172edo

← 171edo

172edo

173edo →

Prime factorization	22 × 43
Step size	6.97674¢
Fifth	101\172 (704.651¢)
Semitones (A1:m2)	19:11 (132.6¢ : 76.74¢)
Dual sharp fifth	101\172 (704.651¢)
Dual flat fifth	100\172 (697.674¢) (→25\43)
Dual major 2nd	29\172 (202.326¢)
Consistency limit	3
Distinct consistency limit	3

172 equal divisions of the octave (abbreviated 172edo or 172ed2), also called 172-tone equal temperament (172tet) or 172 equal temperament (172et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 172 equal parts of about 6.98 ¢ each. Each step represents a frequency ratio of 2^1/172, or the 172nd root of 2.

172edo is inconsistent to the 5-odd-limit and higher limits, with three mappings possible for the 5-limit: ⟨172 273 399] (patent val), ⟨172 272 399] (172b), and ⟨172 273 400] (172c). Using the patent val, it tempers out the semicomma, 2109375/2097152 and 1220703125/1162261467 in the 5-limit; 245/243, 3125/3087, and 2097152/2066715 in the 7-limit, supporting the 7-limit bohpier temperament; 385/384, 896/891, 1331/1323, and 9375/9317 in the 11-limit; 169/168, 352/351, 364/363, and 1716/1715 in the 13-limit, supporting the leapweek temperament. Using the 172f val, 275/273, 640/637, 847/845, and 1575/1573 are tempered out in the 13-limit.

Using the 172c val, it tempers out the diaschisma, 2048/2025 and [1 36 -25⟩ in the 5-limit; 4375/4374, 50421/50000, and 110592/109375 in the 7-limit; 176/175, 896/891, and 1331/1323 in the 11-limit. Using the alternative 172cf val, 196/195, 1716/1715, 2080/2079, 2197/2187, and 2200/2197 are tempered out in the 13-limit. Using the alternative 172cef val, it tempers out 441/440, 1344/1331, and 3388/3375 in the 11-limit; 196/195, 352/351, 832/825, 1001/1000, and 2197/2187 in the 13-limit.

Using the 172b val, it tempers out the unicorn comma, 1594323/1562500 and 2197265625/2147483648 in the 5-limit; 1728/1715, 3645/3584, and 390625/388962 in the 7-limit; 441/440, 1944/1925, 4000/3993, and 4125/4096 in the 11-limit; 625/624, 975/968, 1188/1183, 1287/1280, and 1573/1568 in the 13-limit. Using the alternative 172bdee val, it tempers out 225/224, 118098/117649, and 3176523/3125000 in the 7-limit; 243/242, 1617/1600, 2079/2048, and 117649/117128 in the 11-limit; 351/350, 625/624, 1188/1183, and 1573/1568 in the 13-limit.

Odd harmonics

Approximation of odd harmonics in 172edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	+2.70	-2.59	+0.94	-1.58	-0.16	-3.32	+0.10	-0.30	+2.49	-3.34	-0.37
	Relative (%)	+38.6	-37.2	+13.5	-22.7	-2.2	-47.6	+1.5	-4.4	+35.6	-47.9	-5.3
Steps (reduced)		273 (101)	399 (55)	483 (139)	545 (29)	595 (79)	636 (120)	672 (156)	703 (15)	731 (43)	755 (67)	778 (90)

Subsets and supersets

Since 172 factors into 2² × 43, 172edo has subset edos 2, 4, 43, and 86.