113edo
Theory
113edo is distinctly consistent in the 13-odd-limit with a flat tendency. As a temperament, it tempers out the amity comma and the ampersand in the 5-limit; 225/224, 1029/1024 and 1071875/1062882 in the 7-limit; 243/242, 385/384, 441/440 and 540/539 in the 11-limit; 325/324, 364/363, 729/728, and 1625/1617 in the 13-limit. It notably supports the 5-limit amity temperament, 7-limit amicable temperament, 7- and 11-limit miracle temperament, and 13-limit manna temperament.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -1.07 | -4.01 | -2.45 | +0.89 | -1.59 | +1.24 | -0.17 | -1.73 | +0.51 | +1.87 |
| Relative (%) | +0.0 | -10.1 | -37.8 | -23.1 | +8.4 | -15.0 | +11.7 | -1.6 | -16.3 | +4.8 | +17.6 | |
| Steps (reduced) |
113 (0) |
179 (66) |
262 (36) |
317 (91) |
391 (52) |
418 (79) |
462 (10) |
480 (28) |
511 (59) |
549 (97) |
560 (108) | |
Subsets and supersets
113edo is the 30th prime edo.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-179 113⟩ | [⟨113 179]] | +0.338 | 0.338 | 3.18 |
| 2.3.5 | 1600000/1594323, 34171875/33554432 | [⟨113 179 262]] | +0.801 | 0.712 | 6.70 |
| 2.3.5.7 | 225/224, 1029/1024, 1071875/1062882 | [⟨113 179 262 317]] | +0.820 | 0.617 | 5.81 |
| 2.3.5.7.11 | 225/224, 243/242, 385/384, 980000/970299 | [⟨113 179 262 317 391]] | +0.604 | 0.700 | 6.59 |
| 2.3.5.7.11.13 | 225/224, 243/242, 325/324, 385/384, 1875/1859 | [⟨113 179 262 317 391 418]] | +0.575 | 0.643 | 6.05 |
Rank-2 temperaments
| Periods per 8ve |
Generator (Reduced) |
Cents (Reduced) |
Associated Ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 4\113 | 42.48 | 40/39 | Humorous |
| 1 | 6\113 | 63.72 | 28/27 | Sycamore / betic |
| 1 | 8\113 | 84.96 | 21/20 | Amicable / pseudoamical / pseudoamorous |
| 1 | 11\113 | 116.81 | 15/14~16/15 | Miracle / manna |
| 1 | 13\113 | 138.05 | 27/25 | Quartemka |
| 1 | 22\113 | 233.63 | 8/7 | Slendric |
| 1 | 27\113 | 286.73 | 13/11 | Gamity |
| 1 | 29\113 | 307.96 | 3200/2673 | Familia |
| 1 | 32\113 | 339.82 | 243/200 | Houborizic |
| 1 | 34\113 | 360.06 | 16/13 | Phicordial |
| 1 | 37\113 | 392.92 | 2744/2187 | Emmthird |
| 1 | 47\113 | 499.12 | 4/3 | Gracecordial |
| 1 | 56\113 | 594.69 | 55/39 | Gaster |