289edo
| ← 288edo | 289edo | 290edo → |
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
| 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
Template:Comma basis begin |- | 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 Template:Comma basis end
- 289et has a lower absolute error in the 5-limit than any previous equal temperaments, past 171 and followed by 323.
Rank-2 temperaments
Template:Rank-2 begin
|-
| 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
Template:Rank-2 end
Template:Orf