638edo

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← 637edo638edo639edo →
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