213edo
| ← 212edo | 213edo | 214edo → |
213 equal divisions of the octave (abbreviated 213edo or 213ed2), also called 213-tone equal temperament (213tet) or 213 equal temperament (213et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 213 equal parts of about 5.63 ¢ each. Each step represents a frequency ratio of 21/213, or the 213th root of 2.
213edo is (uniquely) consistent through the 7-odd-limit, but harmonics 3 and 5 are about halfway between its steps. Higher primes are slightly better tuned. Moreover, intervals involving a factor of 5/3 or 15 are quite well approximated. Thus it makes sense to view this as a 2.9.15.7.11.13 subgroup temperament.
The full 13-limit patent val for 213edo is ⟨213 338 495 598 737 788], which tempers out the following commas up to the 13-limit: [3 -10 11⟩ in the 5-limit; [6 -5 -4 4⟩, [10 -11 2 1⟩ and 6144/6125 in the 7-limit; 896/891 in the 11-limit; [12 -7 0 1 0 -1⟩, 325/324, 352/351 and 364/363 in the 13-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +2.27 | +2.42 | +0.19 | -1.09 | +0.79 | -1.09 | -0.94 | +2.09 | +1.08 | +2.46 | +2.71 |
| relative (%) | +40 | +43 | +3 | -19 | +14 | -19 | -17 | +37 | +19 | +44 | +48 | |
| Steps (reduced) |
338 (125) |
495 (69) |
598 (172) |
675 (36) |
737 (98) |
788 (149) |
832 (193) |
871 (19) |
905 (53) |
936 (84) |
964 (112) | |
Subsets and supersets
Since 213 factors into 3 × 71, 213edo contains 3edo and 71edo as its subsets.