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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)^(1/3) 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]].
{{Infobox ET}}
{{ED intro}}


==Factoids about 3EDF==
== Theory ==
3EDF is related to the [[Gamelismic clan|gamelismic temperaments]], which temper out 1029/1024 in the 7-limit.
3edf divides 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 ==
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 ==
== Intervals ==
{| class="wikitable"
{| class="wikitable center-all"
|+
! #
! rowspan="2" |
! Cents
! rowspan="2" |ed31\54
! Approximate JI ratios
! rowspan="2" |ed121/81 (~ed11\19)
! rowspan="2" |ed3/2
! colspan="2" |Pyrite
! rowspan="2" |ed122/81 (~ed13\22)
! rowspan="2" |ed34\57
|-
|-
!(~ed17\29)
| 1
!(~ed10\17)
| 233.99
| [[8/7]]
|-
|-
|1
| 2
|229.63
| 467.97
|231.605
| [[21/16]], [[17/13]]
|233.985
|234.545
|235.285
|236.355
|238.597
|-
|-
|2
| 3
|259.259
| 701.96
|463.211
| exact [[3/2]]
|467.97
|469.091
|470.57
|472.71
|477.193
|-
|3
|688.888
|694.816
|701.995
|703.636
|705.8885
|709.065
|715.7895
|-
|4
|918.5185
|926.421
|935.94
|938.181
|941.141
|945.42
|954.386
|-
|5
|1148.148
|1158.0265
|1169.925
|1172.727
|1176.426
|1181.775
|1192.9825
|-
|6
|1377.778
|1389.632
|1403.91
|1407.272
|1411.711
|1418.13
|1431.579
|}
|}
[[Category:Edf]]
 
[[Category:Edonoi]]
== Music ==
* [https://www.youtube.com/watch?v=0ecvufTJowE Sequences & Chaos] by Bazil Müzik
 
[[Category:Listen]]
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