# 8539edo

← 8538edo | 8539edo | 8540edo → |

**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 2^{1/8539}, 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

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.