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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 (and in the [[15-odd-limit]], only [[15/11]] and its complement are inconsistent). 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.


Since 113edo has a step of 10.6195 cents, it also allows one to use its MOS scales as circulating temperaments. It is the first edo which allows one to use an MOS scale of 90 tones or more as a circulating temperament.
113edo is also notable as a [[31-limit]] system, especially if error on prime 5 is tolerable. In fact, it is consistent in the no-15 no-25 [[29-odd-limit]], which nearly extends to the [[33-odd-limit]], with only two inconsistent interval pairs that both involve 31, being [[31/21]] (50.8% off) and [[31/20]] (55.4% off) and their complements – and serves as a nearly optimal tuning for [[slendric]], in particular a 2.3.7.13.17(.19.23).29 extension of slendric harmonies known as [[euslendric]]. Notably as a slendric system, it is the largest EDO that maps [[64/49]] and [[21/16]] to the same interval consistently.
{| class="wikitable"
 
|+Circulating temperaments in 113edo
=== Prime harmonics ===
!Tones
{{Harmonics in equal|113}}
!Pattern
 
!L:s
=== 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"
|-
|-
|5
! rowspan="2" | [[Subgroup]]
|[[3L 2s]]
! rowspan="2" | [[Comma list]]
|23:22
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
|6
! [[TE error|Absolute]] (¢)
|[[5L 1s]]
! [[TE simple badness|Relative]] (%)
|19:18
|-
|-
|7
| 2.3
|[[1L 6s]]
| {{monzo| -179 113 }}
|17:16
| {{mapping| 113 179 }}
| +0.338
| 0.338
| 3.18
|-
|-
|8
| 2.3.5
|[[1L 7s]]
| 1600000/1594323, 34171875/33554432
|15:14
| {{mapping| 113 179 262 }}
| +0.801
| 0.712
| 6.70
|-
|-
|9
| 2.3.5.7
|[[5L 4s]]
| 225/224, 1029/1024, 1071875/1062882
|13:12
| {{mapping| 113 179 262 317 }}
| +0.820
| 0.617
| 5.81
|-
|-
|10
| 2.3.5.7.11
|[[3L 7s]]
| 225/224, 243/242, 385/384, 980000/970299
|12:11
| {{mapping| 113 179 262 317 391 }}
| +0.604
| 0.700
| 6.59
|-
|-
|11
| 2.3.5.7.11.13
|[[3L 8s]]
| 225/224, 243/242, 325/324, 385/384, 1875/1859
|11:10
| {{mapping| 113 179 262 317 391 418 }}
|-
| +0.575
|12
| 0.643
|[[5L 7s]]
| 6.05
|10:9
|}
|-
 
|13
=== Rank-2 temperaments ===
|[[9L 4s]]
{| class="wikitable center-all left-5"
| rowspan="2" |9:8
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|14
|[[1L 13s]]
|-
|15
|[[7L 8s]]
| rowspan="2" |8:7
|-
|16
|1L 15s
|-
|17
|[[11L 6s]]
| rowspan="2" |7:6
|-
|18
|5L 13s
|-
|19
|18L 1s
| rowspan="4" |6:5
|-
|20
|[[13L 7s]]
|-
|21
|[[8L 13s]]
|-
|22
|[[3L 19s]]
|-
|23
|21L 2s
| rowspan="6" |5:4
|-
|24
|[[17L 7s]]
|-
|25
|13L 12s
|-
|26
|9L 17s
|-
|27
|[[5L 22s]]
|-
|28
|1L 27s
|-
|29
|26L 3s
| rowspan="9" |4:3
|-
|30
|23L 7s
|-
|31
|20L 11s
|-
|32
|17L 15s
|-
|33
|14L 19s
|-
|34
|11L 23s
|-
|35
|8L 27s
|-
|36
|5L 31s
|-
|37
|2L 35s
|-
|38
|37L 1s
| rowspan="19" |3:2
|-
|39
|35L 4s
|-
|40
|33L 7s
|-
|41
|31L 10s
|-
|42
|29L 13s
|-
|43
|27L 16s
|-
|44
|25L 19s
|-
|45
|23L 22s
|-
|46
|21L 25s
|-
|47
|19L 28s
|-
|48
|17L 31s
|-
|49
|15L 34s
|-
|50
|13L 37s
|-
|51
|11L 40s
|-
|52
|9L  43s
|-
|53
|7L 46s
|-
|54
|5L 49s
|-
|55
|3L 52s
|-
|56
|1L 55s
|-
|57
|56L 1s
| rowspan="34" |2:1
|-
|58
|55L 3s
|-
|59
|54L 5s
|-
|60
|53L 7s
|-
|61
|52L 9s
|-
|62
|51L 11s
|-
|63
|50L 13s
|-
|64
|49L 15s
|-
|65
|48L 17s
|-
|66
|47L 19s
|-
|67
|46L 21s
|-
|68
|45L 23s
|-
|69
|44L 25s
|-
|70
|43L 27s
|-
|71
|42L 29s
|-
|72
|41L 31s
|-
|73
|40L 33s
|-
|74
|39L 35s
|-
|75
|38L 37s
|-
|-
|76
! Periods<br />per 8ve
|37L 39s
! Generator*
! Cents*
! Associated<br />ratio*
! Temperaments
|-
|-
|77
| 1
|36L 41s
| 4\113
| 42.48
| 40/39
| [[Humorous]]
|-
|-
|78
| 1
|35L 43s
| 6\113
| 63.72
| 28/27
| [[Sycamore]] / [[betic]]
|-
|-
|79
| 1
|34L 45s
| 8\113
| 84.96
| 21/20
| [[Amicable]] / [[pseudoamical]] / [[pseudoamorous]]
|-
|-
|80
| 1
|33L 47s
| 11\113
| 116.81
| 15/14~16/15
| [[Miracle]] / [[manna]]
|-
|-
|81
| 1
|32L 49s
| 13\113
| 138.05
| 27/25
| [[Quartemka]]
|-
|-
|82
| 1
|31L 51s
| 20\113
| 212.39
| 26/23
| [[Shoal]]
|-
|-
|83
| 1
|30L 53s
| 22\113
| 233.63
| 8/7
| [[Slendric]] / [[euslendric]]
|-
|-
|84
| 1
|29L 55s
| 27\113
| 286.73
| 13/11
| [[Gamity]]
|-
|-
|85
| 1
|28L 57s
| 29\113
| 307.96
| 3200/2673
| [[Familia]]
|-
|-
|86
| 1
|27L 59s
| 32\113
| 339.82
| 243/200
| [[Houborizic]]
|-
|-
|87
| 1
|26L 61s
| 34\113
| 361.06
| 16/13
| [[Phicordial]]
|-
|-
|88
| 1
|25L 63s
| 37\113
| 392.92
| 2744/2187
| [[Emmthird]]
|-
|-
|89
| 1
|24L 65s
| 47\113
| 499.12
| 4/3
| [[Gracecordial]]
|-
|-
|90
| 1
|23L 67s
| 56\113
| 594.69
| 55/39
| [[Gaster temperament|Gaster]]
|}
|}
[[Category:Equal divisions of the octave]]
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
[[Category:Prime EDO]]
[[Category:Theory]]
Retrieved from "https://en.xen.wiki/w/113edo"