116edo: Difference between revisions

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116edo ~~tempers out 20000/19683 (tetracot comma) and 2197265625/2147483648 (wizard comma) in~~ the 5-limit. ~~Using~~ the ~~patent~~ val~~, it tempers~~ out ~~225/224, 15625/15309, and 51200/50421~~ in the 7-~~limit; 385/384~~, ~~540/539, 4000/3993, and 6655/6561 in the~~ 11-~~limit; 169/168, 275/273, 352/351~~, and ~~640/637 in the~~ 13-limit. ~~116edo provides~~ the ~~optimal~~ patent val ~~for~~ [[~~Marvel temperaments~~|~~submajor temperament~~]].

116edo is only [[consistent]] to the [[5-odd-limit]], and is not quite accurate for its size. It can be viewed as splitting [[58edo]]'s step in two, and the [[enfactoring|enfactored]] 116cef [[val]] comes out on top accuracy in the 7-, 11-, and 13-limit. In the 5-limit, however, the [[patent val]] {{val| 116 184 '''269''' }} beats the enfactored 116c val {{val| 116 184 '''270''' }} by a thin margin, and it [[Tempering out|tempers out]] 20000/19683 ([[tetracot comma]]) and 2197265625/2147483648 (wizard comma).

In the 7-, 11- and 13-limit, the patent val {{val| 116 184 '''269''' 326 '''401''' '''429''' }} comes in second best after the enfactored 116cef val {{val| 116 184 '''270''' 326 '''402''' '''430''' }} , and it tempers out [[225/224]], 15625/15309, and 51200/50421 in the 7-limit; [[385/384]], [[540/539]], [[4000/3993]], and 6655/6561 in the 11-limit; [[169/168]], [[275/273]], [[352/351]], and [[640/637]] in the 13-limit. 116edo provides the [[optimal patent val]] for [[submajor]] temperament in the 11- and 13-limit.

=== Prime harmonics ===

[[Category:~~Equal divisions of the octave|###~~]] ~~~~

=== Subsets and supersets ===

Since 116 factors into 2<sup>2</sup> × 29, 116edo has subset edos {{EDOs| 2, 4, 29, and 58 }}. [[232edo]], which doubles it, is a notable tuning.

[[Category:Submajor (temperament)]]

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 {{EDO intro|116}}
-edo tempers out 20000/19683 (tetracot comma) and 2197265625/2147483648 (wizard comma) in the 5-limit. Using the patent val, it tempers out 225/224, 15625/15309, and 51200/50421 in the 7-limit; 385/384, 540/539, 4000/3993, and 6655/6561 in the 11-limit; 169/168, 275/273, 352/351, and 640/637 in the 13-limit. 116edo provides the optimal patent val for [[Marvel temperaments|submajor temperament]].
+edo is only [[consistent]] to the [[5-odd-limit]], and is not quite accurate for its size. It can be viewed as splitting [[58edo]]'s step in two, and the [[enfactoring|enfactored]] 116cef [[val]] comes out on top accuracy in the 7-, 11-, and 13-limit. In the 5-limit, however, the [[patent val]] {{val| 116 184 '''269''' }} beats the enfactored 116c val {{val| 116 184 '''270''' }} by a thin margin, and it [[Tempering out|tempers out]] 20000/19683 ([[tetracot comma]]) and 2197265625/2147483648 (wizard comma).
+In the 7-, 11- and 13-limit, the patent val {{val| 116 184 '''269''' 326 '''401''' '''429''' }} comes in second best after the enfactored 116cef val {{val| 116 184 '''270''' 326 '''402''' '''430''' }} , and it tempers out [[225/224]], 15625/15309, and 51200/50421 in the 7-limit; [[385/384]], [[540/539]], [[4000/3993]], and 6655/6561 in the 11-limit; [[169/168]], [[275/273]], [[352/351]], and [[640/637]] in the 13-limit. 116edo provides the [[optimal patent val]] for [[submajor]] temperament in the 11- and 13-limit.
+=== Prime harmonics ===
 {{Harmonics in equal|116}}
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
+=== Subsets and supersets ===
+Since 116 factors into 2<sup>2</sup> × 29, 116edo has subset edos {{EDOs| 2, 4, 29, and 58 }}. [[232edo]], which doubles it, is a notable tuning.
+[[Category:Submajor (temperament)]]