | 156.522 | 35/32 | ^6D♯, v4E |
| 16 | 166.957 | 11/10 | ^7D♯, v3E |
| 17 | 177.391 | ^8D♯, vvE | |
| 18 | 187.826 | 39/35, 49/44 | D𝄪, vE |
| 19 | 198.261 | 28/25, 55/49 | E |
| 20 | 208.696 | 44/39 | ^E, F♭ |
| 21 | 219.13 | 25/22 | ^^E, v8F |
| 22 | 229.565 | 8/7 | ^3E, v7F |
| 23 | 240 | ^4E, v6F | |
| 24 | 250.435 | ^5E, v5F | |
| 25 | 260.87 | 64/55, 65/56 | ^6E, v4F |
| 26 | 271.304 | 75/64 | ^7E, v3F |
| 27 | 281.739 | 33/28 | ^8E, vvF |
| 28 | 292.174 | 13/11, 77/65 | E♯, vF |
| 29 | 302.609 | 25/21 | F |
| 30 | 313.043 | 6/5 | ^F, G♭♭ |
| 31 | 323.478 | 77/64 | ^^F, v8G♭ |
| 32 | 333.913 | 40/33 | ^3F, v7G♭ |
| 33 | 344.348 | 39/32 | ^4F, v6G♭ |
| 34 | 354.783 | 49/40 | ^5F, v5G♭ |
| 35 | 365.217 | ^6F, v4G♭ | |
| 36 | 375.652 | ^7F, v3G♭ | |
| 37 | 386.087 | 5/4 | ^8F, vvG♭ |
| 38 | 396.522 | 44/35, 49/39, 63/50 | F♯, vG♭ |
| 39 | 406.957 | ^F♯, G♭ | |
| 40 | 417.391 | 14/11, 80/63 | ^^F♯, v8G |
| 41 | 427.826 | 32/25, 50/39 | ^3F♯, v7G |
| 42 | 438.261 | ^4F♯, v6G | |
| 43 | 448.696 | ^5F♯, v5G | |
| 44 | 459.13 | 64/49 | ^6F♯, v4G |
| 45 | 469.565 | 21/16, 55/42 | ^7F♯, v3G |
| 46 | 480 | 33/25 | ^8F♯, vvG |
| 47 | 490.435 | 65/49 | F𝄪, vG |
| 48 | 500.87 | 4/3 | G |
| 49 | 511.304 | 35/26 | ^G, A♭♭ |
| 50 | 521.739 | ^^G, v8A♭ | |
| 51 | 532.174 | ^3G, v7A♭ | |
| 52 | 542.609 | ^4G, v6A♭ | |
| 53 | 553.043 | 11/8 | ^5G, v5A♭ |
| 54 | 563.478 | ^6G, v4A♭ | |
| 55 | 573.913 | 39/28 | ^7G, v3A♭ |
| 56 | 584.348 | 7/5 | ^8G, vvA♭ |
| 57 | 594.783 | 55/39 | G♯, vA♭ |
| 58 | 605.217 | 78/55 | ^G♯, A♭ |
| 59 | 615.652 | 10/7 | ^^G♯, v8A |
| 60 | 626.087 | 56/39 | ^3G♯, v7A |
| 61 | 636.522 | ^4G♯, v6A | |
| 62 | 646.957 | 16/11 | ^5G♯, v5A |
| 63 | 657.391 | ^6G♯, v4A | |
| 64 | 667.826 | ^7G♯, v3A | |
| 65 | 678.261 | 65/44, 77/52 | ^8G♯, vvA |
| 66 | 688.696 | 52/35 | G𝄪, vA |
| 67 | 699.13 | 3/2 | A |
| 68 | 709.565 | ^A, B♭♭ | |
| 69 | 720 | 50/33 | ^^A, v8B♭ |
| 70 | 730.435 | 32/21 | ^3A, v7B♭ |
| 71 | 740.87 | 49/32 | ^4A, v6B♭ |
| 72 | 751.304 | 77/50 | ^5A, v5B♭ |
| 73 | 761.739 | ^6A, v4B♭ | |
| 74 | 772.174 | 25/16, 39/25 | ^7A, v3B♭ |
| 75 | 782.609 | 11/7, 63/40 | ^8A, vvB♭ |
| 76 | 793.043 | A♯, vB♭ | |
| 77 | 803.478 | 35/22, 78/49 | ^A♯, B♭ |
| 78 | 813.913 | 8/5 | ^^A♯, v8B |
| 79 | 824.348 | ^3A♯, v7B | |
| 80 | 834.783 | ^4A♯, v6B | |
| 81 | 845.217 | 80/49 | ^5A♯, v5B |
| 82 | 855.652 | 64/39 | ^6A♯, v4B |
| 83 | 866.087 | 33/20 | ^7A♯, v3B |
| 84 | 876.522 | ^8A♯, vvB | |
| 85 | 886.957 | 5/3 | A𝄪, vB |
| 86 | 897.391 | 42/25 | B |
| 87 | 907.826 | 22/13 | ^B, C♭ |
| 88 | 918.261 | 56/33 | ^^B, v8C |
| 89 | 928.696 | ^3B, v7C | |
| 90 | 939.13 | 55/32 | ^4B, v6C |
| 91 | 949.565 | ^5B, v5C | |
| 92 | 960 | ^6B, v4C | |
| 93 | 970.435 | 7/4 | ^7B, v3C |
| 94 | 980.87 | 44/25 | ^8B, vvC |
| 95 | 991.304 | 39/22 | B♯, vC |
| 96 | 1001.739 | 25/14 | C |
| 97 | 1012.174 | 70/39 | ^C, D♭♭ |
| 98 | 1022.609 | ^^C, v8D♭ | |
| 99 | 1033.043 | 20/11 | ^3C, v7D♭ |
| 100 | 1043.478 | 64/35 | ^4C, v6D♭ |
| 101 | 1053.913 | ^5C, v5D♭ | |
| 102 | 1064.348 | ^6C, v4D♭ | |
| 103 | 1074.783 | 13/7 | ^7C, v3D♭ |
| 104 | 1085.217 | 15/8 | ^8C, vvD♭ |
| 105 | 1095.652 | 49/26, 66/35 | C♯, vD♭ |
| 106 | 1106.087 | ^C♯, D♭ | |
| 107 | 1116.522 | 21/11, 40/21 | ^^C♯, v8D |
| 108 | 1126.957 | 48/25 | ^3C♯, v7D |
| 109 | 1137.391 | 77/40 | ^4C♯, v6D |
| 110 | 1147.826 | 64/33 | ^5C♯, v5D |
| 111 | 1158.261 | 39/20 | ^6C♯, v4D |
| 112 | 1168.696 | 49/25, 55/28, 63/32 | ^7C♯, v3D |
| 113 | 1179.13 | 77/39 | ^8C♯, vvD |
| 114 | 1189.565 | C𝄪, vD | |
| 115 | 1200 | 2/1 | D |