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'''113edo''' is the [[EDO|equal division of the octave]] into 113 parts of 10.6195 [[cent]]s each. It [[tempers out]] 1600000/1594323 and 34171875/33554432 in the [[5-limit]]; 225/224, 1029/1024 and 1071875/1062882 in the [[7-limit]]; 243/242, 385/384, and 441/440 in the [[11-limit]]; 325/324, 364/363, 729/728, and 1625/1617 in the [[13-limit]]. It supports the 5-limit [[Amity family|amity temperament]], 7-limit [[amicable]] temperament, 7- and 11-limit [[Gamelismic clan|miracle temperament]], and 13-limit [[Gamelismic clan|manna temperament]].
{{Infobox ET}}
{{ED intro}}


113edo is the 30th [[prime EDO]].
== Theory ==
113edo is [[consistency|distinctly consistent]] in the [[13-odd-limit]] with a flat tendency. As an equal temperament, it [[tempering out|tempers out]] the [[amity comma]] and the [[ampersand comma]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and 1071875/1062882 in the [[7-limit]]; [[243/242]], [[385/384]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[325/324]], [[364/363]], [[729/728]], and 1625/1617 in the [[13-limit]]. It notably [[support]]s the 5-limit [[amity]] temperament, 7-limit [[amicable]] temperament, 7- and 11-limit [[miracle]] temperament, and 13-limit [[manna]] temperament.


[[Category:Equal divisions of the octave]]
113edo might be notable as a no-fives system, where it is consistent in the [[29-odd-limit]] and serves as a nearly optimal tuning for [[slendric]], in particular a 2.3.7.13.17.29 extension of slendric harmonies known as [[euslendric]].
[[Category:Prime EDO]]
 
[[Category:Theory]]
=== Prime harmonics ===
{{Harmonics in equal|113}}
 
=== Subsets and supersets ===
113edo is the 30th [[prime edo]], following [[109edo]] and before [[127edo]].
 
== Intervals ==
{{Interval table}}
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -179 113 }}
| {{mapping| 113 179 }}
| +0.338
| 0.338
| 3.18
|-
| 2.3.5
| 1600000/1594323, 34171875/33554432
| {{mapping| 113 179 262 }}
| +0.801
| 0.712
| 6.70
|-
| 2.3.5.7
| 225/224, 1029/1024, 1071875/1062882
| {{mapping| 113 179 262 317 }}
| +0.820
| 0.617
| 5.81
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 980000/970299
| {{mapping| 113 179 262 317 391 }}
| +0.604
| 0.700
| 6.59
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 325/324, 385/384, 1875/1859
| {{mapping| 113 179 262 317 391 418 }}
| +0.575
| 0.643
| 6.05
|}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br />per 8ve
! Generator*
! Cents*
! Associated<br />ratio*
! Temperaments
|-
| 1
| 4\113
| 42.48
| 40/39
| [[Humorous]]
|-
| 1
| 6\113
| 63.72
| 28/27
| [[Sycamore]] / [[betic]]
|-
| 1
| 8\113
| 84.96
| 21/20
| [[Amicable]] / [[pseudoamical]] / [[pseudoamorous]]
|-
| 1
| 11\113
| 116.81
| 15/14~16/15
| [[Miracle]] / [[manna]]
|-
| 1
| 13\113
| 138.05
| 27/25
| [[Quartemka]]
|-
| 1
| 22\113
| 233.63
| 8/7
| [[Slendric]]
|-
| 1
| 27\113
| 286.73
| 13/11
| [[Gamity]]
|-
| 1
| 29\113
| 307.96
| 3200/2673
| [[Familia]]
|-
| 1
| 32\113
| 339.82
| 243/200
| [[Houborizic]]
|-
| 1
| 34\113
| 360.06
| 16/13
| [[Phicordial]]
|-
| 1
| 37\113
| 392.92
| 2744/2187
| [[Emmthird]]
|-
| 1
| 47\113
| 499.12
| 4/3
| [[Gracecordial]]
|-
| 1
| 56\113
| 594.69
| 55/39
| [[Gaster temperament|Gaster]]
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
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