4edf: Difference between revisions

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Scale tree is fine, see the description of it i added at the top of the section, "EDF scales can be approximated in EDOs by subdividing diatonic fifths."
-irrelevant shit
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==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.
== Compositions ==
== Compositions ==
* [http://www.seraph.it/dep/det/repetitions1.mp3 Repetitions 1] [https://www.youtube.com/watch?v=XEklMo0tIW0 Repetitions 1 video] by [[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]]

Revision as of 13:17, 7 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

4EDF is the equal division of the just perfect fifth into four parts of 175.489 cents each, corresponding to 6.8380 edo. It is related to the tetracot temperament, which tempers out 20000/19683 in the 5-limit.

Intervals

degree cents value octave-reduced cents value Tetratonic notation
0 C
1 175.489 D
2 350.978 E
3 526.466 F
4 701.955 C
5 877.444 D
6 1052.933 E
second octave
7 1228.421 28.421 F
8 1403.910 203.910 C
nonet
9 1579.399 379.399 D
10 1754.888 554.888 E
11 1930.376 730.376 F
12 2105.865 905.865 C
13 2281.354 1081.354 D
third octave
14 2456.843 56.843 E

Compositions

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