1578edo: Difference between revisions

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The '''1578''' equal division divides the octave into 1578 equal parts of 0.7605 cents each. It is a very strong higher limit system, and is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak, integral and gap edo]]. It is distinctly consistent through the 29 limit, and is the first edo past 311 with a lower 29-limit [[Tenney-Euclidean_temperament_measures#TE simple badness|relative error]]. It is also the lowest past 311 in the 31 limit, the lowest past 581 in the 23 limit, and the lowest past 1178 in the 19 limit. It is also quite strong taken just as an 11-limit system; the only smaller edo with a lower 11-limit relative error is [[342edo|342]].

Some 31 limit or lower ~~superpaticular~~ commas it tempers out are 3249/3248, 3510/3509, 3876/3875, 3969/3968, 4186/4185, 4225/4224, 4641/4640, 4693/4692, 4761/4760, 4901/4900, 4914/4913, 4992/4991, 5083/5082, 5643/5642, 5776/5775, 5832/5831, 5888/5887, 5985/5984, 6175/6174, 6325/6324, 6480/6479, 6656/6655, 6728/6727, 7106/7105, 7425/7424, 7657/7656, 7866/7865, 7889/7888, 8092/8091, 8281/8280, 8464/8463, 8526/8525, 8625/8624, 8671/8670, 8960/8959, 9425/9424, 9801/9800, 9802/9801, 10241/10240, 10557/10556, 10626/10625, 10830/10829, 10881/10880, 11271/11270, 11340/11339, 11781/11780, 12006/12005, 12122/12121, 12168/12167, 12376/12375, 12636/12635, 12673/12672, 13225/13224, 13300/13299, 13311/13310, 13312/13311, 13377/13376, 14365/14364, 14400/14399, 15625/15624, 16929/16928, 19228/19227, 19251/19250, 19344/19343, 19551/19550, 19965/19964, 20736/20735, 21505/21504, 21736/21735, 23276/23275, 23375/23374, 23409/23408, 23716/23715, 23751/23750, 24795/24794, 25025/25024, 25840/25839, 25921/25920, 27000/~~26999..~~. .

1578edo is a very strong higher limit system, and is a [[zeta edo|zeta peak, peak integer, integral and gap edo]]. It is [[consistency|distinctly consistent]] through the [[29-odd-limit]], and is the first [[edo]] past [[311edo]] with a lower 29-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. It is also the lowest past 311edo in the [[31-limit]], the lowest past [[581edo]] in the [[23-limit]], and the lowest past [[1178edo]] in the [[19-limit]]. It is also quite strong taken just as an [[11-limit]] system; the only smaller edo with a lower 11-limit relative error is [[342edo]]. It additionally contains an extremely accurate approximation of [[quarter-comma meantone]] inherited from 789edo.

Some 31-limit or lower superparticular commas it tempers out are 3249/3248, 3510/3509, 3876/3875, 3969/3968, 4186/4185, 4225/4224, 4641/4640, 4693/4692, 4761/4760, 4901/4900, 4914/4913, 4992/4991, 5083/5082, 5643/5642, 5776/5775, 5832/5831, 5888/5887, 5985/5984, 6175/6174, 6325/6324, 6480/6479, 6656/6655, 6728/6727, 7106/7105, 7425/7424, 7657/7656, 7866/7865, 7889/7888, 8092/8091, 8281/8280, 8464/8463, 8526/8525, 8625/8624, 8671/8670, 8960/8959, 9425/9424, 9801/9800, 9802/9801, 10241/10240, 10557/10556, 10626/10625, 10830/10829, 10881/10880, 11271/11270, 11340/11339, 11781/11780, 12006/12005, 12122/12121, 12168/12167, 12376/12375, 12636/12635, 12673/12672, 13225/13224, 13300/13299, 13311/13310, 13312/13311, 13377/13376, 14365/14364, 14400/14399, 15625/15624, 16929/16928, 19228/19227, 19251/19250, 19344/19343, 19551/19550, 19965/19964, 20736/20735, 21505/21504, 21736/21735, 23276/23275, 23375/23374, 23409/23408, 23716/23715, 23751/23750, 24795/24794, 25025/25024, 25840/25839, 25921/25920, 27000/26999… .

=== Prime harmonics ===

=== Subsets and supersets ===

Since 1578 factors into {{factorization|1578}}, 1578edo has subset edos [[2edo|2]], [[3edo|3]], [[6edo|6]], [[263edo|263]], [[526edo|526]], and [[789edo|789]].

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-The '''1578''' equal division divides the octave into 1578 equal parts of 0.7605 cents each. It is a very strong higher limit system, and is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak, integral and gap edo]]. It is distinctly consistent through the 29 limit, and is the first edo past 311 with a lower 29-limit  [[Tenney-Euclidean_temperament_measures#TE simple badness|relative error]]. It is also the lowest past 311 in the 31 limit, the lowest past 581 in the 23 limit, and the lowest past 1178 in the 19 limit. It is also quite strong taken just as an 11-limit system; the only smaller edo with a lower 11-limit relative error is [[342edo|342]].
+{{Infobox ET}}
+{{ED intro}}
-Some 31 limit or lower superpaticular commas it tempers out are 3249/3248, 3510/3509, 3876/3875, 3969/3968, 4186/4185, 4225/4224, 4641/4640, 4693/4692, 4761/4760, 4901/4900, 4914/4913, 4992/4991, 5083/5082, 5643/5642, 5776/5775, 5832/5831, 5888/5887, 5985/5984, 6175/6174, 6325/6324, 6480/6479, 6656/6655, 6728/6727, 7106/7105, 7425/7424, 7657/7656, 7866/7865, 7889/7888, 8092/8091, 8281/8280, 8464/8463, 8526/8525, 8625/8624, 8671/8670, 8960/8959, 9425/9424, 9801/9800, 9802/9801, 10241/10240, 10557/10556, 10626/10625, 10830/10829, 10881/10880, 11271/11270, 11340/11339, 11781/11780, 12006/12005, 12122/12121, 12168/12167, 12376/12375, 12636/12635, 12673/12672, 13225/13224, 13300/13299, 13311/13310, 13312/13311, 13377/13376, 14365/14364, 14400/14399, 15625/15624, 16929/16928, 19228/19227, 19251/19250, 19344/19343, 19551/19550, 19965/19964, 20736/20735, 21505/21504, 21736/21735, 23276/23275, 23375/23374, 23409/23408, 23716/23715, 23751/23750, 24795/24794, 25025/25024, 25840/25839, 25921/25920, 27000/26999... .
+edo is a very strong higher limit system, and is a [[zeta edo|zeta peak, peak integer, integral and gap edo]]. It is [[consistency|distinctly consistent]] through the [[29-odd-limit]], and is the first [[edo]] past [[311edo]] with a lower 29-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. It is also the lowest past 311edo in the [[31-limit]], the lowest past [[581edo]] in the [[23-limit]], and the lowest past [[1178edo]] in the [[19-limit]]. It is also quite strong taken just as an [[11-limit]] system; the only smaller edo with a lower 11-limit relative error is [[342edo]]. It additionally contains an extremely accurate approximation of [[quarter-comma meantone]] inherited from 789edo.
+Some 31-limit or lower superparticular commas it tempers out are 3249/3248, 3510/3509, 3876/3875, 3969/3968, 4186/4185, 4225/4224, 4641/4640, 4693/4692, 4761/4760, 4901/4900, 4914/4913, 4992/4991, 5083/5082, 5643/5642, 5776/5775, 5832/5831, 5888/5887, 5985/5984, 6175/6174, 6325/6324, 6480/6479, 6656/6655, 6728/6727, 7106/7105, 7425/7424, 7657/7656, 7866/7865, 7889/7888, 8092/8091, 8281/8280, 8464/8463, 8526/8525, 8625/8624, 8671/8670, 8960/8959, 9425/9424, 9801/9800, 9802/9801, 10241/10240, 10557/10556, 10626/10625, 10830/10829, 10881/10880, 11271/11270, 11340/11339, 11781/11780, 12006/12005, 12122/12121, 12168/12167, 12376/12375, 12636/12635, 12673/12672, 13225/13224, 13300/13299, 13311/13310, 13312/13311, 13377/13376, 14365/14364, 14400/14399, 15625/15624, 16929/16928, 19228/19227, 19251/19250, 19344/19343, 19551/19550, 19965/19964, 20736/20735, 21505/21504, 21736/21735, 23276/23275, 23375/23374, 23409/23408, 23716/23715, 23751/23750, 24795/24794, 25025/25024, 25840/25839, 25921/25920, 27000/26999… .
+=== Prime harmonics ===
+{{Harmonics in equal|1578|columns=11}}
+=== Subsets and supersets ===
+Since 1578 factors into {{factorization|1578}}, 1578edo has subset edos [[2edo|2]], [[3edo|3]], [[6edo|6]], [[263edo|263]], [[526edo|526]], and [[789edo|789]].