172edo: Difference between revisions
CompactStar (talk | contribs) No edit summary |
Contribution (talk | contribs) No edit summary |
||
| (5 intermediate revisions by 3 users not shown) | |||
| Line 1: | Line 1: | ||
{{Infobox ET}} | {{Infobox ET}} | ||
{{ED intro}} | |||
172edo is in[[consistent]] to the [[5-odd-limit]] and higher limits, with three mappings possible for the 5-limit: {{val| 172 273 399 }} ([[patent val]]), {{val| 172 '''272''' 399 }} (172b), and {{val| 172 273 '''400''' }} (172c). Using the patent val, it [[tempering out|tempers out]] the [[semicomma]], 2109375/2097152 and 1220703125/1162261467 in the 5-limit; [[245/243]], [[3125/3087]], and 2097152/2066715 in the 7-limit, [[support]]ing 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 {{monzo| 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. | |||
Using the 172f val, every 7 steps is an excellent approximation of the ninth no-2 zeta peak - associated with [[39edt]] - in the 15-limit. | |||
=== Odd harmonics === | |||
{{Harmonics in equal|172}} | |||
=== Subsets and supersets === | |||
Since 172 factors into {{factorization|172}}, 172edo has subset edos {{EDOs| 2, 4, 43, and 86 }}. | |||
Latest revision as of 21:12, 7 June 2025
| ← 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.
Using the 172f val, every 7 steps is an excellent approximation of the ninth no-2 zeta peak - associated with 39edt - in the 15-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 (%) | +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.