107edo
| ← 106edo | 107edo | 108edo → |
107 equal divisions of the octave (abbreviated 107edo or 107ed2), also called 107-tone equal temperament (107tet) or 107 equal temperament (107et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 107 equal parts of about 11.2 ¢ each. Each step represents a frequency ratio of 21/107, or the 107th root of 2.
Theory
107edo is inconsistent to the 5-odd-limit and higher limits, and harmonics 3, 5, and 7 are about halfway between its steps. Either the 2.9.5.7 or 2.9.15.21 subgroup can be used. For the full 7-limit, it has four possible mappings: ⟨107 170 248 300] (patent val), ⟨107 169 248 300] (107b), ⟨107 170 249 300] (107c), and ⟨107 170 249 301] (107cd).
Using the patent val, it tempers out 3125/3072 (magic comma) and [28 -22 3⟩ in the 5-limit; 1029/1024, 2240/2187, and 3125/3087 in the 7-limit; 100/99, 1232/1215, and 1331/1323 in the 11-limit.
Using the 107cd val, it tempers out 1728/1715, 4000/3969, and 28672/28125 in the 7-limit; 121/120, 896/891, 1375/1372, and 3168/3125 in the 11-limit.
Using the 107c val, it tempers out 1638400/1594323 (immunity comma) and 1990656/1953125 (valentine comma) in the 5-limit; 126/125, 1029/1024, and 307328/295245 in the 7-limit; 121/120, 176/175, 441/440, and 184877/177147 in the 11-limit.
Using the 107b val, it tempers out 81/80 (syntonic comma) and [-61 -1 27⟩; in the 5-limit; 2401/2400, 2430/2401, and 234375/229376 in the 7-limit; 385/384, 1350/1331, 1375/1372, and 1944/1925 in the 11-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +4.59 | -5.01 | -4.34 | -2.04 | -1.79 | +0.59 | -0.42 | -4.02 | +5.29 | +0.25 | -0.24 |
| Relative (%) | +40.9 | -44.6 | -38.7 | -18.2 | -15.9 | +5.3 | -3.7 | -35.9 | +47.2 | +2.2 | -2.1 | |
| Steps (reduced) |
170 (63) |
248 (34) |
300 (86) |
339 (18) |
370 (49) |
396 (75) |
418 (97) |
437 (9) |
455 (27) |
470 (42) |
484 (56) | |
Subsets and supersets
107edo is the 28th prime edo, following 103edo and before 109edo. 214edo, which doubles it, provides correction for the approximation to harmonics 3, 5, and 7.
Intervals
| Steps | Cents | Approximate Ratios | Ups and Downs Notation (Dual Flat Fifth 62\107) |
Ups and Downs Notation (Dual Sharp Fifth 63\107) |
|---|---|---|---|---|
| 0 | 0 | 1/1 | D | D |
| 1 | 11.215 | ^D, v4E♭♭ | ^D, v5E♭ | |
| 2 | 22.43 | 65/64 | ^^D, v3E♭♭ | ^^D, v4E♭ |
| 3 | 33.645 | 50/49, 56/55 | ^3D, vvE♭♭ | ^3D, v3E♭ |
| 4 | 44.86 | 77/75 | ^4D, vE♭♭ | ^4D, vvE♭ |
| 5 | 56.075 | 33/32 | ^5D, E♭♭ | ^5D, vE♭ |
| 6 | 67.29 | 80/77 | D♯, v5E♭ | ^6D, E♭ |
| 7 | 78.505 | 22/21 | ^D♯, v4E♭ | ^7D, v12E |
| 8 | 89.72 | ^^D♯, v3E♭ | ^8D, v11E | |
| 9 | 100.935 | ^3D♯, vvE♭ | ^9D, v10E | |
| 10 | 112.15 | 16/15 | ^4D♯, vE♭ | ^10D, v9E |
| 11 | 123.364 | 15/14 | ^5D♯, E♭ | ^11D, v8E |
| 12 | 134.579 | 13/12 | D𝄪, v5E | ^12D, v7E |
| 13 | 145.794 | ^D𝄪, v4E | D♯, v6E | |
| 14 | 157.009 | ^^D𝄪, v3E | ^D♯, v5E | |
| 15 | 168.224 | 11/10 | ^3D𝄪, vvE | ^^D♯, v4E |
| 16 | 179.439 | ^4D𝄪, vE | ^3D♯, v3E | |
| 17 | 190.654 | E | ^4D♯, vvE | |
| 18 | 201.869 | 55/49 | ^E, v4F♭ | ^5D♯, vE |
| 19 | 213.084 | ^^E, v3F♭ | E | |
| 20 | 224.299 | ^3E, vvF♭ | ^E, v5F | |
| 21 | 235.514 | 8/7 | ^4E, vF♭ | ^^E, v4F |
| 22 | 246.729 | 15/13, 52/45 | ^5E, F♭ | ^3E, v3F |
| 23 | 257.944 | 65/56 | E♯, v5F | ^4E, vvF |
| 24 | 269.159 | ^E♯, v4F | ^5E, vF | |
| 25 | 280.374 | ^^E♯, v3F | F | |
| 26 | 291.589 | 13/11, 77/65 | ^3E♯, vvF | ^F, v5G♭ |
| 27 | 302.804 | ^4E♯, vF | ^^F, v4G♭ | |
| 28 | 314.019 | F | ^3F, v3G♭ | |
| 29 | 325.234 | ^F, v4G♭♭ | ^4F, vvG♭ | |
| 30 | 336.449 | 63/52 | ^^F, v3G♭♭ | ^5F, vG♭ |
| 31 | 347.664 | 49/40 | ^3F, vvG♭♭ | ^6F, G♭ |
| 32 | 358.879 | 16/13 | ^4F, vG♭♭ | ^7F, v12G |
| 33 | 370.093 | 26/21 | ^5F, G♭♭ | ^8F, v11G |
| 34 | 381.308 | F♯, v5G♭ | ^9F, v10G | |
| 35 | 392.523 | ^F♯, v4G♭ | ^10F, v9G | |
| 36 | 403.738 | ^^F♯, v3G♭ | ^11F, v8G | |
| 37 | 414.953 | 14/11, 33/26 | ^3F♯, vvG♭ | ^12F, v7G |
| 38 | 426.168 | ^4F♯, vG♭ | F♯, v6G | |
| 39 | 437.383 | ^5F♯, G♭ | ^F♯, v5G | |
| 40 | 448.598 | F𝄪, v5G | ^^F♯, v4G | |
| 41 | 459.813 | ^F𝄪, v4G | ^3F♯, v3G | |
| 42 | 471.028 | 21/16 | ^^F𝄪, v3G | ^4F♯, vvG |
| 43 | 482.243 | ^3F𝄪, vvG | ^5F♯, vG | |
| 44 | 493.458 | 65/49 | ^4F𝄪, vG | G |
| 45 | 504.673 | 75/56 | G | ^G, v5A♭ |
| 46 | 515.888 | ^G, v4A♭♭ | ^^G, v4A♭ | |
| 47 | 527.103 | ^^G, v3A♭♭ | ^3G, v3A♭ | |
| 48 | 538.318 | 15/11 | ^3G, vvA♭♭ | ^4G, vvA♭ |
| 49 | 549.533 | 11/8 | ^4G, vA♭♭ | ^5G, vA♭ |
| 50 | 560.748 | ^5G, A♭♭ | ^6G, A♭ | |
| 51 | 571.963 | G♯, v5A♭ | ^7G, v12A | |
| 52 | 583.178 | 7/5 | ^G♯, v4A♭ | ^8G, v11A |
| 53 | 594.393 | 45/32 | ^^G♯, v3A♭ | ^9G, v10A |
| 54 | 605.607 | 64/45 | ^3G♯, vvA♭ | ^10G, v9A |
| 55 | 616.822 | 10/7 | ^4G♯, vA♭ | ^11G, v8A |
| 56 | 628.037 | ^5G♯, A♭ | ^12G, v7A | |
| 57 | 639.252 | G𝄪, v5A | G♯, v6A | |
| 58 | 650.467 | 16/11 | ^G𝄪, v4A | ^G♯, v5A |
| 59 | 661.682 | 22/15 | ^^G𝄪, v3A | ^^G♯, v4A |
| 60 | 672.897 | 65/44 | ^3G𝄪, vvA | ^3G♯, v3A |
| 61 | 684.112 | ^4G𝄪, vA | ^4G♯, vvA | |
| 62 | 695.327 | A | ^5G♯, vA | |
| 63 | 706.542 | ^A, v4B♭♭ | A | |
| 64 | 717.757 | ^^A, v3B♭♭ | ^A, v5B♭ | |
| 65 | 728.972 | 32/21 | ^3A, vvB♭♭ | ^^A, v4B♭ |
| 66 | 740.187 | 75/49 | ^4A, vB♭♭ | ^3A, v3B♭ |
| 67 | 751.402 | 77/50 | ^5A, B♭♭ | ^4A, vvB♭ |
| 68 | 762.617 | A♯, v5B♭ | ^5A, vB♭ | |
| 69 | 773.832 | ^A♯, v4B♭ | ^6A, B♭ | |
| 70 | 785.047 | 11/7, 52/33 | ^^A♯, v3B♭ | ^7A, v12B |
| 71 | 796.262 | ^3A♯, vvB♭ | ^8A, v11B | |
| 72 | 807.477 | ^4A♯, vB♭ | ^9A, v10B | |
| 73 | 818.692 | ^5A♯, B♭ | ^10A, v9B | |
| 74 | 829.907 | 21/13 | A𝄪, v5B | ^11A, v8B |
| 75 | 841.121 | 13/8 | ^A𝄪, v4B | ^12A, v7B |
| 76 | 852.336 | 80/49 | ^^A𝄪, v3B | A♯, v6B |
| 77 | 863.551 | ^3A𝄪, vvB | ^A♯, v5B | |
| 78 | 874.766 | ^4A𝄪, vB | ^^A♯, v4B | |
| 79 | 885.981 | B | ^3A♯, v3B | |
| 80 | 897.196 | ^B, v4C♭ | ^4A♯, vvB | |
| 81 | 908.411 | 22/13 | ^^B, v3C♭ | ^5A♯, vB |
| 82 | 919.626 | 75/44 | ^3B, vvC♭ | B |
| 83 | 930.841 | ^4B, vC♭ | ^B, v5C | |
| 84 | 942.056 | ^5B, C♭ | ^^B, v4C | |
| 85 | 953.271 | 26/15, 45/26 | B♯, v5C | ^3B, v3C |
| 86 | 964.486 | 7/4 | ^B♯, v4C | ^4B, vvC |
| 87 | 975.701 | ^^B♯, v3C | ^5B, vC | |
| 88 | 986.916 | ^3B♯, vvC | C | |
| 89 | 998.131 | ^4B♯, vC | ^C, v5D♭ | |
| 90 | 1009.346 | C | ^^C, v4D♭ | |
| 91 | 1020.561 | ^C, v4D♭♭ | ^3C, v3D♭ | |
| 92 | 1031.776 | 20/11 | ^^C, v3D♭♭ | ^4C, vvD♭ |
| 93 | 1042.991 | ^3C, vvD♭♭ | ^5C, vD♭ | |
| 94 | 1054.206 | ^4C, vD♭♭ | ^6C, D♭ | |
| 95 | 1065.421 | 24/13 | ^5C, D♭♭ | ^7C, v12D |
| 96 | 1076.636 | 28/15 | C♯, v5D♭ | ^8C, v11D |
| 97 | 1087.85 | 15/8 | ^C♯, v4D♭ | ^9C, v10D |
| 98 | 1099.065 | ^^C♯, v3D♭ | ^10C, v9D | |
| 99 | 1110.28 | ^3C♯, vvD♭ | ^11C, v8D | |
| 100 | 1121.495 | 21/11 | ^4C♯, vD♭ | ^12C, v7D |
| 101 | 1132.71 | 77/40 | ^5C♯, D♭ | C♯, v6D |
| 102 | 1143.925 | 64/33 | C𝄪, v5D | ^C♯, v5D |
| 103 | 1155.14 | ^C𝄪, v4D | ^^C♯, v4D | |
| 104 | 1166.355 | 49/25, 55/28 | ^^C𝄪, v3D | ^3C♯, v3D |
| 105 | 1177.57 | ^3C𝄪, vvD | ^4C♯, vvD | |
| 106 | 1188.785 | ^4C𝄪, vD | ^5C♯, vD | |
| 107 | 1200 | 2/1 | D | D |
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [339 -107⟩ | [⟨107 339]] | +0.322 | 0.322 | 2.87 |
| 2.9.5 | 9765625/9565938, [-34 10 1⟩ | [⟨107 339 248]] | +0.933 | 0.904 | 8.06 |
| 2.9.5.7 | 225/224, 84035/82944, [14 -6 7 -4⟩ | [⟨107 339 248 300]] | +1.087 | 0.827 | 7.37 |
| 2.9.5.7.11 | 225/224, 441/440, 26411/26244, 161280/161051 | [⟨107 339 248 300 370]] | +0.973 | 0.774 | 6.90 |
| 2.9.5.7.11.13 | 225/224, 325/324, 441/440, 847/845, 24500/24167 | [⟨107 339 248 300 370 396]] | +0.783 | 0.823 | 7.33 |
| 2.9.5.7.11.13.17 | 170/169, 225/224, 325/324, 441/440, 847/845, 2000/1989 | [⟨107 339 248 300 370 396 437]] | +0.812 | 0.765 | 6.82 |