219edo
| ← 218edo | 219edo | 220edo → |
219 equal divisions of the octave (abbreviated 219edo or 219ed2), also called 219-tone equal temperament (219tet) or 219 equal temperament (219et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 219 equal parts of about 5.48 ¢ each. Each step represents a frequency ratio of 21/219, or the 219th root of 2.
Theory
219edo is inconsistent in the 5-odd-limit as well as higher odd limits. Its approximations to lower harmonics are exceptionally bad: 5, 11, and 13 are about halfway between its steps, and 19 and 23 are off by about a third step. If anything, it can be considered as a 2.3.7.17.29.31 subgroup tuning. One can see that there are much better alternatives to 219edo if the goal is to mimick just intonation, for example 212edo (being a superset of 53edo) or 217edo (being a superset of 31edo).
The patent val for 219edo is ⟨214 347 509 615 758 810], which tempers out the following commas up to the 13-limit: 32805/32768 in the 5-limit; 243/242, 441/440 and 65536/65219 in the 11-limit; 364/363 in the 13-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.00 | -0.59 | +2.73 | +1.04 | +2.11 | -2.17 | -0.85 | -1.62 | +1.86 | +0.56 | +0.17 |
| relative (%) | +0 | -11 | +50 | +19 | +38 | -40 | -15 | -30 | +34 | +10 | +3 | |
| Steps (reduced) |
219 (0) |
347 (128) |
509 (71) |
615 (177) |
758 (101) |
810 (153) |
895 (19) |
930 (54) |
991 (115) |
1064 (188) |
1085 (209) | |
Subsets and supersets
Since 219edo factors into 3 × 73, 219edo contains 3edo and 73edo as its subsets.