# 123edo

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 ← 122edo 123edo 124edo →
Prime factorization 3 × 41
Step size 9.7561¢
Fifth 72\123 (702.439¢) (→24\41)
Semitones (A1:m2) 12:9 (117.1¢ : 87.8¢)
Consistency limit 5
Distinct consistency limit 5

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 21/123, 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

Approximation of prime harmonics in 123edo
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.