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{{Infobox ET}}
{{Infobox ET}}
'''12EDF''' is the [[EDF|equal division of the just perfect fifth]] into 12 parts of 58.49625 [[cent|cents]] each, corresponding to 20.5141 [[edo]] (similar to every second step of [[41edo]]). It is an intersection of [[3edf]]~[[5edo]] and [[4edf]]~[[7edo]] relations, and could pass as both [[20edo]] and [[21edo]], with both relations nearly breaking down by this point. It is related to the [[Tetracot family|dodecacot temperament]], which tempers out 3087/3125 and 10976/10935 in the 7-limit.
'''12EDF''' is the [[EDF|equal division of the just perfect fifth]] into 12 parts of 58.49625 [[cent|cents]] each, corresponding to 20.5141 [[edo]] (similar to every second step of [[41edo]]). It is an intersection of [[3edf]]~[[5edo]] and [[4edf]]~[[7edo]] relations, and could pass as both [[20edo]] and [[21edo]], with both relations nearly breaking down by this point. It is related to the [[Tetracot family#Dodecacot|dodecacot temperament]], which tempers out 3087/3125 and 10976/10935 in the 7-limit.
 
It is a strong [[half-prime subgroup|3/2.5/2.7/2 subgroup]] system, a fact first noted by [[User:CompactStar|CompactStar]], tempering out the commas [[10976/10935]] and [[3125/3087]], although the representation of [[11/2]] is more questionable. [[24edf]] (effectively 41edo) provides a correction for 11/2. It contains the [[macrodiatonic and microdiatonic scales|microdiatonic]] scale that corresponds to 12edo's [[5L 2s|diatonic scale]] with [[2/1]] compressed to [[3/2]].
 
==Harmonics==
{{Harmonics in equal|12|3|2}}
{{Harmonics in equal|12|3|2|start=12|collapsed=1}}


==Intervals==
==Intervals==
Line 16: Line 22:
| | 1
| | 1
| | 58.49625
| | 58.49625
| | 91/88, 88/85
| | [[28/27]], 91/88, 88/85
| |  
| |  
|-
|-
Line 26: Line 32:
| | 3
| | 3
| | 175.48875
| | 175.48875
| | [[21/19]]
| | [[10/9]], [[21/19]]
| |  
| |  
|-
|-
Line 46: Line 52:
| | 7
| | 7
| | 409.47375
| | 409.47375
| | [[19/15]]
| | [[19/15]], [[63/50]]
| |  
| |  
|-
|-
Line 66: Line 72:
| | 11
| | 11
| | 643.4588
| | 643.4588
| |13/9
| | [[13/9]]
| |  
| |  
|-
|-
Line 134: Line 140:
|
|
|}
|}
==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.
Generator range: 57.14286 cents (4\7/12 = 1\21) to 60 cents (3\5/12 = 1\20)
{| class="wikitable center-all"
! colspan="7" |Fifth
!Cents
!Comments
|-
|4\7 || || || || || || ||57.1429||
|-
| || ||  || || || ||27\47||57.4468 ||
|-
|  || ||  || || ||23\40|| ||57.5000||
|-
| || ||  || || ||  ||42\73||57.53425||
|-
| ||  || || || 19\33|| || || 57.{{Overline|57}}||
|-
| || || || || || ||53\92||57.6087||
|-
| || || || || ||34\59|| ||57.6271||
|-
| || || || || || ||49\85||57.6471||
|-
| || || ||15\26|| || || ||57.6923||
|-
| || || || || || ||56\97||57.7320||
|-
| || || || || ||41\71|| ||57.7465||
|-
| || || || || || ||67\116||57.7586||
|-
| || || || ||26\45|| || ||57.{{Overline|7}}||[[Flattone]] is in this region
|-
| || || || || || ||63\109||57.7982||
|-
| || || || || ||37\64|| ||57.8125||
|-
| || || || || || ||48\83||57.8313||
|-
| || ||11\19|| || || || ||57.8947||
|-
| || || || || || ||51\88||57.9{{Overline|54}}||
|-
| || || || || ||40\69|| ||57.9710||
|-
| || || || || || ||69\119||57.9832||
|-
| || || || ||29\50|| || ||58.000||
|-
| || || || || || ||76\131||58.0153||[[Golden meantone]] (696.2145¢)
|-
| || || || || ||47\81|| ||58.0247||
|-
| || || || || || ||65\112||58.0357||
|-
| || || ||18\31|| || || ||58.0645||[[Meantone]] is in this region
|-
| || || || || || ||61\105||58.0952||
|-
| || || || || ||43\74|| ||58.{{Overline|108}}||
|-
| || || || || || ||68\117||58.1967||
|-
| || || || ||25\43|| || ||58.1935||
|-
| || || || || || ||57\98||58.1633||
|-
| || || || || ||32\55|| ||58.{{Overline|18}}||
|-
| || || || || || ||39\67||58.2090||The generator closest to a just [[7/5]] for EDOs less than 2400
|-
| ||7\12|| || || || || ||58.{{Overline|3}}||
|-
| || || || || || ||38\65||58.4615||
|-
| || || || || ||31\53|| ||58.4906||The fifth closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || ||55\94||58,5106||[[Garibaldi]] / [[Cassandra]]
|-
| || || || ||24\41|| || ||58.5366||
|-
| || || || || || ||65\111||58.{{Overline|558}}||
|-
| || || || || ||41\70|| ||58.5714||
|-
| || || || || || ||58\99||58.{{Overline|58}}||
|-
| || || ||17\29|| || || ||58.6207||
|-
| || || || || || ||61\104||58.65385||
|-
| || || || || ||44\75|| ||58.{{Overline|6}}||
|-
| || || || || || ||71\121||58.6777||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||58.69565||[[Neogothic]] is in this region
|-
| || || || || || ||64\109||58.7155||
|-
| || || || || ||37\63|| ||58.7302||
|-
| || || || || || ||47\80||58.7500||
|-
| || ||10\17|| || || || ||58.8235||
|-
| || || || || || ||43\73||58.9041||
|-
| || || || || ||33\56|| ||58.9286||
|-
| || || || || || ||56\95||58.9474||
|-
| || || || ||23\39|| || ||58.9744||
|-
| || || || || || ||59\100||59.0000||
|-
| || || || || ||36\61|| ||59.0164||
|-
| || || || || || ||49\83||59.0361||
|-
| || || ||13\22|| || || ||59.{{Overline|09}}||[[Archy]] is in this region
|-
| || || || || || ||42\71||59.1549||
|-
| || || || || ||29\49|| ||59.1837||
|-
| || || || || || ||45\76||59.2105||
|-
| || || || ||16\27|| || ||59.{{Overline|259}}||
|-
| || || || || || ||35\59||59.3220||
|-
| || || || || ||19\32|| ||59.3750||
|-
| || || || || || ||22\37||59.{{Overline|459}}||
|-
|3\5|| || || || || || ||60.0000||
|}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.
{{todo|expand}}
[[Category:Edf]]
[[Category:Edonoi]]

Latest revision as of 19:20, 1 August 2025

← 11edf 12edf 13edf →
Prime factorization 22 × 3 (highly composite)
Step size 58.4963 ¢ 
Octave 21\12edf (1228.42 ¢) (→ 7\4edf)
Twelfth 33\12edf (1930.38 ¢) (→ 11\4edf)
Consistency limit 3
Distinct consistency limit 3

12EDF is the equal division of the just perfect fifth into 12 parts of 58.49625 cents each, corresponding to 20.5141 edo (similar to every second step of 41edo). It is an intersection of 3edf~5edo and 4edf~7edo relations, and could pass as both 20edo and 21edo, with both relations nearly breaking down by this point. It is related to the dodecacot temperament, which tempers out 3087/3125 and 10976/10935 in the 7-limit.

It is a strong 3/2.5/2.7/2 subgroup system, a fact first noted by CompactStar, tempering out the commas 10976/10935 and 3125/3087, although the representation of 11/2 is more questionable. 24edf (effectively 41edo) provides a correction for 11/2. It contains the microdiatonic scale that corresponds to 12edo's diatonic scale with 2/1 compressed to 3/2.

Harmonics

Approximation of harmonics in 12edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +28.4 +28.4 -1.7 +21.5 -1.7 +24.0 +26.8 -1.7 -8.6 +1.9 +26.8
Relative (%) +48.6 +48.6 -2.8 +36.8 -2.8 +41.0 +45.8 -2.8 -14.6 +3.3 +45.8
Steps
(reduced) 		21
(9) 		33
(9) 		41
(5) 		48
(0) 		53
(5) 		58
(10) 		62
(2) 		65
(5) 		68
(8) 		71
(11) 		74
(2)
Approximation of harmonics in 12edf
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) +5.2 -6.1 -8.6 -3.3 +8.7 +26.8 -8.3 +19.9 -6.1 -28.2 +11.9
Relative (%) +8.9 -10.5 -14.6 -5.7 +14.9 +45.8 -14.3 +33.9 -10.5 -48.1 +20.3
Steps
(reduced) 		76
(4) 		78
(6) 		80
(8) 		82
(10) 		84
(0) 		86
(2) 		87
(3) 		89
(5) 		90
(6) 		91
(7) 		93
(9)

Intervals

degree cents value corresponding
JI intervals 		comments
0 exact 1/1
1 58.49625 28/27, 91/88, 88/85
2 116.9925 15/14
3 175.48875 10/9, 21/19
4 233.9850 8/7
5 292.48125 45/38
6 350.9775 11/9, 27/22
7 409.47375 19/15, 63/50
8 467.9700 21/16
9 526.46625 19/14
10 584.9625 7/5
11 643.4588 13/9
12 701.9550 exact 3/2 just perfect fifth
13 760.45125 273/176, 132/85
14 818.9475 8/5
15 877.44375 63/38
16 935.94 12/7
17 994.43625 135/76
18 1052.9325 11/6, 81/44
19 1111.42875 19/10
20 1169.925 63/32
21 1228.42125 57/28
22 1286.9175 21/10
23 1345.41375 13/6
24 1403.91 exact 9/4
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