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{{Infobox ET}}
{{Infobox ET}}
'''[[EDF|Division of the just perfect fifth]] into 7 equal parts''' (7EDF) is related to [[12edo|12 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 3.3514 cents stretched and the step size is about 100.2793 cents. The patent val has a generally sharp tendency for harmonics up to 21, with the exception for 11 and 13.
{{ED intro}}


Lookalikes: [[12edo]], [[19ED3|19ed3]], [[31ed6]]
== Theory ==
7edf is related to [[12edo]], but with the 3/2 rather than the 2/1 being just, which stretches the octave by 3.3514{{c}}. The patent val has a generally sharp tendency for harmonics up to 21, with the exception for 11 and 13. It forms as a decent approximation to stretched-octave tuning on pianos, since pianos' strings have overtones that tend slightly sharp and are thus often tuned with stretched octaves.
 
=== Harmonics ===
{{Harmonics in equal|7|3|2|prec=2|columns=15}}
 
=== Subsets and supersets ===
7edf is the 4th [[prime equal division|prime edf]], after [[5edf]] and before [[11edf]].


== Intervals ==
== Intervals ==
{| class="wikitable"
{| class="wikitable center-1 right-2 center-3"
|-
|-
! #
! #
! Cents
! Cents
! Approximate ratios
! 12edo notation
! 12edo notation
|-
| 0
| 0
| exact 1/1
| C
|-
|-
| 1
| 1
| 100.2793
| 100.3
| 18/17, 17/16
| C#, Db
| C#, Db
|-
|-
| 2
| 2
| 200.5586
| 200.6
| 9/8
| D
| D
|-
|-
| 3
| 3
| 300.8379
| 300.8
| 19/16, 44/37
| D#, Eb
| D#, Eb
|-
|-
| 4
| 4
| 401.1171
| 401.1
| 63/50
| E
| E
|-
|-
| 5
| 5
| 501.3964
| 501.4
|4/3
| F
| F
|-
|-
| 6
| 6
| 601.6757
| 601.7
| 64/45
| F#, Gb
| F#, Gb
|-
|-
| 7
| 7
| 701.955
| 702.0
| exact 3/2
| G
| G
|-
|-
| 8
| 8
| 802.2343
| 802.2
| 100/63
| G#, Ab
| G#, Ab
|-
|-
| 9
| 9
| 902.5136
| 902.5
| 27/16
| A
| A
|-
|-
| 10
| 10
| 1002.7929
| 1002.8
| 16/9
| A#, Bb
| A#, Bb
|-
|-
| 11
| 11
| 1103.0721
| 1103.1
| 17/9
| B
| B
|-
|-
| 12
| 12
| 1203.3514
| 1203.4
| 2/1
| C
| C
|-
|-
| 13
| 13
| 1303.6307
| 1303.6
| 17/8
| C#, Db
| C#, Db
|-
|-
| 14
| 14
| 1403.91
| 1403.9
| exact 9/4
| D
| D
|}
|}
==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.
== See also ==
* [[12edo]] – relative edo
* [[19edt]] – relative edt
* [[28ed5]] – relative ed5
* [[31ed6]] – relative ed6
* [[34ed7]] – relative ed7
* [[40ed10]] – relative ed10
* [[43ed12]] – relative ed12
* [[76ed80]] – close to the zeta-optimized tuning for 12edo
* [[1ed18/17|AS18/17]] – relative [[AS|ambitonal sequence]]


Generator range: 97.9592 cents (4\7/7 = 4\49) to 102.8571 cents (3\5/7 = 3\35)
{{Todo|expand}}
{| class="wikitable center-all"
! colspan="7" |Fifth
!Cents
! Comments
|-
|4\7|| || ||  || || || || 97.959||
|-
| || || ||  || || ||27\47||98.480||
|-
| || || || || ||23\40|| ||98.571||
|-
| || || || ||  || ||42\73||98.630||
|-
| || || || ||19\33|| || ||98.701||
|-
| || || || || || ||53\92||98.758||
|-
| || || || || ||34\59|| ||98.789||
|-
| || || || || || ||49\85||98.8235||
|-
| || || ||15\26|| || || ||98.901||
|-
| || || || || || ||56\97||98.969||The generator closest to a just [[18/17]] for EDOs less than 1400
|-
| || || || || ||41\71|| ||98.994||
|-
| || || || || || ||67\116||99.015||
|-
| || || || ||26\45|| || ||99.048||[[Flattone]] is in this region
|-
| || || || || || ||63\109||99.083||
|-
| || || || || ||37\64|| ||99.107||
|-
| || || || || || ||48\83||99.139||
|-
| || ||11\19|| || || || ||99.248||
|-
| || || || || || ||51\88||99.351||
|-
| || || || || ||40\69|| ||99.379||
|-
| || || || || || ||69\119||99.400||
|-
| || || || ||29\50|| || ||99.429||
|-
| || || || || || ||76\131||99.455||[[Golden meantone]] (696.2145¢)
|-
| || || || || ||47\81|| ||99.471||
|-
| || || || || || ||65\112||99.490||
|-
| || || ||18\31|| || || ||99.539||[[Meantone]] is in this region
|-
| || || || || || ||61\105||99.592||
|-
| || || || || ||43\74|| ||99.613||The generator closest to a just [[16/9]] for EDOs less than 1400
|-
| || || || || || ||68\117||99.634||
|-
| || || || ||25\43|| || ||99.668||
|-
| || || || || || ||57\98||99.7085||
|-
| || || || || ||32\55|| ||99.740||
|-
| || || || || || ||39\67||99.787||
|-
| ||7\12|| || || || || ||100.000||
|-
| || || || || || ||38\65||100.220||
|-
| || || || || ||31\53|| ||100.2695||The fifth closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || ||55\94||100.304||[[Garibaldi]] / [[Cassandra]]
|-
| || || || ||24\41|| || ||100.348||
|-
| || || || || || ||65\111||100.361||
|-
| || || || || ||41\70|| ||100.408||
|-
| || || || || || ||58\99||100.433||
|-
| || || ||17\29|| || || ||100.493||
|-
| || || || || || ||61\104||100.5495||
|-
| || || || || ||44\75|| ||100.571||
|-
| || || || || || ||71\121||100.590||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||100.621||[[Neogothic]] is in this region
|-
| || || || || || ||64\109||100.655||
|-
| || || || || ||37\63|| ||100.680||
|-
| || || || || || ||47\80||100.714||
|-
| || ||10\17|| || || || ||100.840||
|-
| || || || || || ||43\73||100.9785||
|-
| || || || || ||33\56|| ||101.020||
|-
| || || || || || ||56\95||101.053||
|-
| || || || ||23\39|| || ||101.099||
|-
| || || || || || ||59\100||101.143||
|-
| || || || || ||36\61|| ||101.171||
|-
| || || || || || ||49\83||101.205||
|-
| || || ||13\22|| || || ||101.299||[[Archy]] is in this region
|-
| || || || || || ||42\71||101.4085||
|-
| || || || || ||29\49|| ||101.458||
|-
| || || || || || ||45\76||101.504||
|-
| || || || ||16\27|| || ||101.587||
|-
| || || || || || ||35\59||101.695||
|-
| || || || || ||19\32|| ||101.786||The generator closest to a just [[9/5]] for EDOs less than 1400
|-
| || || || || || ||22\37||101.9305||
|-
|3\5|| || || || || || ||102.857||
|}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:12edo]]
[[Category:Edf]]
[[Category:Edonoi]]
[[Category:todo:improve synopsis]]

Latest revision as of 13:26, 10 June 2025

← 6edf 7edf 8edf →
Prime factorization 7 (prime)
Step size 100.279 ¢ 
Octave 12\7edf (1203.35 ¢)
(convergent)
Twelfth 19\7edf (1905.31 ¢)
(convergent)
Consistency limit 10
Distinct consistency limit 6

7 equal divisions of the perfect fifth (abbreviated 7edf or 7ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 7 equal parts of about 100 ¢ each. Each step represents a frequency ratio of (3/2)1/7, or the 7th root of 3/2.

Theory

7edf is related to 12edo, but with the 3/2 rather than the 2/1 being just, which stretches the octave by 3.3514 ¢. The patent val has a generally sharp tendency for harmonics up to 21, with the exception for 11 and 13. It forms as a decent approximation to stretched-octave tuning on pianos, since pianos' strings have overtones that tend slightly sharp and are thus often tuned with stretched octaves.

Harmonics

Approximation of harmonics in 7edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Error Absolute (¢) +3.35 +3.35 +6.70 +21.51 +6.70 +40.67 +10.05 +6.70 +24.86 -39.87 +10.05 -28.24 +44.02 +24.86 +13.41
Relative (%) +3.3 +3.3 +6.7 +21.4 +6.7 +40.6 +10.0 +6.7 +24.8 -39.8 +10.0 -28.2 +43.9 +24.8 +13.4
Steps
(reduced) 		12
(5) 		19
(5) 		24
(3) 		28
(0) 		31
(3) 		34
(6) 		36
(1) 		38
(3) 		40
(5) 		41
(6) 		43
(1) 		44
(2) 		46
(4) 		47
(5) 		48
(6)

Subsets and supersets

7edf is the 4th prime edf, after 5edf and before 11edf.

Intervals

# Cents Approximate ratios 12edo notation
0 0 exact 1/1 C
1 100.3 18/17, 17/16 C#, Db
2 200.6 9/8 D
3 300.8 19/16, 44/37 D#, Eb
4 401.1 63/50 E
5 501.4 4/3 F
6 601.7 64/45 F#, Gb
7 702.0 exact 3/2 G
8 802.2 100/63 G#, Ab
9 902.5 27/16 A
10 1002.8 16/9 A#, Bb
11 1103.1 17/9 B
12 1203.4 2/1 C
13 1303.6 17/8 C#, Db
14 1403.9 exact 9/4 D

See also

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