243edo
| ← 242edo | 243edo | 244edo → |
243 equal divisions of the octave (abbreviated 243edo or 243ed2), also called 243-tone equal temperament (243tet) or 243 equal temperament (243et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 243 equal parts of about 4.94 ¢ each. Each step represents a frequency ratio of 21/243, or the 243rd root of 2.
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) | |
Subsets and supersets
Since 243 factors into 35, 243edo has subset edos 3, 9, 27, and 81.
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 | [⟨243 385 564 682]] | +0.318 | 0.192 | 3.90 |
- 243et (243e val) has lower absolute errors than any previous equal temperaments in the 19-, 23-limit, and somewhat beyond, despite inconsistency in the corresponding odd limits. In both the 19- and 23-limit, it beats 217 and is only bettered by 270et.
- It is much stronger in the no-11 subgroups of the limits above, holding the record of lowest relative errors until being bettered in the no-11 19-limit by 354et in terms of absolute error and 935et in terms of relative error, and in the no-11 23-limit by 422 in terms of absolute error and 2460 in terms of relative error.
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | 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 |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct