1578edo: Difference between revisions

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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]].

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.

It ~~is also the most~~ accurate ~~edo below 10000 to approximate~~ [[quarter-comma meantone]]~~, with its approximation of {{monzo| 0 0 1/4 }} with a relative error of only 0.06%~~.

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… .

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 {{ED intro}}
-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]].
+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.
-It is also the most accurate edo below 10000 to approximate [[quarter-comma meantone]], with its approximation of {{monzo| 0 0 1/4 }} with a relative error of only 0.06%.
 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… .