2edf: Difference between revisions
m Removing from Category:Edonoi using Cat-a-lot |
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{{Infobox ET}} | {{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 | '''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. | ||
== Factoids about 2edf == | == Factoids about 2edf == | ||
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! # | ! # | ||
! Cents | ! Cents | ||
!JI ratios | |||
|- | |- | ||
| 1 | | 1 | ||
| 350.98 | | 350.98 | ||
|11/9 | |||
|- | |- | ||
| 2 | | 2 | ||
| 701.96 | | 701.96 | ||
|exact 3/2 | |||
|} | |} | ||
==Scale tree== | ==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. | {{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. | If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum. | ||
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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. | 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. | ||