# 613edo

 ← 612edo 613edo 614edo →
Prime factorization 613 (prime)
Step size 1.95759¢
Fifth 359\613 (702.773¢)
Semitones (A1:m2) 61:44 (119.4¢ : 86.13¢)
Dual sharp fifth 359\613 (702.773¢)
Dual flat fifth 358\613 (700.816¢)
Dual major 2nd 104\613 (203.589¢)
Consistency limit 3
Distinct consistency limit 3

613 equal divisions of the octave (abbreviated 613edo or 613ed2), also called 613-tone equal temperament (613tet) or 613 equal temperament (613et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 613 equal parts of about 1.96 ¢ each. Each step represents a frequency ratio of 21/613, or the 613th root of 2.

## Theory

613edo is only consistent to the 3-odd-limit and the harmonic 3 is about halfway its steps. It can be used in the 2.9.5.7.13.19.23.29.31.37 subgroup, tempering out 1521/1520, 875/874, 1863/1862, 38475/38416, 2205/2204, 1520/1519, 186875/186624, 1665/1664 and 194560/194481.

### Odd harmonics

Approximation of odd harmonics in 613edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +0.818 -0.669 +0.179 -0.321 +0.721 -0.723 +0.149 +0.754 +0.040 -0.960 +0.111
Relative (%) +41.8 -34.2 +9.1 -16.4 +36.8 -37.0 +7.6 +38.5 +2.0 -49.1 +5.7
Steps
(reduced)
972
(359)
1423
(197)
1721
(495)
1943
(104)
2121
(282)
2268
(429)
2395
(556)
2506
(54)
2604
(152)
2692
(240)
2773
(321)

### Subsets and supersets

613edo is the 112th prime EDO. 1226edo, which doubles it, gives a good correction to the harmonic 3.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.9 [-1943 613 [613 1943]] 0.0506 0.0506 2.58
2.9.5 32805/32768, [-97 -69 136 [613 1943 1423]] 0.1299 0.1194 6.10
2.9.5.7 32805/32768, 40500000/40353607, [-23 -7 11 7 [613 1943 1423 1721]] 0.0814 0.1331 6.80