829edo
| ← 828edo | 829edo | 830edo → |
829 equal divisions of the octave (abbreviated 829edo or 829ed2), also called 829-tone equal temperament (829tet) or 829 equal temperament (829et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 829 equal parts of about 1.45 ¢ each. Each step represents a frequency ratio of 21/829, or the 829th root of 2.
Theory
829edo is consistent to the 11-odd-limit. The equal temperament tempers out 4375/4374, [-25 6 -3 8⟩, and [16 -9 -8 6⟩ in the 7-limit; 41503/41472, 200704/200475, and 3750705/3748096 in the 11-limit. It supports squarschmidt, senior and acrokleismic.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.096 | +0.176 | -0.430 | +0.190 | +0.486 | +0.714 | +0.678 | -0.048 | -0.385 | -0.042 |
| Relative (%) | +0.0 | +6.6 | +12.2 | -29.7 | +13.1 | +33.5 | +49.3 | +46.8 | -3.3 | -26.6 | -2.9 | |
| Steps (reduced) |
829 (0) |
1314 (485) |
1925 (267) |
2327 (669) |
2868 (381) |
3068 (581) |
3389 (73) |
3522 (206) |
3750 (434) |
4027 (711) |
4107 (791) | |
Subsets and supersets
829edo is the 145th prime edo.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [1314 -829⟩ | [⟨829 1314]] | −0.0302 | 0.0302 | 2.09 |
| 2.3.5 | [39 -29 3⟩, [61 4 -29⟩ | [⟨829 1314 1925]] | −0.0454 | 0.0327 | 2.26 |
| 2.3.5.7 | 4375/4374, [-25 6 -3 8⟩, [16 -9 -8 6⟩ | [⟨829 1314 1925 2327]] | +0.0043 | 0.0906 | 6.25 |
| 2.3.5.7.11 | 4375/4374, 41503/41472, 200704/200475, 3750705/3748096 | [⟨829 1314 1925 2327 2868]] | −0.0076 | 0.0844 | 5.83 |
| 2.3.5.7.11.13 | 4096/4095, 4375/4374, 4459/4455, 47432/47385, 59535/59488 | [⟨829 1314 1925 2327 2868 3068]] | −0.0282 | 0.0898 | 6.20 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 218\829 | 315.561 | 6/5 | Acrokleismic |
| 1 | 223\829 | 322.799 | 3087/2560 | Senior / seniority |
| 1 | 274\829 | 396.622 | 98304/78125 | Squarschmidt |
| 1 | 391\829 | 565.983 | 59049/40960 | Tricot |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct