14348edo: Difference between revisions

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14348edo is a strong 17-limit system, with a lower 17-limit [[relative error]] than any smaller edo aside from [[7033edo|7033]]. It is also distinctly [[consistent]] in the 29-odd-limit, and has a lower 23-limit [[relative error]] than any lower equal temperaments aside from [[2460edo|2460]], [[8269edo|8269]], [[8539edo|8539]] and [[11664edo|11664]]. Besides all that, it is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak, integral and gap edo]], which has to do with its higher limit capability—it has lower relative errors than any smaller equal temperaments in the 41-limit and way beyond. The only inconsistent interval pair in the [[69-odd-limit]] is ([[31/29]], [[58/31]]) with 50.2% relative error~~. [[71083ed31]], a compressed version of 14348edo, is consistent to the 56-[[integer-limit]], though overall stretching the octave is better~~. An EDO of similar size ~~that is consistent this far without stretch~~ (~~to the~~ [[57-odd-limit]]) is [[20567edo]].

14348edo is a strong 17-limit system, with a lower 17-limit [[relative error]] than any smaller edo aside from [[7033edo|7033]]. It is also distinctly [[consistent]] in the 29-odd-limit, and has a lower 23-limit [[relative error]] than any lower equal temperaments aside from [[2460edo|2460]], [[8269edo|8269]], [[8539edo|8539]] and [[11664edo|11664]]. Besides all that, it is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak, integral and gap edo]], which has to do with its higher limit capability—it has lower relative errors than any smaller equal temperaments in the 41-limit and way beyond. The only inconsistent interval pair in the [[69-odd-limit]] is ([[31/29]], [[58/31]]) with 50.2% relative error. An EDO of similar size with full consistency to a very high limit ([[57-odd-limit]]) is [[20567edo]].

=== Prime harmonics ===

{{Harmonics in equal|14348~~|columns=11~~|start=12|collapsed=true|title=Approximation of prime harmonics in 14348edo (continued)}}

{{Harmonics in equal|14348|start=12|collapsed=true|title=Approximation of prime harmonics in 14348edo (continued)}}

=== Subsets and supersets ===

It factors as 2<sup>2</sup> × 17 × 211, so [[17edo|17]], [[34edo|34]], [[68edo|68]] and [[422edo|422]] are all divisors.

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 {{ED intro}}
-edo is a strong 17-limit system, with a lower 17-limit [[relative error]] than any smaller edo aside from [[7033edo|7033]]. It is also distinctly [[consistent]] in the 29-odd-limit, and has a lower 23-limit [[relative error]] than any lower equal temperaments aside from [[2460edo|2460]], [[8269edo|8269]], [[8539edo|8539]] and [[11664edo|11664]]. Besides all that, it is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak, integral and gap edo]], which has to do with its higher limit capability—it has lower relative errors than any smaller equal temperaments in the 41-limit and way beyond. The only inconsistent interval pair in the [[69-odd-limit]] is ([[31/29]], [[58/31]]) with 50.2% relative error. [[71083ed31]], a compressed version of 14348edo, is consistent to the 56-[[integer-limit]], though overall stretching the octave is better. An EDO of similar size that is consistent this far without stretch (to the [[57-odd-limit]]) is [[20567edo]].
+edo is a strong 17-limit system, with a lower 17-limit [[relative error]] than any smaller edo aside from [[7033edo|7033]]. It is also distinctly [[consistent]] in the 29-odd-limit, and has a lower 23-limit [[relative error]] than any lower equal temperaments aside from [[2460edo|2460]], [[8269edo|8269]], [[8539edo|8539]] and [[11664edo|11664]]. Besides all that, it is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak, integral and gap edo]], which has to do with its higher limit capability—it has lower relative errors than any smaller equal temperaments in the 41-limit and way beyond. The only inconsistent interval pair in the [[69-odd-limit]] is ([[31/29]], [[58/31]]) with 50.2% relative error. An EDO of similar size with full consistency to a very high limit ([[57-odd-limit]]) is [[20567edo]].
 === Prime harmonics ===
-{{Harmonics in equal|14348|columns=11}}
+{{Harmonics in equal|14348}}
-{{Harmonics in equal|14348|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 14348edo (continued)}}
+{{Harmonics in equal|14348|start=12|collapsed=true|title=Approximation of prime harmonics in 14348edo (continued)}}
 === Subsets and supersets ===
 It factors as 2<sup>2</sup> × 17 × 211, so [[17edo|17]], [[34edo|34]], [[68edo|68]] and [[422edo|422]] are all divisors.