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| Line 136: |
Line 136: |
| | C | | | C |
| |} | | |} |
| ==Scale tree==
| |
| EDF scales can be approximated in [[EDO]]s by subdividing diatonic fifths. 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.
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|
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| If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum.
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|
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| Generator range: 114.2857 cents (4\7/6 = 2\21) to 120 cents (3\5/6 = 1\10)
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| {| class="wikitable center-all"
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| ! colspan="7" |Fifth
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| !Cents
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| !Comments
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| |-
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| |4\7|| || || || || || ||114.286||
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| |-
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| | || || || || || || 27\47||114.894||
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| |-
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| | || || || || ||23\40|| ||115.000||
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| |-
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| | || || || || || ||42\73||115.0685||
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| |-
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| | || || || || 19\33|| || ||115.{{Overline|15}}||
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| |-
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| | || || || || || || 53\92||115.217||
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| |-
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| | || || || || || 34\59|| ||115.254||
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| |-
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| | || || || || || ||49\85 ||115.294||
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| |-
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| | || || ||15\26|| || || ||115.385||
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| |-
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| | || || || || || ||56\97 ||115.464||
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| |-
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| | || || || || ||41\71 || ||115.493||
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| |-
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| | || || || || || ||67\116 ||115.517||
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| |-
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| | || || || ||26\45|| || ||115.{{Overline|5}}||[[Flattone]] is in this region
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| |-
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| | || || || || || ||63\109||115.596||
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| |-
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| | || || || || ||37\64|| ||115.625||
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| |-
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| | || || || || || ||48\83||115.663||
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| |-
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| | || ||11\19|| || || || ||115.684||
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| |-
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| | || || || || || ||51\88||115.{{Overline|90}}||
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| |-
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| | || || || || ||40\69|| ||115.942||
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| |-
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| | || || || || || ||69\119||115.966||
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| |-
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| | || || || ||29\50|| || ||116.000||
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| |-
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| | || || || || || ||76\131||116.0305||[[Golden meantone]] (696.2145¢)
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| |-
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| | || || || || ||47\81|| ||116.049||
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| |-
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| | || || || || || ||65\112||116.071||
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| |-
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| | || || ||18\31|| || || ||116.129||[[Meantone]] is in this region
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| |-
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| | || || || || || ||61\105||116.1905||
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| |-
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| | || || || || ||43\74|| ||116.{{Overline|216}}||
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| |-
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| | || || || || || ||68\117||116.239||
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| |-
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| | || || || ||25\43|| || ||116.279||
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| |-
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| | || || || || || ||57\98||116.3265||
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| |-
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| | || || || || ||32\55|| ||116.{{Overline|36}}||
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| |-
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| | || || || || || ||39\67||116.418||
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| |-
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| | ||7\12|| || || || || ||116.{{Overline|6}}||
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| |-
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| | || || || || || ||38\65||116.923||
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| |-
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| | || || || || ||31\53|| ||116.981||The fifth closest to a just [[3/2]] for EDOs less than 200
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| |-
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| | || || || || || ||55\94||117.021||[[Garibaldi]] / [[Cassandra]]
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| |-
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| | || || || ||24\41|| || ||117.073||
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| |-
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| | || || || || || ||65\111||117.{{Overline|117}}||
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| |-
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| | || || || || ||41\70|| ||117.143||
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| |-
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| | || || || || || ||58\99||117.{{Overline|17}}||
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| |-
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| | || || ||17\29|| || || ||117.241||
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| |-
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| | || || || || || ||61\104||117.308||
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| |-
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| | || || || || ||44\75|| ||117.{{Overline|3}}||
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| |-
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| | || || || || || ||71\121||117.355||Golden neogothic (704.0956¢)
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| |-
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| | || || || ||27\46|| || ||117.391||[[Neogothic]] is in this region
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| |-
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| | || || || || || ||64\109||117.431||
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| |-
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| | || || || || ||37\63|| ||117.460||
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| |-
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| | || || || || || ||47\80||117.500||
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| |-
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| | || ||10\17|| || || || ||117.647||
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| |-
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| | || || || || || ||43\73||117.808||
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| |-
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| | || || || || ||33\56|| ||117.857||
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| |-
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| | || || || || || ||56\95||117.895||
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| |-
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| | || || || ||23\39|| || ||117.949||
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| |-
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| | || || || || || ||59\100||118.000||
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| |-
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| | || || || || ||36\61|| ||118.033||
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| |-
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| | || || || || || ||49\83||118.072||
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| |-
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| | || || ||13\22|| || || ||118.{{Overline|18}}||[[Archy]] is in this region
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| |-
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| | || || || || || ||42\71||118.310||
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| |-
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| | || || || || ||29\49|| ||118.367||
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| |-
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| | || || || || || ||45\76||118.421||
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| |-
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| | || || || ||16\27|| || ||118.{{Overline|518}}||
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| |-
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| | || || || || || ||35\59||118.644||
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| |-
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| | || || || || ||19\32|| ||118.750||
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| |-
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| | || || || || || ||22\37||118.{{Overline|918}}||The generator closest to a just [[15/14]] for EDOs less than 1200
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| |-
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| |3\5|| || || || || || ||120.000||
| |
| |}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.
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|
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| 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.
| |
| == Compositions == | | == Compositions == |
| * [http://www.seraph.it/dep/det/metashakti.mp3 Metashakti] by [[Carlo Serafini]] | | * [http://www.seraph.it/dep/det/metashakti.mp3 Metashakti] by [[Carlo Serafini]] |
| Prime factorization
|
2 × 3
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| Step size
|
116.993 ¢
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| Octave
|
10\6edf (1169.93 ¢) (→ 5\3edf)
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| Twelfth
|
16\6edf (1871.88 ¢) (→ 8\3edf)
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| Consistency limit
|
3
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| Distinct consistency limit
|
3
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| Special properties
|
|
6EDF is the equal division of the just perfect fifth into six parts of 116.9925 cents each, corresponding to 10.2571 edo. It is related to the miracle temperament, which tempers out 225/224 and 1029/1024 in the 7-limit.
Intervals
| degrees
|
cents ~ cents octave-reduced
|
approximate ratios
|
Neptunian notation
|
| 0
|
0 (perfect unison, 1:1)
|
1/1
|
C
|
| 1
|
117
|
16/15, 15/14
|
C#
|
| 2
|
234
|
8/7
|
Db
|
| 3
|
351
|
11/9, 27/22
|
D
|
| 4
|
468
|
21/16
|
E
|
| 5
|
585
|
7/5, 45/32
|
F
|
| 6
|
702 (just perfect fifth, 3:2)
|
3/2
|
C
|
| 7
|
819
|
8/5, 21/13
|
C#
|
| 8
|
936
|
12/7, 55/32
|
Db
|
| 9
|
1053
|
11/6
|
D
|
| 10
|
1170
|
49/25, 160/81, 2/1
|
E
|
| 11
|
1287 ~ 87
|
|
F
|
| 12
|
1404 ~ 204 (just major whole tone/ninth, 9:4)
|
|
C
|
| 13
|
1521 ~ 321
|
|
C#
|
| 14
|
1638 ~ 438
|
|
Db
|
| 15
|
1755 ~ 555
|
|
D
|
| 16
|
1872 ~ 672
|
|
E
|
| 17
|
1988 ~ 788
|
|
F
|
| 18
|
2106 ~ 906 (Pythagorean major sixth, 27:8)
|
|
C
|
| 19
|
2223 ~ 1023
|
|
C#
|
| 20
|
2340 ~ 1140
|
|
Db
|
| 21
|
2457 ~ 57
|
|
D
|
| 22
|
2574 ~ 174
|
|
E
|
| 23
|
2691 ~ 291
|
|
F
|
| 24
|
2808 ~ 408 (Pythagorean major third, 81:16)
|
|
C
|
Compositions