18edf: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
m Cleanup
ArrowHead294 (talk | contribs)
14,515 edits
mNo edit summary
Line 5: Line 5:
18edf is related to the [[regular temperament]] which [[tempering out|tempers out]] 2401/2400 and 8589934592/8544921875 in the [[7-limit]]; with 5632/5625, 46656/46585, and 166698/166375 in the [[11-limit]], which is supported by [[31edo]], [[369edo]], [[400edo]], [[431edo]], and [[462edo]].
18edf is related to the [[regular temperament]] which [[tempering out|tempers out]] 2401/2400 and 8589934592/8544921875 in the [[7-limit]]; with 5632/5625, 46656/46585, and 166698/166375 in the [[11-limit]], which is supported by [[31edo]], [[369edo]], [[400edo]], [[431edo]], and [[462edo]].


Lookalikes: [[31edo]], [[49edt]], [[72ed5]], [[80ed6]]
Lookalikes: [[31edo]], [[39cET]], [[49edt]], [[72ed5]], [[80ed6]]


=== Harmonics ===
=== Harmonics ===
Line 13: Line 13:
== Intervals ==
== Intervals ==
{| class="wikitable mw-collapsible"
{| class="wikitable mw-collapsible"
|+ Intervals of 18edf
|+ style="font-size: 105%;" | Intervals of 18edf
|-
|-
! | degree
! Degree
! | cents value
! Cents value
! | corresponding <br>JI intervals
! Corresponding<br />JI intervals
! | comments
! Comments
|-
|-
! colspan="2" | 0
! colspan="2" | 0
| | '''exact [[1/1]]'''
| '''exact [[1/1]]'''
| |  
|  
|-
|-
| | 1
| 1
| | 38.9975
| 38.9975
| | 45/44
| 45/44
| |  
|  
|-
|-
| | 2
| 2
| | 77.995
| 77.995
| |  
|  
| |  
|  
|-
|-
| | 3
| 3
| | 116.9925
| 116.9925
| |16/15
| 16/15
| |  
|  
|-
|-
| | 4
| 4
| | 155.99
| 155.99
| | 128/117
| 128/117
| |  
|  
|-
|-
| | 5
| 5
| | 194.9875
| 194.9875
| | 28/25
| 28/25
| |  
|  
|-
|-
| | 6
| 6
| | 233.985
| 233.985
| |8/7
| 8/7
| |  
|  
|-
|-
| | 7
| 7
| | 272.9825
| 272.9825
| |7/6
| 7/6
| |  
|  
|-
|-
| | 8
| 8
| | 311.98
| 311.98
| |6/5
| 6/5
| |  
|  
|-
|-
| | 9
| 9
| | 350.9775
| 350.9775
| | 60/49, 49/40
| 60/49, 49/40
| |  
|  
|-
|-
| | 10
| 10
| | 389.975
| 389.975
| |5/4
| 5/4
| |  
|  
|-
|-
| | 11
| 11
| | 428.9725
| 428.9725
| |9/7
| 9/7
| |  
|  
|-
|-
| | 12
| 12
| | 467.97
| 467.97
| |  
|  
| |  
|  
|-
|-
| | 13
| 13
| | 506.9675
| 506.9675
| | 75/56
| 75/56
| |  
|  
|-
|-
| | 14
| 14
| | 545.965
| 545.965
| |  
|  
| |  
|  
|-
|-
| | 15
| 15
| | 584.9625
| 584.9625
| |  
|  
| |  
|  
|-
|-
| | 16
| 16
| | 623.96
| 623.96
| |  
|  
| |  
|  
|-
|-
| | 17
| 17
| | 662.9575
| 662.9575
| | [[22/15]]
| [[22/15]]
| |  
|  
|-
|-
| | 18
| 18
| | 701.955
| 701.955
| | '''exact [[3/2]]'''
| '''exact [[3/2]]'''
| | just perfect fifth
| just perfect fifth
|-
|-
|19
| 19
|740.9525
| 740.9525
|135/88
| 135/88
|
|  
|-
|-
|20
| 20
|779.95
| 779.95
|
|  
|
|  
|-
|-
|21
| 21
|818.9475
| 818.9475
|8/5
| 8/5
|
|  
|-
|-
|22
| 22
|857.945
| 857.945
|64/39
| 64/39
|
|  
|-
|-
|23
| 23
|896.9425
| 896.9425
|42/25
| 42/25
|
|  
|-
|-
|24
| 24
|935.94
| 935.94
|12/7
| 12/7
|
|  
|-
|-
|25
| 25
|974.9375
| 974.9375
|7/4
| 7/4
|
|  
|-
|-
|26
| 26
|1013.935
| 1013.935
|9/5
| 9/5
|
|  
|-
|-
|27
| 27
|1052.9325
| 1052.9325
|90/49, 147/80
| 90/49, 147/80
|
|  
|-
|-
|28
| 28
|1091.93
| 1091.93
|15/8
| 15/8
|
|  
|-
|-
|29
| 29
|1130.9275
| 1130.9275
|27/14
| 27/14
|
|  
|-
|-
|30
| 30
|1169.925
| 1169.925
|
|  
|
|  
|-
|-
|31
| 31
|1208.9225
| 1208.9225
|225/112
| 225/112
|
|  
|-
|-
|32
| 32
|1247.92
| 1247.92
|
|  
|
|  
|-
|-
|33
| 33
|1286.9175
| 1286.9175
|
|  
|
|  
|-
|-
|34
| 34
|1325.915
| 1325.915
|
|  
|
|  
|-
|-
|35
| 35
|1364.9125
| 1364.9125
|
|  
|
|  
|-
|-
|36
| 36
|1403.91
| 1403.91
|'''exact''' 9/4
| '''exact''' 9/4
|
|  
|}
|}


Line 208: Line 208:
The rank-two regular temperament supported by 31edo and 369edo has three equal divisions of the interval which equals an octave minus the step interval of 18EDF as a generator.
The rank-two regular temperament supported by 31edo and 369edo has three equal divisions of the interval which equals an octave minus the step interval of 18EDF as a generator.


=== 7-limit 31&amp;369 ===
=== 7-limit 31 &amp; 369 ===
Commas: 2401/2400, 8589934592/8544921875
Commas: 2401/2400, 8589934592/8544921875


POTE generator: ~5/4 = 386.997
POTE generator: ~5/4 = 386.997


Mapping: [&lt;1 19 2 7|, &lt;0 -54 1 -13|]
Mapping: [{{map| 1 19 2 7 }}, {{map| 0 -54 1 -13 }}]


EDOs: {{EDOs|31, 369, 400, 431, 462}}
EDOs: {{EDOs|31, 369, 400, 431, 462}}


===11-limit 31&amp;369===
=== 11-limit 31 &amp; 369 ===
Commas: 2401/2400, 5632/5625, 46656/46585
Commas: 2401/2400, 5632/5625, 46656/46585


POTE generator: ~5/4 = 386.999
POTE generator: ~5/4 = 386.999


Mapping: [&lt;1 19 2 7 37|, &lt;0 -54 1 -13 -104|]
Mapping: [{{map| 1 19 2 7 37 }}, {{map| 0 -54 1 -13 -104 }}]


EDOs: 31, 369, 400, 431, 462
EDOs: 31, 369, 400, 431, 462


===13-limit 31&amp;369===
=== 13-limit 31 &amp; 369 ===
Commas: 1001/1000, 1716/1715, 4096/4095, 46656/46585
Commas: 1001/1000, 1716/1715, 4096/4095, 46656/46585


POTE generator: ~5/4 = 387.003
POTE generator: ~5/4 = 387.003


Mapping: [&lt;1 19 2 7 37 -35|, &lt;0 -54 1 -13 -104 120|]
Mapping: [{{map| 1 19 2 7 37 -35 }}, {{map| 0 -54 1 -13 -104 120 }}]


EDOs: 31, 369, 400, 431, 462
EDOs: 31, 369, 400, 431, 462


{{Todo|cleanup|expand|inline=1|comment=say what the temperaments are like and why one would want to use them, and for what}}
{{Todo|cleanup|expand|inline=1|comment=say what the temperaments are like and why one would want to use them, and for what}}

Revision as of 16:47, 21 January 2025

← 17edf 18edf 19edf →
Prime factorization 2 × 32
Step size 38.9975 ¢ 
Octave 31\18edf (1208.92 ¢)
Twelfth 49\18edf (1910.88 ¢)
Consistency limit 4
Distinct consistency limit 4

18 equal divisions of the perfect fifth (abbreviated 18edf or 18ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 18 equal parts of about 39 ¢ each. Each step represents a frequency ratio of (3/2)1/18, or the 18th root of 3/2.

Theory

18edf is related to the regular temperament which tempers out 2401/2400 and 8589934592/8544921875 in the 7-limit; with 5632/5625, 46656/46585, and 166698/166375 in the 11-limit, which is supported by 31edo, 369edo, 400edo, 431edo, and 462edo.

Lookalikes: 31edo, 39cET, 49edt, 72ed5, 80ed6

Harmonics

Approximation of harmonics in 18edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +8.9 +8.9 +17.8 -17.5 +17.8 -15.0 -12.2 +17.8 -8.6 -17.6 -12.2
Relative (%) +22.9 +22.9 +45.8 -44.9 +45.8 -38.6 -31.4 +45.8 -22.0 -45.1 -31.4
Steps
(reduced) 		31
(13) 		49
(13) 		62
(8) 		71
(17) 		80
(8) 		86
(14) 		92
(2) 		98
(8) 		102
(12) 		106
(16) 		110
(2)
Approximation of harmonics in 18edf
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) +5.2 -6.1 -8.6 -3.3 +8.7 -12.2 +11.2 +0.4 -6.1 -8.7 -7.6
Relative (%) +13.3 -15.7 -22.0 -8.5 +22.4 -31.4 +28.6 +0.9 -15.7 -22.2 -19.5
Steps
(reduced) 		114
(6) 		117
(9) 		120
(12) 		123
(15) 		126
(0) 		128
(2) 		131
(5) 		133
(7) 		135
(9) 		137
(11) 		139
(13)

Intervals

Intervals of 18edf
Degree Cents value Corresponding
JI intervals 		Comments
0 exact 1/1
1 38.9975 45/44
2 77.995
3 116.9925 16/15
4 155.99 128/117
5 194.9875 28/25
6 233.985 8/7
7 272.9825 7/6
8 311.98 6/5
9 350.9775 60/49, 49/40
10 389.975 5/4
11 428.9725 9/7
12 467.97
13 506.9675 75/56
14 545.965
15 584.9625
16 623.96
17 662.9575 22/15
18 701.955 exact 3/2 just perfect fifth
19 740.9525 135/88
20 779.95
21 818.9475 8/5
22 857.945 64/39
23 896.9425 42/25
24 935.94 12/7
25 974.9375 7/4
26 1013.935 9/5
27 1052.9325 90/49, 147/80
28 1091.93 15/8
29 1130.9275 27/14
30 1169.925
31 1208.9225 225/112
32 1247.92
33 1286.9175
34 1325.915
35 1364.9125
36 1403.91 exact 9/4

Related regular temperaments

The rank-two regular temperament supported by 31edo and 369edo has three equal divisions of the interval which equals an octave minus the step interval of 18EDF as a generator.

7-limit 31 & 369

Commas: 2401/2400, 8589934592/8544921875

POTE generator: ~5/4 = 386.997

Mapping: [⟨1 19 2 7], ⟨0 -54 1 -13]]

EDOs: 31, 369, 400, 431, 462

11-limit 31 & 369

Commas: 2401/2400, 5632/5625, 46656/46585

POTE generator: ~5/4 = 386.999

Mapping: [⟨1 19 2 7 37], ⟨0 -54 1 -13 -104]]

EDOs: 31, 369, 400, 431, 462

13-limit 31 & 369

Commas: 1001/1000, 1716/1715, 4096/4095, 46656/46585

POTE generator: ~5/4 = 387.003

Mapping: [⟨1 19 2 7 37 -35], ⟨0 -54 1 -13 -104 120]]

EDOs: 31, 369, 400, 431, 462

Todo: cleanup , expand

say what the temperaments are like and why one would want to use them, and for what

Retrieved from "https://en.xen.wiki/index.php?title=18edf&oldid=177557"