18edf: Difference between revisions

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== Intervals ==
== Intervals ==
{| class="wikitable mw-collapsible"
{| class="wikitable center-1 right-2 mw-collapsible"
|+ style="font-size: 105%;" | Intervals of 18edf
|-
|-
! Degree
! #
! Cents value
! Cents
! Corresponding<br />JI intervals
! Approximate ratios
! Comments
! Comments
|-
|-
! colspan="2" | 0
| 0
| '''exact [[1/1]]'''
| 0.0
|
| [[1/1]]
|-
|-
| 1
| 1
| 38.9975
| 39.0
| 45/44
| [[33/32]], [[36/35]], [[49/48]], [[50/49]], [[64/63]]
|
|-
|-
| 2
| 2
| 77.995
| 78.0
|
| [[21/20]], [[22/21]], [[25/24]], [[28/27]]
|  
|-
|-
| 3
| 3
| 116.9925
| 117.0
| 16/15
| [[15/14]], [[16/15]]
|
|-
|-
| 4
| 4
| 155.99
| 156.0
| 128/117
| [[11/10]], [[12/11]]
|
|-
|-
| 5
| 5
| 194.9875
| 195.0
| 28/25
| [[9/8]], [[10/9]]
|
|-
|-
| 6
| 6
| 233.985
| 234.0
| 8/7
| [[8/7]]
|
|-
|-
| 7
| 7
| 272.9825
| 273.0
| 7/6
| [[7/6]]
|
|-
|-
| 8
| 8
| 311.98
| 312.0
| 6/5
| [[6/5]]
|
|-
|-
| 9
| 9
| 350.9775
| 351.0
| 60/49, 49/40
| [[11/9]], [[16/13]]
|
|-
|-
| 10
| 10
| 389.975
| 390.0
| 5/4
| [[5/4]]
|
|-
|-
| 11
| 11
| 428.9725
| 429.0
| 9/7
| [[9/7]], [[14/11]]
|
|-
|-
| 12
| 12
| 467.97
| 468.0
|
| [[13/10]], [[21/16]]
|  
|-
|-
| 13
| 13
| 506.9675
| 507.0
| 75/56
| [[4/3]]
|
|-
|-
| 14
| 14
| 545.965
| 546.0
|
| [[11/8]], [[15/11]]
|  
|-
|-
| 15
| 15
| 584.9625
| 585.0
|
| [[7/5]]
|  
|-
|-
| 16
| 16
| 623.96
| 624.0
|
| [[10/7]]
|  
|-
|-
| 17
| 17
| 662.9575
| 663.0
| [[22/15]]
| [[16/11]], [[22/15]]
|
|-
|-
| 18
| 18
| 701.955
| 702.0
| '''exact [[3/2]]'''
| [[3/2]]
| just perfect fifth
|-
|-
| 19
| 19
| 740.9525
| 741.0
| 135/88
| [[20/13]], [[32/21]]
|
|-
|-
| 20
| 20
| 779.95
| 780.0
|
| [[11/7]], [[14/9]]
|  
|-
|-
| 21
| 21
| 818.9475
| 818.9
| 8/5
| [[8/5]]
|
|-
|-
| 22
| 22
| 857.945
| 857.9
| 64/39
| [[18/11]]
|
|-
|-
| 23
| 23
| 896.9425
| 896.9
| 42/25
| [[5/3]]
|
|-
|-
| 24
| 24
| 935.94
| 935.9
| 12/7
| [[12/7]]
|
|-
|-
| 25
| 25
| 974.9375
| 974.9
| 7/4
| [[7/4]]
|
|-
|-
| 26
| 26
| 1013.935
| 1013.9
| 9/5
| [[9/5]]
|
|-
|-
| 27
| 27
| 1052.9325
| 1052.9
| 90/49, 147/80
| [[11/6]]
|
|-
|-
| 28
| 28
| 1091.93
| 1091.9
| 15/8
| [[15/8]]
|
|-
|-
| 29
| 29
| 1130.9275
| 1130.9
| 27/14
| [[27/14]]
|
|-
|-
| 30
| 30
| 1169.925
| 1169.9
|
| [[35/18]], [[49/25]], [[63/32]]
|  
|-
|-
| 31
| 31
| 1208.9225
| 1208.9
| 225/112
| [[2/1]]
|
|-
|-
| 32
| 32
| 1247.92
| 1247.9
|
| [[33/16]], [[45/22]], [[49/24]], [[55/27]]
|  
|-
|-
| 33
| 33
| 1286.9175
| 1286.9
|
| [[21/10]], [[25/12]]
|  
|-
|-
| 34
| 34
| 1325.915
| 1325.9
|
| [[15/7]]
|  
|-
|-
| 35
| 35
| 1364.9125
| 1364.9
|
| [[11/5]]
|  
|-
|-
| 36
| 36
| 1403.91
| 1403.9
| '''exact''' 9/4
| [[9/4]]
|
|}
|}


Revision as of 12:48, 24 March 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.

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 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +5.2 -6.1 -8.6 -3.3 +8.7 -12.2 +11.2 +0.4 -6.1 -8.7 -7.6 -3.3
Relative (%) +13.3 -15.7 -22.0 -8.5 +22.4 -31.4 +28.6 +0.9 -15.7 -22.2 -19.5 -8.5
Steps
(reduced) 		114
(6) 		117
(9) 		120
(12) 		123
(15) 		126
(0) 		128
(2) 		131
(5) 		133
(7) 		135
(9) 		137
(11) 		139
(13) 		141
(15)

Intervals

# Cents Approximate ratios Comments
0 0.0 1/1
1 39.0 33/32, 36/35, 49/48, 50/49, 64/63
2 78.0 21/20, 22/21, 25/24, 28/27
3 117.0 15/14, 16/15
4 156.0 11/10, 12/11
5 195.0 9/8, 10/9
6 234.0 8/7
7 273.0 7/6
8 312.0 6/5
9 351.0 11/9, 16/13
10 390.0 5/4
11 429.0 9/7, 14/11
12 468.0 13/10, 21/16
13 507.0 4/3
14 546.0 11/8, 15/11
15 585.0 7/5
16 624.0 10/7
17 663.0 16/11, 22/15
18 702.0 3/2
19 741.0 20/13, 32/21
20 780.0 11/7, 14/9
21 818.9 8/5
22 857.9 18/11
23 896.9 5/3
24 935.9 12/7
25 974.9 7/4
26 1013.9 9/5
27 1052.9 11/6
28 1091.9 15/8
29 1130.9 27/14
30 1169.9 35/18, 49/25, 63/32
31 1208.9 2/1
32 1247.9 33/16, 45/22, 49/24, 55/27
33 1286.9 21/10, 25/12
34 1325.9 15/7
35 1364.9 11/5
36 1403.9 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

See also

Retrieved from "https://en.xen.wiki/index.php?title=18edf&oldid=188029"