800edo
| ← 799edo | 800edo | 801edo → |
800 equal divisions of the octave (abbreviated 800edo or 800ed2), also called 800-tone equal temperament (800tet) or 800 equal temperament (800et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 800 equal parts of exactly 1.5 ¢ each. Each step represents a frequency ratio of 21/800, or the 800th root of 2.
Theory
800edo is consistent to the 11-odd-limit, tempering out 5632/5625, 9801/9800, 2401/2400 and 46656/46585 using the patent val. It shares its harmonic 3 with 200edo and 400edo, but due to sharing the harmonics until the 19-limit with 400edo, its harmonics 5 and 11 are about halfway its steps. 800edo enhances the quality of the harmonic 23, making it a strong 2.3.7.17.23.41.43 subgroup tuning, tempering out 5832/5831, 6273/6272, 352256/352107, 16813/16807, 1896129/1895936 and 871235584/870913701.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.045 | +0.686 | +0.174 | +0.682 | -0.528 | +0.045 | -0.513 | +0.226 | -0.577 | -0.536 |
| Relative (%) | +0.0 | +3.0 | +45.8 | +11.6 | +45.5 | -35.2 | +3.0 | -34.2 | +15.0 | -38.5 | -35.7 | |
| Steps (reduced) |
800 (0) |
1268 (468) |
1858 (258) |
2246 (646) |
2768 (368) |
2960 (560) |
3270 (70) |
3398 (198) |
3619 (419) |
3886 (686) |
3963 (763) | |
Subsets and supersets
Since 800 factores into 25 × 52, 800edo has subset edos 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 80, 100, 160, 200, and 400. 1600edo, which doubles it, gives a good correction to the harmonics 5 and 11.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [317 -200⟩ | [⟨800 1268]] | −0.0142 | 0.0142 | 0.95 |
| 2.3.5 | [-7 22 -12⟩, [47 -15 -10⟩ | [⟨800 1268 1858]] | −0.1080 | 0.1331 | 8.87 |
| 2.3.5.7 | 2401/2400, 1959552/1953125, 14348907/14336000 | [⟨800 1268 1858 2246]] | −0.0965 | 0.1170 | 7.80 |
| 2.3.5.7.11 | 5632/5625, 9801/9800, 2401/2400, 46656/46585 | [⟨800 1268 1858 2246 2768]] | −0.1166 | 0.1121 | 7.47 |
Music
- Holy Carp (2025)