# 8269edo

 ← 8268edo 8269edo 8270edo →
Prime factorization 8269 (prime)
Step size 0.14512¢
Fifth 4837\8269 (701.947¢)
Semitones (A1:m2) 783:622 (113.6¢ : 90.26¢)
Consistency limit 27
Distinct consistency limit 27
Special properties

8269 equal divisions of the octave (abbreviated 8269edo or 8269ed2), also called 8269-tone equal temperament (8269tet) or 8269 equal temperament (8269et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 8269 equal parts of about 0.145 ¢ each. Each step represents a frequency ratio of 21/8269, or the 8269th root of 2.

8269edo is both a zeta peak and zeta integral edo, which has to do with the fact that it is a very strong 19- and 23-limit system. It has a lower 19-limit and a lower 23-limit relative error than any smaller division, a lower 19-limit TE logflat badness than any smaller division, and a lower 23-limit logflat badness than any excepting 311, 581, 1578 and 2460. While 8539 has received most of the attention in this size range, 8269 is actually a bit better in the 23-limit and nearly as good in the 19-limit. They are rather like twins, including the fact both are primes. A step of 8269edo has also been similarly proposed as an interval size measure, the major tina.

Some of the simpler commas it tempers out include 123201/123200 in the 13-limit; 194481/194480, 336141/336140 in the 17-limit; 23409/23408, 28900/28899, 43681/43680, 89376/89375 in the 19-limit; and 21505/21504 among others in the 23-limit.

### Prime harmonics

Approximation of prime harmonics in 8269edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37
Error Absolute (¢) +0.0000 -0.0080 -0.0034 -0.0026 -0.0058 +0.0093 -0.0334 -0.0163 -0.0484 +0.0515 -0.0362 +0.0044
Relative (%) +0.0 -5.5 -2.3 -1.8 -4.0 +6.4 -23.0 -11.3 -33.4 +35.5 -24.9 +3.0
Steps
(reduced)
8269
(0)
13106
(4837)
19200
(2662)
23214
(6676)
28606
(3799)
30599
(5792)
33799
(723)
35126
(2050)
37405
(4329)
40171
(7095)
40966
(7890)
43077
(1732)

### Subsets and supersets

8269edo is the 1037th prime edo.