981edo
| ← 980edo | 981edo | 982edo → |
981 equal divisions of the octave (abbreviated 981edo or 981ed2), also called 981-tone equal temperament (981tet) or 981 equal temperament (981et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 981 equal parts of about 1.22 ¢ each. Each step represents a frequency ratio of 21/981, or the 981st root of 2.
Theory
981edo is a good 13- and 17-limit system, consistent to the 17-odd-limit. As an equal temperament, it tempers out 2080/2079, 2401/2400, 2431/2430, 4096/4095, 4225/4224, 4375/4374, 4459/4455 and 4914/4913 in the 17-limit. It provides the optimal patent val for 13-limit ennealimmic, the rank-3 temperament tempering out 2080/2079, 2401/2400, and 4375/4374, and for 13-limit ennealimmia, which also tempers out 4096/4095.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.186 | +0.231 | -0.019 | +0.364 | -0.161 | +0.243 | -0.265 | +0.472 | +0.392 | -0.081 |
| Relative (%) | +0.0 | +15.2 | +18.9 | -1.5 | +29.8 | -13.1 | +19.9 | -21.7 | +38.6 | +32.1 | -6.7 | |
| Steps (reduced) |
981 (0) |
1555 (574) |
2278 (316) |
2754 (792) |
3394 (451) |
3630 (687) |
4010 (86) |
4167 (243) |
4438 (514) |
4766 (842) |
4860 (936) | |
Subsets and supersets
Since 981 factors into primes as 32 × 109, 981edo has subset edos 3, 9, 109, and 327.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [1555 -981⟩ | [⟨981 1555]] | −0.0586 | 0.0586 | 4.79 |
| 2.3.5 | [1 -27 18⟩, [85 -17 -25⟩ | [⟨981 1555 2278]] | −0.0722 | 0.0515 | 4.21 |
| 2.3.5.7 | 2401/2400, 4375/4374, [79 -25 -23 5⟩ | [⟨981 1555 2278 2754]] | −0.0385 | 0.0562 | 4.59 |
| 2.3.5.7.11 | 2401/2400, 4375/4374, 131072/130977, 1771561/1771470 | [⟨981 1555 2278 2754 3394]] | −0.0630 | 0.0545 | 4.46 |
| 2.3.5.7.11.13 | 2080/2079, 2401/2400, 4096/4095, 4375/4374, 1771561/1771470 | [⟨981 1555 2278 2754 3394 3630]] | −0.0453 | 0.0636 | 5.20 |
| 2.3.5.7.11.13.17 | 2080/2079, 2401/2400, 2431/2430, 4096/4095, 4375/4374, 4914/4913 | [⟨981 1555 2278 2754 3394 3630 4010]] | −0.0473 | 0.0591 | 4.83 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 409\981 | 500.306 | 8192/6137 | Protolangwidge |
| 9 | 258\981 (40\981) |
315.60 (48.93) |
6/5 (36/35) |
Ennealimmal / ennealimmia |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct