178edo

← 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 2^1/178, or the 178th root of 2.

178et tempers out 15625/15552 (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.