180edo

← 179edo

180edo

181edo →

Prime factorization	22 × 32 × 5 (highly composite)
Step size	6.66667 ¢
Fifth	105\180 (700 ¢) (→ 7\12)
Semitones (A1:m2)	15:15 (100 ¢ : 100 ¢)
Consistency limit	7
Distinct consistency limit	7

180 equal divisions of the octave (abbreviated 180edo or 180ed2), also called 180-tone equal temperament (180tet) or 180 equal temperament (180et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 180 equal parts of about 6.67 ¢ each. Each step represents a frequency ratio of 2^1/180, or the 180th root of 2.

The equal temperament tempers out 531441/524288 (pythagorean comma) and 30958682112/30517578125 (trisedodge comma) in the 5-limit, as well as 31381059609/30517578125 (mowgli comma) and 274877906944/274658203125 (hemithirds comma); 1029/1024, 3136/3125, and 118098/117649 in the 7-limit.

Using the patent val, it tempers out 540/539, 2835/2816, 4000/3993, and 6912/6875 in the 11-limit; 351/350, 364/363, 1001/1000, and 1701/1690 in the 13-limit. Using the 180e val, it tempers out 385/384, 441/440, 3388/3375, and 216513/214375 in the 11-limit; 351/350, 1188/1183, 1287/1280, 1573/1568, and 3146/3125 in the 13-limit.

Odd harmonics

Approximation of odd harmonics in 180edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	-1.96	+0.35	-2.16	+2.76	+2.02	-0.53	-1.60	+1.71	+2.49	+2.55	-1.61
	Relative (%)	-29.3	+5.3	-32.4	+41.3	+30.2	-7.9	-24.0	+25.7	+37.3	+38.3	-24.1
Steps (reduced)		285 (105)	418 (58)	505 (145)	571 (31)	623 (83)	666 (126)	703 (163)	736 (16)	765 (45)	791 (71)	814 (94)

Subsets and supersets

180edo is the 11th highly composite edo; its nontrivial subsets are: 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, and 90.