3edf: Difference between revisions

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~~'''~~3edf~~'''~~, if the attempt is made to use it as an actual scale, would divide the [[just perfect fifth]] into three equal parts, each of size 233.985 cents, which is to say (3/2)<sup>1/3</sup> as a frequency ratio. It corresponds to 5.1285 [[edo]]. ~~If we want to consider it to~~ be a ~~temperament, it tempers out~~ [[~~16/15~~]], [[~~21/20~~]], [[~~28/27]], [[81/80~~]], and ~~[[256/243]] as well as [[5edo]]~~.

== Theory ==

3edf, if the attempt is made to use it as an actual scale, would divide the [[just perfect fifth]] into three equal parts, each of size 233.985 cents, which is to say (3/2)<sup>1/3</sup> as a frequency ratio. It corresponds to 5.1285 [[edo]].

It can also be treated as a heavily compressed version of [[5edo]], with the octave compressed by about 30 cents. Its [[patent val]] matches that of 5edo up to the [[7-limit]], and thus tempers out the same commas.

=== Harmonics ===

== Factoids about 3edf ==

3edf~~'s step size~~ is ~~close~~ to the [[slendric]] temperament, which tempers out 1029/1024 in the 2.3.7 subgroup~~. It can be used as a~~ slendric ~~tuning~~. It also works well as a tuning for [[Extraclassical tonality|arto and tendo chords.]]

3edf is essentially equivalent to the [[slendric]] temperament, which tempers out 1029/1024 in the 2.3.7 subgroup, without octave repetition, and its step size is the 1/3-comma tuning of the slendric generator (approximated by, for instance, [[41edo|8\41]] and [[200edo|39\200]]). It also works well as a tuning for [[Extraclassical tonality|arto and tendo chords.]]

== Intervals ==

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| 1

| 233.99

|8/7

| [[8/7]]

|-

| 2

| 467.97

|55/42

| [[21/16]], [[17/13]]

|-

| 3

| 701.96

|exact 3/2

| exact [[3/2]]

|}

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 {{Infobox ET}}
-'''3edf''', if the attempt is made to use it as an actual scale, would divide the [[just perfect fifth]] into three equal parts, each of size 233.985 cents, which is to say (3/2)<sup>1/3</sup> as a frequency ratio. It corresponds to 5.1285 [[edo]]. If we want to consider it to be a temperament, it tempers out [[16/15]], [[21/20]], [[28/27]], [[81/80]], and [[256/243]] as well as [[5edo]].
+{{ED intro}}
+== Theory ==
+edf, if the attempt is made to use it as an actual scale, would divide the [[just perfect fifth]] into three equal parts, each of size 233.985 cents, which is to say (3/2)<sup>1/3</sup> as a frequency ratio. It corresponds to 5.1285 [[edo]].
+It can also be treated as a heavily compressed version of [[5edo]], with the octave compressed by about 30 cents. Its [[patent val]] matches that of 5edo up to the [[7-limit]], and thus tempers out the same commas.
+=== Harmonics ===
+{{Harmonics in equal|3|3|2}}
 == Factoids about 3edf ==
-edf's step size is close to the [[slendric]] temperament, which tempers out 1029/1024 in the 2.3.7 subgroup. It can be used as a slendric tuning. It also works well as a tuning for [[Extraclassical tonality|arto and tendo chords.]]
+edf is essentially equivalent to the [[slendric]] temperament, which tempers out 1029/1024 in the 2.3.7 subgroup, without octave repetition, and its step size is the 1/3-comma tuning of the slendric generator (approximated by, for instance, [[41edo|8\41]] and [[200edo|39\200]]). It also works well as a tuning for [[Extraclassical tonality|arto and tendo chords.]]
 == Intervals ==
@@ Line 13: / Line 21: @@
 | 1
 | 233.99
-|8/7
+| [[8/7]]
 |-
 | 2
 | 467.97
-|55/42
+| [[21/16]], [[17/13]]
 |-
 | 3
 | 701.96
-|exact 3/2
+| exact [[3/2]]
 |}