116edo: Difference between revisions

(3 intermediate revisions by 3 users not shown)

Line 1:

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.

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 the [[submajor (temperament)|submajor]] temperament in the 11- and 13-limit.

=== Prime harmonics ===

Line 10:

=== 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.

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

== Intervals ==

[[Category:Submajor (temperament)]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|116}}
+{{ED intro}}
 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.
+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 the [[submajor (temperament)|submajor]] temperament in the 11- and 13-limit.
 === Prime harmonics ===
@@ Line 10: / Line 10: @@
 === 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.
+Since 116 factors into {{factorisation|116}}, 116edo has subset edos {{EDOs| 2, 4, 29, and 58 }}. [[232edo]], which doubles it, is a notable tuning.
+== Intervals ==
+{{Interval table}}
 [[Category:Submajor (temperament)]]