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 11.215 ¢ 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, with four mappings possible for the 7-limit: ⟨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 33554432000/31381059609 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 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.
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 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.
Subsets and supersets
107edo is the 28th prime edo.
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 (%) | +41 | -45 | -39 | -18 | -16 | +5 | -4 | -36 | +47 | +2 | -2 | |
| Steps (reduced) |
170 (63) |
248 (34) |
300 (86) |
339 (18) |
370 (49) |
396 (75) |
418 (97) |
437 (9) |
455 (27) |
470 (42) |
484 (56) | |
Intervals
| Steps | Cents | Ups and downs notation (dual flat fifth 62\107) |
Ups and downs notation (dual sharp fifth 63\107) |
Approximate ratios |
|---|---|---|---|---|
| 0 | 0 | D | D | 1/1 |
| 1 | 11.215 | ^D, v4Ebb | ^D, v5Eb | |
| 2 | 22.4299 | ^^D, v3Ebb | ^^D, v4Eb | 65/64 |
| 3 | 33.6449 | ^3D, vvEbb | ^3D, v3Eb | 50/49, 56/55 |
| 4 | 44.8598 | ^4D, vEbb | ^4D, vvEb | 77/75 |
| 5 | 56.0748 | ^5D, Ebb | ^5D, vEb | 33/32 |
| 6 | 67.2897 | D#, v5Eb | ^6D, Eb | 80/77 |
| 7 | 78.5047 | ^D#, v4Eb | ^7D, v12E | 22/21 |
| 8 | 89.7196 | ^^D#, v3Eb | ^8D, v11E | |
| 9 | 100.935 | ^3D#, vvEb | ^9D, v10E | |
| 10 | 112.15 | ^4D#, vEb | ^10D, v9E | 16/15 |
| 11 | 123.364 | ^5D#, Eb | ^11D, v8E | 15/14 |
| 12 | 134.579 | Dx, v5E | ^12D, v7E | 13/12 |
| 13 | 145.794 | ^Dx, v4E | D#, v6E | |
| 14 | 157.009 | ^^Dx, v3E | ^D#, v5E | |
| 15 | 168.224 | ^3Dx, vvE | ^^D#, v4E | 11/10 |
| 16 | 179.439 | ^4Dx, vE | ^3D#, v3E | |
| 17 | 190.654 | E | ^4D#, vvE | |
| 18 | 201.869 | ^E, v4Fb | ^5D#, vE | 55/49 |
| 19 | 213.084 | ^^E, v3Fb | E | |
| 20 | 224.299 | ^3E, vvFb | ^E, v5F | |
| 21 | 235.514 | ^4E, vFb | ^^E, v4F | 8/7 |
| 22 | 246.729 | ^5E, Fb | ^3E, v3F | 15/13, 52/45 |
| 23 | 257.944 | E#, v5F | ^4E, vvF | 65/56 |
| 24 | 269.159 | ^E#, v4F | ^5E, vF | |
| 25 | 280.374 | ^^E#, v3F | F | |
| 26 | 291.589 | ^3E#, vvF | ^F, v5Gb | 13/11, 77/65 |
| 27 | 302.804 | ^4E#, vF | ^^F, v4Gb | |
| 28 | 314.019 | F | ^3F, v3Gb | |
| 29 | 325.234 | ^F, v4Gbb | ^4F, vvGb | |
| 30 | 336.449 | ^^F, v3Gbb | ^5F, vGb | 63/52 |
| 31 | 347.664 | ^3F, vvGbb | ^6F, Gb | 49/40 |
| 32 | 358.879 | ^4F, vGbb | ^7F, v12G | 16/13 |
| 33 | 370.093 | ^5F, Gbb | ^8F, v11G | 26/21 |
| 34 | 381.308 | F#, v5Gb | ^9F, v10G | |
| 35 | 392.523 | ^F#, v4Gb | ^10F, v9G | |
| 36 | 403.738 | ^^F#, v3Gb | ^11F, v8G | |
| 37 | 414.953 | ^3F#, vvGb | ^12F, v7G | 14/11, 33/26 |
| 38 | 426.168 | ^4F#, vGb | F#, v6G | |
| 39 | 437.383 | ^5F#, Gb | ^F#, v5G | |
| 40 | 448.598 | Fx, v5G | ^^F#, v4G | |
| 41 | 459.813 | ^Fx, v4G | ^3F#, v3G | |
| 42 | 471.028 | ^^Fx, v3G | ^4F#, vvG | 21/16 |
| 43 | 482.243 | ^3Fx, vvG | ^5F#, vG | |
| 44 | 493.458 | ^4Fx, vG | G | 65/49 |
| 45 | 504.673 | G | ^G, v5Ab | 75/56 |
| 46 | 515.888 | ^G, v4Abb | ^^G, v4Ab | |
| 47 | 527.103 | ^^G, v3Abb | ^3G, v3Ab | |
| 48 | 538.318 | ^3G, vvAbb | ^4G, vvAb | 15/11 |
| 49 | 549.533 | ^4G, vAbb | ^5G, vAb | 11/8 |
| 50 | 560.748 | ^5G, Abb | ^6G, Ab | |
| 51 | 571.963 | G#, v5Ab | ^7G, v12A | |
| 52 | 583.178 | ^G#, v4Ab | ^8G, v11A | 7/5 |
| 53 | 594.393 | ^^G#, v3Ab | ^9G, v10A | 45/32 |
| 54 | 605.607 | ^3G#, vvAb | ^10G, v9A | 64/45 |
| 55 | 616.822 | ^4G#, vAb | ^11G, v8A | 10/7 |
| 56 | 628.037 | ^5G#, Ab | ^12G, v7A | |
| 57 | 639.252 | Gx, v5A | G#, v6A | |
| 58 | 650.467 | ^Gx, v4A | ^G#, v5A | 16/11 |
| 59 | 661.682 | ^^Gx, v3A | ^^G#, v4A | 22/15 |
| 60 | 672.897 | ^3Gx, vvA | ^3G#, v3A | 65/44 |
| 61 | 684.112 | ^4Gx, vA | ^4G#, vvA | |
| 62 | 695.327 | A | ^5G#, vA | |
| 63 | 706.542 | ^A, v4Bbb | A | |
| 64 | 717.757 | ^^A, v3Bbb | ^A, v5Bb | |
| 65 | 728.972 | ^3A, vvBbb | ^^A, v4Bb | 32/21 |
| 66 | 740.187 | ^4A, vBbb | ^3A, v3Bb | 75/49 |
| 67 | 751.402 | ^5A, Bbb | ^4A, vvBb | 77/50 |
| 68 | 762.617 | A#, v5Bb | ^5A, vBb | |
| 69 | 773.832 | ^A#, v4Bb | ^6A, Bb | |
| 70 | 785.047 | ^^A#, v3Bb | ^7A, v12B | 11/7, 52/33 |
| 71 | 796.262 | ^3A#, vvBb | ^8A, v11B | |
| 72 | 807.477 | ^4A#, vBb | ^9A, v10B | |
| 73 | 818.692 | ^5A#, Bb | ^10A, v9B | |
| 74 | 829.907 | Ax, v5B | ^11A, v8B | 21/13 |
| 75 | 841.121 | ^Ax, v4B | ^12A, v7B | 13/8 |
| 76 | 852.336 | ^^Ax, v3B | A#, v6B | 80/49 |
| 77 | 863.551 | ^3Ax, vvB | ^A#, v5B | |
| 78 | 874.766 | ^4Ax, vB | ^^A#, v4B | |
| 79 | 885.981 | B | ^3A#, v3B | |
| 80 | 897.196 | ^B, v4Cb | ^4A#, vvB | |
| 81 | 908.411 | ^^B, v3Cb | ^5A#, vB | 22/13 |
| 82 | 919.626 | ^3B, vvCb | B | 75/44 |
| 83 | 930.841 | ^4B, vCb | ^B, v5C | |
| 84 | 942.056 | ^5B, Cb | ^^B, v4C | |
| 85 | 953.271 | B#, v5C | ^3B, v3C | 26/15, 45/26 |
| 86 | 964.486 | ^B#, v4C | ^4B, vvC | 7/4 |
| 87 | 975.701 | ^^B#, v3C | ^5B, vC | |
| 88 | 986.916 | ^3B#, vvC | C | |
| 89 | 998.131 | ^4B#, vC | ^C, v5Db | |
| 90 | 1009.35 | C | ^^C, v4Db | |
| 91 | 1020.56 | ^C, v4Dbb | ^3C, v3Db | |
| 92 | 1031.78 | ^^C, v3Dbb | ^4C, vvDb | 20/11 |
| 93 | 1042.99 | ^3C, vvDbb | ^5C, vDb | |
| 94 | 1054.21 | ^4C, vDbb | ^6C, Db | |
| 95 | 1065.42 | ^5C, Dbb | ^7C, v12D | 24/13 |
| 96 | 1076.64 | C#, v5Db | ^8C, v11D | 28/15 |
| 97 | 1087.85 | ^C#, v4Db | ^9C, v10D | 15/8 |
| 98 | 1099.07 | ^^C#, v3Db | ^10C, v9D | |
| 99 | 1110.28 | ^3C#, vvDb | ^11C, v8D | |
| 100 | 1121.5 | ^4C#, vDb | ^12C, v7D | 21/11 |
| 101 | 1132.71 | ^5C#, Db | C#, v6D | 77/40 |
| 102 | 1143.93 | Cx, v5D | ^C#, v5D | 64/33 |
| 103 | 1155.14 | ^Cx, v4D | ^^C#, v4D | |
| 104 | 1166.36 | ^^Cx, v3D | ^3C#, v3D | 49/25, 55/28 |
| 105 | 1177.57 | ^3Cx, vvD | ^4C#, vvD | |
| 106 | 1188.79 | ^4Cx, vD | ^5C#, vD | |
| 107 | 1200 | D | D | 2/1 |
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 |