1289edo
| ← 1288edo | 1289edo | 1290edo → |
1289 equal divisions of the octave (abbreviated 1289edo or 1289ed2), also called 1289-tone equal temperament (1289tet) or 1289 equal temperament (1289et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1289 equal parts of about 0.931 ¢ each. Each step represents a frequency ratio of 21/1289, or the 1289th root of 2.
Theory
1289edo is consistent to the 9-odd-limit. As an equal temperament, it tempers out [-16 35 -17⟩ (minortone comma) in the 5-limit. Using the patent val, it tempers out 3025/3024, 180224/180075, 2460375/2458624 and 50014503/50000000 in the 11-limit; 1716/1715, 4096/4095, 91125/91091 and 5282739/5281250 in the 13-limit. In the 2.3.13.23.29.31 subgroup it tempers out 19344/19343, in the 2.3.5.7.11.23.31 subgroup 19251/19250.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.016 | +0.032 | +0.297 | -0.193 | +0.124 | +0.242 | +0.392 | +0.120 | +0.058 | +0.038 |
| Relative (%) | +0.0 | -1.7 | +3.5 | +32.0 | -20.7 | +13.3 | +26.0 | +42.1 | +12.9 | +6.2 | +4.1 | |
| Steps (reduced) |
1289 (0) |
2043 (754) |
2993 (415) |
3619 (1041) |
4459 (592) |
4770 (903) |
5269 (113) |
5476 (320) |
5831 (675) |
6262 (1106) |
6386 (1230) | |
Subsets and supersets
1289edo is the 209th prime edo.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-2043 1289⟩ | [⟨1289 2043]] | +0.0049 | 0.0049 | 0.53 |
| 2.3.5 | [-16 35 -17⟩, [91 -12 -31⟩ | [⟨1289 2043 2993]] | −0.0014 | 0.0097 | 1.04 |
| 2.3.5.7 | 2460375/2458624, 78125000/78121827, 12884901888/12867859375 | [⟨1289 2043 2993 3619]] | −0.0275 | 0.0461 | 4.95 |
| 2.3.5.7.11 | 3025/3024, 180224/180075, 2460375/2458624, 50014503/50000000 | [⟨1289 2043 2993 3619 4459]] | −0.0109 | 0.0530 | 5.69 |
| 2.3.5.7.11.13 | 3025/3024, 1716/1715, 4096/4095, 91125/91091, 5282739/5281250 | [⟨1289 2043 2993 3619 4459 4770]] | −0.0146 | 0.0491 | 5.27 |
| 2.3.5.7.11.13.17 | 3025/3024, 1716/1715, 4096/4095, 2500/2499, 37180/37179, 3536379/3536000 | [⟨1289 2043 2993 3619 4459 4770 5269]] | −0.0210 | 0.0481 | 5.17 |
| 2.3.5.7.11.13.17.19 | 3025/3024, 1716/1715, 2376/2375, 4096/4095, 2500/2499, 270864/270725, 75735/75712 | [⟨1289 2043 2993 3619 4459 4770 5269 5476]] | −0.0299 | 0.0508 | 5.46 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 142\1289 | 132.196 | [-38 5 13⟩ | Astro |
| 1 | 196\1289 | 182.467 | 10/9 | Minortone |
| 1 | 238\1289 | 221.567 | 8388608/7381125 | Fortune |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct