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.

{{EDO intro|8269}} 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 temperament measures #TE simple 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 ===

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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.
+{{EDO intro|8269}} 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 temperament measures #TE simple 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}}