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{{Infobox ET}}
{{Infobox ET}}
'''5EDF''' is the [[EDF|equal division of the just perfect fifth]] into five parts of 140.391 [[cent|cents]] each, corresponding to 8.5476 [[edo]]. It is close to the [[Bleu]] generator chain and every second step of [[17edo]]. 4 steps of 5edf is a fraction of a cent away to the seventh harmonic (which is [[112/81]] instead of [[7/4]] since the equave is 3/2), which is an extremely accurate approximation for the size of this scale.
{{ED intro}} It corresponds to 8.5476 [[edo]].  
==Intervals==
 
{| class="wikitable"
== Theory ==
|-
5edf is close to the [[bleu]] [[generator]] chain and every second step of [[17edo]] (also known as [[17ed4]]) 5edf has an extremely accurate approximation of the seventh harmonic for its size.
!degree
 
!cents value
5edf is notable as a relatively basic and easy-to-use nonoctave system. Traditional harmony using major and minor triads is accessible in 5edf, although they are not 5-limit but rather septimal/undecimal in flavor. One must be wary of the 3/2-equivalence paradigm-there is no dominant, and major and minor triads, seventh chords, ninth chords, etc. are all merely voicings of major and minor dyads. Diminished chords also play a more important role than they do traditionally, as unlike the conventional triads, they are not equivalent to dyads, and are somewhat more consonant than in [[12edo]] due to the laxer subtritone.
!octave-reduced cents value
=== Harmonics ===
!approximate ratios
{{Harmonics in equal|5|3|2|columns=15}}
!colspan="2"|[[1L 3s (fifth-equivalent)|Neptunian]] notation
|-
| colspan="2" |0
|
|[[1/1]]
|perfect unison
|C
|-
|1
|140.391
|
|[[13/12]], [[49/45]]
|augmented unison, minor second
|C#, Db
|-
|2
|280.782
|
|[[75/64]], [[20/17]], [[13/11]]
|major second, minor third
|D, Eb
|-
|3
|421.173
|
|[[14/11]], [[23/18]]
|major third, diminished fourth
|E, Fb
|-
|4
|561.564
|
|[[11/8]], [[18/13]], [[25/18]]
|perfect fourth
|F
|-
|5
|701.955
|
|[[3/2]]
|perfect fifth
|C
|-
|6
|842.346
|
|[[21/13]], [[13/8]], [[18/11]]
|augmented fifth, minor sixth
|C#, Db
|-
|7
|982.737
|
|[[7/4]], [[30/17]]
|major sixth, minor seventh
|D, Eb
|
|-
|8
|1123.128
|
|
|major seventh, minor octave
|E, Fb
|
|-
|9
|1263.519
|63.519
|
|major octave
|F
|-
|10
|1403.910
|203.910
|
|
|C
|-
|11
|1544.301
|344.301
|
|
|C#, Db
|-
|12
|1684.692
|484.692
|
|
|D, Eb
|-
|13
|1825.083
|625.083
|
|
|E
|-
|14
|1965.474
|765.474
|
|
|F
|-
|15
|2105.865
|905.865
|
|
|C
|-
|16
|2246.256
|1046.256
|
|
|C#, Db
|-
|17
|2386.647
|1186.647
|
|
|D
|}
==Scale tree==
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.
=== Subsets and supersets ===
5edf is the 3rd [[prime equal division|prime edf]], after [[3edf]] and before [[7edf]].


Generator range: 137.1429 cents (4\7/5 = 4\35) to 144 cents (3\5/5 = 3\25)
== Intervals ==
{| class="wikitable center-all"
{| class="wikitable center-1 right-2"
! colspan="7" |Fifth
!Cents
!Comments
|-
|-
|4\7|| || || || || || ||137.143||
! #
! Cents
! Approximate ratios
! colspan="2"| [[1L 3s (fifth-equivalent)|Neptunian]] notation
|-
|-
| || || || || || ||27\47||137.872 ||
| 0
| 0.0
| [[1/1]]
| perfect unison
| C
|-
|-
| || || || || ||23\40|| || 138.000||
| 1
| 140
| [[13/12]], [[49/45]]
| augmented unison, minor second
| C#, Db
|-
|-
| || || || || || ||42\73||138.082||
| 2
| 281
| [[13/11]], [[20/17]], [[75/64]]
| major second, minor third
| D, Eb
|-
|-
| || || || ||19\33|| || ||138.{{Overline|18}}||
| 3
| 421
| [[14/11]], [[23/18]]
| major third, diminished fourth
| E, Fb
|-
|-
| || || || || || ||53\92||138.261||
| 4
| 562
| [[11/8]], [[18/13]], [[25/18]]
| perfect fourth
| F
|-
|-
| || || || || ||34\59|| ||138.305||
| 5
| 702
| [[3/2]]
| perfect fifth
| C
|-
|-
| || || || || || ||49\85||138.353||
| 6
| 842
| [[13/8]], [[18/11]], [[21/13]]
| augmented fifth, minor sixth
| C#, Db
|-
|-
| || || ||15\26|| || || ||138.4615||
| 7
| 983
| [[7/4]], [[30/17]]
| major sixth, minor seventh
| D, Eb
|-
|-
| || || || || || ||56\97||138.557||
| 8
| 1123
| 44/23
| major seventh, minor octave
| E, Fb
|-
|-
| || || || || ||41\71|| ||138.5915||The generator closest to a just [[13/12]] for EDOs less than 1000
| 9
| 1264
| 83/40
| major octave
| F
|-
|-
| || || || || || ||67\116||138.621||
| 10
|-
| 1404
| || || || ||26\45|| || ||138.{{Overline|6}}||[[Flattone]] is in this region
| [[9/4]]
|-
| major ninth
| || || || || || ||63\109||138.716||
| C
|-
|}
| || || || || ||37\64|| ||138.750||
|-
| || || || || || ||48\83||138.795||
|-
| || ||11\19|| || || || ||138.947||
|-
| || || || || || ||51\88||139.{{Overline|09}}||
|-
| || || || || ||40\69|| ||139.130||
|-
| || || || || || ||69\119||139.160||
|-
| || || || ||29\50|| || ||139.200||
|-
| || || || || || ||76\131||139.237||[[Golden meantone]] (696.2145¢)
|-
| || || || || ||47\81|| ||139.{{Overline|259}}||
|-
| || || || || || ||65\112||139.286||
|-
| || || ||18\31|| || || ||139.355||[[Meantone]] is in this region
|-
| || || || || || ||61\105||139.429||
|-
| || || || || ||43\74|| ||139.{{Overline|459}}||
|-
| || || || || || ||68\117||139.487||
|-
| || || || ||25\43|| || ||139.535||
|-
| || || || || || ||57\98||139.592||
|-
| || || || || ||32\55|| ||139.{{Overline|63}}||
|-
| || || || || || ||39\67||139.7015||
|-
| ||7\12|| || || || || ||140.000||
|-
| || || || || || ||38\65||140.308||
|-
| || || || || ||31\53|| ||140.377||The fifth closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || ||55\94||140.4255||[[Garibaldi]] / [[Cassandra]]
|-
| || || || ||24\41|| || ||140.488||
|-
| || || || || || ||65\111||140.{{Overline|540}}||
|-
| || || || || ||41\70|| ||140.571||
|-
| || || || || || ||58\99||140.{{Overline|60}}||
|-
| || || ||17\29|| || || ||140.690||
|-
| || || || || || ||61\104||140.769||
|-
| || || || || ||44\75|| ||140.800||
|-
| || || || || || ||71\121||140.826||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||140.870||[[Neogothic]] is in this region
|-
| || || || || || ||64\109||140.917||
|-
| || || || || ||37\63|| ||140.952||
|-
| || || || || || ||47\80||141.000||
|-
| || ||10\17|| || || || ||141.1765||
|-
| || || || || || ||43\73||141.370||
|-
| || || || || ||33\56|| ||141.429||
|-
| || || || || || ||56\95||141.474||
|-
| || || || ||23\39|| || ||141.5385||
|-
| || || || || || ||59\100||141.600||
|-
| || || || || ||36\61|| ||141.639||
|-
| || || || || || ||49\83||141.687||
|-
| || || ||13\22|| || || ||141.{{Overline|81}}||[[Archy]] is in this region
|-
| || || || || || ||42\71||141.972||
|-
| || || || || ||29\49|| ||142.041||
|-
| || || || || || ||45\76||142.105||
|-
| || || || ||16\27|| || ||142.{{Overline|2}}||
|-
| || || || || || ||35\59||142.373||
|-
| || || || || ||19\32|| ||142.500||
|-
| || || || || || ||22\37||142.{{Overline|702}}||
|-
|3\5|| || || || || || ||144.000||
|}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.
{{Todo|expand}}
[[Category:Edf]]
[[Category:Edonoi]]

Latest revision as of 00:38, 10 August 2025

← 4edf 5edf 6edf →
Prime factorization 5 (prime)
Step size 140.391 ¢ 
Octave 9\5edf (1263.52 ¢)
Twelfth 14\5edf (1965.47 ¢)
Consistency limit 3
Distinct consistency limit 3

5 equal divisions of the perfect fifth (abbreviated 5edf or 5ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 5 equal parts of about 140 ¢ each. Each step represents a frequency ratio of (3/2)1/5, or the 5th root of 3/2. It corresponds to 8.5476 edo.

Theory

5edf is close to the bleu generator chain and every second step of 17edo (also known as 17ed4) 5edf has an extremely accurate approximation of the seventh harmonic for its size.

5edf is notable as a relatively basic and easy-to-use nonoctave system. Traditional harmony using major and minor triads is accessible in 5edf, although they are not 5-limit but rather septimal/undecimal in flavor. One must be wary of the 3/2-equivalence paradigm-there is no dominant, and major and minor triads, seventh chords, ninth chords, etc. are all merely voicings of major and minor dyads. Diminished chords also play a more important role than they do traditionally, as unlike the conventional triads, they are not equivalent to dyads, and are somewhat more consonant than in 12edo due to the laxer subtritone.

Harmonics

Approximation of harmonics in 5edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Error Absolute (¢) +63.5 +63.5 -13.4 +21.5 -13.4 +0.6 +50.2 -13.4 -55.4 +60.4 +50.2 +52.0 +64.1 -55.4 -26.7
Relative (%) +45.2 +45.2 -9.5 +15.3 -9.5 +0.4 +35.7 -9.5 -39.4 +43.0 +35.7 +37.0 +45.6 -39.4 -19.0
Steps
(reduced) 		9
(4) 		14
(4) 		17
(2) 		20
(0) 		22
(2) 		24
(4) 		26
(1) 		27
(2) 		28
(3) 		30
(0) 		31
(1) 		32
(2) 		33
(3) 		33
(3) 		34
(4)

Subsets and supersets

5edf is the 3rd prime edf, after 3edf and before 7edf.

Intervals

# Cents Approximate ratios Neptunian notation
0 0.0 1/1 perfect unison C
1 140 13/12, 49/45 augmented unison, minor second C#, Db
2 281 13/11, 20/17, 75/64 major second, minor third D, Eb
3 421 14/11, 23/18 major third, diminished fourth E, Fb
4 562 11/8, 18/13, 25/18 perfect fourth F
5 702 3/2 perfect fifth C
6 842 13/8, 18/11, 21/13 augmented fifth, minor sixth C#, Db
7 983 7/4, 30/17 major sixth, minor seventh D, Eb
8 1123 44/23 major seventh, minor octave E, Fb
9 1264 83/40 major octave F
10 1404 9/4 major ninth C
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