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{{Infobox ET}}
{{Infobox ET}}
'''[[EDF|Division of the just perfect fifth]] into 21 equal parts''' (21EDF) is related to [[36edo|36 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 3.3514 cents stretched and the step size is about 33.4264 cents. Unlike 36edo, it is only consistent up to the [[3-odd-limit|4-integer-limit]], with discrepancy for the 5th harmonic.
{{ED intro}}


Lookalikes: [[36edo]], [[57edt]]
21EDF is related to [[36edo]], but with the 3/2 rather than the 2/1 being just, which stretches the octave by about 3.3514 cents. Unlike 36edo, it is only consistent up to the 4-[[integer-limit]], with discrepancy for the 5th harmonic.


=Approximations=
Lookalikes: [[36edo]], [[57edt]], [[93ed6]], [[101ed7]], [[129ed12]]
==3-limit (Pythagorean) approximations (same as 7edf):==
== Theory ==
21edf acts as a stretched version of 36edo, though under most circumstances the stretch is more than ideal. If used as a fifth-based system, the chord 6:7:8(:9) may act as the fundamental chord of the system. The most important comma tempered out by this system is [[1029/1024]], and the related temperament is a fifth-based version of [[slendric]] with a 1/3-fifth period representing [[8/7]] and a generator of about a sixth-tone. One generator up or down from this period gives [[7/6]] and [[9/8]] respectively. Other edfs supporting this temperament include [[24edf]] and [[45edf]].
== Approximations ==
=== Harmonics ===
{{Harmonics in equal|21|3|2|prec=2|columns=8}}
{{Harmonics in equal|21|3|2|prec=2|columns=8|start=9|title=contd.}}
 
=== 3-limit (Pythagorean) approximations (same as 7edf): ===
2/1 = 1200 cents; 36 degrees of 21edf = 1203.3514... cents.
2/1 = 1200 cents; 36 degrees of 21edf = 1203.3514... cents.


Line 21: Line 28:


128/81 = 792.180... cents; 24 degrees of 21edf = 802.2342... cents.
128/81 = 792.180... cents; 24 degrees of 21edf = 802.2342... cents.
==7-limit approximations:==
 
===7 only:===
=== 7-limit approximations: ===
==== 7 only: ====
7/4 = 968.826... cents; 29 degrees of 21edf = 969.3664... cents.
7/4 = 968.826... cents; 29 degrees of 21edf = 969.3664... cents.


Line 30: Line 38:


64/49 = 462.348... cents; 14 degrees of 21edf = 467.97... cents.
64/49 = 462.348... cents; 14 degrees of 21edf = 467.97... cents.
===3 and 7:===
 
==== 3 and 7: ====
7/6 = 266.871... cents; 8 degrees of 21edf = 267.4114... cents.
7/6 = 266.871... cents; 8 degrees of 21edf = 267.4114... cents.


Line 59: Line 68:
63/32 = 1172.736... cents; 35 degrees of 21edf = 1169.925... cents.
63/32 = 1172.736... cents; 35 degrees of 21edf = 1169.925... cents.


The following table gives an overview of all degrees of 36edo.
== Intervals ==
{| class="wikitable"
The following table gives an overview of all degrees of 21edf.
 
{| class="wikitable mw-collapsible" style="text-align: center;"
|+ style="font-size: 105%;" | Intervals of 21edf
|-
|-
! |Degree
! Degree
! |Size
! Size<br />in [[Cent|cents]]
in [[Cent|cents]]
! Approximate<br />ratios of 2.3.7
! |Approximate
! Additional ratios<br />of 2.3.7.13.17
ratios of 2.3.7
! |Additional ratios
of 2.3.7.13.17
|-
|-
| colspan="2" style="text-align:center;" |0
| colspan="2" | 0
| style="text-align:center;" |1/1
| 1/1
| style="text-align:center;" |
|  
|-
|-
| style="text-align:center;" |1
| 1
| style="text-align:right;" |33.4264
| style="text-align: right;" | 33.4264
| style="text-align:center;" |64/63, [[49/48]]
| 64/63, [[49/48]]
| style="text-align:center;" |
|  
|-
|-
| style="text-align:center;" |2
| 2
| style="text-align:right;" |66.8529
| style="text-align: right;" | 66.8529
| style="text-align:center;" |[[28/27]]
| [[28/27]]
| style="text-align:center;" |
|  
|-
|-
| style="text-align:center;" |3
| 3
| style="text-align:right;" |100.2793
| style="text-align: right;" | 100.2793
| style="text-align:center;" |256/243
| 256/243
| style="text-align:center;" |[[17/16]], [[18/17]]
| [[17/16]], [[18/17]]
|-
|-
| style="text-align:center;" |4
| 4
| style="text-align:right;" |133.7057
| style="text-align: right;" | 133.7057
| style="text-align:center;" |243/224
| 243/224
| style="text-align:center;" |[[14/13]], [[13/12]]
| [[14/13]], [[13/12]]
|-
|-
| style="text-align:center;" |5
| 5
| style="text-align:right;" |167.1321
| style="text-align: right;" | 167.1321
| style="text-align:center;" |[[54/49]]
| [[54/49]]
| style="text-align:center;" |
|  
|-
|-
| style="text-align:center;" |6
| 6
| style="text-align:right;" |200.5586
| style="text-align: right;" | 200.5586
| style="text-align:center;" |[[9/8]]
| [[9/8]]
| style="text-align:center;" |
|  
|-
|-
| style="text-align:center;" |7
| 7
| style="text-align:right;" |233.985
| style="text-align: right;" | 233.985
| style="text-align:center;" |[[8/7]]
| [[8/7]]
| style="text-align:center;" |
|  
|-
|-
| style="text-align:center;" |8
| 8
| style="text-align:right;" |267.4114
| style="text-align: right;" | 267.4114
| style="text-align:center;" |[[7/6]]
| [[7/6]]
| style="text-align:center;" |
|  
|-
|-
| style="text-align:center;" |9
| 9
| style="text-align:right;" |300.8379
| style="text-align: right;" | 300.8379
| style="text-align:center;" |[[32/27]]
| [[32/27]]
| style="text-align:center;" |
|  
|-
|-
| style="text-align:center;" |10
| 10
| style="text-align:right;" |334.2643
| style="text-align: right;" | 334.2643
| style="text-align:center;" |98/81
| 98/81
| style="text-align:center;" |[[17/14]]
| [[17/14]]
|-
|-
| style="text-align:center;" |11
| 11
| style="text-align:right;" |367.6907
| style="text-align: right;" | 367.6907
| style="text-align:center;" |243/196
| 243/196
| style="text-align:center;" |[[16/13]], [[26/21]], [[21/17]]
| [[16/13]], [[26/21]], [[21/17]]
|-
|-
| style="text-align:center;" |12
| 12
| style="text-align:right;" |401.1171
| style="text-align: right;" | 401.1171
| style="text-align:center;" |[[81/64]]
| [[81/64]]
| style="text-align:center;" |
|  
|-
|-
| style="text-align:center;" |13
| 13
| style="text-align:right;" |434.5436
| style="text-align: right;" | 434.5436
| style="text-align:center;" |[[9/7]]
| [[9/7]]
| style="text-align:center;" |
|  
|-
|-
| style="text-align:center;" |14
| 14
| style="text-align:right;" |467.97
| style="text-align: right;" | 467.97
| style="text-align:center;" |[[64/49]], [[21/16]]
| [[64/49]], [[21/16]]
| style="text-align:center;" |[[17/13]]
| [[17/13]]
|-
|-
| style="text-align:center;" |15
| 15
| style="text-align:right;" |501.3964
| style="text-align: right;" | 501.3964
| style="text-align:center;" |[[4/3]]
| [[4/3]]
| style="text-align:center;" |
|  
|-
|-
| style="text-align:center;" |16
| 16
| style="text-align:right;" |534.8229
| style="text-align: right;" | 534.8229
| style="text-align:center;" |[[49/36]]
| [[49/36]]
| style="text-align:center;" |
|  
|-
|-
| style="text-align:center;" |17
| 17
| style="text-align:right;" |568.2493
| style="text-align: right;" | 568.2493
| style="text-align:center;" |
|  
| style="text-align:center;" |[[18/13]]
| [[18/13]]
|-
|-
| style="text-align:center;" |18
| 18
| style="text-align:right;" |601.6757
| style="text-align: right;" | 601.6757
| style="text-align:center;" |
|  
| style="text-align:center;" |
|  
|-
|-
| style="text-align:center;" |19
| 19
| style="text-align:right;" |635.1021
| style="text-align: right;" | 635.1021
| style="text-align:center;" |
|  
| style="text-align:center;" |[[13/9]]
| [[13/9]]
|-
|-
| style="text-align:center;" |20
| 20
| style="text-align:right;" |668.5286
| style="text-align: right;" | 668.5286
| style="text-align:center;" |72/49
| 72/49
| style="text-align:center;" |
|  
|-
|-
| style="text-align:center;" |21
| 21
| style="text-align:right;" |701.955
| style="text-align: right;" | 701.955
| style="text-align:center;" |[[3/2]]
| [[3/2]]
| style="text-align:center;" |
|  
|-
|-
| style="text-align:center;" |22
| 22
| style="text-align:right;" |735.3814
| style="text-align: right;" | 735.3814
| style="text-align:center;" |[[49/32]], [[32/21]]
| [[49/32]], [[32/21]]
| style="text-align:center;" |[[26/17]]
| [[26/17]]
|-
|-
| style="text-align:center;" |23
| 23
| style="text-align:right;" |768.8079
| style="text-align: right;" | 768.8079
| style="text-align:center;" |[[14/9]]
| [[14/9]]
| style="text-align:center;" |
|  
|-
|-
| style="text-align:center;" |24
| 24
| style="text-align:right;" |802.2343
| style="text-align: right;" | 802.2343
| style="text-align:center;" |[[128/81]]
| [[128/81]]
| style="text-align:center;" |
|  
|-
|-
| style="text-align:center;" |25
| 25
| style="text-align:right;" |835.6607
| style="text-align: right;" | 835.6607
| style="text-align:center;" |392/243
| 392/243
| style="text-align:center;" |[[13/8]], [[21/13]], [[34/21]]
| [[13/8]], [[21/13]], [[34/21]]
|-
|-
| style="text-align:center;" |26
| 26
| style="text-align:right;" |869.0871
| style="text-align: right;" | 869.0871
| style="text-align:center;" |81/49
| 81/49
| style="text-align:center;" |[[28/17]]
| [[28/17]]
|-
|-
| style="text-align:center;" |27
| 27
| style="text-align:right;" |902.5136
| style="text-align: right;" | 902.5136
| style="text-align:center;" |[[27/16]]
| [[27/16]]
| style="text-align:center;" |
|  
|-
|-
| style="text-align:center;" |28
| 28
| style="text-align:right;" |935.94
| style="text-align: right;" | 935.94
| style="text-align:center;" |[[12/7]]
| [[12/7]]
| style="text-align:center;" |
|  
|-
|-
| style="text-align:center;" |29
| 29
| style="text-align:right;" |969.3664
| style="text-align: right;" | 969.3664
| style="text-align:center;" |[[7/4]]
| [[7/4]]
| style="text-align:center;" |
|  
|-
|-
| style="text-align:center;" |30
| 30
| style="text-align:right;" |1002.7929
| style="text-align: right;" | 1002.7929
| style="text-align:center;" |[[16/9]]
| [[16/9]]
| style="text-align:center;" |
|  
|-
|-
| style="text-align:center;" |31
| 31
| style="text-align:right;" |1036.2193
| style="text-align: right;" | 1036.2193
| style="text-align:center;" |49/27
| 49/27
| style="text-align:center;" |
|  
|-
|-
| style="text-align:center;" |32
| 32
| style="text-align:right;" |1069.6457
| style="text-align: right;" | 1069.6457
| style="text-align:center;" |448/243
| 448/243
| style="text-align:center;" |[[13/7]], [[24/13]]
| [[13/7]], [[24/13]]
|-
|-
| style="text-align:center;" |33
| 33
| style="text-align:right;" |1103.0721
| style="text-align: right;" | 1103.0721
| style="text-align:center;" |[[243/128]]
| [[243/128]]
| style="text-align:center;" |[[32/17]], [[17/9]]
| [[32/17]], [[17/9]]
|-
|-
| style="text-align:center;" |34
| 34
| style="text-align:right;" |1136.4986
| style="text-align: right;" | 1136.4986
| style="text-align:center;" |[[27/14]]
| [[27/14]]
| style="text-align:center;" |
|  
|-
|-
| style="text-align:center;" |35
| 35
| style="text-align:right;" |1169.925
| style="text-align: right;" | 1169.925
| style="text-align:center;" |63/32, 96/49
| 63/32, 96/49
| style="text-align:center;" |
|  
|-
|-
| style="text-align:center;" |36
| 36
| style="text-align:right;" |1203.3514
| style="text-align: right;" | 1203.3514
| style="text-align:center;" |2/1
| 2/1
| style="text-align:center;" |
|  
|-
|-
|37
| 37
|1236.7779
| 1236.7779
|128/63, 49/24
| 128/63, 49/24
|
|  
|-
|-
|38
| 38
|1270.2043
| 1270.2043
|56/27
| 56/27
|
|  
|-
|-
|39
| 39
|1303.6307
| 1303.6307
|512/243
| 512/243
|17/8, 36/17
| 17/8, 36/17
|-
|-
|40
| 40
|1337.05715
| 1337.05715
|243/112
| 243/112
|28/13, 13/6
| 28/13, 13/6
|-
|-
|41
| 41
|1370.4836
| 1370.4836
|108/49
| 108/49
|
|  
|-
|-
|42
| 42
|1403.91
| 1403.91
|9/4
| 9/4
|
|  
|}
|}
== See also ==
* [[36edo]] – relative edo
* [[57edt]] – relative edt
* [[93ed6]] – relative ed6
* [[101ed7]] – relative ed7
* [[129ed12]] – relative ed12, close to the zeta-optimized tuning for 36edo
{{todo|expand}}
[[Category:36edo]]
Retrieved from "https://en.xen.wiki/w/21edf"