# 138edo

← 137edo | 138edo | 139edo → |

**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 2^{1/138}, 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

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.