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{{Infobox ET}}
{{Infobox ET}}
{{ED intro}}


'''11EDF''' is the [[EDF|equal division of the just perfect fifth]] into 11 parts of 63.8141 [[cent|cents]] each, corresponding to 18.8046 [[edo]] (similar to every fifth step of [[94edo]]). It is almost identical to [[19edo]] and nearly identical to [[Carlos Beta]].
== Theory ==
11edf corresponds to 18.8046…[[edo]]. It is similar to [[19edo]], and nearly identical to [[Carlos Beta]]. Unlike 19edo, which is [[consistent]] to the [[integer limit|10-integer-limit]], 11edf is only consistent to the 7-integer-limit.  


==Intervals==
While the fifth is just, the fourth is very sharp and significantly less accurate than in 19edo. At 510.51{{c}}, it is 12.47{{c}} sharper than just and 3.7{{c}} flat of that of [[7edo]].
{| class="wikitable"
 
11edf represents the upper bound of the [[phoenix]] tuning range. It benefits from all the desirable properties of phoenix tuning systems.
 
=== Harmonics ===
{{Harmonics in equal|11|3|2|intervals=integer|columns=11}}
{{Harmonics in equal|11|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 11edf (continued)}}
 
=== Subsets and supersets ===
11edf is the fifth [[prime equal division|prime edf]], past [[7edf]] and before [[13edf]]. It does not contain any nontrivial subset edfs.
 
== Intervals ==
{| class="wikitable center-1 right-2"
|-
|-
! | degree
! #
! | cents value
! Cents
! | corresponding <br>JI intervals
! Approximate ratios
! | comments
|-
|-
| colspan="2"| 0
| 0
| | '''exact [[1/1]]'''
| 0.0
| |
| [[1/1]]
|-
|-
| | 1
| 1
| | 63.8141
| 63.8
| | ([[28/27]]), ([[27/26]])
| [[21/20]], [[25/24]], [[27/26]], [[28/27]]
| |
|-
|-
| | 2
| 2
| | 127.6282
| 127.6
| |[[14/13]]
| [[13/12]], [[14/13]], [[15/14]], [[16/15]]
| |
|-
|-
| | 3
| 3
| | 191.4423
| 191.4
| |
| [[9/8]], [[10/9]]
| |
|-
|-
| | 4
| 4
| | 255.2564
| 255.3
| |
| [[7/6]], ''[[8/7]]''
| |
|-
|-
| | 5
| 5
| | 319.07045
| 319.1
| | 6/5
| [[6/5]]
| |
|-
|-
| | 6
| 6
| | 382.8845
| 382.9
| | 5/4
| [[5/4]]
| |
|-
|-
| | 7
| 7
| | 446.6986
| 446.7
| |
| [[9/7]]
| |
|-
|-
| | 8
| 8
| | 510.5127
| 510.5
| |
| [[4/3]]
| |
|-
|-
| | 9
| 9
| | 574.3268
| 574.3
| |39/28
| [[7/5]]
| |
|-
|-
| | 10
| 10
| | 638.1409
| 638.1
| |([[13/9]])
| [[13/9]]
| |
|-
|-
| | 11
| 11
| | 701.955
| 702.0
| |'''exact [[3/2]]'''
| [[3/2]]
| | just perfect fifth
|-
|-
| | 12
| 12
| | 765.7691
| 765.8
| |14/9, 81/52
| [[14/9]]
| |
|-
|-
| | 13
| 13
| | 828.5732
| 828.6
| |21/13
| [[8/5]], [[13/8]], [[21/13]]
| |
|-
|-
| | 14
| 14
| | 893.3973
| 893.4
| |
| [[5/3]]
| |
|-
|-
| | 15
| 15
| | 956.2114
| 956.2
| |
| [[7/4]]
| |
|-
|-
| | 16
| 16
| | 1020.0255
| 1020.0
| | 9/5
| [[9/5]]
| |
|-
|-
| | 17
| 17
| | 1084.8395
| 1084.8
| | 15/8
| [[15/8]]
| |
|-
|-
| | 18
| 18
| | 1148.6536
| 1148.7
| |
| [[27/14]], [[35/18]]
| |
|-
|-
| | 19
| 19
| | 1211.4677
| 1211.5
| |
| [[2/1]]
| |
|-
|-
| | 20
| 20
| | 1276.2816
| 1276.3
| | 117/56
| [[21/10]], [[25/12]], [[27/13]]
| |
|-
|-
| | 21
| 21
| | 1340.0959
| 1340.1
| | 13/6
| [[13/6]]
| |
|-
|-
| | 22
| 22
| | 1403.91
| 1403.9
| |'''exact''' 9/4
| [[9/4]]
| |
|}
|}
==Scale tree==
If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking [[Mediant|"freshman sums"]] of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO.


If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum.
== Music ==
 
; [[Francium]]
Generator range: 62.33766 cents (4\7/11 = 4\77) to 65.{{Overline|45}} cents (3\5/11 = 3\55)
* "McGarfyGarf" from ''Microtonal Six-Dimensional Cats'' (2025) – [https://open.spotify.com/track/2iaicUkq6EcjcGM8RioFZj Spotify] | [https://francium223.bandcamp.com/track/mcgarfygarf Bandcamp] | [https://www.youtube.com/watch?v=sI8X6PNOiXE YouTube]
{| class="wikitable center-all"
! colspan="7" |Fifth
!Cents
!Comments
|-
|4\7|| || || ||  || || ||62.3377||
|-
| ||  || || || || ||27\47||62.66925 ||
|-
|  || || ||  || ||23\40|| ||62.{{Overline|72}}||
|-
| || ||  || || || ||42\73||62.7646||
|-
|  || || || ||19\33|| || ||62.8099||
|-
| || || || || || || 53\92||62.84585 ||
|-
| || ||  || || || 34\59|| || 62.86595||
|-
| || || || ||  || || 49\85|| 62.8877||
|-
|  || ||  ||15\26|| ||  || ||62.9371||
|-
| || || || || ||  ||56\97 ||62.9803||The generator closest to a just [[28/27]] for EDOs less than 200
|-
| || ||  || ||  ||41\71 || ||62.9962 ||
|-
| || || || || || ||67\116||63.0094||
|-
| || || || ||26\45|| || ||63.{{Overline|03}}||[[Flattone]] is in this region
|-
| || || || || || ||63\109||63.0525||
|-
| || || || || ||37\64|| ||63.06{{Overline|81}}||
|-
| || || || || || ||48\83||63.0887||
|-
| || ||11\19|| || || || ||63.1579||
|-
| || || || || || ||51\88||63.2231||
|-
| || || || || ||40\69|| ||63.2411||
|-
| || || || || || ||69\119||63.2544||
|-
| || || || ||29\50|| || ||63.{{Overline|27}}||
|-
| || || || || || ||76\131||63.2893||[[Golden meantone]] (696.2145¢)
|-
| || || || || ||47\81|| ||63.2997||
|-
| || || || || || ||65\112||63.3117||
|-
| || || ||18\31|| || || ||63.3431||[[Meantone]] is in this region
|-
| || || || || || ||61\105||63.3766||
|-
| || || || || ||43\74|| ||63.3907||
|-
| || || || || || ||68\117||63.4033||
|-
| || || || ||25\43|| || ||63.42495||
|-
| || || || || || ||57\98||63.4508||
|-
| || || || || ||32\55|| ||63.{{Overline|45}}||
|-
| || || || || || ||39\67||63.5007||
|-
| ||7\12|| || || || || ||63.{{Overline|63}}||
|-
| || || || || || ||38\65||63.7762||
|-
| || || || || ||31\53|| ||63.8079||The fifth closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || ||55\94||63.8298||[[Garibaldi]] / [[Cassandra]]
|-
| || || || ||24\41|| || ||63.8581||
|-
| || || || || || ||65\111||63.8821||
|-
| || || || || ||41\70|| ||63.8951||
|-
| || || || || || ||58\99||63.91185||
|-
| || || ||17\29|| || || ||63.9499||
|-
| || || || || || ||61\104||63.8960||
|-
| || || || || ||44\75|| ||64.000||
|-
| || || || || || ||71\121||64.0120||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||64.0316||[[Neogothic]] is in this region
|-
| || || || || || ||64\109||64.0534||
|-
| || || || || ||37\63|| ||64.0693||
|-
| || || || || || ||47\80||64.{{Overline|09}}||
|-
| || ||10\17|| || || || ||64.1711||
|-
| || || || || || ||43\73||64.2590||
|-
| || || || || ||33\56|| ||64.2857||
|-
| || || || || || ||56\95||64.3062||
|-
| || || || ||23\39|| || ||64.3357||
|-
| || || || || || ||59\100||64.{{Overline|36}}||
|-
| || || || || ||36\61|| ||64.3815||
|-
| || || || || || ||49\83||64.4031||
|-
| || || ||13\22|| || || ||64.4628||[[Archy]] is in this region
|-
| || || || || || ||42\71||64.53265||
|-
| || || || || ||29\49|| ||64.5640||
|-
| || || || || || ||45\76||64.5933||
|-
| || || || ||16\27|| || ||64.{{Overline|64}}||
|-
| || || || || || ||35\59||64.71495||
|-
| || || || || ||19\32|| ||64.7{{Overline|72}}||
|-
| || || || || || ||22\37||64.{{Overline|864}}||
|-
|3\5|| || || || || || ||65.{{Overline|45}}||
|}Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.


Tunings below 7\12 on this chart are called "positive tunings" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth tunings such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 13\22 through 16\27, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.
== See also ==
[[Category:edf]]
* [[19edo]] – relative edo
[[Category:Todo:expand]]
* [[30edt]] – relative edt
[[Category:Stub]]
* [[49ed6]] – relative ed6
* [[53ed7]] – relative ed7
* [[68ed12]] – relative ed12
* [[93ed30]] – relative ed30
Retrieved from "https://en.xen.wiki/w/11edf"