8269edo
| ← 8268edo | 8269edo | 8270edo → |
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.
Theory
8269edo is a very strong 19- and 23-limit system, with a lower 19-limit and a lower 23-limit relative error than any smaller edo. It is both a zeta peak and zeta integral edo.
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. The sum of 8269 and 8539 is 16808, which does well in not just the 23-limit, but also the 31-limit. The difference between 8269 and 8539 is 270, which does well in the 13-limit, and also beyond. 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
| 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) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0044 | +0.0584 | +0.0315 | +0.0152 | -0.0253 | +0.0616 | -0.0388 | -0.0713 | -0.0376 | +0.0494 | +0.0054 |
| Relative (%) | +3.0 | +40.2 | +21.7 | +10.5 | -17.4 | +42.5 | -26.7 | -49.2 | -25.9 | +34.1 | +3.7 | |
| Steps (reduced) |
43077 (1732) |
44302 (2957) |
44870 (3525) |
45931 (4586) |
47364 (6019) |
48644 (7299) |
49041 (7696) |
50160 (546) |
50852 (1238) |
51184 (1570) |
52126 (2512) | |
Subsets and supersets
8269edo is the 1037th prime edo.