206edo
| ← 205edo | 206edo | 207edo → |
(semiconvergent)
206 equal divisions of the octave (abbreviated 206edo or 206ed2), also called 206-tone equal temperament (206tet) or 206 equal temperament (206et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 206 equal parts of about 5.83 ¢ each. Each step represents a frequency ratio of 21/206, or the 206th root of 2.
Theory
206edo is closely related to 103edo, but the patent vals differ on the mapping for 3, 11, and 19.
It is inconsistent to the 5-odd-limit and higher limits, with four mappings possible for the 19-limit:
- ⟨206 327 478 578 713 762 842 875] (patent val)
- ⟨206 326 478 578 713 762 842 875] (206b)
- ⟨206 326 478 578 712 762 842 875] (206be)
- ⟨206 327 479 579 713 763 842 875] (206cdf)
Using the patent val, it tempers out 48828125/47775744 (sycamore comma) and 32768000000/31381059609 in the 5-limit; 4000/3969, 84035/82944, and 458752/455625 in the 7-limit; 385/384, 2401/2376, 6875/6804, and 9375/9317 in the 11-limit; 352/351, 1575/1573, 1625/1617, 2197/2178, and 4096/4095 in the 13-limit; 289/288, 375/374, 442/441, 715/714, and 2000/1989 in the 17-limit; 190/189, 361/360, 665/663, 969/968, 1235/1232, and 1375/1368 in the 19-limit.
Using the 206b val, it tempers out 78732/78125 (sensipent comma) and 34171875/33554432 (ampersand) in the 5-limit; 225/224, 1029/1024, and 177147/175000 in the 7-limit; 4375/4356, 9801/9800, 15309/15125, and 73728/73205 in the 11-limit; 351/350, 364/363, 625/624, 1701/1690, and 31213/30976 in the 13-limit; 273/272, 833/832, 850/847, 1089/1088, 1225/1224, and 1458/1445 in the 17-limit; 210/209, 495/494, 729/722, 1235/1232, and 1445/1444 in the 19-limit.
Using the 206be val, it tempers out 243/242, 385/384, 441/440, and 43923/43750 in the 11-limit; 351/350, 625/624, 847/845, 1001/1000, and 1573/1568 in the 13-limit; 273/272, 375/374, 561/560, 715/714, and 833/832 in the 17-limit; 363/361 and 729/722 in the 19-limit.
Using the 206cdf val, it tempers out 2048/2025 (diaschisma), and [4 -45 29⟩ in the 5-limit; 4375/4374, 110592/109375, and 235298/234375 in the 7-limit; 176/175, 896/891, and 1331/1323 in the 11-limit; 640/637, 847/845, 1001/1000, and 2197/2187 in the 13-limit; 136/135, 256/255, 561/560, and 1275/1274 in the 17-limit; 190/189, 476/475, 608/605, 836/833, and 969/968 in the 19-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +2.90 | -1.85 | -1.84 | -0.03 | +2.08 | -1.69 | +1.05 | -0.10 | -0.43 | +1.06 | +0.85 |
| relative (%) | +50 | -32 | -32 | -0 | +36 | -29 | +18 | -2 | -7 | +18 | +15 | |
| Steps (reduced) |
327 (121) |
478 (66) |
578 (166) |
653 (35) |
713 (95) |
762 (144) |
805 (187) |
842 (18) |
875 (51) |
905 (81) |
932 (108) | |
Subsets and supersets
206edo contains 2edo and 103edo as subsets. 412edo, which doubles it, provides an excellent correction to the approximation of harmonic 3.