581edo: Difference between revisions

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== Theory ==

581edo is a very strong 19- and 23-limit system, distinctly [[consistent]] to the [[25-odd-limit]]. It tempers out [[2401/2400]] in the 7-limit, [[3025/3024]], [[19712/19683]], 151263/151250 in the 11-limit, and [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]] in the 13-limit. It [[support]]s and gives a good tuning for [[newt]], the 41 & 229 microtemperament, which has a neutral thirds generator. It is the first division after 270 with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]], and the first past 311 with a lower 23-limit relative error, and not until [[1578edo|1578]] do we reach a lower 23-limit relative error.

581edo is a very strong 19- and 23-limit system, distinctly [[consistent]] to the [[25-odd-limit]]. It tempers out [[2401/2400]] in the 7-limit, [[3025/3024]], [[19712/19683]], 151263/151250 in the 11-limit, and [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]] in the 13-limit. It [[support]]s and gives a good tuning for [[newt]], the 41 & 229 microtemperament, which has a neutral thirds generator.

=== Prime harmonics ===

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* 581et is the first equal temperament after [[270edo|270]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]], and the first after [[311edo|311]] with a lower 23-limit relative error. It is only bettered by [[742edo|742]] in terms of either 19-limit absolute error or 19-limit relative error, by [[718edo|718]] in terms of 23-limit absolute error, and not until [[1578edo|1578]] do we reach a lower 23-limit relative error.

* 581et is also notable in the 17-limit, where it has a lower absolute error than any previous equal temperaments, past [[494edo|494]] and followed by 742.

=== Rank-2 temperaments ===

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 == Theory ==
-edo is a very strong 19- and 23-limit system, distinctly [[consistent]] to the [[25-odd-limit]]. It tempers out [[2401/2400]] in the 7-limit, [[3025/3024]], [[19712/19683]], 151263/151250 in the 11-limit, and [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]] in the 13-limit. It [[support]]s and gives a good tuning for [[newt]], the 41 &amp; 229 microtemperament, which has a neutral thirds generator. It is the first division after 270 with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]], and the first past 311 with a lower 23-limit relative error, and not until [[1578edo|1578]] do we reach a lower 23-limit relative error.
+edo is a very strong 19- and 23-limit system, distinctly [[consistent]] to the [[25-odd-limit]]. It tempers out [[2401/2400]] in the 7-limit, [[3025/3024]], [[19712/19683]], 151263/151250 in the 11-limit, and [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]] in the 13-limit. It [[support]]s and gives a good tuning for [[newt]], the 41 &amp; 229 microtemperament, which has a neutral thirds generator.
 === Prime harmonics ===
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 |}
+* 581et is the first equal temperament after [[270edo|270]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]], and the first after [[311edo|311]] with a lower 23-limit relative error. It is only bettered by [[742edo|742]] in terms of either 19-limit absolute error or 19-limit relative error, by [[718edo|718]] in terms of 23-limit absolute error, and not until [[1578edo|1578]] do we reach a lower 23-limit relative error.
+* 581et is also notable in the 17-limit, where it has a lower absolute error than any previous equal temperaments, past [[494edo|494]] and followed by 742.
 === Rank-2 temperaments ===