282edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|282}} | |||
== Theory == | == Theory == | ||
282edo is the smallest | 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 [[Stearnsmic clan #Garistearn|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 chord]]s 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 === | === Prime harmonics === | ||
{{Harmonics in equal|282|columns=11}} | {{Harmonics in equal|282|columns=11}} | ||
=== Subsets and supersets === | |||
Since 282 factors into 2 × 3 × 47, it has subset edos {{EDOs| 2, 3, 47, 94, and 141 }}. | |||
== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" | Subgroup | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning Error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
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{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per | ! Periods<br>per 8ve | ||
! Generator<br>( | ! Generator<br>(Reduced) | ||
! Cents<br>( | ! Cents<br>(Reduced) | ||
! Associated<br> | ! Associated<br>Ratio | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
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|} | |} | ||
[[Category:Septisuperfourth]] | [[Category:Septisuperfourth]] | ||
[[Category:Jupiter]] | [[Category:Jupiter]] | ||
Revision as of 11:50, 11 May 2023
| ← 281edo | 282edo | 283edo → |
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. 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 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 (%) | +0.0 | +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) | |
Subsets and supersets
Since 282 factors into 2 × 3 × 47, it 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 |
Rank-2 temperaments
| Periods per 8ve |
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 |