4edf: Difference between revisions

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This notation can be said to be equalized of any 4-note MOS
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{{Infobox ET}}
{{Infobox ET}}
'''4EDF''' is the [[EDF|equal division of the just perfect fifth]] into four parts of 175.489 [[cent|cents]] each, corresponding to 6.8380 [[edo]]. It is related to the [[Tetracot family|tetracot temperament]], which tempers out 20000/19683 in the 5-limit.
{{ED intro}} It corresponds to 6.8380[[edo]].  
 
== Theory ==
4edf is related to the [[tetracot]] temperament, which [[tempering out|tempers out]] [[20000/19683]] in the 5-limit.
 
=== Harmonics ===
{{Harmonics in equal|4|3|2}}
 
== Intervals ==
== Intervals ==


{| class="wikitable"
{| class="wikitable center-1 right-2 center-3"
|-
|-
! degree
! #
! cents value
! Cents
! octave-reduced cents value
! Tetratonic<br>Notation
! Notation
|-
|-
| colspan="2" | 0
| 0
|
| 0.0
| C
| C
|-
|-
| 1
| 1
| 175.489
| 175.5
|
| D
| D
|-
|-
| 2
| 2
| 350.978
| 351.0
|
| E
| E
|-
|-
| 3
| 3
| 526.466
| 526.5
|
| F  
| F  
|-
|-
| 4
| 4
| 701.955
| 702.0
|
| C
| C
|-
|-
| 5
| 5
| 877.444
| 877.4
|
| D
| D
|-
|-
| 6
| 6
| 1052.933
| 1052.9
|
| E
| E
|-
! colspan="4" | second octave
|-
|-
| 7
| 7
| 1228.421
| 1228.4
| 28.421
| F
| F
|-
|-
| 8
| 8
| 1403.910
| 1403.9
| 203.910
| C
| C
|-
! colspan="4" |nonet
|-
|-
| 9
| 9
| 1579.399
| 1579.4
| 379.399
| D
| D
|-
|-
| 10
| 10
| 1754.888
| 1754.9
| 554.888
| E
| E
|-
|-
| 11
| 11
| 1930.376
| 1930.4
| 730.376
| F
| F
|-
|-
| 12
| 12
| 2105.865
| 2105.9
| 905.865
| C
| C
|-
|-
| 13
| 13
| 2281.354
| 2281.4
| 1081.354
| D
| D
|-
! colspan="4" | third octave
|-
|-
| 14
| 14
| 2456.843
| 2456.8
| 56.843
| E
| E
|}
|}
==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.
Generator range: 171.4286 cents (4\7/4 = 1\7) to 180 cents (3\5/4 = 3\20)
{| class="wikitable center-all"
! colspan="7" |Fifth
! Cents
! Comments
|-
|4\7|| || || ||  || || ||171.429||
|-
| || || || ||  || ||27\47||172.340 ||
|-
| || || || || ||23\40|| ||172.500||
|-
| || || ||  || || ||42\73||172.603||
|-
| || || || ||19\33 || || ||172.{{Overline|72}}||
|-
| || || || || ||  ||53\92||172.826 ||
|-
| || || ||  || ||34\59|| || 172.881||
|-
| || || || || || || 49\85||172.941||
|-
| || || || 15\26|| || || ||173.076||
|-
| || ||  || || ||  ||56\97||173.196||
|-
| ||  || || || ||41\71|| ||173.239||
|-
| || || || || || ||67\116 || 173.276||
|-
| || || || ||26\45|| || ||173.{{Overline|3}}||[[Flattone]] is in this region
|-
| || || || || || ||63\109||173.3945||
|-
| || || || || ||37\64|| ||173.4375||
|-
| || || || || || ||48\83||173.494||
|-
| || ||11\19|| || || || ||173.684||
|-
| || || || || || ||51\88||173.8{{Overline|63}}||
|-
| || || || || ||40\69|| ||173.913||
|-
| || || || || || ||69\119||173.950||
|-
| || || || ||29\50|| || ||174.000||
|-
| || || || || || ||76\131||174.046||[[Golden meantone]] (696.2145¢)
|-
| || || || || ||47\81|| ||174.{{Overline|074}}||
|-
| || || || || || ||65\112||174.107||
|-
| || || ||18\31|| || || ||174.193||[[Meantone]] is in this region
|-
| || || || || || ||61\105||174.286||
|-
| || || || || ||43\74|| ||174.{{Overline|324}}||
|-
| || || || || || ||68\117||174.359||
|-
| || || || ||25\43|| || ||174.419||
|-
| || || || || || ||57\98||174.490||
|-
| || || || || ||32\55|| ||174.{{Overline|54}}||
|-
| || || || || || ||39\67||174.627||
|-
| ||7\12|| || || || || ||175.000||
|-
| || || || || || ||38\65||175.385||
|-
| || || || || ||31\53|| ||175.472||The fifth closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || ||55\94||175.532||[[Garibaldi]] / [[Cassandra]]
|-
| || || || ||24\41|| || ||175.610||
|-
| || || || || || ||65\111||175.{{Overline|675}}||
|-
| || || || || ||41\70|| ||175.714||
|-
| || || || || || ||58\99||175.{{Overline|75}}||
|-
| || || ||17\29|| || || ||175.862||
|-
| || || || || || ||61\104||175.9615||
|-
| || || || || ||44\75|| ||176.000||
|-
| || || || || || ||71\121||176.033||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||176.087||[[Neogothic]] is in this region
|-
| || || || || || ||64\109||176.147||
|-
| || || || || ||37\63|| ||176.1905||
|-
| || || || || || ||47\80||176.250||
|-
| || ||10\17|| || || || ||176.471||
|-
| || || || || || ||43\73||176.712||
|-
| || || || || ||33\56|| ||176.786||
|-
| || || || || || ||56\95||176.842||
|-
| || || || ||23\39|| || ||176.923||
|-
| || || || || || ||59\100||177.000||
|-
| || || || || ||36\61|| ||177.049||
|-
| || || || || || ||49\83||177.108||
|-
| || || ||13\22|| || || ||177.{{Overline|27}}||[[Archy]] is in this region
|-
| || || || || || ||42\71||177.648||
|-
| || || || || ||29\49|| ||177.551||
|-
| || || || || || ||45\76||177.532||
|-
| || || || ||16\27|| || ||177.{{Overline|7}}||
|-
| || || || || || ||35\59||177.966||
|-
| || || || || ||19\32|| ||178.125||
|-
| || || || || || ||22\37||178.{{Overline|378}}||
|-
|3\5|| || || || || || ||180.000||
|}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.
== Music ==
== Compositions ==
; [[Carlo Serafini]]
* [http://www.seraph.it/dep/det/repetitions1.mp3 Repetitions 1] [https://www.youtube.com/watch?v=XEklMo0tIW0 Repetitions 1 video] by [[Carlo Serafini]]
* ''Repetitions 1'' – [http://www.seraph.it/dep/det/repetitions1.mp3 play] | [https://www.youtube.com/watch?v=XEklMo0tIW0 YouTube]


[[Category:Edf]]
[[Category:Listen]]
[[Category:Listen]]
[[Category:Nonoctave]]

Latest revision as of 06:09, 10 May 2024

← 3edf 4edf 5edf →
Prime factorization 22
Step size 175.489 ¢ 
Octave 7\4edf (1228.42 ¢)
(semiconvergent)
Twelfth 11\4edf (1930.38 ¢)
(semiconvergent)
Consistency limit 6
Distinct consistency limit 5
Special properties

4 equal divisions of the perfect fifth (abbreviated 4edf or 4ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 4 equal parts of about 175 ¢ each. Each step represents a frequency ratio of (3/2)1/4, or the 4th root of 3/2. It corresponds to 6.8380edo.

Theory

4edf is related to the tetracot temperament, which tempers out 20000/19683 in the 5-limit.

Harmonics

Approximation of harmonics in 4edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +28.4 +28.4 +56.8 +21.5 +56.8 -34.5 +85.3 +56.8 +49.9 +60.4 +85.3
Relative (%) +16.2 +16.2 +32.4 +12.3 +32.4 -19.7 +48.6 +32.4 +28.5 +34.4 +48.6
Steps
(reduced) 		7
(3) 		11
(3) 		14
(2) 		16
(0) 		18
(2) 		19
(3) 		21
(1) 		22
(2) 		23
(3) 		24
(0) 		25
(1)

Intervals

# Cents Tetratonic
Notation
0 0.0 C
1 175.5 D
2 351.0 E
3 526.5 F
4 702.0 C
5 877.4 D
6 1052.9 E
7 1228.4 F
8 1403.9 C
9 1579.4 D
10 1754.9 E
11 1930.4 F
12 2105.9 C
13 2281.4 D
14 2456.8 E

Music

Carlo Serafini
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