78005edo
| ← 78004edo | 78005edo | 78006edo → |
78005 equal divisions of the octave (abbreviated 78005edo or 78005ed2), also called 78005-tone equal temperament (78005tet) or 78005 equal temperament (78005et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 78005 equal parts of about 0.0154 ¢ each. Each step represents a frequency ratio of 21/78005, or the 78005th root of 2.
While 78005edo is distinctly consistent through the 21-odd-limit, its notability stems from the fact that it is a very strong 5-limit system, with lower 5-limit relative error than any smaller edo, and lower 5-limit TE logflat badness than any smaller edo excepting 4296. The equal temperament tempers out [232 -183 25⟩, [324 8 -145⟩, [92 191 -170⟩, [140 -374 195⟩, the selenia [-433 -137 280⟩, and the quark [-573 237 85⟩.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00000 | +0.00000 | -0.00002 | +0.00430 | +0.00056 | +0.00307 | +0.00709 | +0.00637 | -0.00693 | +0.00296 | -0.00128 |
| Relative (%) | +0.0 | +0.0 | -0.1 | +27.9 | +3.7 | +20.0 | +46.1 | +41.4 | -45.0 | +19.2 | -8.3 | |
| Steps (reduced) |
78005 (0) |
123635 (45630) |
181122 (25112) |
218988 (62978) |
269853 (35838) |
288653 (54638) |
318843 (6823) |
331360 (19340) |
352860 (40840) |
378947 (66927) |
386452 (74432) | |
Subsets and supersets
78005edo contains 15601edo, from which the approximation of the 3rd harmonic is derived.