|
|
| Line 134: |
Line 134: |
| | | | | |
| |} | | |} |
| ==Scale tree==
| |
|
| |
| EDFs can be approximated by subdividing diatonic fifths between 4\7 and 3\5.
| |
|
| |
| 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.
| |
| [[Category:Edf]] | | [[Category:Edf]] |
| [[Category:Edonoi]] | | [[Category:Edonoi]] |
| 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.
Intervals
| degree
|
cents value
|
corresponding
JI intervals
|
comments
|
| 0
|
exact 1/1
|
|
| 1
|
58.49625
|
91/88, 88/85
|
|
| 2
|
116.9925
|
15/14
|
|
| 3
|
175.48875
|
21/19
|
|
| 4
|
233.9850
|
8/7
|
|
| 5
|
292.48125
|
45/38
|
|
| 6
|
350.9775
|
11/9, 27/22
|
|
| 7
|
409.47375
|
19/15
|
|
| 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
|
|