20ed11/5: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
'''20ed11/5''' is the [[ed11/5|equal division of the neutral ninth]] into 20 parts of 68.3 cents each.
'''20ed11/5''' is the [[ed11/5|equal division of the neutral ninth]] into 20 parts of 68.3 cents each.
== Temperaments ==
== Temperaments ==
20ed11/5 tempers out 2093663/2088025 and 54925/54043 in the no-twos-or-threes 17-limit which means it supports [[salmon]] temperament, which can be viewed as the no-twos-or-threes-analog of [[meantone]]. However, its [[13/11]] approximation is quite flat (closer to a subminor third) which affects the quality of the 11:13:17 chord, one of the most consonant no-twos-or-threes chords.
20ed11/5 tempers out 2093663/2088025 and 54925/54043 in the no-twos-or-threes 17-limit which means it supports [[catfish]] temperament, which can be viewed as the no-twos-or-threes-analog of [[meantone]]. However, its [[13/11]] approximation is quite flat (closer to a subminor third) which affects the quality of the 11:13:17 chord, one of the most consonant no-twos-or-threes chords.
== Intervals ==
== Intervals ==
{|class="wikitable"
{|class="wikitable"
Line 11: Line 9:
!Cents
!Cents
!Approximate ratios
!Approximate ratios
![[Catfish]] nonatonic notation (J = 1/1)
|-
|-
|0
|0
|0.0
|0.0
|[[1/1]]
|[[1/1]]
|J
|-
|-
|1
|1
|68.3
|68.3
|[[28/27]], [[27/26]], [[26/25]], [[25/24]]
|[[28/27]], [[27/26]], [[26/25]], [[25/24]]
|J#, Kb
|-
|-
|2
|2
|136.5
|136.5
|[[14/13]], [[27/25]], [[13/12]]
|[[14/13]], [[27/25]], [[13/12]]
|K
|-
|-
|3
|3
|204.8
|204.8
|[[9/8]]
|[[9/8]]
|K#, Lb
|-
|-
|4
|4
|273.0
|273.0
|[[7/6]], [[75/64]]
|[[7/6]], [[75/64]]
|L
|-
|-
|5
|5
|341.3
|341.3
|[[17/14]], [[39/32]], [[11/9]]
|[[17/14]], [[39/32]], [[11/9]]
|L#, Mb
|-
|-
|6
|6
|409.5
|409.5
|[[14/11]],[[81/64]], [[19/15]]
|[[14/11]],[[81/64]], [[19/15]]
|M
|-
|-
|7
|7
|477.8
|477.8
|[21/16]], [[33/25]]
|[[21/16]], [[33/25]]
|M#, Nb
|-
|-
|8
|8
|546.0
|546.0
|[[11/8]], [[15/11]], [[48/35]]
|[[11/8]], [[15/11]], [[48/35]]
|N
|-
|-
|9
|9
|614.3
|614.3
|[[10/7]], [[121/85]]
|[[10/7]], [[121/85]]
|N#
|-
|-
|10
|10
|682.5
|682.5
|[[3/2]], [[40/27]]
|[[3/2]], [[40/27]]
|Ob
|-
|-
|11
|11
|750.1
|750.1
|[[17/11]], [[49/32]]
|[[17/11]], [[49/32]]
|O
|-
|-
|12
|12
|819.0
|819.0
|[[8/5]], [[21/13]]
|[[8/5]], [[21/13]]
|O#, Pb
|-
|-
|13
|13
|887.3
|887.3
|[[5/3]], [[38/23]]
|[[5/3]], [[38/23]]
|P
|-
|-
|14
|14
|955.5
|955.5
|[[7/4]], [[26/15]]
|[[7/4]], [[26/15]]
|P#, Qb
|-
|-
|15
|15
|1023.8
|1023.8
|[[9/5]], [[29/16]], [[20/11]]
|[[9/5]], [[29/16]], [[20/11]]
|Q
|-
|-
|16
|16
|1092.0
|1092.0
|[[15/8]], [[32/17]]
|[[15/8]], [[32/17]]
|Q#, Rb
|-
|-
|17
|17
|1160.3
|1160.3
|[[49/25]], [[35/18]]
|[[49/25]], [[35/18]]
|R
|-
|-
|18
|18
|1228.5
|1228.5
|[[2/1]]
|[[2/1]]
|R#
|-
|-
|19
|19
|1296.8
|1296.8
|[[15/7]]
|[[15/7]]
|Jb
|-
|-
|20
|20
|1365.0
|1365.0
|[[11/5]]
|[[11/5]]
|J
|}
|}
== Harmonics ==
{{Harmonics in equal
| steps = 20
| num = 11
| denom = 5
}}
{{Harmonics in equal
| steps = 20
| num = 11
| denom = 5
| start = 12
| collapsed = 1
}}


[[Category:Nonoctave]]
[[Category:Nonoctave]]
[[Category:Equal-step tunings]]

Latest revision as of 01:43, 5 October 2024

← 19ed11/5 20ed11/5 21ed11/5 →
Prime factorization 22 × 5
Step size 68.2502 ¢ 
Octave 18\20ed11/5 (1228.5 ¢) (→ 9\10ed11/5)
Twelfth 28\20ed11/5 (1911.01 ¢) (→ 7\5ed11/5)
Consistency limit 3
Distinct consistency limit 3

20ed11/5 is the equal division of the neutral ninth into 20 parts of 68.3 cents each.

Temperaments

20ed11/5 tempers out 2093663/2088025 and 54925/54043 in the no-twos-or-threes 17-limit which means it supports catfish temperament, which can be viewed as the no-twos-or-threes-analog of meantone. However, its 13/11 approximation is quite flat (closer to a subminor third) which affects the quality of the 11:13:17 chord, one of the most consonant no-twos-or-threes chords.

Intervals

# Cents Approximate ratios Catfish nonatonic notation (J = 1/1)
0 0.0 1/1 J
1 68.3 28/27, 27/26, 26/25, 25/24 J#, Kb
2 136.5 14/13, 27/25, 13/12 K
3 204.8 9/8 K#, Lb
4 273.0 7/6, 75/64 L
5 341.3 17/14, 39/32, 11/9 L#, Mb
6 409.5 14/11,81/64, 19/15 M
7 477.8 21/16, 33/25 M#, Nb
8 546.0 11/8, 15/11, 48/35 N
9 614.3 10/7, 121/85 N#
10 682.5 3/2, 40/27 Ob
11 750.1 17/11, 49/32 O
12 819.0 8/5, 21/13 O#, Pb
13 887.3 5/3, 38/23 P
14 955.5 7/4, 26/15 P#, Qb
15 1023.8 9/5, 29/16, 20/11 Q
16 1092.0 15/8, 32/17 Q#, Rb
17 1160.3 49/25, 35/18 R
18 1228.5 2/1 R#
19 1296.8 15/7 Jb
20 1365.0 11/5 J

Harmonics

Approximation of harmonics in 20ed11/5
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +28.5 +9.1 -11.2 +11.9 -30.7 -24.6 +17.3 +18.1 -27.8 +11.9 -2.2
Relative (%) +41.8 +13.3 -16.5 +17.5 -45.0 -36.0 +25.3 +26.5 -40.7 +17.5 -3.2
Steps
(reduced) 		18
(18) 		28
(8) 		35
(15) 		41
(1) 		45
(5) 		49
(9) 		53
(13) 		56
(16) 		58
(18) 		61
(1) 		63
(3)
Approximation of harmonics in 20ed11/5
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) -4.3 +3.9 +21.0 -22.5 +9.1 -21.6 +21.3 +0.7 -15.5 -27.8 +31.7
Relative (%) -6.2 +5.8 +30.8 -32.9 +13.3 -31.7 +31.1 +1.0 -22.7 -40.7 +46.5
Steps
(reduced) 		65
(5) 		67
(7) 		69
(9) 		70
(10) 		72
(12) 		73
(13) 		75
(15) 		76
(16) 		77
(17) 		78
(18) 		80
(0)
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