105edo
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Prime factorization
3 × 5 × 7
Step size
11.4286¢
Fifth
61\105 (697.143¢)
Semitones (A1:m2)
7:10 (80¢ : 114.3¢)
Dual sharp fifth
62\105 (708.571¢)
Dual flat fifth
61\105 (697.143¢)
Dual major 2nd
18\105 (205.714¢) (→6\35)
Consistency limit
3
Distinct consistency limit
3
| ← 104edo | 105edo | 106edo → |
105 equal divisions of the octave (abbreviated 105edo or 105ed2), also called 105-tone equal temperament (105tet) or 105 equal temperament (105et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 105 equal parts of about 11.429 ¢ each. Each step represents a frequency ratio of 21/105, or the 105th root of 2.
Theory
105edo is most notable as a tuning of meantone and in particular higher-limit extensions of meantone, such as grosstone and Huygens. It tempers out 81/80 in the 5-limit; 81/80, 126/125 and hence 225/224 in the 7-limit; 99/98, 176/175 and 441/440 in the 11-limit; and if we want to push that far, 144/143 in the 13-limit. This is the sharper fifth mapping of 11-limit meantone (aka huygens rather than meanpop), for which it gives the optimal patent val, and provides a good tuning for the 13-limit extension, though 74edo is in that case the optimal patent val. 105edo's meantone fifth is nearly identical to the CTE generator for meantone.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | -4.81 | +2.26 | +2.60 | +1.80 | -2.75 | +5.19 | -2.55 | -2.10 | -0.37 | -2.21 | +0.30 |
| relative (%) | -42 | +20 | +23 | +16 | -24 | +45 | -22 | -18 | -3 | -19 | +3 | |
| Steps (reduced) |
166 (61) |
244 (34) |
295 (85) |
333 (18) |
363 (48) |
389 (74) |
410 (95) |
429 (9) |
446 (26) |
461 (41) |
475 (55) | |
Subsets and supersets
105 is the product of 3 × 5 × 7, the three smallest odd primes, with other divisors being 15, 21 and 35.
Intervals
- Main article: Table of 105edo intervals
| Steps | Cents | Ups and downs notation (dual flat fifth 61\105) |
Ups and downs notation (dual sharp fifth 62\105) |
Approximate ratios |
|---|---|---|---|---|
| 0 | 0 | D | D | 1/1 |
| 1 | 11.4286 | ^D, vvEbb | ^D, v4Eb | |
| 2 | 22.8571 | ^^D, vEbb | ^^D, v3Eb | 78/77 |
| 3 | 34.2857 | ^3D, Ebb | ^3D, vvEb | 50/49, 56/55 |
| 4 | 45.7143 | ^4D, v6Eb | ^4D, vEb | 40/39, 77/75 |
| 5 | 57.1429 | ^5D, v5Eb | ^5D, Eb | |
| 6 | 68.5714 | ^6D, v4Eb | ^6D, v13E | 26/25, 80/77 |
| 7 | 80 | D#, v3Eb | ^7D, v12E | 21/20, 22/21 |
| 8 | 91.4286 | ^D#, vvEb | ^8D, v11E | |
| 9 | 102.857 | ^^D#, vEb | ^9D, v10E | 52/49 |
| 10 | 114.286 | ^3D#, Eb | ^10D, v9E | 16/15 |
| 11 | 125.714 | ^4D#, v6E | ^11D, v8E | 14/13 |
| 12 | 137.143 | ^5D#, v5E | ^12D, v7E | |
| 13 | 148.571 | ^6D#, v4E | ^13D, v6E | 12/11 |
| 14 | 160 | Dx, v3E | D#, v5E | |
| 15 | 171.429 | ^Dx, vvE | ^D#, v4E | |
| 16 | 182.857 | ^^Dx, vE | ^^D#, v3E | 39/35 |
| 17 | 194.286 | E | ^3D#, vvE | 28/25 |
| 18 | 205.714 | ^E, vvFb | ^4D#, vE | 44/39 |
| 19 | 217.143 | ^^E, vFb | E | |
| 20 | 228.571 | ^3E, Fb | ^E, v4F | 8/7 |
| 21 | 240 | ^4E, v6F | ^^E, v3F | 55/48 |
| 22 | 251.429 | ^5E, v5F | ^3E, vvF | |
| 23 | 262.857 | ^6E, v4F | ^4E, vF | 64/55 |
| 24 | 274.286 | E#, v3F | F | 75/64 |
| 25 | 285.714 | ^E#, vvF | ^F, v4Gb | |
| 26 | 297.143 | ^^E#, vF | ^^F, v3Gb | |
| 27 | 308.571 | F | ^3F, vvGb | |
| 28 | 320 | ^F, vvGbb | ^4F, vGb | 77/64 |
| 29 | 331.429 | ^^F, vGbb | ^5F, Gb | |
| 30 | 342.857 | ^3F, Gbb | ^6F, v13G | 39/32 |
| 31 | 354.286 | ^4F, v6Gb | ^7F, v12G | 49/40 |
| 32 | 365.714 | ^5F, v5Gb | ^8F, v11G | |
| 33 | 377.143 | ^6F, v4Gb | ^9F, v10G | |
| 34 | 388.571 | F#, v3Gb | ^10F, v9G | 5/4 |
| 35 | 400 | ^F#, vvGb | ^11F, v8G | |
| 36 | 411.429 | ^^F#, vGb | ^12F, v7G | |
| 37 | 422.857 | ^3F#, Gb | ^13F, v6G | 32/25 |
| 38 | 434.286 | ^4F#, v6G | F#, v5G | 50/39, 77/60 |
| 39 | 445.714 | ^5F#, v5G | ^F#, v4G | |
| 40 | 457.143 | ^6F#, v4G | ^^F#, v3G | 13/10 |
| 41 | 468.571 | Fx, v3G | ^3F#, vvG | 21/16, 55/42 |
| 42 | 480 | ^Fx, vvG | ^4F#, vG | |
| 43 | 491.429 | ^^Fx, vG | G | 65/49 |
| 44 | 502.857 | G | ^G, v4Ab | 75/56 |
| 45 | 514.286 | ^G, vvAbb | ^^G, v3Ab | 35/26 |
| 46 | 525.714 | ^^G, vAbb | ^3G, vvAb | |
| 47 | 537.143 | ^3G, Abb | ^4G, vAb | 15/11 |
| 48 | 548.571 | ^4G, v6Ab | ^5G, Ab | 11/8 |
| 49 | 560 | ^5G, v5Ab | ^6G, v13A | |
| 50 | 571.429 | ^6G, v4Ab | ^7G, v12A | 39/28 |
| 51 | 582.857 | G#, v3Ab | ^8G, v11A | 7/5 |
| 52 | 594.286 | ^G#, vvAb | ^9G, v10A | 55/39 |
| 53 | 605.714 | ^^G#, vAb | ^10G, v9A | 78/55 |
| 54 | 617.143 | ^3G#, Ab | ^11G, v8A | 10/7, 63/44 |
| 55 | 628.571 | ^4G#, v6A | ^12G, v7A | 56/39 |
| 56 | 640 | ^5G#, v5A | ^13G, v6A | |
| 57 | 651.429 | ^6G#, v4A | G#, v5A | 16/11 |
| 58 | 662.857 | Gx, v3A | ^G#, v4A | 22/15 |
| 59 | 674.286 | ^Gx, vvA | ^^G#, v3A | |
| 60 | 685.714 | ^^Gx, vA | ^3G#, vvA | 52/35 |
| 61 | 697.143 | A | ^4G#, vA | |
| 62 | 708.571 | ^A, vvBbb | A | |
| 63 | 720 | ^^A, vBbb | ^A, v4Bb | |
| 64 | 731.429 | ^3A, Bbb | ^^A, v3Bb | 32/21 |
| 65 | 742.857 | ^4A, v6Bb | ^3A, vvBb | 20/13 |
| 66 | 754.286 | ^5A, v5Bb | ^4A, vBb | |
| 67 | 765.714 | ^6A, v4Bb | ^5A, Bb | 39/25 |
| 68 | 777.143 | A#, v3Bb | ^6A, v13B | 25/16 |
| 69 | 788.571 | ^A#, vvBb | ^7A, v12B | |
| 70 | 800 | ^^A#, vBb | ^8A, v11B | |
| 71 | 811.429 | ^3A#, Bb | ^9A, v10B | 8/5 |
| 72 | 822.857 | ^4A#, v6B | ^10A, v9B | |
| 73 | 834.286 | ^5A#, v5B | ^11A, v8B | |
| 74 | 845.714 | ^6A#, v4B | ^12A, v7B | 80/49 |
| 75 | 857.143 | Ax, v3B | ^13A, v6B | 64/39 |
| 76 | 868.571 | ^Ax, vvB | A#, v5B | |
| 77 | 880 | ^^Ax, vB | ^A#, v4B | |
| 78 | 891.429 | B | ^^A#, v3B | |
| 79 | 902.857 | ^B, vvCb | ^3A#, vvB | |
| 80 | 914.286 | ^^B, vCb | ^4A#, vB | |
| 81 | 925.714 | ^3B, Cb | B | 75/44 |
| 82 | 937.143 | ^4B, v6C | ^B, v4C | 55/32 |
| 83 | 948.571 | ^5B, v5C | ^^B, v3C | |
| 84 | 960 | ^6B, v4C | ^3B, vvC | |
| 85 | 971.429 | B#, v3C | ^4B, vC | 7/4 |
| 86 | 982.857 | ^B#, vvC | C | |
| 87 | 994.286 | ^^B#, vC | ^C, v4Db | 39/22 |
| 88 | 1005.71 | C | ^^C, v3Db | 25/14 |
| 89 | 1017.14 | ^C, vvDbb | ^3C, vvDb | 70/39 |
| 90 | 1028.57 | ^^C, vDbb | ^4C, vDb | |
| 91 | 1040 | ^3C, Dbb | ^5C, Db | |
| 92 | 1051.43 | ^4C, v6Db | ^6C, v13D | 11/6 |
| 93 | 1062.86 | ^5C, v5Db | ^7C, v12D | |
| 94 | 1074.29 | ^6C, v4Db | ^8C, v11D | 13/7 |
| 95 | 1085.71 | C#, v3Db | ^9C, v10D | 15/8 |
| 96 | 1097.14 | ^C#, vvDb | ^10C, v9D | 49/26 |
| 97 | 1108.57 | ^^C#, vDb | ^11C, v8D | |
| 98 | 1120 | ^3C#, Db | ^12C, v7D | 21/11, 40/21 |
| 99 | 1131.43 | ^4C#, v6D | ^13C, v6D | 25/13, 77/40 |
| 100 | 1142.86 | ^5C#, v5D | C#, v5D | |
| 101 | 1154.29 | ^6C#, v4D | ^C#, v4D | 39/20 |
| 102 | 1165.71 | Cx, v3D | ^^C#, v3D | 49/25, 55/28 |
| 103 | 1177.14 | ^Cx, vvD | ^3C#, vvD | 77/39 |
| 104 | 1188.57 | ^^Cx, vD | ^4C#, vD | |
| 105 | 1200 | D | D | 2/1 |
15-odd-limit interval mappings
| Interval, complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 15/11, 22/15 | 0.192 | 1.7 |
| 7/5, 10/7 | 0.345 | 3.0 |
| 11/6, 12/11 | 2.066 | 18.1 |
| 5/4, 8/5 | 2.258 | 19.8 |
| 15/8, 16/15 | 2.554 | 22.4 |
| 13/7, 14/13 | 2.584 | 22.6 |
| 7/4, 8/7 | 2.603 | 22.8 |
| 11/8, 16/11 | 2.747 | 24.0 |
| 13/10, 20/13 | 2.929 | 25.6 |
| 3/2, 4/3 | 4.812 | 42.1 |
| 11/10, 20/11 | 5.004 | 43.8 |
| 15/14, 28/15 | 5.157 | 45.1 |
| 13/8, 16/13 | 5.187 | 45.4 |
| 11/7, 14/11 | 5.349 | 46.8 |
| 11/9, 18/11 | 6.878 | 60.2 |
| 5/3, 6/5 | 7.070 | 61.9 |
| 7/6, 12/7 | 7.415 | 64.9 |
| 15/13, 26/15 | 7.741 | 67.7 |
| 13/11, 22/13 | 7.933 | 69.4 |
| 9/8, 16/9 | 9.624 | 84.2 |
| 13/12, 24/13 | 9.999 | 87.5 |
| 9/5, 10/9 | 11.882 | 104.0 |
| 9/7, 14/9 | 12.227 | 107.0 |
| 13/9, 18/13 | 14.811 | 129.6 |