414edo: Difference between revisions

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== Theory ==

414edo is closely related to [[207edo]], but the [[patent val]]s differ on the mapping for [[harmonic]] [[5/1|5]]~~. It is [[consistent]] to the [[17-odd-limit]] with a flat tendency for most of the harmonics, making for a good full 17-limit system~~. It [[tempers out]] {{monzo| -36 11 8 }} (submajor comma) and {{monzo| 1 -27 18 }} ([[ennealimma]]) in the 5-limit; [[2401/2400]], [[4375/4374]], and {{monzo| -37 4 12 1 }} in the 7-limit; [[3025/3024]], [[9801/9800]], [[41503/41472]], and 1265625/1261568 in the 11-limit; [[625/624]], [[729/728]], [[1575/1573]], [[2200/2197]], and 26411/26364 in the 13-limit; [[833/832]], [[1089/1088]], [[1225/1224]], [[1275/1274]], and [[1701/1700]] in the 17-limit. It [[support]]s the 11-limit [[hemiennealimmal]] and the 13-limit [[quatracot]].

414edo is [[consistent]] to the [[17-odd-limit]] with a flat tendency for most of the [[harmonic]]s, making for a good full [[17-limit]] system. It is closely related to [[207edo]], but the [[patent val]]s differ on the mapping for [[harmonic]] [[5/1|5]]. It [[tempering out|tempers out]] {{monzo| -36 11 8 }} (submajor comma) and {{monzo| 1 -27 18 }} ([[ennealimma]]) in the 5-limit; [[2401/2400]], [[4375/4374]], and {{monzo| -37 4 12 1 }} in the 7-limit; [[3025/3024]], [[9801/9800]], [[41503/41472]], and 1265625/1261568 in the 11-limit; [[625/624]], [[729/728]], [[1575/1573]], [[2200/2197]], and 26411/26364 in the 13-limit; [[833/832]], [[1089/1088]], [[1225/1224]], [[1275/1274]], and [[1701/1700]] in the 17-limit. It [[support]]s the 11-limit [[hemiennealimmal]] and the 13-limit [[quatracot]].

=== Prime harmonics ===

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=== Subsets and supersets ===

Since 414 factors into ~~{{factorization|414}}~~, 414edo has subset edos {{EDOs| 2, 3, 6, 9, 18, 23, 46, 69, 138, and 207 }}.

Since 414 factors into 2 × 3<sup>2</sup> × 23, 414edo has subset edos {{EDOs| 2, 3, 6, 9, 18, 23, 46, 69, 138, and 207 }}.

== Regular temperament properties ==

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 == Theory ==
-edo is closely related to [[207edo]], but the [[patent val]]s differ on the mapping for [[harmonic]] [[5/1|5]]. It is [[consistent]] to the [[17-odd-limit]] with a flat tendency for most of the harmonics, making for a good full 17-limit system. It [[tempers out]] {{monzo| -36 11 8 }} (submajor comma) and {{monzo| 1 -27 18 }} ([[ennealimma]]) in the 5-limit; [[2401/2400]], [[4375/4374]], and {{monzo| -37 4 12 1 }} in the 7-limit; [[3025/3024]], [[9801/9800]], [[41503/41472]], and 1265625/1261568 in the 11-limit; [[625/624]], [[729/728]], [[1575/1573]], [[2200/2197]], and 26411/26364 in the 13-limit; [[833/832]], [[1089/1088]], [[1225/1224]], [[1275/1274]], and [[1701/1700]] in the 17-limit. It [[support]]s the 11-limit [[hemiennealimmal]] and the 13-limit [[quatracot]].
+edo is [[consistent]] to the [[17-odd-limit]] with a flat tendency for most of the [[harmonic]]s, making for a good full [[17-limit]] system. It is closely related to [[207edo]], but the [[patent val]]s differ on the mapping for [[harmonic]] [[5/1|5]]. It [[tempering out|tempers out]] {{monzo| -36 11 8 }} (submajor comma) and {{monzo| 1 -27 18 }} ([[ennealimma]]) in the 5-limit; [[2401/2400]], [[4375/4374]], and {{monzo| -37 4 12 1 }} in the 7-limit; [[3025/3024]], [[9801/9800]], [[41503/41472]], and 1265625/1261568 in the 11-limit; [[625/624]], [[729/728]], [[1575/1573]], [[2200/2197]], and 26411/26364 in the 13-limit; [[833/832]], [[1089/1088]], [[1225/1224]], [[1275/1274]], and [[1701/1700]] in the 17-limit. It [[support]]s the 11-limit [[hemiennealimmal]] and the 13-limit [[quatracot]].
 === Prime harmonics ===
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 === Subsets and supersets ===
-Since 414 factors into {{factorization|414}}, 414edo has subset edos {{EDOs| 2, 3, 6, 9, 18, 23, 46, 69, 138, and 207 }}.
+Since 414 factors into 2 × 3<sup>2</sup> × 23, 414edo has subset edos {{EDOs| 2, 3, 6, 9, 18, 23, 46, 69, 138, and 207 }}.
 == Regular temperament properties ==