7edt: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
CompactStar (talk | contribs)
5,020 edits
No edit summary
+subsets and supersets; cleanup; -redundant categories
 
(11 intermediate revisions by 3 users not shown)
Line 1: Line 1:
{{Infobox ET}}
{{Infobox ET}}
'''7edt''' (short for '''7''' '''e'''qual '''d'''ivision of '''t'''ritave) divides the interval [[3/1]] it into 7 equal parts of 271.708 [[cent]]s each, corresponding to 4.4165 edo.
{{ED intro}}


__FORCETOC__
== Theory ==
== Properties ==
Since one step of 7edt approximates a [[7/6]] subminor third (4.84{{c}} sharp) quite nicely, three steps are almost exactly [[8/5]] (tempering out [[1728/1715]], the orwellisma), and four steps are very nearly [[15/8]] (tempering out [[2430/2401]], the nuwell comma). 7edt is the lowest equal division of the tritave to accurately approximate some [[7-limit]] harmony, along with some elements of the [[11-limit]], such as the [[11/8]] major fourth. Seven steps make up a tritave, meaning that 7edt tempers out 839808/823543, the eric comma.
The step size is very close to the 271.509 cents of 7-limit [[Orwell|orwell temperament]] and also close to the 271.426 cents of [[11-limit]] orwell. It is almost identical to 12\53, the [[53edo]] orwell generator which is 271.698 cents. 7edt is a good tuning for [[Electra]] temperament, being the equalized version of it's [[4L 3s (tritave-equivalent)|4L 3s<3/1>]] MOS scale.


== Scale degrees of 7edt ==
Due to the proximity of the step size with 7/6, 7edt supports [[orwell]] temperament. One step of 7edt is almost identical to 12\53, the [[53edo]] orwell generator, at about 271.698 cents. 7edt is also a good tuning for [[Electra]] temperament, with two steps of 7edt being a close approximation to [[15/11]].


{| class="wikitable"
=== Harmonics ===
! Degrees
{{Harmonics in equal|7|3|1|columns=15}}
 
=== Subsets and supersets ===
7edt is the 4th [[prime equal division|prime edt]], after [[5edt]] and before [[11edt]].
 
== Intervals ==
{| class="wikitable center-1 right-2 right-3"
|-
! #
! Cents
! Cents
!hekts
! [[Hekt]]s
! Approximate Ratio
! Approximate ratios
! [[Electra]] notation<br>({{nowrap|J {{=}} 1/1}})
|-
|-
! colspan="3" | 0
| 0
| 0
| 0
| [[1/1]]
| [[1/1]]
| J
|-
|-
| 1
| 1
| 271.708
| 272
|185.714
| 186
| [[7/6]]
| [[7/6]]
| K
|-
|-
| 2
| 2
| 543.416
| 543
|371.429
| 371
| [[15/11]], [[11/8]]
| [[11/8]], [[15/11]]
| L
|-
|-
| 3
| 3
| 815.124
| 815
|557.143
| 557
| [[8/5]]
| [[8/5]]
| M
|-
|-
| 4
| 4
| 1086.831
| 1087
|742.857
| 743
| [[15/8]]
| [[15/8]]
| N
|-
|-
| 5
| 5
| 1358.539
| 1359
|928.571
| 929
| [[11/5]] ([[11/10]] plus an octave)
| [[11/5]]
| O
|-
|-
| 6
| 6
| 1630.247
| 1630
|1114.286
| 1114
| [[18/7]] ([[9/7]] plus an octave)
| [[18/7]]
| P
|-
|-
| 7
| 7
| 1901.955
| 1902
|1300
| 1300
| [[3/1]]
| [[3/1]]
| J
|}
|}


Since one step of 7edt is a sharp subminor ([[7/6]]) third, three steps are almost exactly [[8/5]], four steps are very nearly [[15/8]] and six steps are a bit flat of [[18/7]], 7edt is the lowest equal division of the tritave to accurately approximate some [[7-limit]] harmony. Seven steps make up a tritave, meaning that 7edt tempers out 839808/823543, the [[eric]] [[comma]].
[[Category:Orwell]]
 
[[Category:Subminor third]]
== 7n-edt Family ==
* [[14edt]]
* [[21edt]]
* [[28edt]]
* [[56edt]]
 
[[category:macrotonal]]
[[Category:53edo]]
[[Category:orwell]]
[[Category:subminor third]]
[[Category:Edt]]

Latest revision as of 13:40, 23 February 2025

← 6edt 7edt 8edt →
Prime factorization 7 (prime)
Step size 271.708 ¢ 
Octave 4\7edt (1086.83 ¢)
Consistency limit 3
Distinct consistency limit 3

7 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 7edt or 7ed3), is a nonoctave tuning system that divides the interval of 3/1 into 7 equal parts of about 272 ¢ each. Each step represents a frequency ratio of 31/7, or the 7th root of 3.

Theory

Since one step of 7edt approximates a 7/6 subminor third (4.84 ¢ sharp) quite nicely, three steps are almost exactly 8/5 (tempering out 1728/1715, the orwellisma), and four steps are very nearly 15/8 (tempering out 2430/2401, the nuwell comma). 7edt is the lowest equal division of the tritave to accurately approximate some 7-limit harmony, along with some elements of the 11-limit, such as the 11/8 major fourth. Seven steps make up a tritave, meaning that 7edt tempers out 839808/823543, the eric comma.

Due to the proximity of the step size with 7/6, 7edt supports orwell temperament. One step of 7edt is almost identical to 12\53, the 53edo orwell generator, at about 271.698 cents. 7edt is also a good tuning for Electra temperament, with two steps of 7edt being a close approximation to 15/11.

Harmonics

Approximation of harmonics in 7edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Error Absolute (¢) -113 +0 +45 -69 -113 -108 -68 +0 +89 -76 +45 -93 +50 -69 +91
Relative (%) -41.7 +0.0 +16.7 -25.5 -41.7 -39.9 -25.0 +0.0 +32.9 -27.9 +16.7 -34.3 +18.5 -25.5 +33.4
Steps
(reduced) 		4
(4) 		7
(0) 		9
(2) 		10
(3) 		11
(4) 		12
(5) 		13
(6) 		14
(0) 		15
(1) 		15
(1) 		16
(2) 		16
(2) 		17
(3) 		17
(3) 		18
(4)

Subsets and supersets

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

Intervals

# Cents Hekts Approximate ratios Electra notation
(J = 1/1)
0 0 0 1/1 J
1 272 186 7/6 K
2 543 371 11/8, 15/11 L
3 815 557 8/5 M
4 1087 743 15/8 N
5 1359 929 11/5 O
6 1630 1114 18/7 P
7 1902 1300 3/1 J
Retrieved from "https://en.xen.wiki/index.php?title=7edt&oldid=182807"