282edo
| ← 281edo | 282edo | 283edo → |
282 equal divisions of the octave (abbreviated 282edo or 282ed2), also called 282-tone equal temperament (282tet) or 282 equal temperament (282et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 282 equal parts of about 4.26 ¢ each. Each step represents a frequency ratio of 21/282, or the 282nd root of 2.
Theory
282edo is the smallest edo distinctly 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.
The equal temperament 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 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 | +0.17 | +0.92 | +1.39 | +1.87 | +2.03 | +1.43 | +0.36 | +1.51 | +0.21 | -0.35 |
| relative (%) | +0 | +4 | +22 | +33 | +44 | +48 | +34 | +8 | +36 | +5 | -8 | |
| Steps (reduced) |
282 (0) |
447 (165) |
655 (91) |
792 (228) |
976 (130) |
1044 (198) |
1153 (25) |
1198 (70) |
1276 (148) |
1370 (242) |
1397 (269) | |
Subsets and supersets
Since 282 factors into 2 × 3 × 47, 282edo has subset edos 2, 3, 47, 94, and 141.
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 |
- 282et has a lower relative error than any previous equal temperaments in the 23-limit, past 270 and before 311.
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 13\282 | 55.32 | 33/32 | Escapade |
| 1 | 133\282 | 565.96 | 4096/2835 | 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 (7-limit) |
| 6 | 51\282 (4\282) |
217.02 (17.02) |
17/15 (105/104) |
Stearnscape |
| 6 | 80\282 (14\282) |
340.43 (59.57) |
162/133 (88/85) |
Semiseptichrome |
| 6 | 117\282 (23\282) |
497.87 (97.87) |
4/3 (128/121) |
Sextile |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct