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{{Infobox ET}}
{{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 ==
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 ===
{{Harmonics in equal|3|3|2}}


== Factoids about 3edf ==
== Factoids about 3edf ==
3edf's step size is close to the [[slendric]] temperament, which tempers out 1029/1024 in the 2.3.7 subgroup.
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 ==
Line 9: Line 17:
! #
! #
! Cents
! Cents
!Approximate JI Ratios
|-
|-
| 1
| 1
| 233.99
| 233.99
| [[8/7]]
|-
|-
| 2
| 2
| 467.97
| 467.97
| [[21/16]], [[17/13]]
|-
|-
| 3
| 3
| 701.96
| 701.96
| exact [[3/2]]
|}
|}
==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.


If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum.
== Music ==
 
* [https://www.youtube.com/watch?v=0ecvufTJowE Sequences & Chaos] by Bazil Müzik
Generator range: 228.5714 cents (4\7/3 = 4\21) to 240 cents (3\5/3 = 1\5)
{| class="wikitable center-all"
! colspan="7" |Fifth
!Cents
!Comments
|-
|4\7|| || || || || || ||228.571||
|-
| || || || || || ||27\47||229.787||
|-
| || || || || ||23\40|| ||230.000||
|-
| || || || || || ||42\73||230.137||
|-
| || || || ||19\33|| || ||230.{{Overline|30}}||
|-
| || || || || || ||53\92||230.435||
|-
| || || || || ||34\59|| ||230.5085||
|-
| || || || || || ||49\85||230.588||
|-
| || || ||15\26|| || || ||230.769||
|-
| || || || || || ||56\97||230.928||
|-
| || || || || ||41\71|| ||230.986||
|-
| || || || || || ||67\116||231.0345||
|-
| || || || ||26\45|| || ||231.{{Overline|1}}||[[Flattone]] is in this region
|-
| || || || || || ||63\109||231.193||The generator closest to a just [[8/7]] for EDOs less than 600
|-
| || || || || ||37\64|| ||231.25||
|-
| || || || || || ||48\83||231.325||
|-
| || ||11\19|| || || || ||231.579||
|-
| || || || || || ||51\88||231.{{Overline|81}}||
|-
| || || || || ||40\69|| ||231.884||
|-
| || || || || || ||69\119||231.933||
|-
| || || || ||29\50|| || ||232.000||
|-
| || || || || || ||76\131||232.061||[[Golden meantone]] (696.2145¢)
|-
| || || || || ||47\81|| ||232.099||
|-
| || || || || || ||65\112||232.143||
|-
| || || ||18\31|| || || ||232.258||[[Meantone]] is in this region
|-
| || || || || || ||61\105||232.381||
|-
| || || || || ||43\74|| ||232.{{Overline|432}}||
|-
| || || || || || ||68\117||232.479||
|-
| || || || ||25\43|| || ||232.558||
|-
| || || || || || ||57\98||232.653||
|-
| || || || || ||32\55|| ||232.{{Overline|72}}||
|-
| || || || || || ||39\67||232.836||
|-
| ||7\12|| || || || || ||233.{{Overline|3}}||
|-
| || || || || || ||38\65||233.846||
|-
| || || || || ||31\53|| ||233.962||The fifth closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || ||55\94||234.043||[[Garibaldi]] / [[Cassandra]]
|-
| || || || ||24\41|| || ||234.146||
|-
| || || || || || ||65\111||234.{{Overline|234}}||
|-
| || || || || ||41\70|| ||234.286||
|-
| || || || || || ||58\99||234.{{Overline|34}}||
|-
| || || ||17\29|| || || ||234.483||
|-
| || || || || || ||61\104||234.615||
|-
| || || || || ||44\75|| ||234.{{Overline|6}}||
|-
| || || || || || ||71\121||234.711||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||234.783||[[Neogothic]] is in this region
|-
| || || || || || ||64\109||234.862||
|-
| || || || || ||37\63|| ||234.921||
|-
| || || || || || ||47\80||235.000||
|-
| || ||10\17|| || || || ||235.294||
|-
| || || || || || ||43\73||235.616||
|-
| || || || || ||33\56|| ||235.714||
|-
| || || || || || ||56\95||235.7895||
|-
| || || || ||23\39|| || ||235.897||
|-
| || || || || || ||59\100||236.000||
|-
| || || || || ||36\61|| ||236.066||
|-
| || || || || || ||49\83||236.145||
|-
| || || ||13\22|| || || ||236.{{Overline|36}}||[[Archy]] is in this region
|-
| || || || || || ||42\71||236.620||
|-
| || || || || ||29\49|| ||236.735||
|-
| || || || || || ||45\76||236.842||
|-
| || || || ||16\27|| || ||237.{{Overline|037}}||
|-
| || || || || || ||35\59||237.288||
|-
| || || || || ||19\32|| ||237.500||
|-
| || || || || || ||22\37||237.{{Overline|837}}||
|-
|3\5|| || || || || || ||240.000||
|}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:Listen]]
[[Category:Edf]]
[[Category:Edonoi]]

Latest revision as of 19:43, 20 March 2026

← 2edf 3edf 4edf →
Prime factorization 3 (prime)
Step size 233.985 ¢ 
Octave 5\3edf (1169.93 ¢)
(convergent)
Twelfth 8\3edf (1871.88 ¢)
(convergent)
Consistency limit 10
Distinct consistency limit 4

3 equal divisions of the perfect fifth (abbreviated 3edf or 3ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 3 equal parts of about 234 ¢ each. Each step represents a frequency ratio of (3/2)1/3, or the cube root of 3/2.

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)1/3 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

Approximation of harmonics in 3edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -30 -30 -60 +22 -60 -93 -90 -60 -9 +60 -90
Relative (%) -12.9 -12.9 -25.7 +9.2 -25.7 -39.8 -38.6 -25.7 -3.7 +25.8 -38.6
Steps
(reduced) 		5
(2) 		8
(2) 		10
(1) 		12
(0) 		13
(1) 		14
(2) 		15
(0) 		16
(1) 		17
(2) 		18
(0) 		18
(0)

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, 8\41 and 39\200). It also works well as a tuning for arto and tendo chords.

Intervals

# Cents Approximate JI Ratios
1 233.99 8/7
2 467.97 21/16, 17/13
3 701.96 exact 3/2

Music

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