888edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
888edo is in[[consistent]] to the [[5-odd-limit]] and [[3/1|harmonic 3]] is about halfway between its steps | 888edo is in[[consistent]] to the [[5-odd-limit]] and [[3/1|harmonic 3]] is about halfway between its steps. Otherwise it is excellent in approximating harmonics [[5/1|5]], [[7/1|7]], [[9/1|9]], [[11/1|11]], and [[13/1|13]], making it suitable for a 2.9.5.7.11.13 [[subgroup]] interpretation. The equal temperament [[Tempering out|tempers out]] [[4096/4095]], [[6656/6655]], [[9801/9800]], [[10648/10647]], 105644/105625, 151263/151200, and 250047/250000 in the above subgroup. | ||
=== Odd harmonics === | === Odd harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
Since 888 factors into | Since 888 factors into {{factorization|888}}, 888edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 12, 24, 37, 74, 111, 148, 222, 296, and 444 }}. 1776edo, which doubles it, provides a good correction for harmonic 3. | ||
Latest revision as of 23:00, 20 February 2025
| ← 887edo | 888edo | 889edo → |
888 equal divisions of the octave (abbreviated 888edo or 888ed2), also called 888-tone equal temperament (888tet) or 888 equal temperament (888et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 888 equal parts of about 1.35 ¢ each. Each step represents a frequency ratio of 21/888, or the 888th root of 2.
888edo is inconsistent to the 5-odd-limit and harmonic 3 is about halfway between its steps. Otherwise it is excellent in approximating harmonics 5, 7, 9, 11, and 13, making it suitable for a 2.9.5.7.11.13 subgroup interpretation. The equal temperament tempers out 4096/4095, 6656/6655, 9801/9800, 10648/10647, 105644/105625, 151263/151200, and 250047/250000 in the above subgroup.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.604 | +0.173 | +0.093 | +0.144 | +0.033 | +0.013 | -0.431 | +0.450 | -0.216 | -0.511 | +0.104 |
| Relative (%) | -44.7 | +12.8 | +6.9 | +10.7 | +2.5 | +1.0 | -31.9 | +33.3 | -16.0 | -37.8 | +7.7 | |
| Steps (reduced) |
1407 (519) |
2062 (286) |
2493 (717) |
2815 (151) |
3072 (408) |
3286 (622) |
3469 (805) |
3630 (78) |
3772 (220) |
3900 (348) |
4017 (465) | |
Subsets and supersets
Since 888 factors into 23 × 3 × 37, 888edo has subset edos 2, 3, 4, 6, 8, 12, 24, 37, 74, 111, 148, 222, 296, and 444. 1776edo, which doubles it, provides a good correction for harmonic 3.