# 213edo

 ← 212edo 213edo 214edo →
Prime factorization 3 × 71
Step size 5.6338¢
Fifth 125\213 (704.225¢)
Semitones (A1:m2) 23:14 (129.6¢ : 78.87¢)
Dual sharp fifth 125\213 (704.225¢)
Dual flat fifth 124\213 (698.592¢)
Dual major 2nd 36\213 (202.817¢) (→12\71)
Consistency limit 7
Distinct consistency limit 7

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

Approximation of odd harmonics in 213edo
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 (%) +40.3 +42.9 +3.3 -19.4 +14.1 -19.4 -16.8 +37.0 +19.1 +43.6 +48.1
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.