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=Equal division of the perfect fifth (3/2) into 6 equal parts=
{{Infobox ET}}
(a.k.a. "6th root of 3/2")
{{ED intro}} It corresponds to 10.2571 [[edo]].  


The single step of 6-edf is 116.9925 cents. This is very close to the size of the [[Regular_Temperaments#miracle|miracle]] generator, and thus this scale is close to the "nonoctave miracle" scale...
== Theory ==
6edf is related to the [[miracle]] temperament, which [[tempering out|tempers out]] [[225/224]] and [[1029/1024]] in the 7-limit.


==Compositions==
=== Harmonics ===
[http://www.seraph.it/dep/det/metashakti.mp3 Metashakti] by [[Carlo_Serafini|Carlo Serafini]]
{{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
! [[1L 3s (fifth-equivalent)|Neptunian]]<br>Notation
|-
|-
| | 0
| 0
| | 0 (perfect unison, 1:1)
| 0
| [[1/1]]
| C
|-
|-
| | 1
| 1
| | 117
| 117
| [[16/15]], [[15/14]]
| C#
|-
|-
| | 2
| 2
| | 234
| 234
| [[8/7]]
| Db
|-
|-
| | 3
| 3
| | 351
| 351
| [[11/9]], [[27/22]]
| D
|-
|-
| | 4
| 4
| | 468
| 468
| [[21/16]]
| E
|-
|-
| | 5
| 5
| | 585
| 585
| [[7/5]], [[45/32]]
| F
|-
|-
| | 6
| 6
| | 702 (just perfect fifth, 3:2)
| 702
| [[3/2]]
| C
|-
|-
| | 7
| 7
| | 819
| 819
| [[8/5]], [[21/13]]
| C#
|-
|-
| | 8
| 8
| | 936
| 936
| [[12/7]], [[55/32]]
| Db
|-
|-
| | 9
| 9
| | 1053
| 1053
| [[11/6]]
| D
|-
|-
| | 10
| 10
| | 1170
| 1170
| [[49/25]], [[160/81]], [[2/1]]
| E
|-
|-
| | 11
| 11
| | 1287 ~ 87
| 1287
|  
| F
|-
|-
| | 12
| 12
| | 1404 ~ 204 (just major whole tone/ninth, 9:4)
| 1404
| 9/4
| C
|-
|-
| | 13
| 13
| | 1521 ~ 321
| 1521
|
| C#
|-
|-
| | 14
| 14
| | 1638 ~ 438
| 1638
|
| Db
|-
|-
| | 15
| 15
| | 1755 ~ 555
| 1755
|
| D
|-
|-
| | 16
| 16
| | 1872 ~ 672
| 1872
|
| E
|-
|-
| | 17
| 17
| | 1988 ~ 788
| 1988
|
| F
|-
|-
| | 18
| 18
| | 2106 ~ 906 (Pythagorean major sixth, 27:8)
| 2106
| 27/8
| C
|-
|-
| | 19
| 19
| | 2223 ~ 1023
| 2223
|
| C#
|-
|-
| | 20
| 20
| | 2340 ~ 1140
| 2340
|
| Db
|-
|-
| | 21
| 21
| | 2457 ~ 57
| 2457
|
| D
|-
|-
| | 22
| 22
| | 2574 ~ 174
| 2574
|
| E
|-
|-
| | 23
| 23
| | 2691 ~ 291
| 2691
|
| F
|-
|-
| | 24
| 24
| | 2808 ~ 408 (Pythagorean major third, 81:16)
| 2808
| 81/16
| C
|}
|}
[[Category:edf]]
 
[[Category:edonoi]]
== Music ==
[[Category:listen]]
; [[Carlo Serafini]]
* [http://www.seraph.it/dep/det/metashakti.mp3 ''Metashakti'']
 
; [[XэнкøрX]]
* [https://youtu.be/OSQljL4ANf8 "The Blame Game"] from ''State of the World (XLP)'' (2023)
 
[[Category:Listen]]
 
{{todo|expand}}

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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