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{{Infobox ET}}
{{Infobox ET}}
The '''113 equal divisions of the octave''' ('''113edo'''), or the '''113(-tone) equal temperament''' ('''113tet''', '''113et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 113 parts of about 10.6 [[cent]]s each.
{{ED intro}}


== Theory ==
== Theory ==
113edo is distinctly [[consistent]] in the [[13-odd-limit]] with a flat tendency. As a temperament, it [[tempers out]] the [[amity comma]] and the [[ampersand]] 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.
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.


113edo is the 30th [[prime EDO]].
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.


=== Prime harmonics ===
=== Prime harmonics ===
{{Primes in edo|113}}
{{Harmonics in equal|113}}
 
=== Subsets and supersets ===
113edo is the 30th [[prime edo]], following [[109edo]] and before [[127edo]].
 
== Intervals ==
{{Interval table}}


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
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| 2.3
| 2.3
| {{monzo| -179 113 }}
| {{monzo| -179 113 }}
| [{{val| 113 179 }}]
| {{mapping| 113 179 }}
| +0.338
| +0.338
| 0.338
| 0.338
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| 2.3.5
| 2.3.5
| 1600000/1594323, 34171875/33554432
| 1600000/1594323, 34171875/33554432
| [{{val| 113 179 262 }}]
| {{mapping| 113 179 262 }}
| +0.801
| +0.801
| 0.712
| 0.712
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| 2.3.5.7
| 2.3.5.7
| 225/224, 1029/1024, 1071875/1062882
| 225/224, 1029/1024, 1071875/1062882
| [{{val| 113 179 262 317 }}]
| {{mapping| 113 179 262 317 }}
| +0.820
| +0.820
| 0.617
| 0.617
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| 2.3.5.7.11
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 980000/970299
| 225/224, 243/242, 385/384, 980000/970299
| [{{val| 113 179 262 317 391 }}]
| {{mapping| 113 179 262 317 391 }}
| +0.604
| +0.604
| 0.700
| 0.700
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 225/224, 243/242, 325/324, 385/384, 1875/1859
| 225/224, 243/242, 325/324, 385/384, 1875/1859
| [{{val| 113 179 262 317 391 418 }}]
| {{mapping| 113 179 262 317 391 418 }}
| +0.575
| +0.575
| 0.643
| 0.643
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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per octave
|-
! Generator<br>(reduced)
! Periods<br />per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
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| 27/25
| 27/25
| [[Quartemka]]
| [[Quartemka]]
|-
| 1
| 20\113
| 212.39
| 26/23
| [[Shoal]]
|-
|-
| 1
| 1
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| 233.63
| 233.63
| 8/7
| 8/7
| [[Slendric]]
| [[Slendric]] / [[euslendric]]
|-
|-
| 1
| 1
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| 339.82
| 339.82
| 243/200
| 243/200
| [[Amity]] / [[houborizic]]
| [[Houborizic]]
|-
|-
| 1
| 1
| 34\113
| 34\113
| 360.06
| 361.06
| 16/13
| 16/13
| [[Phicordial]]
| [[Phicordial]]
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| [[Gaster temperament|Gaster]]
| [[Gaster temperament|Gaster]]
|}
|}
 
<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
== Scales ==
Since 113edo has a step of 10.6195 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}. It is the first edo which allows one to use an MOS scale of 90 tones or more as a circulating temperament.
 
{| class="wikitable"
|+Circulating temperaments in 113edo
!Tones
!Pattern
!L:s
|-
|5
|[[3L 2s]]
|23:22
|-
|6
|[[5L 1s]]
|19:18
|-
|7
|[[1L 6s]]
|17:16
|-
|8
|[[1L 7s]]
|15:14
|-
|9
|[[5L 4s]]
|13:12
|-
|10
|[[3L 7s]]
|12:11
|-
|11
|[[3L 8s]]
|11:10
|-
|12
|[[5L 7s]]
|10:9
|-
|13
|[[9L 4s]]
| rowspan="2" |9:8
|-
|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
|37L 39s
|-
|77
|36L 41s
|-
|78
|35L 43s
|-
|79
|34L 45s
|-
|80
|33L 47s
|-
|81
|32L 49s
|-
|82
|31L 51s
|-
|83
|30L 53s
|-
|84
|29L 55s
|-
|85
|28L 57s
|-
|86
|27L 59s
|-
|87
|26L 61s
|-
|88
|25L 63s
|-
|89
|24L 65s
|-
|90
|23L 67s
|}
 
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Prime EDO]]
Retrieved from "https://en.xen.wiki/w/113edo"