718edo: Difference between revisions

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== Theory ==

718edo is distinctly [[consistent]] in the [[23-odd-limit]]. It is closely related to [[359edo]], but the mapping differs for [[5/1|5]], [[13/1|13]], [[17/1|17]] and [[31/1|31]].

718edo is distinctly [[consistent]] in the [[23-odd-limit]], and does well enough in the 31-limit. It is closely related to [[359edo]], but the mapping differs for [[5/1|5]], [[13/1|13]], [[17/1|17]] and [[31/1|31]].

As does 359et, 718et tempers out the 359-comma in the 3-limit, rendering a very accurate perfect fifth. In the 5-limit it tempers out the gammic comma, {{monzo| -29 -11 20 }}, and the [[monzisma]], {{monzo| 54 -37 2 }}. In the 7-limit it tempers out [[4375/4374]]; in the 11-limit [[3025/3024]], [[9801/9800]] and [[131072/130977]]; in the 13-limit [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]]; in the 17-limit [[1275/1274]], [[2025/2023]]; in the 19-limit [[2432/2431]], 4200/4199 and 5985/5984; and in the 23-limit 2024/2023, 2025/2024, 2185/2184, 3060/3059. It supports [[gammic]], [[monzismic]] and [[abigail]].

As does 359et, 718et tempers out the 359-comma in the 3-limit, rendering a very accurate perfect fifth. In the 5-limit it tempers out the gammic comma, {{monzo| -29 -11 20 }}, and the [[monzisma]], {{monzo| 54 -37 2 }}. In the 7-limit it tempers out [[4375/4374]]; in the 11-limit [[3025/3024]], [[9801/9800]] and [[131072/130977]]; in the 13-limit [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]]; in the 17-limit [[1275/1274]], [[2025/2023]]; in the 19-limit [[2432/2431]], 3250/3249, 4200/4199 and 5985/5984; and in the 23-limit 2024/2023, 2025/2024, 2185/2184, 3060/3059. It supports [[gammic]], [[monzismic]] and [[abigail]].

=== Prime harmonics ===

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* 718et has a lower [[relative error]] in the 23-limit than any previous equal temperaments, past [[581edo|581]] and followed by [[742edo|742i]].

* 718et has a lower [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] in the 23-limit than any previous equal temperaments, past [[581edo|581]] and followed by [[742edo|742i]].

=== Rank-2 temperaments ===

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 == Theory ==
-edo is distinctly [[consistent]] in the [[23-odd-limit]]. It is closely related to [[359edo]], but the mapping differs for [[5/1|5]], [[13/1|13]], [[17/1|17]] and [[31/1|31]].
+edo is distinctly [[consistent]] in the [[23-odd-limit]], and does well enough in the 31-limit. It is closely related to [[359edo]], but the mapping differs for [[5/1|5]], [[13/1|13]], [[17/1|17]] and [[31/1|31]].
-As does 359et, 718et tempers out the 359-comma in the 3-limit, rendering a very accurate perfect fifth. In the 5-limit it tempers out the gammic comma, {{monzo| -29 -11 20 }}, and the [[monzisma]], {{monzo| 54 -37 2 }}. In the 7-limit it tempers out [[4375/4374]]; in the 11-limit [[3025/3024]], [[9801/9800]] and [[131072/130977]]; in the 13-limit [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]]; in the 17-limit [[1275/1274]], [[2025/2023]]; in the 19-limit [[2432/2431]], 4200/4199 and 5985/5984; and in the 23-limit 2024/2023, 2025/2024, 2185/2184, 3060/3059. It supports [[gammic]], [[monzismic]] and [[abigail]].
+As does 359et, 718et tempers out the 359-comma in the 3-limit, rendering a very accurate perfect fifth. In the 5-limit it tempers out the gammic comma, {{monzo| -29 -11 20 }}, and the [[monzisma]], {{monzo| 54 -37 2 }}. In the 7-limit it tempers out [[4375/4374]]; in the 11-limit [[3025/3024]], [[9801/9800]] and [[131072/130977]]; in the 13-limit [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]]; in the 17-limit [[1275/1274]], [[2025/2023]]; in the 19-limit [[2432/2431]], 3250/3249, 4200/4199 and 5985/5984; and in the 23-limit 2024/2023, 2025/2024, 2185/2184, 3060/3059. It supports [[gammic]], [[monzismic]] and [[abigail]].
 === Prime harmonics ===
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 |}
-* 718et has a lower [[relative error]] in the 23-limit than any previous equal temperaments, past [[581edo|581]] and followed by [[742edo|742i]].
+* 718et has a lower [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] in the 23-limit than any previous equal temperaments, past [[581edo|581]] and followed by [[742edo|742i]].
 === Rank-2 temperaments ===