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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 17<sup>2</sup>
{{ED intro}}
| Step size = 4.15225¢
| Fifth = 169\289 (701.73¢)
| Semitones = 27:22 (112.11¢ : 91.35¢)
| Consistency = 9
}}
The '''289 equal divisions of the octave''' ('''289edo'''), or the '''289(-tone) equal temperament''' ('''289tet''', '''289et''') when viewed from a [[regular temperament]] perspective, divides the octave into 289 equal parts of about 4.15 [[cent]]s each.
==Theory==
289edo is the [[optimal patent val]] for [[13-limit]] [[History (temperament)|history]] temperament, which tempers out [[364/363]], [[441/440]] and [[676/675]], and provides a good tuning for the 11-limit version also, and is also the optimal patent val for [[sextilififths]] in both the 11- and 13-limit. It is uniquely [[consistent]] in the 9-odd-limit, and tempers out the [[schisma]], 32805/32768 in the 5-limit; [[4375/4374]] and 65625/65536 in the 7-limit; 441/440 and [[4000/3993]] in the 11-limit; and 364/363, 676/675, [[1001/1000]], [[1575/1573]] and [[2080/2079]] in the 13-limit.


Since 289 is square of 17<!-- How specifically the square property of 289 ensures it supports chlorine? -->, 289 = 17 × 17, 289edo [[support]]s the [[chlorine]] temperament, which tempers out the [[septendecima]] {{monzo|-52 -17 34}} and the ragisma 4375/4374.
== Theory ==
289edo is a strong 5-limit system with decent [[11-limit|11-]] and [[13-limit]] interpretations despite in[[consistency]] in the [[13-odd-limit]]. As an equal temperament, it [[tempering out|tempers out]] the [[schisma]], 32805/32768 in the [[5-limit]]; [[4375/4374]] and [[65625/65536]] in the [[7-limit]]; [[441/440]] and [[4000/3993]] in the 11-limit; and [[364/363]], [[676/675]], [[1001/1000]], [[1575/1573]] and [[2080/2079]] in the 13-limit.
 
It is the [[optimal patent val]] for the [[13-limit]] rank-5 temperament tempering out 364/363, and the 13-limit [[history (temperament)|history]] temperament, which tempers out 364/363, 441/440 and 676/675. It provides a good tuning for the 11-limit version also. It is also the optimal patent val for [[sextilifourths]], [[quintaschis]], and [[quincy]] in both the 11- and 13-limit.  


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|289}}
{{Harmonics in equal|289}}


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
=== Subsets and supersets ===
289 is 17 squared. In light of containing [[17edo]] as a subset, 289edo [[support]]s the [[chlorine]] temperament, which tempers out the [[septendecima]] {{monzo| -52 -17 34 }} and the ragisma 4375/4374.
 
== 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
| {{monzo| -458 289 }}
| {{mapping| 289 458 }}
| +0.0709
| 0.0710
| 1.71
|-
| 2.3.5
| 32805/32768, {{monzo| 7 41 -31 }}
| {{mapping| 289 458 671 }}
| +0.0695
| 0.0580
| 1.40
|-
| 2.3.5.7
| 4375/4374, 32805/32768, 235298/234375
| {{mapping| 289 458 671 811 }}
| +0.1725
| 0.1854
| 4.46
|-
| 2.3.5.7.11
| 441/440, 4000/3993, 4375/4374, 32805/32768
| {{mapping| 289 458 671 811 1000 }}
| +0.0841
| 0.2423
| 5.83
|-
| 2.3.5.7.11.13
| 364/363, 441/440, 676/675, 4375/4374, 19773/19712
| {{mapping| 289 458 671 811 1000 1069 }}
| +0.1500
| 0.2657
| 6.40
|}
* 289et has a lower absolute error in the 5-limit than any previous equal temperaments, past [[171edo|171]] and followed by [[323edo|323]].
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br />per 8ve
! Generator*
! Cents*
! Associated<br />ratio*
! Temperaments
|-
| 1
| 4\289
| 16.61
| 100/99
| [[Quincy]]
|-
| 1
| 13\289
| 53.98
| 33/32
| [[Tridecafifths]]
|-
| 1
| 20\289
| 83.04
| 21/20
| [[Sextilifourths]]
|-
| 1
| 24\289
| 99.65
| 18/17
| [[Quintaschis]]
|-
| 1
| 76\289
| 315.57
| 6/5
| [[Acrokleismic]]
|-
| 1
| 86\289
| 357.09
| 768/625
| [[Dodifo]]
|-
| 1
| 108\289
| 448.44
| 35/27
| [[Semidimfourth]]
|-
| 1
| 120\289
| 498.27
| 4/3
| [[Pontiac]]
|-
| 1
| 135\289
| 560.55
| 864/625
| [[Whoosh]]
|-
| 17
| 93\289<br />(8\289)
| 386.16<br />(33.22)
| {{monzo| -23 5 9 -2 }}<br />(100352/98415)
| [[Chlorine]]
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
[[Category:History (temperament)]]
[[Category:History (temperament)]]
[[Category:Sextilififths]]
[[Category:Minor minthmic]]
[[Category:Quincy]]
[[Category:Quintaschis]]
[[Category:Sextilifourths]]
Retrieved from "https://en.xen.wiki/w/289edo"