112edo
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Prime factorization
24 × 7
Step size
10.7143¢
Fifth
66\112 (707.143¢) (→33\56)
Semitones (A1:m2)
14:6 (150¢ : 64.29¢)
Dual sharp fifth
66\112 (707.143¢) (→33\56)
Dual flat fifth
65\112 (696.429¢)
Dual major 2nd
19\112 (203.571¢)
Consistency limit
3
Distinct consistency limit
3
| ← 111edo | 112edo | 113edo → |
112 equal divisions of the octave (abbreviated 112edo or 112ed2), also called 112-tone equal temperament (112tet) or 112 equal temperament (112et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 112 equal parts of about 10.7 ¢ each. Each step represents a frequency ratio of 21/112, or the 112th root of 2.
Theory
112edo has two great perfect fifths, the lower of which approximates 1/4-comma meantone (just a tad lower), and the upper of which- the patent fifth- is identical to the perfect fifth of 56edo, a great inverse gentle fifth where +5 fifths gives a near-just 28:27 while -8 fifths gives a near-just 32:39 (identical to 2 degrees of 7edo) and +9 fifths gives a close approximation to 17:21.
One can form a 17-tone circle by taking 15 large fifths and 2 small fifths, as above, which gives some nice interval shadings a wee bit different from 17edo, but sharing a similar structure.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +5.19 | -0.60 | -4.54 | -0.34 | -4.89 | -4.81 | +4.59 | +2.19 | +2.49 | +0.65 | +3.87 |
| relative (%) | +48 | -6 | -42 | -3 | -46 | -45 | +43 | +20 | +23 | +6 | +36 | |
| Steps (reduced) |
178 (66) |
260 (36) |
314 (90) |
355 (19) |
387 (51) |
414 (78) |
438 (102) |
458 (10) |
476 (28) |
492 (44) |
507 (59) | |
Intervals
"| Steps | Cents | Ups and downs notation (dual flat fifth 65\112) |
Ups and downs notation (dual sharp fifth 66\112) |
Approximate ratios |
|---|---|---|---|---|
| 0 | 0 | D | D | 1/1 |
| 1 | 10.7143 | ^D, v3E♭♭ | ^D, v5E♭ | |
| 2 | 21.4286 | ^^D, vvE♭♭ | ^^D, v4E♭ | |
| 3 | 32.1429 | ^3D, vE♭♭ | ^3D, v3E♭ | 56/55 |
| 4 | 42.8571 | ^4D, E♭♭ | ^4D, vvE♭ | 40/39 |
| 5 | 53.5714 | ^5D, v6E♭ | ^5D, vE♭ | 33/32 |
| 6 | 64.2857 | ^6D, v5E♭ | ^6D, E♭ | 26/25 |
| 7 | 75 | D♯, v4E♭ | ^7D, v13E | |
| 8 | 85.7143 | ^D♯, v3E♭ | ^8D, v12E | 21/20 |
| 9 | 96.4286 | ^^D♯, vvE♭ | ^9D, v11E | 55/52 |
| 10 | 107.143 | ^3D♯, vE♭ | ^10D, v10E | 52/49 |
| 11 | 117.857 | ^4D♯, E♭ | ^11D, v9E | |
| 12 | 128.571 | ^5D♯, v6E | ^12D, v8E | 14/13 |
| 13 | 139.286 | ^6D♯, v5E | ^13D, v7E | |
| 14 | 150 | D𝄪, v4E | D♯, v6E | |
| 15 | 160.714 | ^D𝄪, v3E | ^D♯, v5E | |
| 16 | 171.429 | ^^D𝄪, vvE | ^^D♯, v4E | |
| 17 | 182.143 | ^3D𝄪, vE | ^3D♯, v3E | 49/44 |
| 18 | 192.857 | E | ^4D♯, vvE | 28/25 |
| 19 | 203.571 | ^E, v3F♭ | ^5D♯, vE | 55/49 |
| 20 | 214.286 | ^^E, vvF♭ | E | |
| 21 | 225 | ^3E, vF♭ | ^E, v5F | 25/22 |
| 22 | 235.714 | ^4E, F♭ | ^^E, v4F | |
| 23 | 246.429 | ^5E, v6F | ^3E, v3F | |
| 24 | 257.143 | ^6E, v5F | ^4E, vvF | 65/56 |
| 25 | 267.857 | E♯, v4F | ^5E, vF | |
| 26 | 278.571 | ^E♯, v3F | F | 75/64 |
| 27 | 289.286 | ^^E♯, vvF | ^F, v5G♭ | 13/11, 77/65 |
| 28 | 300 | ^3E♯, vF | ^^F, v4G♭ | 25/21 |
| 29 | 310.714 | F | ^3F, v3G♭ | |
| 30 | 321.429 | ^F, v3G♭♭ | ^4F, vvG♭ | |
| 31 | 332.143 | ^^F, vvG♭♭ | ^5F, vG♭ | 40/33 |
| 32 | 342.857 | ^3F, vG♭♭ | ^6F, G♭ | 39/32 |
| 33 | 353.571 | ^4F, G♭♭ | ^7F, v13G | |
| 34 | 364.286 | ^5F, v6G♭ | ^8F, v12G | |
| 35 | 375 | ^6F, v5G♭ | ^9F, v11G | |
| 36 | 385.714 | F♯, v4G♭ | ^10F, v10G | 5/4 |
| 37 | 396.429 | ^F♯, v3G♭ | ^11F, v9G | 44/35 |
| 38 | 407.143 | ^^F♯, vvG♭ | ^12F, v8G | |
| 39 | 417.857 | ^3F♯, vG♭ | ^13F, v7G | 14/11 |
| 40 | 428.571 | ^4F♯, G♭ | F♯, v6G | 32/25, 50/39 |
| 41 | 439.286 | ^5F♯, v6G | ^F♯, v5G | |
| 42 | 450 | ^6F♯, v5G | ^^F♯, v4G | 13/10 |
| 43 | 460.714 | F𝄪, v4G | ^3F♯, v3G | |
| 44 | 471.429 | ^F𝄪, v3G | ^4F♯, vvG | 21/16 |
| 45 | 482.143 | ^^F𝄪, vvG | ^5F♯, vG | 33/25 |
| 46 | 492.857 | ^3F𝄪, vG | G | 65/49 |
| 47 | 503.571 | G | ^G, v5A♭ | |
| 48 | 514.286 | ^G, v3A♭♭ | ^^G, v4A♭ | 35/26 |
| 49 | 525 | ^^G, vvA♭♭ | ^3G, v3A♭ | |
| 50 | 535.714 | ^3G, vA♭♭ | ^4G, vvA♭ | |
| 51 | 546.429 | ^4G, A♭♭ | ^5G, vA♭ | |
| 52 | 557.143 | ^5G, v6A♭ | ^6G, A♭ | |
| 53 | 567.857 | ^6G, v5A♭ | ^7G, v13A | |
| 54 | 578.571 | G♯, v4A♭ | ^8G, v12A | 7/5 |
| 55 | 589.286 | ^G♯, v3A♭ | ^9G, v11A | |
| 56 | 600 | ^^G♯, vvA♭ | ^10G, v10A | |
| 57 | 610.714 | ^3G♯, vA♭ | ^11G, v9A | |
| 58 | 621.429 | ^4G♯, A♭ | ^12G, v8A | 10/7 |
| 59 | 632.143 | ^5G♯, v6A | ^13G, v7A | |
| 60 | 642.857 | ^6G♯, v5A | G♯, v6A | |
| 61 | 653.571 | G𝄪, v4A | ^G♯, v5A | |
| 62 | 664.286 | ^G𝄪, v3A | ^^G♯, v4A | |
| 63 | 675 | ^^G𝄪, vvA | ^3G♯, v3A | 65/44 |
| 64 | 685.714 | ^3G𝄪, vA | ^4G♯, vvA | 52/35 |
| 65 | 696.429 | A | ^5G♯, vA | |
| 66 | 707.143 | ^A, v3B♭♭ | A | |
| 67 | 717.857 | ^^A, vvB♭♭ | ^A, v5B♭ | 50/33 |
| 68 | 728.571 | ^3A, vB♭♭ | ^^A, v4B♭ | 32/21 |
| 69 | 739.286 | ^4A, B♭♭ | ^3A, v3B♭ | |
| 70 | 750 | ^5A, v6B♭ | ^4A, vvB♭ | 20/13 |
| 71 | 760.714 | ^6A, v5B♭ | ^5A, vB♭ | |
| 72 | 771.429 | A♯, v4B♭ | ^6A, B♭ | 25/16, 39/25 |
| 73 | 782.143 | ^A♯, v3B♭ | ^7A, v13B | 11/7 |
| 74 | 792.857 | ^^A♯, vvB♭ | ^8A, v12B | |
| 75 | 803.571 | ^3A♯, vB♭ | ^9A, v11B | 35/22 |
| 76 | 814.286 | ^4A♯, B♭ | ^10A, v10B | 8/5 |
| 77 | 825 | ^5A♯, v6B | ^11A, v9B | |
| 78 | 835.714 | ^6A♯, v5B | ^12A, v8B | |
| 79 | 846.429 | A𝄪, v4B | ^13A, v7B | |
| 80 | 857.143 | ^A𝄪, v3B | A♯, v6B | 64/39 |
| 81 | 867.857 | ^^A𝄪, vvB | ^A♯, v5B | 33/20 |
| 82 | 878.571 | ^3A𝄪, vB | ^^A♯, v4B | |
| 83 | 889.286 | B | ^3A♯, v3B | |
| 84 | 900 | ^B, v3C♭ | ^4A♯, vvB | 42/25 |
| 85 | 910.714 | ^^B, vvC♭ | ^5A♯, vB | 22/13 |
| 86 | 921.429 | ^3B, vC♭ | B | |
| 87 | 932.143 | ^4B, C♭ | ^B, v5C | |
| 88 | 942.857 | ^5B, v6C | ^^B, v4C | |
| 89 | 953.571 | ^6B, v5C | ^3B, v3C | |
| 90 | 964.286 | B♯, v4C | ^4B, vvC | |
| 91 | 975 | ^B♯, v3C | ^5B, vC | 44/25 |
| 92 | 985.714 | ^^B♯, vvC | C | |
| 93 | 996.429 | ^3B♯, vC | ^C, v5D♭ | |
| 94 | 1007.14 | C | ^^C, v4D♭ | 25/14 |
| 95 | 1017.86 | ^C, v3D♭♭ | ^3C, v3D♭ | |
| 96 | 1028.57 | ^^C, vvD♭♭ | ^4C, vvD♭ | |
| 97 | 1039.29 | ^3C, vD♭♭ | ^5C, vD♭ | |
| 98 | 1050 | ^4C, D♭♭ | ^6C, D♭ | |
| 99 | 1060.71 | ^5C, v6D♭ | ^7C, v13D | |
| 100 | 1071.43 | ^6C, v5D♭ | ^8C, v12D | 13/7 |
| 101 | 1082.14 | C♯, v4D♭ | ^9C, v11D | |
| 102 | 1092.86 | ^C♯, v3D♭ | ^10C, v10D | 49/26 |
| 103 | 1103.57 | ^^C♯, vvD♭ | ^11C, v9D | |
| 104 | 1114.29 | ^3C♯, vD♭ | ^12C, v8D | 40/21 |
| 105 | 1125 | ^4C♯, D♭ | ^13C, v7D | |
| 106 | 1135.71 | ^5C♯, v6D | C♯, v6D | 25/13 |
| 107 | 1146.43 | ^6C♯, v5D | ^C♯, v5D | 64/33 |
| 108 | 1157.14 | C𝄪, v4D | ^^C♯, v4D | 39/20 |
| 109 | 1167.86 | ^C𝄪, v3D | ^3C♯, v3D | 55/28 |
| 110 | 1178.57 | ^^C𝄪, vvD | ^4C♯, vvD | |
| 111 | 1189.29 | ^3C𝄪, vD | ^5C♯, vD | |
| 112 | 1200 | D | D | 2/1 |