172edo
| ← 171edo | 172edo | 173edo → |
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 21/172, 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
| 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 (%) | +39 | -37 | +13 | -23 | -2 | -48 | +1 | -4 | +36 | -48 | -5 | |
| 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.