178edo
| ← 177edo | 178edo | 179edo → |
178 equal divisions of the octave (abbreviated 178edo or 178ed2), also called 178-tone equal temperament (178tet) or 178 equal temperament (178et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 178 equal parts of about 6.74 ¢ each. Each step represents a frequency ratio of 21/178, or the 178th root of 2.
178et tempers out 15625/15555 (kleisma) and [-49 28 2⟩ in the 5-limit. Using the patent val, it tempers out 225/224, 4375/4374, and 40960000/40353607 in the 7-limit; 243/242, 3025/3024, 4375/4356, and 16896/16807 in the 11-limit; 640/637, 1188/1183, 1625/1617, 1716/1715, and 4096/4095 in the 13-limit. Using the 178def val, it tempers out 10976/10935, 33075/32768, and 50421/50000 in the 7-limit; 441/440, 3388/3375, 4125/4096, and 8019/8000 in the 11-limit; 325/324, 625/624, 847/845, 1287/1280, and 1573/1568 in the 13-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.00 | -0.83 | -2.04 | +1.96 | +1.49 | +2.17 | +2.91 | -0.88 | -1.31 | +1.88 | +1.03 |
| relative (%) | +0 | -12 | -30 | +29 | +22 | +32 | +43 | -13 | -19 | +28 | +15 | |
| Steps (reduced) |
178 (0) |
282 (104) |
413 (57) |
500 (144) |
616 (82) |
659 (125) |
728 (16) |
756 (44) |
805 (93) |
865 (153) |
882 (170) | |
Subsets and supersets
Since 178 factors into 2 × 89, 178edo contains 2edo and 89edo as its subsets.