14348edo: Difference between revisions

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The 14348 division divides the octave into 14348 equal parts of 0.083635 cents each. It 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 limit, and has a lower 23-limit relative error than any lower division 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 edo]]. It factors as 2^2 * 17 * 211, so [[17edo|17]], [[34edo|34]], [[68edo|68]] and [[422edo|422]] are all divisors.

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

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. Thus the full [[67-limit]] interpretation using the patent val is obvious, though it may be preferable to omit either prime [[29/1|29]] or [[31/1|31]].

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)}}

=== Subsets and supersets ===

14348 factors into primes as 2<sup>2</sup> × 17 × 211, so [[17edo|17]], [[34edo|34]], [[68edo|68]] and [[422edo|422]] are all subsets.

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-The 14348 division divides the octave into 14348 equal parts of 0.083635 cents each. It 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 limit, and has a lower 23-limit relative error than any lower division 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 edo]]. It factors as 2^2 * 17 * 211, so [[17edo|17]], [[34edo|34]], [[68edo|68]] and [[422edo|422]] are all divisors.
+{{Infobox ET}}
+{{ED intro}}
-[[Category:Equal divisions of the octave|#####]] <!-- 5-digit number -->
+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. Thus the full [[67-limit]] interpretation using the patent val is obvious, though it may be preferable to omit either prime [[29/1|29]] or [[31/1|31]].
+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|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 14348edo (continued)}}
+=== Subsets and supersets ===
+factors into primes as 2<sup>2</sup> × 17 × 211, so [[17edo|17]], [[34edo|34]], [[68edo|68]] and [[422edo|422]] are all subsets.