110edo
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Prime factorization
2 × 5 × 11
Step size
10.9091¢
Fifth
64\110 (698.182¢) (→32\55)
Semitones (A1:m2)
8:10 (87.27¢ : 109.1¢)
Dual sharp fifth
65\110 (709.091¢) (→13\22)
Dual flat fifth
64\110 (698.182¢) (→32\55)
Dual major 2nd
19\110 (207.273¢)
Consistency limit
5
Distinct consistency limit
5
| ← 109edo | 110edo | 111edo → |
110 equal divisions of the octave (abbreviated 110edo or 110ed2), also called 110-tone equal temperament (110tet) or 110 equal temperament (110et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 110 equal parts of about 10.9 ¢ each. Each step represents a frequency ratio of 21/110, or the 110th root of 2.
Theory
The equal temperament tempers out 15625/15552 (kleisma) and 3486784401/3355443200 in the 5-limit. Using the patent val, it tempers out 1728/1715, 3125/3087, and 3645/3584 in the 7-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -3.77 | -4.50 | +2.08 | +3.36 | +5.05 | -0.53 | +2.64 | +4.14 | -2.97 | -1.69 | +4.45 |
| Relative (%) | -34.6 | -41.2 | +19.1 | +30.8 | +46.3 | -4.8 | +24.2 | +37.9 | -27.2 | -15.5 | +40.8 | |
| Steps (reduced) |
174 (64) |
255 (35) |
309 (89) |
349 (19) |
381 (51) |
407 (77) |
430 (100) |
450 (10) |
467 (27) |
483 (43) |
498 (58) | |
Subsets and supersets
Since 110 factors into 2 × 5 × 11, 110edo has subset edos 2, 5, 10, 11, 22, and 55.
Intervals
| Steps | Cents | Approximate ratios | Ups and downs notation (Dual flat fifth 64\110) |
Ups and downs notation (Dual sharp fifth 65\110) |
|---|---|---|---|---|
| 0 | 0 | 1/1 | D | D |
| 1 | 10.9 | ^D, vE♭♭ | ^D, v4E♭ | |
| 2 | 21.8 | ^^D, E♭♭ | ^^D, v3E♭ | |
| 3 | 32.7 | ^3D, ^E♭♭ | ^3D, vvE♭ | |
| 4 | 43.6 | 39/38, 40/39, 41/40, 42/41 | ^4D, ^^E♭♭ | ^4D, vE♭ |
| 5 | 54.5 | 32/31, 33/32 | v3D♯, ^3E♭♭ | ^5D, E♭ |
| 6 | 65.5 | vvD♯, v4E♭ | ^6D, ^E♭ | |
| 7 | 76.4 | 23/22 | vD♯, v3E♭ | ^7D, ^^E♭ |
| 8 | 87.3 | 20/19, 41/39 | D♯, vvE♭ | v7D♯, ^3E♭ |
| 9 | 98.2 | 37/35 | ^D♯, vE♭ | v6D♯, ^4E♭ |
| 10 | 109.1 | 33/31 | ^^D♯, E♭ | v5D♯, ^5E♭ |
| 11 | 120 | ^3D♯, ^E♭ | v4D♯, ^6E♭ | |
| 12 | 130.9 | 41/38 | ^4D♯, ^^E♭ | v3D♯, ^7E♭ |
| 13 | 141.8 | 38/35 | v3D𝄪, ^3E♭ | vvD♯, v7E |
| 14 | 152.7 | 47/43 | vvD𝄪, v4E | vD♯, v6E |
| 15 | 163.6 | vD𝄪, v3E | D♯, v5E | |
| 16 | 174.5 | 21/19, 31/28 | D𝄪, vvE | ^D♯, v4E |
| 17 | 185.5 | 39/35 | ^D𝄪, vE | ^^D♯, v3E |
| 18 | 196.4 | 37/33, 47/42 | E | ^3D♯, vvE |
| 19 | 207.3 | ^E, vF♭ | ^4D♯, vE | |
| 20 | 218.2 | 42/37 | ^^E, F♭ | E |
| 21 | 229.1 | 8/7 | ^3E, ^F♭ | ^E, v4F |
| 22 | 240 | ^4E, ^^F♭ | ^^E, v3F | |
| 23 | 250.9 | 37/32 | v3E♯, ^3F♭ | ^3E, vvF |
| 24 | 261.8 | 43/37 | vvE♯, v4F | ^4E, vF |
| 25 | 272.7 | 41/35 | vE♯, v3F | F |
| 26 | 283.6 | 33/28 | E♯, vvF | ^F, v4G♭ |
| 27 | 294.5 | ^E♯, vF | ^^F, v3G♭ | |
| 28 | 305.5 | 31/26, 37/31 | F | ^3F, vvG♭ |
| 29 | 316.4 | 6/5 | ^F, vG♭♭ | ^4F, vG♭ |
| 30 | 327.3 | 29/24, 35/29 | ^^F, G♭♭ | ^5F, G♭ |
| 31 | 338.2 | 17/14 | ^3F, ^G♭♭ | ^6F, ^G♭ |
| 32 | 349.1 | ^4F, ^^G♭♭ | ^7F, ^^G♭ | |
| 33 | 360 | 16/13 | v3F♯, ^3G♭♭ | v7F♯, ^3G♭ |
| 34 | 370.9 | 26/21 | vvF♯, v4G♭ | v6F♯, ^4G♭ |
| 35 | 381.8 | vF♯, v3G♭ | v5F♯, ^5G♭ | |
| 36 | 392.7 | F♯, vvG♭ | v4F♯, ^6G♭ | |
| 37 | 403.6 | 24/19 | ^F♯, vG♭ | v3F♯, ^7G♭ |
| 38 | 414.5 | 33/26, 47/37 | ^^F♯, G♭ | vvF♯, v7G |
| 39 | 425.5 | ^3F♯, ^G♭ | vF♯, v6G | |
| 40 | 436.4 | ^4F♯, ^^G♭ | F♯, v5G | |
| 41 | 447.3 | 22/17 | v3F𝄪, ^3G♭ | ^F♯, v4G |
| 42 | 458.2 | 43/33 | vvF𝄪, v4G | ^^F♯, v3G |
| 43 | 469.1 | 21/16, 38/29 | vF𝄪, v3G | ^3F♯, vvG |
| 44 | 480 | F𝄪, vvG | ^4F♯, vG | |
| 45 | 490.9 | ^F𝄪, vG | G | |
| 46 | 501.8 | G | ^G, v4A♭ | |
| 47 | 512.7 | 35/26, 39/29, 43/32 | ^G, vA♭♭ | ^^G, v3A♭ |
| 48 | 523.6 | 23/17, 42/31 | ^^G, A♭♭ | ^3G, vvA♭ |
| 49 | 534.5 | ^3G, ^A♭♭ | ^4G, vA♭ | |
| 50 | 545.5 | ^4G, ^^A♭♭ | ^5G, A♭ | |
| 51 | 556.4 | 40/29 | v3G♯, ^3A♭♭ | ^6G, ^A♭ |
| 52 | 567.3 | 25/18, 43/31 | vvG♯, v4A♭ | ^7G, ^^A♭ |
| 53 | 578.2 | vG♯, v3A♭ | v7G♯, ^3A♭ | |
| 54 | 589.1 | G♯, vvA♭ | v6G♯, ^4A♭ | |
| 55 | 600 | 41/29 | ^G♯, vA♭ | v5G♯, ^5A♭ |
| 56 | 610.9 | 37/26, 47/33 | ^^G♯, A♭ | v4G♯, ^6A♭ |
| 57 | 621.8 | ^3G♯, ^A♭ | v3G♯, ^7A♭ | |
| 58 | 632.7 | 36/25 | ^4G♯, ^^A♭ | vvG♯, v7A |
| 59 | 643.6 | 29/20 | v3G𝄪, ^3A♭ | vG♯, v6A |
| 60 | 654.5 | 35/24 | vvG𝄪, v4A | G♯, v5A |
| 61 | 665.5 | 47/32 | vG𝄪, v3A | ^G♯, v4A |
| 62 | 676.4 | 31/21, 34/23 | G𝄪, vvA | ^^G♯, v3A |
| 63 | 687.3 | ^G𝄪, vA | ^3G♯, vvA | |
| 64 | 698.2 | A | ^4G♯, vA | |
| 65 | 709.1 | ^A, vB♭♭ | A | |
| 66 | 720 | 47/31 | ^^A, B♭♭ | ^A, v4B♭ |
| 67 | 730.9 | 29/19, 32/21 | ^3A, ^B♭♭ | ^^A, v3B♭ |
| 68 | 741.8 | 43/28 | ^4A, ^^B♭♭ | ^3A, vvB♭ |
| 69 | 752.7 | 17/11 | v3A♯, ^3B♭♭ | ^4A, vB♭ |
| 70 | 763.6 | vvA♯, v4B♭ | ^5A, B♭ | |
| 71 | 774.5 | vA♯, v3B♭ | ^6A, ^B♭ | |
| 72 | 785.5 | A♯, vvB♭ | ^7A, ^^B♭ | |
| 73 | 796.4 | 19/12 | ^A♯, vB♭ | v7A♯, ^3B♭ |
| 74 | 807.3 | ^^A♯, B♭ | v6A♯, ^4B♭ | |
| 75 | 818.2 | ^3A♯, ^B♭ | v5A♯, ^5B♭ | |
| 76 | 829.1 | 21/13 | ^4A♯, ^^B♭ | v4A♯, ^6B♭ |
| 77 | 840 | 13/8 | v3A𝄪, ^3B♭ | v3A♯, ^7B♭ |
| 78 | 850.9 | vvA𝄪, v4B | vvA♯, v7B | |
| 79 | 861.8 | 28/17 | vA𝄪, v3B | vA♯, v6B |
| 80 | 872.7 | 43/26 | A𝄪, vvB | A♯, v5B |
| 81 | 883.6 | 5/3 | ^A𝄪, vB | ^A♯, v4B |
| 82 | 894.5 | 47/28 | B | ^^A♯, v3B |
| 83 | 905.5 | ^B, vC♭ | ^3A♯, vvB | |
| 84 | 916.4 | ^^B, C♭ | ^4A♯, vB | |
| 85 | 927.3 | 41/24 | ^3B, ^C♭ | B |
| 86 | 938.2 | ^4B, ^^C♭ | ^B, v4C | |
| 87 | 949.1 | v3B♯, ^3C♭ | ^^B, v3C | |
| 88 | 960 | vvB♯, v4C | ^3B, vvC | |
| 89 | 970.9 | 7/4 | vB♯, v3C | ^4B, vC |
| 90 | 981.8 | 37/21 | B♯, vvC | C |
| 91 | 992.7 | ^B♯, vC | ^C, v4D♭ | |
| 92 | 1003.6 | C | ^^C, v3D♭ | |
| 93 | 1014.5 | ^C, vD♭♭ | ^3C, vvD♭ | |
| 94 | 1025.5 | 38/21, 47/26 | ^^C, D♭♭ | ^4C, vD♭ |
| 95 | 1036.4 | ^3C, ^D♭♭ | ^5C, D♭ | |
| 96 | 1047.3 | ^4C, ^^D♭♭ | ^6C, ^D♭ | |
| 97 | 1058.2 | 35/19 | v3C♯, ^3D♭♭ | ^7C, ^^D♭ |
| 98 | 1069.1 | vvC♯, v4D♭ | v7C♯, ^3D♭ | |
| 99 | 1080 | vC♯, v3D♭ | v6C♯, ^4D♭ | |
| 100 | 1090.9 | C♯, vvD♭ | v5C♯, ^5D♭ | |
| 101 | 1101.8 | ^C♯, vD♭ | v4C♯, ^6D♭ | |
| 102 | 1112.7 | 19/10 | ^^C♯, D♭ | v3C♯, ^7D♭ |
| 103 | 1123.6 | 44/23 | ^3C♯, ^D♭ | vvC♯, v7D |
| 104 | 1134.5 | ^4C♯, ^^D♭ | vC♯, v6D | |
| 105 | 1145.5 | 31/16 | v3C𝄪, ^3D♭ | C♯, v5D |
| 106 | 1156.4 | 39/20, 41/21 | vvC𝄪, v4D | ^C♯, v4D |
| 107 | 1167.3 | vC𝄪, v3D | ^^C♯, v3D | |
| 108 | 1178.2 | C𝄪, vvD | ^3C♯, vvD | |
| 109 | 1189.1 | ^C𝄪, vD | ^4C♯, vD | |
| 110 | 1200 | 2/1 | D | D |
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