216edo
| ← 215edo | 216edo | 217edo → |
216 equal divisions of the octave (abbreviated 216edo or 216ed2), also called 216-tone equal temperament (216tet) or 216 equal temperament (216et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 216 equal parts of about 5.56 ¢ each. Each step represents a frequency ratio of 21/216, or the 216th root of 2.
216edo is inconsistent to the 5-odd-limit and higher limits, with many mappings possible for the 13-limit. The following four will be discussed here: ⟨216 342 502 606 747 799] (patent val), ⟨216 343 502 607 748 800] (216bdef), ⟨216 342 501 606 747 799] (216c), and ⟨216 342 502 607 747 799] (216d).
The 216c val is enfactored in the 11-limit, and it happens to be of the best accuracy. Like 72, it tempers out 15625/15552 and 531441/524288 in the 5-limit; 225/224, 1029/1024, and 4375/4374 in the 7-limit; 243/242, 385/384, 441/440, and 4000/3993 in the 11-limit. However, it tempers out 2200/2197 and 2205/2197 in the 13-limit, and practically corrects the approximate 13th harmonic to as fit as it can be.
The 216bdef val chooses the sharp mapping for each of the harmonics, so it is the opposite of 216c in terms of tuning. It tempers out 2048/2025 and [1 -46 31⟩ in the 5-limit; 3136/3125, 4000/3969, and 40353607/39858075 in the 7-limit; 2560/2541, 3025/3024, 3388/3375, and 12005/11979 in the 11-limit; 325/324, 364/363, 640/637, and 1716/1715 in the 13-limit.
Using the patent val, it tempers out 531441/524288 and 1990656/1953125 in the 5-limit; 126/125, 1029/1024, and 118098/117649 in the 7-limit; 243/242, 3388/3375, 41503/41472, and 43923/43904 in the 11-limit; 676/675, 847/845, 1287/1280, 1701/1690, and 1716/1715 in the 13-limit.
Using the 216d val, it tempers out 2430/2401, 3136/3125, and 531441/524288 in the 7-limit; 176/175, 243/242, 1375/1372, and 131769/131072 in the 11-limit; 676/675, 1188/1183, 1287/1280, and 3042/3025 in the 13-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | -1.96 | +2.58 | -2.16 | +1.65 | -1.32 | -1.64 | +0.62 | +0.60 | +2.49 | +1.44 | -0.50 |
| relative (%) | -35 | +46 | -39 | +30 | -24 | -29 | +11 | +11 | +45 | +26 | -9 | |
| Steps (reduced) |
342 (126) |
502 (70) |
606 (174) |
685 (37) |
747 (99) |
799 (151) |
844 (196) |
883 (19) |
918 (54) |
949 (85) |
977 (113) | |
Subsets and supersets
Since 216 factors into 23 × 33, 216edo has subset edos 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, and 108.