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


'''22EDF''' is the [[EDF|equal division of the just perfect fifth]] into 22 parts of 31.907 [[cent|cents]] each, corresponding to 37.6092 [[edo]] (similar to every fifth step of [[188edo]]).
== 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.
! | degree
 
! | cents value
=== Harmonics ===
! | corresponding <br>JI intervals
{{Harmonics in equal|11|3|2|intervals=integer|columns=11}}
! | comments
{{Harmonics in equal|11|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 11edf (continued)}}
|-
 
| colspan="2"| 0
=== Subsets and supersets ===
| | '''exact [[1/1]]'''
11edf is the fifth [[prime equal division|prime edf]], past [[7edf]] and before [[13edf]]. It does not contain any nontrivial subset edfs.
| |  
 
|-
== Intervals ==
| | 1
{| class="wikitable center-1 right-2"
| | 31.907
| | [[55/54]]
| |
|-
| | 2
| | 63.8141
| | ([[28/27]]), ([[27/26]])
| |
|-
| | 3
| | 95.7211
| |
| |
|-
| | 4
| | 127.6282
| | [[14/13]]
| |
|-
|-
| | 5
! #
| | 159.5352
! Cents
| | 57/52
! Approximate ratios
| |
|-
|-
| | 6
| 0
| | 191.4423
| 0.0
| |
| [[1/1]]
| |
|-
|-
| | 7
| 1
| | 223.3493
| 63.8
| |8/7
| [[21/20]], [[25/24]], [[27/26]], [[28/27]]
| |
|-
|-
| | 8
| 2
| | 255.2564
| 127.6
| |
| [[13/12]], [[14/13]], [[15/14]], [[16/15]]
| |
|-
|-
| | 9
| 3
| | 287.1634
| 191.4
| |13/11
| [[9/8]], [[10/9]]
| |
|-
|-
| | 10
| 4
| | 319.0705
| 255.3
| |6/5
| [[7/6]], ''[[8/7]]''
| |
|-
|-
| | 11
| 5
| | 350.9775
| 319.1
| | 60/49, 49/40
| [[6/5]]
| |
|-
|-
| | 12
| 6
| | 382.8845
| 382.9
| |5/4
| [[5/4]]
| |
|-
|-
| | 13
| 7
| | 414.7916
| 446.7
| |14/11
| [[9/7]]
| |
|-
|-
| | 14
| 8
| | 446.6986
| 510.5
| |
| [[4/3]]
| |
|-
|-
| | 15
| 9
| | 478.6057
| 574.3
| |
| [[7/5]]
| |
|-
|-
| | 16
| 10
| | 510.5127
| 638.1
| |
| [[13/9]]
| |
|-
|-
| | 17
| 11
| | 542.4198
| 702.0
| | [[26/19]]
| [[3/2]]
| |
|-
|-
| | 18
| 12
| | 574.3268
| 765.8
| | 39/28
| [[14/9]]
| |
|-
|-
| | 19
| 13
| | 606.2339
| 828.6
| |64/45
| [[8/5]], [[13/8]], [[21/13]]
| |
|-
|-
| | 20
| 14
| | 638.1409
| 893.4
| | ([[13/9]])
| [[5/3]]
| |
|-
|-
| | 21
| 15
| | 670.048
| 956.2
| | 81/55
| [[7/4]]
| |
|-
|-
| | 22
| 16
| | 701.955
| 1020.0
| | '''exact [[3/2]]'''
| [[9/5]]
| | just perfect fifth
|-
|-
|23
| 17
|733.862
| 1084.8
|55/36
| [[15/8]]
|
|-
|-
|24
| 18
|765.7691
| 1148.7
|14/9, 81/52
| [[27/14]], [[35/18]]
|
|-
|-
|25
| 19
|797.6761
| 1211.5
|
| [[2/1]]
|
|-
|-
|26
| 20
|828.5732
| 1276.3
|21/13
| [[21/10]], [[25/12]], [[27/13]]
|
|-
|-
|27
| 21
|861.4902
| 1340.1
|171/104
| [[13/6]]
|
|-
|-
|28
| 22
|893.3973
| 1403.9
|
| [[9/4]]
|
|-
|29
|925.3043
|12/7
|
|-
|30
|956.2114
|
|
|-
|31
|988.1184
|39/22
|
|-
|32
|1020.0255
|9/5
|
|-
|33
|1052.9235
|90/49, 147/80
|
|-
|34
|1084.8395
|15/8
|
|-
|35
|1116.7466
|21/11
|
|-
|36
|1148.6536
|
|
|-
|37
|1180.5607
|
|
|-
|38
|1211.4677
|
|
|-
|39
|1244.3748
|39/19
|
|-
|40
|1276.2816
|117/56
|
|-
|41
|1308.1889
|32/15
|
|-
|42
|1340.0959
|13/6
|
|-
|43
|1372.003
|243/110
|
|-
|44
|1403.91
|'''exact''' 9/4
|
|}
|}


[[Category:edf]]
== Music ==
[[Category:Todo:expand]]
; [[Francium]]
[[Category:Stub]]
* "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]
 
== See also ==
* [[19edo]] – relative edo
* [[30edt]] – relative edt
* [[49ed6]] – relative ed6
* [[53ed7]] – relative ed7
* [[68ed12]] – relative ed12
* [[93ed30]] – relative ed30

Latest revision as of 19:34, 31 May 2025

← 10edf 11edf 12edf →
Prime factorization 11 (prime)
Step size 63.8141 ¢ 
Octave 19\11edf (1212.47 ¢)
Twelfth 30\11edf (1914.42 ¢)
Consistency limit 7
Distinct consistency limit 7

11 equal divisions of the perfect fifth (abbreviated 11edf or 11ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 11 equal parts of about 63.8 ¢ each. Each step represents a frequency ratio of (3/2)1/11, or the 11th root of 3/2.

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 10-integer-limit, 11edf is only consistent to the 7-integer-limit.

While the fifth is just, the fourth is very sharp and significantly less accurate than in 19edo. At 510.51 ¢, it is 12.47 ¢ sharper than just and 3.7 ¢ flat of that of 7edo.

11edf represents the upper bound of the phoenix tuning range. It benefits from all the desirable properties of phoenix tuning systems.

Harmonics

Approximation of harmonics in 11edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +12.5 +12.5 +24.9 +21.5 +24.9 +13.3 -26.4 +24.9 -29.8 -3.4 -26.4
Relative (%) +19.5 +19.5 +39.1 +33.7 +39.1 +20.9 -41.4 +39.1 -46.8 -5.3 -41.4
Steps
(reduced) 		19
(8) 		30
(8) 		38
(5) 		44
(0) 		49
(5) 		53
(9) 		56
(1) 		60
(5) 		62
(7) 		65
(10) 		67
(1)
Approximation of harmonics in 11edf (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +26.5 +25.8 -29.8 -13.9 +8.7 -26.4 +7.6 -17.4 +25.8 +9.1 -4.1 -13.9
Relative (%) +41.5 +40.4 -46.8 -21.8 +13.7 -41.4 +11.9 -27.2 +40.4 +14.2 -6.4 -21.8
Steps
(reduced) 		70
(4) 		72
(6) 		73
(7) 		75
(9) 		77
(0) 		78
(1) 		80
(3) 		81
(4) 		83
(6) 		84
(7) 		85
(8) 		86
(9)

Subsets and supersets

11edf is the fifth prime edf, past 7edf and before 13edf. It does not contain any nontrivial subset edfs.

Intervals

# Cents Approximate ratios
0 0.0 1/1
1 63.8 21/20, 25/24, 27/26, 28/27
2 127.6 13/12, 14/13, 15/14, 16/15
3 191.4 9/8, 10/9
4 255.3 7/6, 8/7
5 319.1 6/5
6 382.9 5/4
7 446.7 9/7
8 510.5 4/3
9 574.3 7/5
10 638.1 13/9
11 702.0 3/2
12 765.8 14/9
13 828.6 8/5, 13/8, 21/13
14 893.4 5/3
15 956.2 7/4
16 1020.0 9/5
17 1084.8 15/8
18 1148.7 27/14, 35/18
19 1211.5 2/1
20 1276.3 21/10, 25/12, 27/13
21 1340.1 13/6
22 1403.9 9/4

Music

Francium

See also

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