8269edo

From Xenharmonic Wiki
Jump to navigation Jump to search
← 8268edo8269edo8270edo →
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 division. 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.

Prime harmonics

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

Subsets and supersets

8269edo is the 1037th prime edo.