282edo
| ← 281edo | 282edo | 283edo → |
The 282 equal divisions of the octave (282edo), or the 282(-tone) equal temperament (282tet, 282et) when viewed from a regular temperament perspective, is the equal division of the octave into 282 parts of about 4.26 cents each.
Theory
282edo is the smallest equal temperament uniquely consistent through to the 23-odd-limit, and also the smallest consistent to the 29-odd-limit. It shares the same 3rd, 7th, and 13th harmonics with 94edo (282 = 3 × 94), as well as 11/10 and 20/17 (supporting the garistearn temperament). It has a distinct sharp tendency for odd harmonics up to 29. It tempers out 6144/6125 (porwell), 118098/117649 (stearnsma), and 250047/250000 (landscape comma) in the 7-limit, and 540/539 and 5632/5625 in the 11-limit, so that it provides the optimal patent val for the jupiter temperament; it also tempers out 4000/3993 and 234375/234256, providing the optimal patent val for septisuperfourth temperament. In the 13-limit, it tempers out 729/728, 1575/1573, 1716/1715, 2080/2079, and 10648/10647. It allows essentially tempered chords including swetismic chords, squbemic chords, and petrmic triad 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 | +0.17 | +0.92 | +1.39 | +1.87 | +2.03 | +1.43 | +0.36 | +1.51 | +0.21 | -0.35 |
| Relative (%) | +0.0 | +4.1 | +21.6 | +32.6 | +44.0 | +47.6 | +33.5 | +8.4 | +35.6 | +4.9 | -8.3 | |
| Steps (reduced) |
282 (0) |
447 (165) |
655 (91) |
792 (228) |
976 (130) |
1044 (198) |
1153 (25) |
1198 (70) |
1276 (148) |
1370 (242) |
1397 (269) | |
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | [32 -7 -9⟩, [-7 22 -12⟩ | [⟨282 447 655]] | -0.1684 | 0.1671 | 3.93 |
| 2.3.5.7 | 6144/6125, 118098/117649, 250047/250000 | [⟨282 447 655 792]] | -0.2498 | 0.2020 | 4.75 |
| 2.3.5.7.11 | 540/539, 4000/3993, 5632/5625, 137781/137500 | [⟨282 447 655 792 976]] | -0.3081 | 0.2151 | 5.06 |
| 2.3.5.7.11.13 | 540/539, 729/728, 1575/1573, 2200/2197, 3584/3575 | [⟨282 447 655 792 976 1044]] | -0.3480 | 0.2156 | 5.07 |
| 2.3.5.7.11.13.17 | 540/539, 729/728, 936/935, 1156/1155, 1575/1573, 2200/2197 | [⟨282 447 655 792 976 1044 1153]] | -0.3481 | 0.1996 | 4.69 |
| 2.3.5.7.11.13.17.19 | 456/455, 540/539, 729/728, 936/935, 969/968, 1156/1155, 1575/1573 | [⟨282 447 655 792 976 1044 1153 1198]] | -0.3152 | 0.2061 | 4.84 |
| 2.3.5.7.11.13.17.19.23 | 456/455, 540/539, 729/728, 760/759, 936/935, 969/968, 1156/1155, 1288/1287 | [⟨282 447 655 792 976 1044 1153 1198 1276]] | -0.3173 | 0.1944 | 4.57 |
Rank-2 temperaments
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 13\282 | 55.32 | 33/32 | Escapade |
| 1 | 133\282 | 565.96 | 4096/2835 | Tricot / trident (282ef) |
| 2 | 13\282 | 55.32 | 33/32 | Septisuperfourth |
| 2 | 43\282 | 182.98 | 10/9 | Unidecmic |
| 3 | 33\282 | 140.43 | 243/224 | Septichrome |
| 3 | 37\282 | 157.45 | 35/32 | Nessafof |
| 6 | 51\282 (4\282) |
217.02 (17.02) |
567/500 (245/243) |
Stearnscape |
| 6 | 117\282 (23\282) |
497.87 (97.87) |
4/3 (128/121) |
Sextile |