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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
|
|  
|}
|}
==Scale tree==
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.
== 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


Generator range: 32.65306 cents (4\7/21 = 4\147) to 34.28571 cents (3\5/21 = 1\35)
{{todo|expand}}
{| class="wikitable center-all"
! colspan="7" |Fifth
!Cents
!Comments
|-
| 4\7|| || ||  || || || || 32.6531||
|-
|  || || || || || ||27\47 ||32.82675||
|-
|  || || || || ||23\40|| ||32.8571||
|-
| || || ||  ||  || ||42\73||32.8767||
|-
| ||  || || ||19\33|| || ||32.9004||
|-
| || || || || || ||53\92 ||32.91925||
|-
|  || || || || ||34\59|| ||32.9298||
|-
| || || || || || ||49\85||32.9412 ||
|-
|  || || ||15\26|| || || ||33.9670||
|-
| || || || || ||  || 56\97||32.9897||
|-
| || || || ||  ||41\71|| || 32.9980||
|-
| || || || || ||  ||67\116|| 33.0049||
|-
|  || || || ||26\45|| || ||33.0159||[[Flattone]] is in this region
|-
| || || || ||  || ||63\109||33.0275 ||
|-
|  || || || ||  ||37\64|| || 33.0357||
|-
| || ||  ||  || || ||48\83||33.0465||
|-
| || ||11\19||  || || || ||33.0827 ||
|-
| || ||  ||  || ||  ||51\88||33.1167 ||
|-
| || || || || ||40\69|| ||33.1263||
|-
| || || || || || || 69\119||33.13325 ||
|-
| || || || ||29\50||  || ||33.1429||
|-
| || || ||  || || ||76\131||33.1516 ||[[Golden meantone]] (696.2145¢)
|-
| ||  || || || ||47\81|| ||33.1570 ||
|-
| || ||  || || || ||65\112||33.1633 ||
|-
| || || ||18\31|| ||  || ||33.1797||[[Meantone]] is in this region
|-
| || || || || || || 61\105||33.1973||
|-
| || ||  || || ||43\74|| ||33.2046||
|-
| || || || || || ||68\117 ||33.2112||
|-
| || || || || 25\43|| || ||33.2226||
|-
| || || || ||  || ||57\98 ||33.23615 ||
|-
| || || || || ||32\55|| ||33.24675||
|-
| || ||  ||  || || ||39\67||33.2623||
|-
| ||7\12|| || || || || ||33.{{Overline|3}}||
|-
|  || || ||  ||  || ||38\65||33.4066||
|-
| || || || || ||31\53|| ||33.4232 ||The fifth closest to a just [[3/2]] for EDOs less than 200
|-
| ||  || || || || ||55\94||33.43465||[[Garibaldi]] / [[Cassandra]]
|-
| || ||  || ||24\41|| || ||33.4495||
|-
|  || || || || || ||65\111||33.4620||
|-
| || || ||  || ||41\70|| ||33.4694||
|-
| || || || ||  || ||58\99||33.4776||
|-
| || || || 17\29|| || || ||33.4975||
|-
| || || || || || ||61\104||33.5165||
|-
| || || || || ||44\75||  ||33.5328||
|-
| || || || || || ||71\121||33.5301||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||33.5404 ||[[Neogothic]] is in this region
|-
| || || || || || ||64\109|| 33.5518||
|-
| || || || || ||37\63|| ||33.5601||
|-
| || || || ||  || ||47\80||33.5714||
|-
|  || ||10\17|| || || || ||33.61345 ||The generator closest to a just [[17/14]] for EDOs less than 4200
|-
| || || || || || ||43\73||33.6595 ||
|-
| ||  || || || ||33\56|| ||33.6735||
|-
| ||  || || || || || 56\95||33.6842||
|-
| || ||  || ||23\39|| || ||33.6996||
|-
| || || || || || ||59\100||33.7143||
|-
| || || || || ||36\61|| ||33.72365||
|-
| || || || || || ||49\83||33.7349||
|-
| || || ||13\22|| || || ||33.7662||[[Archy]] is in this region
|-
| || || || || || ||42\71||33.8028 ||
|-
| || || || || ||29\49 || ||33.8192||
|-
| || || || || || ||45\76 ||33.8346||
|-
| || || || ||16\27|| || ||33.8624||
|-
| || ||  || || || ||35\59||33.8983||
|-
| || || || || ||19\32|| || 33.9286||
|-
| || || || || || ||22\37|| 33.9768||
|-
|3\5|| || || || || || ||34.2857||
|}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:36edo]]
[[Category:Edf]]
[[Category:Edonoi]]
Retrieved from "https://en.xen.wiki/w/21edf"