# 123edo

← 122edo | 123edo | 124edo → |

**123 equal divisions of the octave** (abbreviated **123edo** or **123ed2**), also called **123-tone equal temperament** (**123tet**) or **123 equal temperament** (**123et**) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 123 equal parts of about 9.76 ¢ each. Each step represents a frequency ratio of 2^{1/123}, or the 123rd root of 2.

123 = 3 × 41, and 123edo shares its fifth with 41edo. The equal temperament tempers out 1990656/1953125 (valentine comma), 67108864/66430125 (misty comma), and [-13 17 -6⟩ (graviton) in the 5-limit; 126/125, 1029/1024 and 537824/531441 in the 7-limit; 243/242, 896/891, 2401/2376, and 3388/3375 in the 11-limit; 196/195, 351/350, 832/825, 1575/1573, and 2197/2178 in the 13-limit. It provides the optimal patent val for the gravid temperament.

Given its inconsistency to the 7-odd-limit and higher odd limits, the mapping ⟨123 195 286 **346**] (123d) is also possible for the 7-limit. Using the 123d val, it tempers out 2430/2401, 3136/3125, and 5120/5103 in the 7-limit; 176/175, 243/242, 1375/1372, and 2560/2541 in the 11-limit; 169/168, 364/363, 640/637, 729/728, and 832/825 in the 13-limit. Using the 123df val, it tempers out 144/143, 351/350, 352/351, and 847/845 in the 13-limit.

Using the 123ce val, it tempers out 1331/1323 in the 11-limit, as well as 225/224, 245/243, and 1029/1024; 275/273, 352/351, 847/845, 1573/1568, and 3185/3168 in the 13-limit. Using the 123e val, it tempers out 121/120, 176/175, and 441/440 in the 11-limit; 196/195, 351/350, 352/351, 1287/1280, and 2197/2187 in the 13-limit.

5 steps of the 123be val is related to Triple BP.

### Prime harmonics

Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Error | Absolute (¢) | +0.00 | +0.48 | +3.93 | -2.97 | +4.78 | -1.50 | +2.36 | -4.83 | -3.88 | +4.57 | -3.57 |

Relative (%) | +0.0 | +5.0 | +40.3 | -30.5 | +49.0 | -15.4 | +24.2 | -49.5 | -39.8 | +46.8 | -36.6 | |

Steps (reduced) |
123 (0) |
195 (72) |
286 (40) |
345 (99) |
426 (57) |
455 (86) |
503 (11) |
522 (30) |
556 (64) |
598 (106) |
609 (117) |

### Subsets and supersets

Since 123 factors into 3 × 41, 123edo contains 3edo and 41edo as its subsets.