282edo: Difference between revisions
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The 7-limit commas were wrong (they were 152et's, lol); +categories |
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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 [[EDO|equal division of the octave]] into 282 parts of 4. | 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 [[EDO|equal division of the octave]] into 282 parts of about 4.26 [[cent]]s each. | ||
== Theory == | == 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 [[Stearnsmic clan #Garistearn|garistearn]] temperament). It has a distinct sharp tendency for odd harmonics up to 29. It tempers out | 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 [[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]] and [[2080/2079]]. | ||
=== Prime harmonics === | === Prime harmonics === | ||
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| [[Sextile]] | | [[Sextile]] | ||
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[[Category:Equal divisions of the octave]] | [[Category:Equal divisions of the octave]] | ||
[[Category:29-limit]] | [[Category:29-limit]] | ||
[[Category:Septisuperfourth]] | |||
[[Category:Jupiter]] | |||
Revision as of 15:03, 27 December 2021
| ← 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 and 2080/2079.
Prime harmonics
Script error: No such module "primes_in_edo".
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 |