299edo
| ← 298edo | 299edo | 300edo → |
299 equal divisions of the octave (abbreviated 299edo or 299ed2), also called 299-tone equal temperament (299tet) or 299 equal temperament (299et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 299 equal parts of about 4.01 ¢ each. Each step represents a frequency ratio of 21/299, or the 299th root of 2.
Theory
In the 5-limit, 299et tempers out the kleisma, 15625/15552, in the 7-limit 10976/10935, in the 11-limit 385/384; and in the 13-limit 325/324, 625/624 and 676/675. It provides the optimal patent val for the 13-limit rank-3 enlil temperament, and the rank-4 temperament tempering out 325/324 and 385/384.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.39 | -1.03 | -1.60 | -1.49 | -1.73 | -0.61 | -0.52 | +1.83 | +1.86 | -1.22 |
| Relative (%) | +0.0 | +9.6 | -25.7 | -39.9 | -37.0 | -43.1 | -15.1 | -13.0 | +45.5 | +46.4 | -30.5 | |
| Steps (reduced) |
299 (0) |
474 (175) |
694 (96) |
839 (241) |
1034 (137) |
1106 (209) |
1222 (26) |
1270 (74) |
1353 (157) |
1453 (257) |
1481 (285) | |
Subsets and supersets
Since 299 factors into 13 × 23, 299edo contains 13edo and 23edo as subsets.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [474 -299⟩ | [⟨299 474]] | −0.1218 | 0.1218 | 3.04 |
| 2.3.5 | 15625/15552, [80 -49 -1⟩ | [⟨299 474 694]] | +0.0665 | 0.2844 | 7.09 |
| 2.3.5.7 | 10976/10935, 15625/15552, 823543/819200 | [⟨299 474 694 839]] | +0.1925 | 0.3291 | 8.20 |
| 2.3.5.7.11 | 385/384, 6250/6237, 10976/10935, 12005/11979 | [⟨299 474 694 839 1034]] | +0.2399 | 0.3092 | 7.70 |
| 2.3.5.7.11.13 | 325/324, 385/384, 625/624, 10648/10647, 10976/10935 | [⟨299 474 694 839 1034 1106]] | +0.2779 | 0.2948 | 7.34 |
| 2.3.5.7.11.13.17 | 325/324, 385/384, 595/594, 625/624, 2058/2057, 8624/8619 | [⟨299 474 694 839 1034 1106 1222]] | +0.2595 | 0.2767 | 6.89 |
| 2.3.5.7.11.13.17.19 | 325/324, 343/342, 385/384, 595/594, 625/624, 1216/1215, 1445/1444 | [⟨299 474 694 839 1034 1106 1222 1270]] | +0.2424 | 0.2627 | 6.54 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 79\299 | 317.06 | 6/5 | Hanson |
| 1 | 124\299 | 497.66 | 4/3 | Cotoneum (7-limit) |
| 1 | 124\299 | 505.69 | 75/56 | Marfifths |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct