15ed7/3: Difference between revisions

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{{Stub}}
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{{Infobox ET}}
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{{ED intro}} The chord [0 5 9]\15ed7/3, closely resembling the 5-limit major triad of [[15edo]], is a close approximation in this tuning system to [[3:4:5]]. A potentially tempered 15ed7/3 chain with octaves as the equivalence results in [[passion]] temperament, with 3:4:5:7 located on the same position that it is in 15ed7/3.
{{ED intro}}
 
== Theory ==
The chord [0 5 9]\15ed7/3, closely resembling the 5-limit major triad of [[15edo]], is a close approximation in this tuning system to [[3:4:5]]. A potentially tempered 15ed7/3 chain with octaves as the equivalence results in [[passion]] temperament, with 3:4:5:7 located on the same position that it is in 15ed7/3.
 
=== Harmonics ===
{{Harmonics in equal|15|7|3|columns=11}}
{{Harmonics in equal|15|7|3|columns=12|start=12|collapsed=1|title=Approximation of harmonics in 11ed7/3 (continued)}}


== Intervals ==
== Intervals ==
{| class="wikitable"
{| class="wikitable center-1 right-2"
!Degrees
! #
!ed7/3
! Cents
|-
|-
|1
| 1
|97.7914
| 97.8
|-
|-
|2
| 2
|195.5828
| 195.6
|-
|-
|3
| 3
|293.3742
| 293.4
|-
|-
|4
| 4
|391.1656
| 391.2
|-
|-
|5
| 5
|488.957
| 489.0
|-
|-
|6
| 6
|586.7484
| 586.7
|-
|-
|7
| 7
|684.5398
| 684.5
|-
|-
|8
| 8
|782.33115
| 782.3
|-
|-
|9
| 9
|880.1225
| 880.1
|-
|-
|10
| 10
|977.9139
| 977.9
|-
|-
|11
| 11
|1075.7053
| 1075.7
|-
|-
|12
| 12
|1173.4967
| 1173.5
|-
|-
|13
| 13
|1271.2881
| 1271.3
|-
|-
|14
| 14
|1369.0795
| 1369.1
|-
|-
|15
| 15
|1466.8709
| 1466.9
|}
|}
== Harmonics ==
{{Harmonics in equal
| steps = 15
| num = 7
| denom = 3
}}
{{Harmonics in equal
| steps = 15
| num = 7
| denom = 3
| start = 12
| collapsed = 1
}}

Latest revision as of 18:03, 14 May 2026

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← 14ed7/3 15ed7/3 16ed7/3 →
Prime factorization 3 × 5
Step size 97.7914 ¢ 
Octave 12\15ed7/3 (1173.5 ¢) (→ 4\5ed7/3)
Twelfth 19\15ed7/3 (1858.04 ¢)
Consistency limit 3
Distinct consistency limit 3

15 equal divisions of 7/3 (abbreviated 15ed7/3) is a nonoctave tuning system that divides the interval of 7/3 into 15 equal parts of about 97.8 ¢ each. Each step represents a frequency ratio of (7/3)1/15, or the 15th root of 7/3.

Theory

The chord [0 5 9]\15ed7/3, closely resembling the 5-limit major triad of 15edo, is a close approximation in this tuning system to 3:4:5. A potentially tempered 15ed7/3 chain with octaves as the equivalence results in passion temperament, with 3:4:5:7 located on the same position that it is in 15ed7/3.

Harmonics

Approximation of harmonics in 15ed7/3
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -26.5 -43.9 +44.8 -48.2 +27.4 -43.9 +18.3 +10.0 +23.1 -44.1 +0.9
Relative (%) -27.1 -44.9 +45.8 -49.2 +28.0 -44.9 +18.7 +10.2 +23.7 -45.1 +0.9
Steps
(reduced) 		12
(12) 		19
(4) 		25
(10) 		28
(13) 		32
(2) 		34
(4) 		37
(7) 		39
(9) 		41
(11) 		42
(12) 		44
(14)
Approximation of harmonics in 11ed7/3 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -39.9 +27.4 +5.7 -8.2 -15.4 -16.5 -12.4 -3.4 +10.0 +27.2 +48.0 -25.6
Relative (%) -40.8 +28.0 +5.8 -8.4 -15.7 -16.9 -12.6 -3.4 +10.2 +27.8 +49.1 -26.2
Steps
(reduced) 		45
(0) 		47
(2) 		48
(3) 		49
(4) 		50
(5) 		51
(6) 		52
(7) 		53
(8) 		54
(9) 		55
(10) 		56
(11) 		56
(11)

Intervals

# Cents
1 97.8
2 195.6
3 293.4
4 391.2
5 489.0
6 586.7
7 684.5
8 782.3
9 880.1
10 977.9
11 1075.7
12 1173.5
13 1271.3
14 1369.1
15 1466.9
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