8269edo: Difference between revisions

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== 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]].

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 [[Riemann zeta function #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'''.

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 == 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]].
+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 [[Riemann zeta function #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'''.