8269edo: Difference between revisions
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8269edo is both a [[The Riemann zeta function and tuning #Zeta | == Theory == | ||
8269edo is both a [[The Riemann zeta function and tuning #Zeta edo lists|zeta peak and zeta integral edo]], which has to do with the fact that it is a very strong [[19-limit|19-]] and [[23-limit]] system. It has a lower 19-limit and a lower 23-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any smaller division, a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|TE logflat badness]] than any smaller division, and a lower 23-limit logflat badness than any excepting [[311edo|311]], [[581edo|581]], [[1578edo|1578]] and [[2460edo|2460]]. While [[8539edo|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 [[tempering out|tempers out]] include [[123201/123200]] in the 13-limit; [[194481/194480]], [[336141/336140]] in the 17-limit; 23409/23408, 27456/27455, 28900/28899, 43681/43680, 89376/89375 in the 19-limit; and 21505/21504 among others in the 23-limit. | Some of the simpler commas it [[tempering out|tempers out]] include [[123201/123200]] in the 13-limit; [[194481/194480]], [[336141/336140]] in the 17-limit; 23409/23408, 27456/27455, 28900/28899, 43681/43680, 89376/89375 in the 19-limit; and 21505/21504 among others in the 23-limit. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|8269|columns= | {{Harmonics in equal|8269|columns=9}} | ||
{{Harmonics in equal|8269|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 8269edo (continued)}} | |||
=== Subsets and supersets === | === Subsets and supersets === | ||
Latest revision as of 14:21, 11 March 2025
| ← 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 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, 27456/27455, 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 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0000 | -0.0080 | -0.0034 | -0.0026 | -0.0058 | +0.0093 | -0.0334 | -0.0163 | -0.0484 |
| Relative (%) | +0.0 | -5.5 | -2.3 | -1.8 | -4.0 | +6.4 | -23.0 | -11.3 | -33.4 | |
| Steps (reduced) |
8269 (0) |
13106 (4837) |
19200 (2662) |
23214 (6676) |
28606 (3799) |
30599 (5792) |
33799 (723) |
35126 (2050) |
37405 (4329) | |
| Harmonic | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 | 61 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0515 | -0.0362 | +0.0044 | +0.0584 | +0.0315 | +0.0152 | -0.0253 | +0.0616 | -0.0388 |
| Relative (%) | +35.5 | -24.9 | +3.0 | +40.2 | +21.7 | +10.5 | -17.4 | +42.5 | -26.7 | |
| Steps (reduced) |
40171 (7095) |
40966 (7890) |
43077 (1732) |
44302 (2957) |
44870 (3525) |
45931 (4586) |
47364 (6019) |
48644 (7299) |
49041 (7696) | |
Subsets and supersets
8269edo is the 1037th prime edo.