9ed5/3: Difference between revisions

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'''9ED5/3''' is the [[EdVI|equal division of the just major sixth]] into sixteen parts of 98.2621 [[cent|cents]] each, corresponding to 12.2122 [[edo]]. It is very closely related to the [[Archytas clan|passion temperament]] and a golden tuning of [[15edX]].
{{Infobox ET}}
'''9ed5/3''' is the [[Ed5/3|equal division of the just major sixth]] into nine parts of 98.2621 [[cent]]s each, corresponding to 12.2122[[edo]]. It is very closely related to the [[Passion family#Passion|passion temperament]].


{| class="wikitable"
==Intervals==
{| class="wikitable right-2 center-3"
|+
|+
!Degrees
!Degrees
!15ed(11φ+5\9φ+4)~9ed(5/3)
!Cents
!18/17et
!5/3.4/3.7/3 interpretation
|-
|-
|1
|1
|98.25665
|98.2621
98.2621
|[[16/15]], [[21/20]]
|98.9546
|-
|-
|2
|2
|196.5133
|196.5241
196.5241
|[[28/25]]
|197.9092
|-
|-
|3
|3
|294.7699
|294.7862
294.7862
|[[25/21]]
|296.8638
|-
|-
|4
|4
|393.0266
|393.0483
393.0483
|[[5/4]]
|395.8184
|-
|-
|5
|5
|491.2832
|491.3104
491.3104
|[[4/3]]
|494.773
|-
|-
|6
|6
|589.5399
|589.5725
589.5725
|[[7/5]]
|593.72755
|-
|-
|7
|7
|687.7965
|687.83455
687.83455
|[[112/75]]
|692.68215
|-
|-
|8
|8
|786.0532
|786.0966
786.0966
|[[25/16]], [[80/63]]
|791.6367
|-
|-
|9
|'''9'''
|884.3098
|'''884.3587'''
884.3587
|'''[[5/3]] (just)'''
|890.5913
|-
|-
|10
|10
|982.56645
|982.6207
982.6207
|[[7/4]], [[16/9]]
|989.5459
|-
|-
|11
|11
|1080.8231
|1080.8828
1080.8828
|[[28/15]]
|1088.5005
|-
|-
|12
|12
|1179.07976
|1179.14495
1179.14495
|[[49/25]]
|1187.4551
|-
|-
|13
|13
|1277.3364
|1277.407
1277.407
|[[25/12]]
|1286.4097
|-
|-
|14
|14
|1375.593
|1375.6691
1375.6691
|[[20/9]]
|1385.3643
|-
|-
|15
|15
|1473.8497
|1473.9312
1473.9312
|[[7/3]]
|1484.3189
|}
|}
[[Category:EdVI]]
 
The interval interpretations listed belong to the generator chain of [[septimal passion]] without octaves.
 
== Harmonics ==
{{Harmonics in equal|9|5|3}}
{{Harmonics in equal|9|5|3|collapsed=1|start=12}}
 
 
{{todo|inline=1|improve synopsis|expand}}
[[Category:Nonoctave]]
[[Category:Nonoctave]]

Latest revision as of 14:20, 24 April 2025

← 8ed5/3 9ed5/3 10ed5/3 →
Prime factorization 32
Step size 98.2621 ¢ 
Octave 12\9ed5/3 (1179.14 ¢) (→ 4\3ed5/3)
Twelfth 19\9ed5/3 (1866.98 ¢)
Consistency limit 5
Distinct consistency limit 5

9ed5/3 is the equal division of the just major sixth into nine parts of 98.2621 cents each, corresponding to 12.2122edo. It is very closely related to the passion temperament.

Intervals

Degrees Cents 5/3.4/3.7/3 interpretation
1 98.2621 16/15, 21/20
2 196.5241 28/25
3 294.7862 25/21
4 393.0483 5/4
5 491.3104 4/3
6 589.5725 7/5
7 687.83455 112/75
8 786.0966 25/16, 80/63
9 884.3587 5/3 (just)
10 982.6207 7/4, 16/9
11 1080.8828 28/15
12 1179.14495 49/25
13 1277.407 25/12
14 1375.6691 20/9
15 1473.9312 7/3

The interval interpretations listed belong to the generator chain of septimal passion without octaves.

Harmonics

Approximation of harmonics in 9ed5/3
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -20.9 -35.0 -41.7 -35.0 +42.4 -27.9 +35.7 +28.3 +42.4 -24.3 +21.6
Relative (%) -21.2 -35.6 -42.4 -35.6 +43.2 -28.4 +36.3 +28.8 +43.2 -24.7 +22.0
Steps
(reduced) 		12
(3) 		19
(1) 		24
(6) 		28
(1) 		32
(5) 		34
(7) 		37
(1) 		39
(3) 		41
(5) 		42
(6) 		44
(8)
Approximation of harmonics in 9ed5/3
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) -18.7 -48.8 +28.3 +14.8 +8.1 +7.5 +12.1 +21.6 +35.4 -45.2 -23.9
Relative (%) -19.1 -49.6 +28.8 +15.1 +8.3 +7.6 +12.3 +22.0 +36.0 -46.0 -24.3
Steps
(reduced) 		45
(0) 		46
(1) 		48
(3) 		49
(4) 		50
(5) 		51
(6) 		52
(7) 		53
(8) 		54
(0) 		54
(0) 		55
(1)


Todo: improve synopsis , expand
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