103edo
The 103 equal divisions of the octave (103edo), or the 103(-tone) equal temperament (103tet, 103et) when viewed from a regular temperament perspective, is the equal division of the octave into 103 steps of about 11.7 cents each.
Theory
103edo is a good miracle tuning, especially for the 7- and 13-limit and benediction and hemisecordite, two of the 13-limit extensions of miracle. It tempers out 78732/78125 in the 5-limit; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the 11-limit; 351/350 and 847/845 in the 13-limit. In the 13-limit it provides the optimal patent val for marvel temperament as well as benediction and hemisecordite.
103edo is the 27th prime edo.
In 103edo, all intervals in the 17-odd-limit are consistent with only the exception of 9/8 and its octave complement which barely misses.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -2.93 | -1.85 | -1.84 | -3.75 | -1.69 | -0.10 | +5.40 | +0.85 | -4.33 | -3.29 |
| Relative (%) | +0.0 | -25.1 | -15.9 | -15.8 | -32.1 | -14.5 | -0.9 | +46.3 | +7.3 | -37.2 | -28.2 | |
| Steps (reduced) |
103 (0) |
163 (60) |
239 (33) |
289 (83) |
356 (47) |
381 (72) |
421 (9) |
438 (26) |
466 (54) |
500 (88) |
510 (98) | |
Intervals
| Degree | Cents | Approximate Ratios |
|---|---|---|
| 1 | 11.650 | 81/80, 126/125 |
| 2 | 23.301 | 65/64, 66/65, 78/77 |
| 3 | 34.951 | 49/48, 50/49, 64/63 |
| 4 | 46.602 | 33/32, 35/34, 36/35 |
| 5 | 58.252 | 27/26, 34/33 |
| 6 | 69.903 | 25/24, 26/25, 28/27 |
| 7 | 81.553 | 21/20, 22/21 |
| 8 | 93.204 | 18/17 |
| 9 | 104.854 | 17/16 |
| 10 | 116.505 | 15/14, 16/15 |
| 11 | 128.155 | 14/13 |
| 12 | 139.806 | 13/12 |
| 13 | 151.456 | 12/11 |
| 14 | 163.107 | 11/10 |
| 15 | 174.757 | 72/65 |
| 16 | 186.408 | 10/9 |
| 17 | 198.058 | 9/8 |
| 18 | 209.708 | |
| 19 | 221.359 | 17/15, 25/22 |
| 20 | 233.010 | 8/7 |
| 21 | 244.660 | 15/13 |
| 22 | 256.311 | |
| 23 | 267.961 | 7/6 |
| 24 | 279.712 | 20/17 |
| 25 | 291.262 | 13/11 |
| 26 | 303.013 | 25/21 |
| 27 | 314.563 | 6/5 |
| 28 | 326.214 | 63/52, 65/54 |
| 29 | 337.864 | 17/14, 39/32 |
| 30 | 349.615 | 11/9, 27/22 |
| 31 | 361.165 | 16/13, 21/17 |
| 32 | 372.816 | 26/21, 81/65 |
| 33 | 384.466 | 5/4 |
| 34 | 396.117 | 44/35 |
| 35 | 407.767 | 33/26 |
| 36 | 419.417 | 14/11 |
| 37 | 431.068 | 9/7 |
| 38 | 442.708 | 22/17 |
| 39 | 454.369 | 13/10 |
| 40 | 466.019 | 17/13, 21/16 |
| 41 | 477.670 | |
| 42 | 489.320 | 65/49 |
| 43 | 500.971 | 4/3 |
| 44 | 512.621 | 27/20 |
| 45 | 524.272 | 65/48 |
| 46 | 535.922 | 15/11 |
| 47 | 547.573 | 11/8 |
| 48 | 559.223 | 18/13 |
| 49 | 570.874 | 25/18 |
| 50 | 582.524 | 7/5 |
| 51 | 594.175 | 24/17 |
| … | … | … |
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-163 103⟩ | [⟨103 166]] | +0.923 | 0.924 | 7.92 |
| 2.3.5 | 78732/78125, 34171875/33554432 | [⟨103 166 239]] | +0.881 | 0.757 | 6.49 |
| 2.3.5.7 | 225/224, 1029/1024, 78732/78125 | [⟨103 166 239 289]] | +0.824 | 0.663 | 5.68 |
| 2.3.5.7.11 | 225/224, 243/242, 385/384, 43923/43750 | [⟨103 166 239 289 356]] | +0.876 | 0.602 | 5.16 |
| 2.3.5.7.11.13 | 225/224, 243/242, 351/350, 385/384, 847/845 | [⟨103 166 239 289 356 381]] | +0.806 | 0.571 | 4.90 |
| 2.3.5.7.11.13.17 | 225/224, 243/242, 273/272, 351/350, 375/374, 847/845 | [⟨103 166 239 289 356 381 421]] | +0.694 | 0.595 | 5.10 |
Rank-2 temperaments
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 3\103 | 34.951 | 1990656/1953125 | Gammic (5-limit) |
| 1 | 5\103 | 58.252 | 27/26 | Hemisecordite |
| 1 | 9\103 | 104.854 | 17/16 | Septendesemi |
| 1 | 10\103 | 116.505 | 15/14~16/15 | Miracle / benediction |
| 1 | 16\103 | 186.408 | 10/9 | Mintone |
| 1 | 20\103 | 233.010 | 8/7 | Slendric |
| 1 | 21\103 | 244.660 | 15/13 | Subsemifourth |
| 1 | 26\103 | 303.013 | 25/21 | Quinmite |
| 1 | 31\103 | 361.165 | 16/13 | Phicordial |
| 1 | 37\103 | 431.06 | 77/60 | Lockerbie |
| 1 | 38\103 | 442.708 | 162/125 | Sensipent / sensei |
| 1 | 39\103 | 454.369 | 13/10 | Fibo |
| 1 | 40\103 | 466.019 | 55/42 | Hemiseptisix |
| 1 | 42\103 | 489.320 | 65/49 | Catafourth |
| 1 | 45\103 | 524.272 | 65/48 | Widefourth |
| 1 | 47\103 | 547.573 | 11/8 | Heinz |
| 1 | 48\103 | 559.223 | 242/175 | Tritriple |
| 1 | 50\103 | 582.524 | 7/5 | Neptune |
Scales
13-limit temperaments
| Marvel and Benediction | Hemisecordite | ||||
|---|---|---|---|---|---|
| Degree | cents | Difference from 72edo | Degree | cents | Difference from 62edo |
| 1 | 11.6505 | -5.016¢ | 2 | 23.301 | 3.946¢ |
| 3 | 34.9515 | 1.618¢ | 3 | 34.9515 | -3.758¢ |
| 4 | 46.602 | -3.398¢ | 5 | 58.252 | 0.188¢ |
| 6 | 69.903 | 3.236¢ | 7 | 81.553 | 4.134¢ |
| 7 | 81.553 | -1.78¢ | 8 | 93.204 | -3.57¢ |
| 9 | 104.854 | 4.854¢ | 10 | 116.505 | 0.376¢ |
| 10 | 116.5045 | -0.162¢ | 12 | 139.806 | 4.322¢ |
| 11 | 128.155 | -5.178¢ | 13 | 151.456 | -3.382¢ |
| 13 | 151.456¢ | 1.456¢ | 15 | 174.757 | 0.563¢ |
| 14 | 163.107¢ | -3.56¢ | 17 | 198.058 | 4.51¢ |
| 16 | 186.408 | 3.074¢ | 18 | 209.709 | -3.1945¢ |
| 17 | 198.058 | -1.942¢ | 20 | 233.01 | 0.751¢ |
| 19 | 221.359 | 4.693¢ | 22 | 256.311 | 4.698¢ |
| 20 | 233.01 | -0.324¢ | 23 | 267.961 | -3.007¢ |
| 21 | 244.66 | -5.34¢ | 25 | 291.262 | 0.94¢ |
| 23 | 267.961 | 1.2945¢ | 27 | 314.563 | 4.886¢ |
| 24 | 279.612 | -3.722¢ | 28 | 326.214 | -2.819¢ |
| 26 | 302.913 | 2.913¢ | 30 | 349.515 | 1.1275¢ |
| 27 | 314.563 | -2.104¢ | 32 | 372.8155 | 5.074¢ |
| 29 | 337.864 | 4.531¢ | 33 | 384.466 | -2.631¢ |
| 30 | 349.515 | -0.485¢ | 35 | 407.767 | 1.315¢ |
| 31 | 361.165 | -5.502¢ | 37 | 431.068 | 5.2615¢ |
| 33 | 384.466 | 1.133¢ | 38 | 442.718 | -2.443¢ |
| 34 | 396.1165 | -3.8835¢ | 40 | 466.0190 | 1.503¢ |
| 36 | 419.4175 | 2.751¢ | 42 | 489.32 | 5.449¢ |
| 37 | 431.068 | -2.265¢ | 43 | 500.971 | -2.255¢ |
| 39 | 454.369 | 4.369¢ | 45 | 524.272 | 1.691¢ |
| 40 | 466.019 | -0.647¢ | 47 | 547.573 | 5.637¢ |
| 41 | 477.67 | -5.663¢ | 48 | 559.223 | -2.067¢ |
| 43 | 500.971 | 0.971¢ | 50 | 582.524 | 1.879¢ |
| 44 | 512.621¢ | -4.045¢ | 52 | 605.825 | 5.825¢ |
| 46 | 535.922¢ | 2.589¢ | 53 | 617.476 | -1.879¢ |
| 47 | 547.573¢ | -2.427¢ | 55 | 640.777¢ | 2.067 |
| 49 | 570.874¢ | 4.207¢ | 56 | 652.427 | -5.637¢ |
| 50 | 582.524 | -0.809¢ | 58 | 675.728 | -1.691 |
| 52 | 605.825 | 5.825¢ | 60 | 699.029 | 2.255¢ |
| 53 | 617.475 | 0.809¢ | 61 | 710.68 | -5.449¢ |
| 54 | 629.126¢ | -4.207¢ | 63 | 733.981 | -1.503¢ |
| 56 | 652.427¢ | 2.427¢ | 65 | 757.282 | 2.443¢ |
| 57 | 664.078 | -2.589¢ | 66 | 768.932 | -5.2615¢ |
| 59 | 687.379 | 4.045¢ | 68 | 792.233 | -1.315¢ |
| 60 | 699.029 | -0.971¢ | 70 | 815.534 | 2.631¢ |
| 62 | 722.33 | 5.663¢ | 71 | 827.1845 | -5.074¢ |
| 63 | 733.981 | 0.647¢ | 73 | 850.485 | -1.1275¢ |
| 64 | 745.631 | -4.369¢ | 75 | 873.786 | 2.819¢ |
| 66 | 768.932 | 2.265¢ | 76 | 885.437 | -4.886¢ |
| 67 | 780.5825 | -2.751¢ | 78 | 908.738 | -0.94¢ |
| 69 | 803.8835 | 3.8835¢ | 80 | 932.039 | 3.007¢ |
| 70 | 815.534 | -1.133¢ | 81 | 943.689 | -4.698¢ |
| 72 | 838.835 | 5.501¢ | 83 | 966.99 | -0.752¢ |
| 73 | 850.485 | 0.485¢ | 85 | 990.291 | 3.1945¢ |
| 74 | 862.136 | -4.531¢ | 86 | 1001.942 | -4.51¢ |
| 76 | 885.439 | 2.104¢ | 88 | 1025.243 | -0.564¢ |
| 77 | 897.087 | -2.913¢ | 90 | 1048.544 | 3.382¢ |
| 79 | 920.388 | 3.722¢ | 91 | 1060.194 | -4.322¢ |
| 80 | 932.039 | -1.2945¢ | 93 | 1083.495 | -0.376¢ |
| 82 | 955.34 | 5.34¢ | 95 | 1106.796 | 3.57¢ |
| 83 | 966.99 | 0.324¢ | 96 | 1118.447¢ | -4.134¢ |
| 84 | 978.641 | -4.693¢ | 98 | 1141.748 | -0.188¢ |
| 86 | 1001.942 | 1.942¢ | 100 | 1165.0485 | 3.758¢ |
| 87 | 1013.592 | -3.074¢ | 101 | 1176.699 | -3.946¢ |
| 89 | 1036.893 | 3.56¢ | |||
| 90 | 1048.544 | -1.456¢ | |||
| 92 | 1071.845 | 5.178¢ | |||
| 93 | 1083.495 | 0.162¢ | |||
| 94 | 1095.146 | -4.854¢ | |||
| 96 | 1118.447 | 1.78¢ | |||
| 97 | 1130.097 | -3.236¢ | |||
| 99 | 1153.398 | 3.398¢ | |||
| 100 | 1165.0485 | -1.618¢ | |||
| 102 | 1188.3495 | 5.016¢ | |||