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{{Infobox ET}}
{{Infobox ET}}
{{ED intro}} It corresponds to 18.8046[[edo]], is is similar to [[19edo]], and nearly identical to [[Carlos Beta]].
{{ED intro}}
 
== Theory ==
11edf corresponds to 18.8046…[[edo]]. It is similar to [[19edo]], and nearly identical to [[Carlos Beta]].


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]].
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]].


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


== Harmonics ==
=== Harmonics ===
{{Harmonics in equal|11|3|2|prec=2|columns=15}}
{{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)}}


== Intervals ==
== Intervals ==
{| class="wikitable"
{| class="wikitable center-1 right-2"
|-
|-
! Degree
! #
! Cent 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]]
|
|}
|}
{{todo|expand}}

Revision as of 12:56, 30 March 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.

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)

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
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