# 289edo

 ← 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. The equal temperament 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 sextilififths, 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

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 Sextilififths
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 it is distinct