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{{Infobox ET}}
'''2edf''', if the attempt is made to use it as an actual scale, would divide the [[just perfect fifth]] into two equal parts, each of size 350.9775 [[cent]]s, which is to say sqrt (3/2) as a frequency ratio. It corresponds to 3.4190 [[edo]]. If we want to consider it to be a temperament, it tempers out [[6/5]], [[9/7]], [[32/27]], and [[81/80]] in the patent val.
'''2edf''', if the attempt is made to use it as an actual scale, would divide the [[just perfect fifth]] into two equal parts, each of size 350.9775 [[cent]]s, which is to say sqrt (3/2) as a frequency ratio. It corresponds to 3.4190 [[edo]]. If we want to consider it to be a temperament, it tempers out [[6/5]], [[9/7]], [[32/27]], and [[81/80]] in the patent val.


Line 10: Line 11:
! #
! #
! Cents
! Cents
!JI ratios
|-
|-
| 1
| 1
| 350.98
| 350.98
|11/9
|-
|-
| 2
| 2
| 701.96
| 701.96
|exact 3/2
|}
|}


[[Category:Edf]]
==Scale tree==
[[Category:Edonoi]]
{{todo|correct maths|review|inline=1|text=The text and the table incoherently mix up EDO and EDF calculations. This section should also be moved to a more appropriate page.}}
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: 342.8571 cents (4\7/2 = 2\7) to 360 cents (3\5/2 = 3/10)
{| class="wikitable center-all"
! colspan="7" |Fifth
!Cents
!Comments
|-
|4\7|| || || || || || ||342.857||
|-
| || || || || || ||27\47||344.681||
|-
| || || || || ||23\40|| ||345.000||
|-
| || || || || || ||42\73||345.2055||
|-
| || || || ||19\33|| || ||345.{{Overline|45}}||
|-
| || || || || || ||53\92||345.652||
|-
| || || || || ||34\59|| ||345.763||
|-
| || || || || || ||49\85||345.882||
|-
| || || ||15\26|| || || ||346.153||
|-
| || || || || || ||56\97||346.392||
|-
| || || || || ||41\71|| ||346.479||
|-
| || || || || || ||67\116||346.551||
|-
| || || || ||26\45|| || ||346.{{Overline|6}}||[[Flattone]] is in this region
|-
| || || || || || ||63\109||346.789||
|-
| || || || || ||37\64|| ||346.875||
|-
| || || || || || ||48\83||346.988||
|-
| || ||11\19|| || || || ||347.368||The generator closest to a just [[11/9]] for EDOs less than 200
|-
| || || || || || ||51\88||347.{{Overline|72}}||
|-
| || || || || ||40\69|| ||347.826||
|-
| || || || || || ||69\119||347.899||
|-
| || || || ||29\50|| || ||348.000||
|-
| || || || || || ||76\131||348.092||[[Golden meantone]] (696.2145¢)
|-
| || || || || ||47\81|| ||348.{{Overline|148}}||
|-
| || || || || || ||65\112||348.214||
|-
| || || ||18\31|| || || ||348.387||[[Meantone]] is in this region
|-
| || || || || || ||61\105||348.571||
|-
| || || || || ||43\74|| ||348.{{Overline|648}}||
|-
| || || || || || ||68\117||348.718||
|-
| || || || ||25\43|| || ||348.837||
|-
| || || || || || ||57\98||348.980||
|-
| || || || || ||32\55|| ||349.{{Overline|09}}||
|-
| || || || || || ||39\67||349.254||
|-
| ||7\12|| || || || || ||350.000||
|-
| || || || || || ||38\65||350.769||
|-
| || || || || ||31\53|| ||350.843||The fifth closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || ||55\94||351.064||[[Garibaldi]] / [[Cassandra]]
|-
| || || || ||24\41|| || ||351.2195||
|-
| || || || || || ||65\111||351.{{Overline|351}}||
|-
| || || || || ||41\70|| ||351.429||
|-
| || || || || || ||58\99||351.{{Overline|51}}||
|-
| || || ||17\29|| || || ||351.724||
|-
| || || || || || ||61\104||351.923||
|-
| || || || || ||44\75|| ||352.000||
|-
| || || || || || ||71\121||352.066||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||352.174||[[Neogothic]] is in this region
|-
| || || || || || ||64\109||352.294||
|-
| || || || || ||37\63|| ||352.381||
|-
| || || || || || ||47\80||352.500||
|-
| || ||10\17|| || || || ||352.941||
|-
| || || || || || ||43\73||353.425||
|-
| || || || || ||33\56|| ||353.571||
|-
| || || || || || ||56\95||353.684||
|-
| || || || ||23\39|| || ||353.846||
|-
| || || || || || ||59\100||354.000||
|-
| || || || || ||36\61|| ||354.098||
|-
| || || || || || ||49\83||354.217||
|-
| || || ||13\22|| || || ||354.{{Overline|54}}||[[Archy]] is in this region
|-
| || || || || || ||42\71||354.930||
|-
| || || || || ||29\49|| ||355.102||
|-
| || || || || || ||45\76||355.263||
|-
| || || || ||16\27|| || ||355.{{Overline|5}}||
|-
| || || || || || ||35\59||355.932||
|-
| || || || || ||19\32|| ||356.250||
|-
| || || || || || ||22\37||356.{{Overline|756}}||
|-
|3\5|| || || || || || ||360.000||
|}
 
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.
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