7ed5: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
'''[[Ed5|Division of the 5th harmonic]] into 7 equal parts''' (7ed5) is related to [[3edo|3 edo]], but with the 5/1 rather than the 2/1 being just. The octave is about 5.8656 cents compressed and the step size is about 398.0448 cents. It is present (though possibly tempered) in any [[regular temperament]] which tempers out [[441/440]] and 244515348/244140625 in the [[11-limit]], such as [[equal temperament]]s [[3edo|3]], [[12edo|12]], [[15edo|15]], [[175edo|175]], [[190edo|190]], [[202edo|202]], and [[217edo]].
{{ED intro}}


Due to [[Kirnberger's atom]], its step is 100.0002¢ flat of [[4/3]].
== Theory ==
7ed5 is related to [[3edo]], but with the 5/1 rather than the 2/1 being just. The octave is about 6 cents compressed and the step size is about 398 cents. It is present (though possibly tempered) in any [[regular temperament]] which [[tempering out|tempers out]] [[441/440]] and 244515348/244140625 in the [[11-limit]], such as [[equal temperament]]s [[3edo|3]], [[12edo|12]], [[15edo|15]], [[175edo|175]], [[190edo|190]], [[202edo|202]], and [[217edo]].


== Harmonics ==
Due to [[Kirnberger's atom]], its step is 100.0002¢ flat of [[4/3]]{{clarify}}.
{{Harmonics in equal
 
| steps = 7
=== Harmonics ===
| num = 5
{{Harmonics in equal|7|5|1}}
| denom = 1
{{Harmonics in equal|7|5|1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 7ed5 (contined)}}
}}
{{Harmonics in equal
| steps = 7
| num = 5
| denom = 1
| start = 12
| collapsed = 1
}}


== Intervals ==
== Intervals ==
 
{| class="wikitable center-1 right-2"
{| class="wikitable"
|-
|-
! | degree
! #
! | cents value
! Cents
! | corresponding <br>JI intervals
! Approximate ratios
! | comments
|-
|-
| | 0
| 0
| | 0.0000
| 0
| | '''exact [[1/1]]'''
| [[1/1]]
| |
|-
|-
| | 1
| 1
| | 398.0448
| 398
| | 34/27
| [[5/4]], 34/27
| | pseudo-[[5/4]]
|-
|-
| | 2
| 2
| | 796.0896
| 796
| | [[19/12]]
| [[19/12]]
| |
|-
|-
| | 3
| 3
| | 1194.1344
| 1194
| | 255/128
| [[2/1]], 255/128
| | pseudo-[[octave]]
|-
|-
| | 4
| 4
| | 1592.1793
| 1592
| | 128/51
| [[5/2]], 128/51
| | pseudo-[[5/2]]
|-
|-
| | 5
| 5
| | 1990.2241
| 1990
| | [[30/19|60/19]]
| 60/19
| |
|-
|-
| | 6
| 6
| | 2388.2689
| 2388
| | 135/34
| [[4/1]], 135/34
| | pseudo-[[4/1]]
|-
|-
| | 7
| 7
| | 2786.3137
| 2786
| | '''exact [[5/1]]'''
| [[5/1]]
| | just major third plus two octaves
|}
|}
[[Category:Ed5]]
[[Category:Edonoi]]

Revision as of 08:28, 23 January 2025

← 6ed5 7ed5 8ed5 →
Prime factorization 7 (prime)
Step size 398.045 ¢ 
Octave 3\7ed5 (1194.13 ¢)
(convergent)
Twelfth 5\7ed5 (1990.22 ¢)
(semiconvergent)
Consistency limit 6
Distinct consistency limit 4

7 equal divisions of the 5th harmonic (abbreviated 7ed5) is a nonoctave tuning system that divides the interval of 5/1 into 7 equal parts of about 398 ¢ each. Each step represents a frequency ratio of 51/7, or the 7th root of 5.

Theory

7ed5 is related to 3edo, but with the 5/1 rather than the 2/1 being just. The octave is about 6 cents compressed and the step size is about 398 cents. It is present (though possibly tempered) in any regular temperament which tempers out 441/440 and 244515348/244140625 in the 11-limit, such as equal temperaments 3, 12, 15, 175, 190, 202, and 217edo.

Due to Kirnberger's atom, its step is 100.0002¢ flat of 4/3[clarification needed].

Harmonics

Approximation of harmonics in 7ed5
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -6 +88 -12 +0 +82 -184 -18 +177 -6 -171 +77
Relative (%) -1.5 +22.2 -2.9 +0.0 +20.7 -46.3 -4.4 +44.4 -1.5 -42.9 +19.2
Steps
(reduced) 		3
(3) 		5
(5) 		6
(6) 		7
(0) 		8
(1) 		8
(1) 		9
(2) 		10
(3) 		10
(3) 		10
(3) 		11
(4)
Approximation of harmonics in 7ed5 (contined)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -62 -190 +88 -23 -128 +171 +77 -12 -96 -177 +144 +71
Relative (%) -15.6 -47.8 +22.2 -5.9 -32.3 +42.9 +19.4 -2.9 -24.2 -44.4 +36.3 +17.8
Steps
(reduced) 		11
(4) 		11
(4) 		12
(5) 		12
(5) 		12
(5) 		13
(6) 		13
(6) 		13
(6) 		13
(6) 		13
(6) 		14
(0) 		14
(0)

Intervals

# Cents Approximate ratios
0 0 1/1
1 398 5/4, 34/27
2 796 19/12
3 1194 2/1, 255/128
4 1592 5/2, 128/51
5 1990 60/19
6 2388 4/1, 135/34
7 2786 5/1
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