# 138edo

 ← 137edo 138edo 139edo →
Prime factorization 2 × 3 × 23
Step size 8.69565¢
Fifth 81\138 (704.348¢) (→27\46)
Semitones (A1:m2) 15:9 (130.4¢ : 78.26¢)
Consistency limit 3
Distinct consistency limit 3

138 equal divisions of the octave (abbreviated 138edo or 138ed2), also called 138-tone equal temperament (138tet) or 138 equal temperament (138et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 138 equal parts of about 8.7 ¢ each. Each step represents a frequency ratio of 21/138, or the 138th root of 2.

138 = 3 × 46, and 138edo shares its fifth with 46edo. Unlike 46edo, it is inconsistent to the 5-odd-limit and higher limits, with three mappings possible for the 13-limit: 138 219 320 387 477 511] (patent val), 138 219 321 388 478 511] (138cde), and 138 218 320 387 477 510] (138bf). The last mapping uses an alternative flat fifth from 69edo.

Using the patent val, it tempers out 1953125/1889568 (shibboleth comma) and 67108864/66430125 (misty comma) in the 5-limit; 875/864, 1029/1024, and 1647086/1594323 in the 7-limit; 896/891, 1331/1323, 1375/1372, and 2401/2376 in the 11-limit; 196/195, 275/273, and 1575/1573 in the 13-limit.

The 138cde val is enfactored in the 5-limit, with the same tuning as 46edo, tempering out the diaschisma, 2048/2025 and the sensipent comma, 78732/78125. However, it tempers out 1728/1715, 10976/10935, and 250047/250000 in the 7-limit; 176/175, 540/539, 896/891, and 85184/84375 in the 11-limit; 351/350, 352/351, 364/363, 640/637, and 2197/2187 in the 13-limit, supporting the echidna temperament and giving an excellent tuning.

The 138bf val is also enfactored in the 5-limit, with the same tuning as 69edo, tempering out the syntonic comma, 81/80 and [-41 1 17. However, it tempers out 2401/2400, 2430/2401, and 9765625/9633792 in the 7-limit; 385/384, 1375/1372, 1944/1925, and 9375/9317 in the 11-limit, supporting the cuboctahedra temperament; 625/624, 975/968, 1001/1000, and 1188/1183 in the 13-limit.

138edo can be treated as the 2.7/5.11/5.13/3 subgroup temperament, which tempers out 24192/24167, 1449459/1449175, and 75000000/74942413.

### Odd harmonics

Approximation of odd harmonics in 138edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +2.39 -3.71 -3.61 -3.91 -3.49 +2.95 -1.31 -0.61 -1.86 -1.22 -2.19
Relative (%) +27.5 -42.6 -41.5 -45.0 -40.2 +33.9 -15.1 -7.0 -21.4 -14.0 -25.2
Steps
(reduced)
219
(81)
320
(44)
387
(111)
437
(23)
477
(63)
511
(97)
539
(125)
564
(12)
586
(34)
606
(54)
624
(72)

### Subsets and supersets

Since 138 factors into 2 × 3 × 23, 138edo has subset edos 2, 3, 6, 23, 46, and 69.