86edo: Difference between revisions

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'''86edo''' is the [[EDO|equal division of the octave]] into 86 parts of 13.9535 cents each. 86 = 2 * 43, and the patent val is a contorted [[43edo|43]] in the 5-limit. In the 7-limit the patent val tempers out 6144/6125, so that it supports mohajira temperament. In the 11-limit it tempers out 245/242, 540/539 and 4000/3993, and in the 13-limit 144/143, 196/195 and 676/675. It provides the optimal patent val for the 13-limit 9&86 temperament tempering out 144/143, 196/195, 245/242 and 676/675.

~~86edo~~ is ~~closely related to~~ the [[~~delta scale~~]], ~~which is~~ the ~~equal division of~~ the [[16/~~15|classic diatonic semitone~~]] ~~into eight parts of~~ 13.~~9664 cents each~~.

== Theory ==

Since {{nowrap|86 {{=}} 2 × 43}}, and the [[patent val]] is a [[contorted]] [[43edo]] in the 5-limit. In the 7-limit the [[patent val]] [[tempering out|tempers out]] 6144/6125, so that it [[support]]s the [[mohajira]] temperament. In the 11-limit it tempers out [[245/242]], [[540/539]] and [[4000/3993]], and in the 13-limit [[144/143]], [[196/195]] and [[676/675]]. It provides the optimal patent val for the 13-limit 9 & 86 temperament tempering out 144/143, 196/195, 245/242 and 676/675.

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

It is perhaps more interesting to consider the alternative 86e val, which tempers out [[121/120]] and [[243/242]] and [[support]]s 11-limit mohajira. The 86de val is a less good entry for 11-limit [[migration]]. In any case, this tuning is between [[31edo]] and [[55edo]], and replaces 43edo's lopsided placement of [[11/9]] and [[27/22]] with a true neutral third.

86edo is closely related to the [[8ed16/15|delta scale]], which is the equal division of the [[16/15|classic diatonic semitone]] into eight parts of 13.9664{{c}} each.

=== Odd harmonics ===

=== Subsets and supersets ===

86edo contains [[2edo]] and [[43edo]] as subsets. [[258edo]], which triples it, is a notable tuning.

== Interval table ==

== Instruments ==

A [[Lumatone mapping for 86edo]] is available.

== Music ==

; [[Bryan Deister]]

* [https://www.youtube.com/shorts/5_W323Iea18 ''microtonal improvisation in 86edo''] (2025)

== See also ==

* [[343edo#343ed16|343ed16]] (octave-stretched version of 86edo)

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-'''86edo''' is the [[EDO|equal division of the octave]] into 86 parts of 13.9535 cents each. 86 = 2 * 43, and the patent val is a contorted [[43edo|43]] in the 5-limit. In the 7-limit the patent val tempers out 6144/6125, so that it supports mohajira temperament. In the 11-limit it tempers out 245/242, 540/539 and 4000/3993, and in the 13-limit 144/143, 196/195 and 676/675. It provides the optimal patent val for the 13-limit 9&amp;86 temperament tempering out 144/143, 196/195, 245/242 and 676/675.
+{{Infobox ET}}
+{{ED intro}}
-edo is closely related to the [[delta scale]], which is the equal division of the [[16/15|classic diatonic semitone]] into eight parts of 13.9664 cents each.
+== Theory ==
+Since {{nowrap|86 {{=}} 2 × 43}}, and the [[patent val]] is a [[contorted]] [[43edo]] in the 5-limit. In the 7-limit the [[patent val]] [[tempering out|tempers out]] 6144/6125, so that it [[support]]s the [[mohajira]] temperament. In the 11-limit it tempers out [[245/242]], [[540/539]] and [[4000/3993]], and in the 13-limit [[144/143]], [[196/195]] and [[676/675]]. It provides the optimal patent val for the 13-limit 9 &amp; 86 temperament tempering out 144/143, 196/195, 245/242 and 676/675.
-[[Category:Equal divisions of the octave]]
+It is perhaps more interesting to consider the alternative 86e val, which tempers out [[121/120]] and [[243/242]] and [[support]]s 11-limit mohajira. The 86de val is a less good entry for 11-limit [[migration]]. In any case, this tuning is between [[31edo]] and [[55edo]], and replaces 43edo's lopsided placement of [[11/9]] and [[27/22]] with a true neutral third.
+edo is closely related to the [[8ed16/15|delta scale]], which is the equal division of the [[16/15|classic diatonic semitone]] into eight parts of 13.9664{{c}} each.
+=== Odd harmonics ===
+{{Harmonics in equal|86}}
+=== Subsets and supersets ===
+edo contains [[2edo]] and [[43edo]] as subsets. [[258edo]], which triples it, is a notable tuning.
+== Interval table ==
+{{Interval table}}
+== Instruments ==
+A [[Lumatone mapping for 86edo]] is available.
+== Music ==
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/5_W323Iea18 ''microtonal improvisation in 86edo''] (2025)
+== See also ==
+* [[343edo#343ed16|343ed16]] (octave-stretched version of 86edo)