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.

[[~~Category:Equal divisions~~ of the ~~octave~~|~~####~~]]

== Theory ==

8269edo is a very strong [[19-limit|19-]] and [[23-limit]] system, with a lower 19-limit and a lower 23-limit [[Tenney–Euclidean temperament measures #TE simple badness|relative error]] than any smaller edo. It is both a [[the Riemann zeta function and tuning #Zeta edo lists|zeta peak and zeta integral edo]].

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 sum of 8269 and 8539 is [[16808edo|16808]], which does well in not just the 23-limit, but also the 31-limit. The difference between 8269 and 8539 is [[270edo|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 [[tempering out|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 ===

{{Harmonics in equal|8269|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 8269edo (continued)}}

=== Subsets and supersets ===

8269edo is the 1037th [[prime edo]].

== Music ==

; [[Francium]]

* "Details To Follow" from ''Naughty Girl Era'' (2024) – [https://open.spotify.com/track/12kSjGHgwE4J6AMA4Pxd5d Spotify] | [https://francium223.bandcamp.com/track/details-to-follow Bandcamp] | [https://www.youtube.com/watch?v=a9HqvhEXvBk YouTube] – totziensmic in 8269edo

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 {{Infobox ET}}
-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.
+{{ED intro}}
-[[Category:Equal divisions of the octave|####]] <!-- 4-digit number -->
+== Theory ==
+edo is a very strong [[19-limit|19-]] and [[23-limit]] system, with a lower 19-limit and a lower 23-limit [[Tenney–Euclidean temperament measures #TE simple badness|relative error]] than any smaller edo. It is both a [[the Riemann zeta function and tuning #Zeta edo lists|zeta peak and zeta integral edo]].
+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 sum of 8269 and 8539 is [[16808edo|16808]], which does well in not just the 23-limit, but also the 31-limit. The difference between 8269 and 8539 is [[270edo|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 [[tempering out|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 ===
+{{Harmonics in equal|8269|columns=11}}
+{{Harmonics in equal|8269|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 8269edo (continued)}}
+=== Subsets and supersets ===
+edo is the 1037th [[prime edo]].
+== Music ==
+; [[Francium]]
+* "Details To Follow" from ''Naughty Girl Era'' (2024) – [https://open.spotify.com/track/12kSjGHgwE4J6AMA4Pxd5d Spotify] | [https://francium223.bandcamp.com/track/details-to-follow Bandcamp] | [https://www.youtube.com/watch?v=a9HqvhEXvBk YouTube] – totziensmic in 8269edo