296edo
| ← 295edo | 296edo | 297edo → |
296 equal divisions of the octave (abbreviated 296edo or 296ed2), also called 296-tone equal temperament (296tet) or 296 equal temperament (296et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 296 equal parts of about 4.05 ¢ each. Each step represents a frequency ratio of 21/296, or the 296th root of 2.
Theory
In the 5-limit, 296et not only tempers out the semicomma of 5-limit orwell (orson) temperament, 2109375/2097152, it also provides its optimal patent val, and tempers out the minortone comma, [-16 35 -17⟩. It is also an interesting temperament in higher limits, being distinctly consistent through to the 15-odd-limit. In the 7-limit it tempers out 4375/4374 (ragisma), 16875/16807 (mirkwai), and 118098/117649 (stearnsma), supporting 7-limit octoid and sabric. In the 11-limit, 540/539, 1375/1372, 3025/3024, 4000/3993, 6250/6237 and 9801/9800; in the 13-limit, 625/624, 729/728, 1575/1573, 1716/1715, 2080/2079, and 6656/6655, so that it also supports the 11- and 13-limit versions of octoid. It allows swetismic chords and squbemic chords in the 13-odd-limit, in addition to nicolic chords in the 15-odd-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.60 | -1.18 | +0.09 | +0.03 | -1.34 | +0.45 | -1.57 | +0.10 | +0.15 | -1.79 |
| Relative (%) | +0.0 | -14.9 | -29.1 | +2.3 | +0.8 | -33.0 | +11.1 | -38.7 | +2.6 | +3.8 | -44.2 | |
| Steps (reduced) |
296 (0) |
469 (173) |
687 (95) |
831 (239) |
1024 (136) |
1095 (207) |
1210 (26) |
1257 (73) |
1339 (155) |
1438 (254) |
1466 (282) | |
Subsets and supersets
Since 296 factors into 23 × 37, 296edo has subset edos 2, 4, 8, 37, 74 and 148.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-469 296⟩ | [⟨296 469]] | +0.1904 | 0.1905 | 4.70 |
| 2.3.5 | 2109375/2097152, [-16 35 -17⟩ | [⟨296 469 687]] | +0.2962 | 0.2158 | 5.32 |
| 2.3.5.7 | 4375/4374, 16875/16807, 2100875/2097152 | [⟨296 469 687 831]] | +0.2138 | 0.2350 | 5.80 |
| 2.3.5.7.11 | 540/539, 1375/1372, 4000/3993, 2100875/2097152 | [⟨296 469 687 831 1024]] | +0.1691 | 0.2284 | 5.63 |
| 2.3.5.7.11.13 | 540/539, 625/624, 729/728, 1375/1372, 15379/15360 | [⟨296 469 687 831 1024 1095]] | +0.2012 | 0.2206 | 5.44 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 45\296 | 182.43 | 10/9 | Mitonic |
| 1 | 67\296 | 271.62 | 75/64 | Sabric |
| 1 | 105\296 | 425.68 | 2625/2048 | Rainwell |
| 2 | 57\296 | 231/08 | 8/7 | Orga |
| 8 | 144\296 (4\296) |
583.78 (16.22) |
7/5 (126/125) |
Octoid |
| 37 | 67\296 (3\296) |
271.62 (12.16) |
117/100 (?) |
Dzelic |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct