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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==
EDF scales can be approximated in [[EDO]]s by subdividing diatonic fifths. 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 fifth 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]]

Latest revision as of 11:56, 26 June 2025

← 12edf 13edf 14edf →
Prime factorization 13 (prime)
Step size 53.9965 ¢ 
Octave 22\13edf (1187.92 ¢)
Twelfth 35\13edf (1889.88 ¢)
Consistency limit 4
Distinct consistency limit 4

13 equal divisions of the perfect fifth (abbreviated 13edf or 13ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 13 equal parts of about 54 ¢ each. Each step represents a frequency ratio of (3/2)1/13, or the 13th root of 3/2.

Theory

13edf corresponds to 22.2236edo. It is nearly identical to every ninth step of 200edo, but not quite similar to 22edo; the octave is compressed by 12.076 ¢, a deviation that is small but significant enough to create a discrepancy for the 7th and 11th harmonics.

Harmonics

Approximation of prime harmonics in 13edf
Harmonic 2 3 5 7 11 13 17 19
Error Absolute (¢) -12.1 -12.1 +21.5 -21.0 +6.4 -12.8 +8.7 -21.8
Relative (%) -22.4 -22.4 +39.8 -39.0 +11.9 -23.7 +16.2 -40.4
Steps
(reduced) 		22
(9) 		35
(9) 		52
(0) 		62
(10) 		77
(12) 		82
(4) 		91
(0) 		94
(3)
Approximation of prime harmonics in 13edf
Harmonic 23 29 31 37 41 43 47 53
Error Absolute (¢) +25.4 +2.0 -5.4 +12.3 -3.5 +22.1 -23.9 -15.9
Relative (%) +47.0 +3.8 -10.0 +22.7 -6.4 +40.9 -44.3 -29.5
Steps
(reduced) 		101
(10) 		108
(4) 		110
(6) 		116
(12) 		119
(2) 		121
(4) 		123
(6) 		127
(10)

Intervals

Intervals of 13edf
Degree Cents Corresponding
JI intervals 		Comments
0 exact 1/1
1 53.9965 33/32 pseudo-25/24
2 107.9931 17/16, 117/110, 16/15
3 161.9896 11/10
4 215.9862 17/15
5 269.9827 7/6
6 323.9792 77/64 pseudo-6/5
7 377.9758 56/45 pseudo-5/4
8 431.9723 9/7
9 485.9688 45/34 pseudo-4/3
10 539.9654 15/11
11 593.9619 55/39, 24/17
12 647.9585 16/11
13 701.9550 exact 3/2 just perfect fifth
14 755.9515 99/64
15 809.9481 51/32, 8/5
16 863.9446 33/20
17 917.9412 17/10
18 971.9377 7/4
19 1025.9342 29/16 pseudo-9/5
20 1079.9308 28/15 pseudo-15/8
21 1133.9273 52/27, 27/14
22 1187.9238 135/68 pseudo-octave
23 1241.9204 45/22
24 1295.9169 19/9, 36/17
25 1349.9135 24/11
26 1403.9100 exact 9/4 pythagorean major ninth
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