# 638edo

 ← 637edo 638edo 639edo →
Prime factorization 2 × 11 × 29
Step size 1.88088¢
Fifth 373\638 (701.567¢)
Semitones (A1:m2) 59:49 (111¢ : 92.16¢)
Consistency limit 11
Distinct consistency limit 11

638 equal divisions of the octave (abbreviated 638edo or 638ed2), also called 638-tone equal temperament (638tet) or 638 equal temperament (638et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 638 equal parts of about 1.88 ¢ each. Each step represents a frequency ratio of 21/638, or the 638th root of 2.

## Theory

The equal temperament tempers out the minortone comma, [-16 35 -17, in the 5-limit, 4375/4374 in the 7-limit, 3025/3024, 9801/9800, and 43923/43904, in the 11-limit; and 625/624, 729/728, 1575/1573, 2200/2197 and 4225/4224 in the 13-limit. It supplies the optimal patent val for quatracot, the 224 & 414 temperament.

### Odd harmonics

Approximation of odd harmonics in 638edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -0.388 -0.734 -0.174 -0.775 -0.221 +0.225 +0.759 +0.374 -0.334 -0.561 -0.061
Relative (%) -20.6 -39.0 -9.2 -41.2 -11.7 +11.9 +40.4 +19.9 -17.8 -29.9 -3.3
Steps
(reduced)
1011
(373)
1481
(205)
1791
(515)
2022
(108)
2207
(293)
2361
(447)
2493
(579)
2608
(56)
2710
(158)
2802
(250)
2886
(334)

### Subsets and supersets

Since 638 factors as 2 × 11 × 29, 638edo has subset edos 2, 11, 22, 29, 58, and 319.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [-1011 638 [638 1011]] +0.1223 0.1223 6.50
2.3.5 [-51 19 9, [-16 35 -17 [638 1011 1481]] +0.1869 0.1353 7.19
2.3.5.7 4375/4374, 2100875/2097152, [-11 5 11 -8 [638 1011 1481 1791]] +0.1556 0.1291 6.86
2.3.5.7.11 3025/3024, 4375/4374, 825000/823543, 1265625/1261568 [638 1011 1481 1791 2207]] +0.1373 0.1212 6.44
2.3.5.7.11.13 625/624, 729/728, 1575/1573, 2200/2197, 823680/823543 [638 1011 1481 1791 2207 2361]] +0.1043 0.1330 7.07

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 97\638 182.45 10/9 Mitonic
1 313\638 588.71 7/5 Untriton
2 94\638 176.80 448/405 Quatracot

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct