243edo
The 243 equal division of the octave (243edo) divides the octave into 243 equal parts of 4.938 cents each.
Theory
243et tempers out the semicomma (i.e. the 5-limit orwell comma) 2109375/2097152 in the 5-limit, and 2401/2400 and 4375/4374 in the 7-limit.
Using the patent val, it tempers out 243/242, 441/440, and 540/539 in the 11-limit, and provides the optimal patent val for the ennealimnic temperament. In the 13-limit it tempers out 364/363, 625/624, 729/728, and 2080/2079, and provides the optimal temperament for 13-limit ennealimnic and the rank-3 jovial temperament, and in the 17-limit it tempers out 375/374 and 595/594 and provides the optimal patent val for 17-limit ennealimnic.
Using the alternative val 243e ⟨241 385 564 682 840], with an lower error, it tempers out 385/384, 1375/1372, 8019/8000, and 14641/14580, and in the 13-limit, 625/624, 729/728, 847/845, 1001/1000, and 1716/1715. It provides a good tuning for fibo.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.72 | -1.13 | -0.92 | +1.77 | -1.02 | -1.25 | -1.22 | -1.11 | -2.42 | +0.64 |
| Relative (%) | +0.0 | -14.6 | -22.9 | -18.7 | +35.8 | -20.7 | -25.3 | -24.6 | -22.6 | -48.9 | +13.0 | |
| Steps (reduced) |
243 (0) |
385 (142) |
564 (78) |
682 (196) |
841 (112) |
899 (170) |
993 (21) |
1032 (60) |
1099 (127) |
1180 (208) |
1204 (232) | |
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-385 243⟩ | [⟨243 385]] | +0.227 | 0.227 | 4.60 |
| 2.3.5 | 2109375/2097152, [1 -27 18⟩ | [⟨243 385 564]] | +0.314 | 0.222 | 4.50 |
| 2.3.5.7 | 2401/2400, 4375/4374, 2109375/2097152 | [⟨241 385 564 682]] | +0.318 | 0.192 | 3.90 |
243et (243e val) has a lower absolute error than any previous equal temperaments in the 19-limit, even though it is inconsistent. The same subgroup is only better tuned by 270et. It is much stronger in the no-11 19-limit, with a lower relative error than any previous equal temperaments. The next equal temperament doing better in this subgroup is 354et in terms of absolute error and 935et in terms of relative error.
Rank-2 temperaments
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 11\243 | 54.32 | 405/392 | Quinwell |
| 1 | 47\243 | 232.10 | 8/7 | Quadrawell |
| 1 | 55\243 | 271.60 | 75/64 | Sabric |
| 1 | 64\243 | 316.05 | 6/5 | Counterkleismic |
| 1 | 92\243 | 454.32 | 13/10 | Fibo |
| 9 | 64\243 (10\243) |
316.05 (49.38) |
6/5 (36/35) |
Ennealimmal |