988edo
| ← 987edo | 988edo | 989edo → |
988 equal divisions of the octave (abbreviated 988edo or 988ed2), also called 988-tone equal temperament (988tet) or 988 equal temperament (988et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 988 equal parts of about 1.21 ¢ each. Each step represents a frequency ratio of 21/988, or the 988th root of 2.
Theory
988edo is enfactored in the 17-limit, with the same tuning as 494edo, which is notable for being a zeta edo. If considered in the 19-limit, it provides a good correction for the 19th harmonic over 494edo. A comma basis for 988edo in the 19-limit is {1156/1155, 1275/1274, 1445/1444, 1716/1715, 2080/2079, 2431/2430, 4096/4095}. An alternate mapping for 17 would be the 988g val, where it tempers out 2025/2023, 13013/13005, 15625/15606, 31213/31212.
In addition, in the 988ccd val provides a tuning that is extremely close to the POTE tuning for quadritikleismic temperament in the 7-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.069 | -0.079 | +0.405 | +0.099 | -0.042 | -0.502 | +0.058 | -0.339 | +0.382 | +0.309 |
| Relative (%) | +0.0 | +5.7 | -6.5 | +33.3 | +8.2 | -3.4 | -41.3 | +4.8 | -27.9 | +31.5 | +25.4 | |
| Steps (reduced) |
988 (0) |
1566 (578) |
2294 (318) |
2774 (798) |
3418 (454) |
3656 (692) |
4038 (86) |
4197 (245) |
4469 (517) |
4800 (848) |
4895 (943) | |
Higher limits
988edo provides excellent approximations for harmonics 2, 3, 5, 11, 13, 19, 37, 43, 47, 53, and 59, and reasonable approximations for harmonics 23, 29, 31, and 41, making it a strong higher-limit system.
In the 2.5.11.13.19.29.31 it supports period-52 temperament called french deck, with the tempering out of 6656/6655 inherited from 494edo.
988edo is similar to 2016edo in the fact that both tune well the 2.5.11.13.19.41.47 subgroup. The result is the 988 & 2016 temperament, which reaches 13/8 in four generators and has a comma basis {7943/7942, 322465/322373, 16777475/16777216, 22151168/22150865, 12998046875/12994428928}.
Subsets and supersets
Since 988 factors into 22 × 13 × 19, 988edo has subset edos 2, 4, 13, 19, 26, 38, 52, 76, 247, and 494.
One step of 988edo is named semisqub, given the strong relation to 494edo and the fact that 1 step of 494edo is called a squb.
Regular temperament properties
Rank-2 temperaments
Note: 17-limit temperaments supported by 494edo are not included.
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 4 | 261\988 (14\988) |
317.004 (17.004) |
6/5 (126/125) |
Quadritikleismic (988ccd) |
| 19 | 141\988 (37\988) |
171.255 (44.939) |
6545/5928 (?) |
Kalium |
| 52 | 325\988 (2\988) |
394.736 (2.429) |
134560000/107132311 (?) |
French deck |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct