8269edo: Difference between revisions
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The '''8269 division''' divides the octave into 8269 equal parts of 0.14512 cents each. It 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- and 23-limit division. 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_metrics#Logflat TE badness| TE loglfat badness]] than any smaller division, and a lower 23-limit logflat badness than any excepting 311, 581, 1578 and 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. | The '''8269 division''' divides the octave into 8269 equal parts of 0.14512 cents each. It 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- and 23-limit division. 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_metrics#Logflat TE badness| TE loglfat badness]] than any smaller division, and a lower 23-limit logflat badness than any excepting 311, 581, 1578 and 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. | ||
=== Prime harmonics === | |||
{{Harmonics in equal|8269}} | |||
Revision as of 11:25, 10 February 2023
| ← 8268edo | 8269edo | 8270edo → |
The 8269 division divides the octave into 8269 equal parts of 0.14512 cents each. It 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 loglfat 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.
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) | |