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The '''289 equal divisions of the octave''' ('''289edo''') divides the octave into 289 equal parts of 4.15225 [[cent]]s each. 289edo is the [[optimal patent val]] for [[13-limit]] [[History (temperament)|history]] temperament, which tempers out 364/363, 441/440 and 1001/1000, 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.
{{Infobox ET}}
{{ED intro}}


Since 289 is square of 17, 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]]
=== 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]]

Latest revision as of 20:49, 1 April 2025

← 288edo 289edo 290edo →
Prime factorization 172
Step size 4.15225 ¢ 
Fifth 169\289 (701.73 ¢)
Semitones (A1:m2) 27:22 (112.1 ¢ : 91.35 ¢)
Consistency limit 9
Distinct consistency limit 9

289 equal divisions of the octave (abbreviated 289edo or 289ed2), also called 289-tone equal temperament (289tet) or 289 equal temperament (289et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 289 equal parts of about 4.15 ¢ each. Each step represents a frequency ratio of 21/289, or the 289th root of 2.

Theory

289edo is a strong 5-limit system with decent 11- and 13-limit interpretations despite inconsistency in the 13-odd-limit. As an equal temperament, it 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, 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

Approximation of prime harmonics in 289edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.22 -0.15 -1.35 +0.93 -1.77 -1.15 +1.45 -1.28 +0.18 +0.99
Relative (%) +0.0 -5.4 -3.7 -32.6 +22.4 -42.7 -27.7 +34.9 -30.9 +4.3 +23.7
Steps
(reduced) 		289
(0) 		458
(169) 		671
(93) 		811
(233) 		1000
(133) 		1069
(202) 		1181
(25) 		1228
(72) 		1307
(151) 		1404
(248) 		1432
(276)

Subsets and supersets

289 is 17 squared. In light of containing 17edo as a subset, 289edo supports the chlorine temperament, which tempers out the septendecima [-52 -17 34⟩ and the ragisma 4375/4374.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-458 289⟩ [⟨289 458]] +0.0709 0.0710 1.71
2.3.5 32805/32768, [7 41 -31⟩ [⟨289 458 671]] +0.0695 0.0580 1.40
2.3.5.7 4375/4374, 32805/32768, 235298/234375 [⟨289 458 671 811]] +0.1725 0.1854 4.46
2.3.5.7.11 441/440, 4000/3993, 4375/4374, 32805/32768 [⟨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 [⟨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 171 and followed by 323.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
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
(8\289) 		386.16
(33.22) 		[-23 5 9 -2⟩
(100352/98415) 		Chlorine

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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