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{{Infobox ET}}
{{Infobox ET}}
'''6EDF''' is the [[EDF|equal division of the just perfect fifth]] into six parts of 116.9925 [[cent|cents]] each, corresponding to 10.2571 [[edo]]. It is related to the [[Gamelismic clan|miracle temperament]], which tempers out 225/224 and 1029/1024 in the 7-limit.
{{ED intro}} It corresponds to 10.2571 [[edo]].  
 
== Theory ==
6edf is related to the [[miracle]] temperament, which [[tempering out|tempers out]] [[225/224]] and [[1029/1024]] in the 7-limit.
 
=== Harmonics ===
{{Harmonics in equal|6|3|2}}


== Intervals ==
== Intervals ==


{| class="wikitable"
{| class="wikitable center-1 right-2 center-4"
|-
|-
! degrees
! #
! cents ~ cents octave-reduced
! Cents
! approximate ratios
! Approximate Ratios
! [[1L 3s (fifth-equivalent)|Neptunian]] notation
! [[1L 3s (fifth-equivalent)|Neptunian]]<br>Notation
|-
|-
| 0
| 0
| 0 (perfect unison, 1:1)
| 0
| [[1/1]]
| [[1/1]]
| C
| C
Line 42: Line 48:
|-
|-
| 6
| 6
| 702 (just perfect fifth, 3:2)
| 702
| [[3/2]]
| [[3/2]]
| C
| C
Line 67: Line 73:
|-
|-
| 11
| 11
| 1287 ~ 87
| 1287
|  
|  
| F
| F
|-
|-
| 12
| 12
| 1404 ~ 204 (just major whole tone/ninth, 9:4)
| 1404
|
| 9/4
| C  
| C  
|-
|-
| 13
| 13
| 1521 ~ 321
| 1521
|
|
| C#
| C#
|-
|-
| 14
| 14
| 1638 ~ 438
| 1638
|
|
| Db
| Db
|-
|-
| 15
| 15
| 1755 ~ 555
| 1755
|
|
| D
| D
|-
|-
| 16
| 16
| 1872 ~ 672
| 1872
|
|
| E
| E
|-
|-
| 17
| 17
| 1988 ~ 788
| 1988
|
|
| F
| F
|-
|-
| 18
| 18
| 2106 ~ 906 (Pythagorean major sixth, 27:8)
| 2106
|
| 27/8
| C
| C
|-
|-
| 19
| 19
| 2223 ~ 1023
| 2223
|
|
| C#
| C#
|-
|-
| 20
| 20
| 2340 ~ 1140
| 2340
|
|
| Db
| Db
|-
|-
| 21
| 21
| 2457 ~ 57
| 2457
|
|
| D
| D
|-
|-
| 22
| 22
| 2574 ~ 174
| 2574
|
|
| E
| E
|-
|-
| 23
| 23
| 2691 ~ 291
| 2691
|
|
| F
| F
|-
|-
| 24
| 24
| 2808 ~ 408 (Pythagorean major third, 81:16)
| 2808
|
| 81/16
| C
| C
|}
|}
==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.
== Music ==
; [[Carlo Serafini]]
* [http://www.seraph.it/dep/det/metashakti.mp3 ''Metashakti'']


Generator range: 114.2857 cents (4\7/6 = 2\21) to 120 cents (3\5/6 = 1\10)
; [[XэнкøрX]]
{| class="wikitable center-all"
* [https://youtu.be/OSQljL4ANf8 "The Blame Game"] from ''State of the World (XLP)'' (2023)
! colspan="7" |Fifth
!Cents
!Comments
|-
|4\7|| || || || || || ||114.286||
|-
|  || || || || || || 27\47||114.894||
|-
| || || || || ||23\40|| ||115.000||
|-
| || || || || ||  ||42\73||115.0685||
|-
|  || || || || 19\33|| || ||115.{{Overline|15}}||
|-
| || ||  || || || || 53\92||115.217||
|-
|  || || || || || 34\59||  ||115.254||
|-
| ||  || || || ||  ||49\85 ||115.294||
|-
| ||  || ||15\26|| || || ||115.385||
|-
|  || || || || || ||56\97 ||115.464||
|-
| || || || ||  ||41\71 || ||115.493||
|-
| || || || || || ||67\116 ||115.517||
|-
|  || ||  || ||26\45|| || ||115.{{Overline|5}}||[[Flattone]] is in this region
|-
| || || || || || ||63\109||115.596||
|-
| || || || || ||37\64|| ||115.625||
|-
| || || || || || ||48\83||115.663||
|-
| || ||11\19|| || || || ||115.684||
|-
| || || || || || ||51\88||115.{{Overline|90}}||
|-
| || || || || ||40\69|| ||115.942||
|-
| || || || || || ||69\119||115.966||
|-
| || || || ||29\50|| || ||116.000||
|-
| || || || || || ||76\131||116.0305||[[Golden meantone]] (696.2145¢)
|-
| || || || || ||47\81|| ||116.049||
|-
| || || || || || ||65\112||116.071||
|-
| || || ||18\31|| || || ||116.129||[[Meantone]] is in this region
|-
| || || || || || ||61\105||116.1905||
|-
| || || || || ||43\74|| ||116.{{Overline|216}}||
|-
| || || || || || ||68\117||116.239||
|-
| || || || ||25\43|| || ||116.279||
|-
| || || || || || ||57\98||116.3265||
|-
| || || || || ||32\55|| ||116.{{Overline|36}}||
|-
| || || || || || ||39\67||116.418||
|-
| ||7\12|| || || || || ||116.{{Overline|6}}||
|-
| || || || || || ||38\65||116.923||
|-
| || || || || ||31\53|| ||116.981||The fifth closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || ||55\94||117.021||[[Garibaldi]] / [[Cassandra]]
|-
| || || || ||24\41|| || ||117.073||
|-
| || || || || || ||65\111||117.{{Overline|117}}||
|-
| || || || || ||41\70|| ||117.143||
|-
| || || || || || ||58\99||117.{{Overline|17}}||
|-
| || || ||17\29|| || || ||117.241||
|-
| || || || || || ||61\104||117.308||
|-
| || || || || ||44\75|| ||117.{{Overline|3}}||
|-
| || || || || || ||71\121||117.355||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||117.391||[[Neogothic]] is in this region
|-
| || || || || || ||64\109||117.431||
|-
| || || || || ||37\63|| ||117.460||
|-
| || || || || || ||47\80||117.500||
|-
| || ||10\17|| || || || ||117.647||
|-
| || || || || || ||43\73||117.808||
|-
| || || || || ||33\56|| ||117.857||
|-
| || || || || || ||56\95||117.895||
|-
| || || || ||23\39|| || ||117.949||
|-
| || || || || || ||59\100||118.000||
|-
| || || || || ||36\61|| ||118.033||
|-
| || || || || || ||49\83||118.072||
|-
| || || ||13\22|| || || ||118.{{Overline|18}}||[[Archy]] is in this region
|-
| || || || || || ||42\71||118.310||
|-
| || || || || ||29\49|| ||118.367||
|-
| || || || || || ||45\76||118.421||
|-
| || || || ||16\27|| || ||118.{{Overline|518}}||
|-
| || || || || || ||35\59||118.644||
|-
| || || || || ||19\32|| ||118.750||
|-
| || || || || || ||22\37||118.{{Overline|918}}||The generator closest to a just [[15/14]] for EDOs less than 1200
|-
|3\5|| || || || || || ||120.000||
|}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]]
== Compositions ==
* [http://www.seraph.it/dep/det/metashakti.mp3 Metashakti] by [[Carlo Serafini]]


[[Category:Edf]]
{{todo|expand}}
[[Category:Nonoctave]]
[[Category:Listen]]

Latest revision as of 08:55, 19 December 2024

← 5edf 6edf 7edf →
Prime factorization 2 × 3
Step size 116.993 ¢ 
Octave 10\6edf (1169.93 ¢) (→ 5\3edf)
Twelfth 16\6edf (1871.88 ¢) (→ 8\3edf)
Consistency limit 3
Distinct consistency limit 3
Special properties

6 equal divisions of the perfect fifth (abbreviated 6edf or 6ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 6 equal parts of about 117 ¢ each. Each step represents a frequency ratio of (3/2)1/6, or the 6th root of 3/2. It corresponds to 10.2571 edo.

Theory

6edf is related to the miracle temperament, which tempers out 225/224 and 1029/1024 in the 7-limit.

Harmonics

Approximation of harmonics in 6edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -30.1 -30.1 +56.8 +21.5 +56.8 +24.0 +26.8 +56.8 -8.6 -56.6 +26.8
Relative (%) -25.7 -25.7 +48.6 +18.4 +48.6 +20.5 +22.9 +48.6 -7.3 -48.4 +22.9
Steps
(reduced) 		10
(4) 		16
(4) 		21
(3) 		24
(0) 		27
(3) 		29
(5) 		31
(1) 		33
(3) 		34
(4) 		35
(5) 		37
(1)

Intervals

# Cents Approximate Ratios Neptunian
Notation
0 0 1/1 C
1 117 16/15, 15/14 C#
2 234 8/7 Db
3 351 11/9, 27/22 D
4 468 21/16 E
5 585 7/5, 45/32 F
6 702 3/2 C
7 819 8/5, 21/13 C#
8 936 12/7, 55/32 Db
9 1053 11/6 D
10 1170 49/25, 160/81, 2/1 E
11 1287 F
12 1404 9/4 C
13 1521 C#
14 1638 Db
15 1755 D
16 1872 E
17 1988 F
18 2106 27/8 C
19 2223 C#
20 2340 Db
21 2457 D
22 2574 E
23 2691 F
24 2808 81/16 C

Music

Carlo Serafini
XэнкøрX
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