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{{Infobox ET}}
{{Infobox ET}}
'''13EDF''' is the [[EDF|equal division of the just perfect fifth]] into 13 parts of 53.9965 [[cent|cents]] each, corresponding to 22.2236 [[edo]]. It is nearly identical to every ninth step of [[200edo]].
{{ED intro}}


==Intervals==
== Theory ==
{| class="wikitable"
13edf corresponds to 22.2236[[edo]]. It is nearly identical to every ninth step of [[200edo]], but not quite similar to [[22edo]]; the octave is compressed by 12.076{{c}}, a deviation that is small but significant enough to create a discrepancy for the [[7/1|7th]] and [[11/1|11th]] harmonics.
 
== Harmonics ==
{{Harmonics in equal|13|3|2|intervals=prime|columns=8}}
{{Harmonics in equal|13|3|2|start=9|intervals=prime|columns=8}}
 
== Intervals ==
{| class="wikitable mw-collapsible"
|+ style="font-size: 105%;" | Intervals of 13edf
|-
|-
! | degree
! Degree
! | cents value
! Cents
! | corresponding <br>JI intervals
! Corresponding<br />JI intervals
! | comments
! Comments
|-
|-
! colspan="2" | 0
! colspan="2" | 0
| | '''exact [[1/1]]'''
| '''exact [[1/1]]'''
| |  
|  
|-
|-
| | 1
| 1
| | 53.9965
| 53.9965
| | 33/32
| 33/32
| | pseudo-[[25/24]]
| pseudo-[[25/24]]
|-
|-
| | 2
| 2
| | 107.9931
| 107.9931
| | [[17/16]], 117/110, [[16/15]]
| [[17/16]], 117/110, [[16/15]]
| |  
|  
|-
|-
| | 3
| 3
| | 161.9896
| 161.9896
| | [[11/10]]
| [[11/10]]
| |  
|  
|-
|-
| | 4
| 4
| | 215.9862
| 215.9862
| | [[17/15]]
| [[17/15]]
| |  
|  
|-
|-
| | 5
| 5
| | 269.9827
| 269.9827
| | [[7/6]]
| [[7/6]]
| |  
|  
|-
|-
| | 6
| 6
| | 323.9792
| 323.9792
| | [[77/64]]
| [[77/64]]
| | pseudo-[[6/5]]
| pseudo-[[6/5]]
|-
|-
| | 7
| 7
| | 377.9758
| 377.9758
| | 56/45
| 56/45
| | pseudo-[[5/4]]
| pseudo-[[5/4]]
|-
|-
| | 8
| 8
| | 431.9723
| 431.9723
| | [[9/7]]
| [[9/7]]
| |  
|  
|-
|-
| | 9
| 9
| | 485.9688
| 485.9688
| | 45/34
| 45/34
| | pseudo-[[4/3]]
| pseudo-[[4/3]]
|-
|-
| | 10
| 10
| | 539.9654
| 539.9654
| | [[15/11]]
| [[15/11]]
| |  
|  
|-
|-
| | 11
| 11
| | 593.9619
| 593.9619
| | 55/39, [[24/17]]
| 55/39, [[24/17]]
| |  
|  
|-
|-
| | 12
| 12
| | 647.9585
| 647.9585
| | [[16/11]]
| [[16/11]]
| |  
|  
|-
|-
| | 13
| 13
| | 701.9550
| 701.9550
| | '''exact [[3/2]]'''
| '''exact [[3/2]]'''
| | just perfect fifth
| just perfect fifth
|-
|-
| | 14
| 14
| | 755.9515
| 755.9515
| | 99/64
| 99/64
| |  
|  
|-
|-
| | 15
| 15
| | 809.9481
| 809.9481
| | 51/32, [[8/5]]
| 51/32, [[8/5]]
| |  
|  
|-
|-
| | 16
| 16
| | 863.9446
| 863.9446
| | 33/20
| 33/20
| |  
|  
|-
|-
| | 17
| 17
| | 917.9412
| 917.9412
| | [[17/10]]
| [[17/10]]
| |  
|  
|-
|-
| | 18
| 18
| | 971.9377
| 971.9377
| | [[7/4]]
| [[7/4]]
| |  
|  
|-
|-
| | 19
| 19
| | 1025.9342
| 1025.9342
| | [[29/16]]
| [[29/16]]
| | pseudo-[[9/5]]
| pseudo-[[9/5]]
|-
|-
| | 20
| 20
| | 1079.9308
| 1079.9308
| | [[28/15]]
| [[28/15]]
| | pseudo-[[15/8]]
| pseudo-[[15/8]]
|-
|-
| | 21
| 21
| | 1133.9273
| 1133.9273
| | 52/27, [[27/14]]
| 52/27, [[27/14]]
| |  
|  
|-
|-
| | 22
| 22
| | 1187.9238
| 1187.9238
| | 135/68
| 135/68
| | pseudo-[[octave]]
| pseudo-[[octave]]
|-
|-
| | 23
| 23
| | 1241.9204
| 1241.9204
| | [[45/44|45/22]]
| [[45/44|45/22]]
| |  
|  
|-
|-
| | 24
| 24
| | 1295.9169
| 1295.9169
| | [[19/18|19/9]], [[18/17|36/17]]
| [[19/18|19/9]], [[18/17|36/17]]
| |  
|  
|-
|-
| | 25
| 25
| | 1349.9135
| 1349.9135
| | [[12/11|24/11]]
| [[12/11|24/11]]
| |  
|  
|-
|-
| | 26
| 26
| | 1403.9100
| 1403.9100
| | '''exact [[9/4]]'''
| '''exact [[9/4]]'''
| | pythagorean major ninth
| pythagorean major ninth
|}
|}
==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.
{{stub}}
 
Generator range: 52.74725 cents (4\7/13 = 4\91) to 55.{{Overline|384615}} cents (3\5/13 = 3\65)
{| class="wikitable center-all"
! colspan="7" |Fifth
! Cents
!Comments
|-
|4\7|| || || || || || ||52.74725
|
|-
| || ||  || || || || 27\47||53.0278 ||
|-
| || ||  || ||  ||23\40|| ||53.0769||
|-
| || || || || ||  ||42\73|| 53.1085||
|-
| ||  || || ||19\33|| || ||53.14685||
|-
| ||  || || || || ||53\92 ||53.1773 ||
|-
| || || ||  || ||34\59|| ||53.1943||
|-
| || ||  || || ||  ||49\85||53.2123 ||
|-
| ||  || ||15\26|| || || ||53.2544||
|-
| || || || ||  || || 56\97||53.2910||
|-
| || || || || ||41\71||  ||53.3044||
|-
| || || || || || ||67\116|| 53.31565||
|-
|  || || || ||26\45||  || ||53.{{Overline|3}}||[[Flattone]] is in this region
|-
| || ||  || || || ||63\109||53.35215||
|-
| || || || || || 37\64||  ||53.3654||
|-
|  || || || || || || 48\83||53.3828||
|-
| || ||11\19 || || || || ||53.4413||
|-
| || ||  || || || || 51\88||53.4965||
|-
| ||  || || || ||40\69|| ||53.5117||
|-
| || || || || || ||69\119||53.52295||
|-
| || || || ||29\50|| || ||53.5385||
|-
| || || || || || ||76\131||53.55255||[[Golden meantone]] (696.2145¢)
|-
| || || || || ||47\81|| ||53.56125||
|-
| || || || || || ||65\112||53.5714||
|-
| || || ||18\31|| || || ||53.5980||[[Meantone]] is in this region
|-
| || || || || || ||61\105||53.6264||
|-
| || || || || ||43\74|| ||53.63825||
|-
| || || || || || ||68\117||53.6489||
|-
| || || || ||25\43|| || ||53.6673||
|-
| || || || || || ||57\98||53.6892||
|-
| || || || || ||32\55|| ||53.7063||
|-
| || || || || || ||39\67||53.73135||
|-
| ||7\12|| || || || || ||53.84615||
|-
| || || || || || ||38\65||53.9645||
|-
| || || || || ||31\53|| ||53.9913||The generator closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || ||55\94||54.0098||[[Garibaldi]] / [[Cassandra]]
|-
| || || || ||24\41|| || ||54.0338||
|-
| || || || || || ||65\111||54.{{Overline|054}}||
|-
| || || || || ||41\70|| ||54.0659||
|-
| || || || || || ||58\99||54.07925||
|-
| || || ||17\29|| || || ||54.1114||
|-
| || || || || || ||61\104||54.1420||
|-
| || || || || ||44\75|| ||54.15385||
|-
| || || || || || ||71\121||54.1640||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||54.1806||[[Neogothic]] is in this region
|-
| || || || || || ||64\109||54.1990||
|-
| || || || || ||37\63|| ||54.21245||
|-
| || || || || || ||47\80||54.2308||
|-
| || ||10\17|| || || || ||54.2986||
|-
| || || || || || ||43\73||54.3730||
|-
| || || || || ||33\56|| ||54.3956||
|-
| || || || || || ||56\95||54.4130||
|-
| || || || ||23\39|| || ||54.4734||
|-
| || || || || || ||59\100||54.4615||
|-
| || || || || ||36\61|| ||54.4767||
|-
| || || || || || ||49\83||54.4949||
|-
| || || ||13\22|| || || ||54.{{Overline|54}}||[[Archy]] is in this region
|-
| || || || || || ||42\71||54.60455||
|-
| || || || || ||29\49|| ||54.6311||
|-
| || || || || || ||45\76||54.6558||
|-
| || || || ||16\27|| || ||54.70085||
|-
| || || || || || ||35\59||54.7588||
|-
| || || || || ||19\32|| ||54.8077||
|-
| || || || || || ||22\37||54.88565||
|-
|3\5|| || || || || || ||55.3846||
|}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.
[[Category:22edo]]
[[Category:Edf]]
[[Category:Edonoi]]
Retrieved from "https://en.xen.wiki/w/13edf"