289edo
| ← 288edo | 289edo | 290edo → |
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
| 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