|
|
| Line 137: |
Line 137: |
| |D | | |D |
| |} | | |} |
| ==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.
| |
|
| |
| If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum.
| |
|
| |
| Generator range: 137.1429 cents (4\7/5 = 4\35) to 144 cents (3\5/5 = 3\25)
| |
| {| class="wikitable center-all"
| |
| ! colspan="7" |Fifth
| |
| !Cents
| |
| !Comments
| |
| |-
| |
| |4\7|| || || || || || ||137.143||
| |
| |-
| |
| | || || || || || ||27\47||137.872 ||
| |
| |-
| |
| | || || || || ||23\40|| || 138.000||
| |
| |-
| |
| | || || || || || ||42\73||138.082||
| |
| |-
| |
| | || || || ||19\33|| || ||138.{{Overline|18}}||
| |
| |-
| |
| | || || || || || ||53\92||138.261||
| |
| |-
| |
| | || || || || ||34\59|| ||138.305||
| |
| |-
| |
| | || || || || || ||49\85||138.353||
| |
| |-
| |
| | || || ||15\26|| || || ||138.4615||
| |
| |-
| |
| | || || || || || ||56\97||138.557||
| |
| |-
| |
| | || || || || ||41\71|| ||138.5915||The generator closest to a just [[13/12]] for EDOs less than 1000
| |
| |-
| |
| | || || || || || ||67\116||138.621||
| |
| |-
| |
| | || || || ||26\45|| || ||138.{{Overline|6}}||[[Flattone]] is in this region
| |
| |-
| |
| | || || || || || ||63\109||138.716||
| |
| |-
| |
| | || || || || ||37\64|| ||138.750||
| |
| |-
| |
| | || || || || || ||48\83||138.795||
| |
| |-
| |
| | || ||11\19|| || || || ||138.947||
| |
| |-
| |
| | || || || || || ||51\88||139.{{Overline|09}}||
| |
| |-
| |
| | || || || || ||40\69|| ||139.130||
| |
| |-
| |
| | || || || || || ||69\119||139.160||
| |
| |-
| |
| | || || || ||29\50|| || ||139.200||
| |
| |-
| |
| | || || || || || ||76\131||139.237||[[Golden meantone]] (696.2145¢)
| |
| |-
| |
| | || || || || ||47\81|| ||139.{{Overline|259}}||
| |
| |-
| |
| | || || || || || ||65\112||139.286||
| |
| |-
| |
| | || || ||18\31|| || || ||139.355||[[Meantone]] is in this region
| |
| |-
| |
| | || || || || || ||61\105||139.429||
| |
| |-
| |
| | || || || || ||43\74|| ||139.{{Overline|459}}||
| |
| |-
| |
| | || || || || || ||68\117||139.487||
| |
| |-
| |
| | || || || ||25\43|| || ||139.535||
| |
| |-
| |
| | || || || || || ||57\98||139.592||
| |
| |-
| |
| | || || || || ||32\55|| ||139.{{Overline|63}}||
| |
| |-
| |
| | || || || || || ||39\67||139.7015||
| |
| |-
| |
| | ||7\12|| || || || || ||140.000||
| |
| |-
| |
| | || || || || || ||38\65||140.308||
| |
| |-
| |
| | || || || || ||31\53|| ||140.377||The fifth closest to a just [[3/2]] for EDOs less than 200
| |
| |-
| |
| | || || || || || ||55\94||140.4255||[[Garibaldi]] / [[Cassandra]]
| |
| |-
| |
| | || || || ||24\41|| || ||140.488||
| |
| |-
| |
| | || || || || || ||65\111||140.{{Overline|540}}||
| |
| |-
| |
| | || || || || ||41\70|| ||140.571||
| |
| |-
| |
| | || || || || || ||58\99||140.{{Overline|60}}||
| |
| |-
| |
| | || || ||17\29|| || || ||140.690||
| |
| |-
| |
| | || || || || || ||61\104||140.769||
| |
| |-
| |
| | || || || || ||44\75|| ||140.800||
| |
| |-
| |
| | || || || || || ||71\121||140.826||Golden neogothic (704.0956¢)
| |
| |-
| |
| | || || || ||27\46|| || ||140.870||[[Neogothic]] is in this region
| |
| |-
| |
| | || || || || || ||64\109||140.917||
| |
| |-
| |
| | || || || || ||37\63|| ||140.952||
| |
| |-
| |
| | || || || || || ||47\80||141.000||
| |
| |-
| |
| | || ||10\17|| || || || ||141.1765||
| |
| |-
| |
| | || || || || || ||43\73||141.370||
| |
| |-
| |
| | || || || || ||33\56|| ||141.429||
| |
| |-
| |
| | || || || || || ||56\95||141.474||
| |
| |-
| |
| | || || || ||23\39|| || ||141.5385||
| |
| |-
| |
| | || || || || || ||59\100||141.600||
| |
| |-
| |
| | || || || || ||36\61|| ||141.639||
| |
| |-
| |
| | || || || || || ||49\83||141.687||
| |
| |-
| |
| | || || ||13\22|| || || ||141.{{Overline|81}}||[[Archy]] is in this region
| |
| |-
| |
| | || || || || || ||42\71||141.972||
| |
| |-
| |
| | || || || || ||29\49|| ||142.041||
| |
| |-
| |
| | || || || || || ||45\76||142.105||
| |
| |-
| |
| | || || || ||16\27|| || ||142.{{Overline|2}}||
| |
| |-
| |
| | || || || || || ||35\59||142.373||
| |
| |-
| |
| | || || || || ||19\32|| ||142.500||
| |
| |-
| |
| | || || || || || ||22\37||142.{{Overline|702}}||
| |
| |-
| |
| |3\5|| || || || || || ||144.000||
| |
| |}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.
| |
|
| |
| 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:Edonoi]]
| |
| Prime factorization
|
5 (prime)
|
| Step size
|
140.391 ¢
|
| Octave
|
9\5edf (1263.52 ¢)
|
| Twelfth
|
14\5edf (1965.47 ¢)
|
| Consistency limit
|
3
|
| Distinct consistency limit
|
3
|
5EDF is the equal division of the just perfect fifth into five parts of 140.391 cents each, corresponding to 8.5476 edo. It is close to the Bleu generator chain and every second step of 17edo. 4 steps of 5edf is a fraction of a cent away to the seventh harmonic (which is 112/81 instead of 7/4 since the equave is 3/2), which is an extremely accurate approximation for the size of this scale.
Intervals
| degree
|
cents value
|
octave-reduced cents value
|
approximate ratios
|
Neptunian notation
|
| 0
|
|
1/1
|
perfect unison
|
C
|
| 1
|
140.391
|
|
13/12, 49/45
|
augmented unison, minor second
|
C#, Db
|
| 2
|
280.782
|
|
75/64, 20/17, 13/11
|
major second, minor third
|
D, Eb
|
| 3
|
421.173
|
|
14/11, 23/18
|
major third, diminished fourth
|
E, Fb
|
| 4
|
561.564
|
|
11/8, 18/13, 25/18
|
perfect fourth
|
F
|
| 5
|
701.955
|
|
3/2
|
perfect fifth
|
C
|
| 6
|
842.346
|
|
21/13, 13/8, 18/11
|
augmented fifth, minor sixth
|
C#, Db
|
| 7
|
982.737
|
|
7/4, 30/17
|
major sixth, minor seventh
|
D, Eb
|
|
| 8
|
1123.128
|
|
|
major seventh, minor octave
|
E, Fb
|
|
| 9
|
1263.519
|
63.519
|
|
major octave
|
F
|
| 10
|
1403.910
|
203.910
|
|
|
C
|
| 11
|
1544.301
|
344.301
|
|
|
C#, Db
|
| 12
|
1684.692
|
484.692
|
|
|
D, Eb
|
| 13
|
1825.083
|
625.083
|
|
|
E
|
| 14
|
1965.474
|
765.474
|
|
|
F
|
| 15
|
2105.865
|
905.865
|
|
|
C
|
| 16
|
2246.256
|
1046.256
|
|
|
C#, Db
|
| 17
|
2386.647
|
1186.647
|
|
|
D
|