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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 4.2553 [[cent]]s each.
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 4.2553 [[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|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 16875/16807, 19683/19600 and 65625/65536 in the 7-limit, and 540/539 and 5632/5625 in the 11-limit, so that it provides the [[Optimal_patent_val|optimal patent val]] for [[Porwell_family|jupiter temperament]]; it also tempers out 4000/3993 and 234375/234256, providing the optimal patent val for [[Porwell temperaments #Septisuperfourth|septisuperfourth]] temperament. In the 13-limit, it tempers out 729/728, 1575/1573, 1716/1715 and 2080/2079.
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 16875/16807, [[19683/19600]] and 65625/65536 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 ===
{{Primes in edo|edo=282|columns=10}}
{{Primes in edo|edo=282|columns=10}}
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| {{monzo| 32 -7 -9 }}, {{monzo| -7 22 -12 }}
| [{{val| 282 447 655 }}]
| -0.1684
| 0.1671
| 3.93
|-
| 2.3.5.7
| 6144/6125, 118098/117649, 250047/250000
| [{{val| 282 447 655 792 }}]
| -0.2498
| 0.2020
| 4.75
|-
| 2.3.5.7.11
| 540/539, 4000/3993, 5632/5625, 137781/137500
| [{{val| 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
| [{{val| 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
| [{{val| 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
| [{{val| 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
| [{{val| 282 447 655 792 976 1044 1153 1198 1276 }}]
| -0.3173
| 0.1944
| 4.57
|}


[[Category:Equal divisions of the octave]]
[[Category:Equal divisions of the octave]]
[[Category:29-limit]]
[[Category:29-limit]]

Revision as of 18:18, 26 December 2021

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 4.2553 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 16875/16807, 19683/19600 and 65625/65536 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
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