# 309edo

 ← 308edo 309edo 310edo →
Prime factorization 3 × 103
Step size 3.8835¢
Fifth 181\309 (702.913¢)
Semitones (A1:m2) 31:22 (120.4¢ : 85.44¢)
Consistency limit 3
Distinct consistency limit 3

309 equal divisions of the octave (abbreviated 309edo or 309ed2), also called 309-tone equal temperament (309tet) or 309 equal temperament (309et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 309 equal parts of about 3.88 ¢ each. Each step represents a frequency ratio of 21/309, or the 309th root of 2.

309edo is inconsistent to the 5-odd-limit, with four possible mappings in the 7-limit:

• 309 490 717 867] (patent val)
• 309 490 718 867] (309c)
• 309 490 717 868] (309d)
• 309 490 718 868] (309cd)

Using the patent val, it tempers out 5120/5103, 117649/116640, 390625/387072, 537824/531441, 823543/819200 and 1953125/1928934 in the 7-limit, supporting the hemifamity temperament.

Using the 309c val, it tempers out 4375/4374, 153664/151875, 395136/390625, 537824/531441, 2100875/2097152 and 5250987/5242880 in the 7-limit, supporting mitonic and ragismic.

Using the 309d val, it tempers out 2109375/2097152 in the 5-limit; 4000/3969, 420175/419904, 829440/823543, 1071875/1062882 and 9765625/9633792 in the 7-limit. It supports the octagari temperament.

Using the 309cd val, it tempers out 67108864/66430125 in the 5-limit; 3136/3125, 5120/5103, 250047/250000, 458752/455625, 49009212/48828125 and 725594112/720600125 in the 7-limit. It supports misty, hemifamity and landscape.

Additionally, the 309b val 309 489 717 867] (309b) is enfactored 103edo.

### Prime harmonics

Approximation of prime harmonics in 309edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.96 -1.85 -1.84 +0.14 -1.69 -0.10 +1.52 +0.85 -0.45 +0.60
Relative (%) +0.0 +24.7 -47.6 -47.3 +3.6 -43.6 -2.6 +39.0 +21.9 -11.6 +15.3
Steps
(reduced)
309
(0)
490
(181)
717
(99)
867
(249)
1069
(142)
1143
(216)
1263
(27)
1313
(77)
1398
(162)
1501
(265)
1531
(295)

### Subsets and supersets

309 factors into 3 × 103, with 3edo and 103edo as its subset edos. 927edo, which triples it, gives a good correction to the harmonic 5.