789edo
| ← 788edo | 789edo | 790edo → |
789 equal divisions of the octave (abbreviated 789edo or 789ed2), also called 789-tone equal temperament (789tet) or 789 equal temperament (789et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 789 equal parts of about 1.52 ¢ each. Each step represents a frequency ratio of 21/789, or the 789th root of 2.
789edo is notable for an extremely good approximation of the 2.5.7 subgroup, unbeaten until 5902edo; notably, it is the denominator of logarithmic semiconvergents to both 5/4 and 7/5, and remarkably tempers out the Don Page comma between the intervals of 128/125 and 50/49, mapping each to 27 and 23 steps respectively, and additionally tempers out 281484423828125/281474976710656 ([-48 0 11 8⟩), supporting exodia, the subgroup restriction of mohajira. It also has a very accurate representation of the 17th harmonic (due to tempering out S49/S50 = 2000033/2000000, equating 17/16 to three 50/49 intervals) and has a good 9th and 23rd harmonic as well; there is a common flat tendency allowing consistency to high distance in the 2.9.5.7.33.17.23 subgroup.
1578edo, which doubles it, provides good corrections for the 3rd and 11th harmonics, making for a very strong 11-limit and higher-limit system.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.707 | -0.002 | -0.005 | -0.108 | -0.748 | +0.537 | +0.705 | -0.012 | +0.586 | +0.702 | -0.137 |
| Relative (%) | +46.5 | -0.1 | -0.3 | -7.1 | -49.2 | +35.3 | +46.3 | -0.8 | +38.5 | +46.2 | -9.0 | |
| Steps (reduced) |
1251 (462) |
1832 (254) |
2215 (637) |
2501 (134) |
2729 (362) |
2920 (553) |
3083 (716) |
3225 (69) |
3352 (196) |
3466 (310) |
3569 (413) | |