# 178edo

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 ← 177edo 178edo 179edo →
Prime factorization 2 × 89
Step size 6.74157¢
Fifth 104\178 (701.124¢) (→52\89)
Semitones (A1:m2) 16:14 (107.9¢ : 94.38¢)
Consistency limit 5
Distinct consistency limit 5

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

Approximation of prime harmonics in 178edo
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 (%) +0.0 -12.3 -30.3 +29.1 +22.1 +32.2 +43.2 -13.1 -19.4 +27.9 +15.3
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.