# 8539edo

 ← 8538edo 8539edo 8540edo →
Prime factorization 8539 (prime)
Step size 0.140532¢
Fifth 4995\8539 (701.956¢)
Semitones (A1:m2) 809:642 (113.7¢ : 90.22¢)
Consistency limit 27
Distinct consistency limit 27
Special properties

8539 equal divisions of the octave (abbreviated 8539edo or 8539ed2), also called 8539-tone equal temperament (8539tet) or 8539 equal temperament (8539et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 8539 equal parts of about 0.141 ¢ each. Each step represents a frequency ratio of 21/8539, or the 8539th root of 2.

While it may strike many people as too large to be practical, 8539edo has seen actual use as a bookkeeping device to keep track of higher-limit intervals which have been allowed to freely modulate, and has been proposed as a unit of interval measure, the tina. This is because it is a very strong higher-limit system, distinctly consistent through the 27-odd-limit. It is a strict zeta tuning, and is also the first non-trivial edo to be consistent in the 27-odd-prime-sum-limit. In the 13-limit, the only smaller systems with a lower logflat badness are 72, 270, 494, 5585 and 6079; in the 17-limit, that becomes 72, 494, 1506, 3395 and 7033. In the 19-limit, where it really shines, nothing beats it in terms of logflat badness until 20203.

Some of the simpler commas it tempers out include 123201/123200 in the 13-limit; 28561/28560, 31213/31212, 37180/37179 in the 17-limit; 27456/27455, 43681/43680, 89376/89375 in the 19-limit; 19551/19550, 21736/21735, 25025/25024, 43264/43263 among others in the 23-limit.

### Prime harmonics

Approximation of prime harmonics in 8539edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37
Error Absolute (¢) +0.0000 +0.0007 +0.0079 -0.0005 -0.0122 -0.0077 +0.0218 -0.0075 +0.0428 -0.0421 +0.0165 +0.0671
Relative (%) +0.0 +0.5 +5.6 -0.4 -8.7 -5.5 +15.5 -5.3 +30.4 -30.0 +11.8 +47.8
Steps
(reduced)
8539
(0)
13534
(4995)
19827
(2749)
23972
(6894)
29540
(3923)
31598
(5981)
34903
(747)
36273
(2117)
38627
(4471)
41482
(7326)
42304
(8148)
44484
(1789)

### Subsets and supersets

8539edo is the 1065th prime edo. On that basis, the tina as a unit of measure could be criticized; however, some people prefer primes for this sort of job, as they do not imply a preference for one smaller edo over another.