19edf: Difference between revisions

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19edf falls exactly halfway between 32 and 33 edos. It tempers out the same commas as 65edo with the addition of <-27/19 65/19| (1.425¢) resulting from its inexact 4/1.
{{Infobox ET}}


==Intervals==
== Theory ==
19edf corresponds to 32.4807 [[edo]] (similar to every second step of [[65edo]]). It tempers out the same commas as 65edo with the addition of {{monzo| -103/19 65/19 }} (1.425{{c}}) resulting from its inexact 4/1. It is not as similar to [[32edo]] as [[13edf]] and [[16edf]] are to [[22edo]] and [[27edo]].


{| class="wikitable"
== Harmonics ==
{{Harmonics in equal|19|3|2}}
{{Harmonics in equal|19|3|2|start=12|collapsed=1}}
 
== Intervals ==
{| class="wikitable mw-collapsible"
|+ style="font-size: 105%;" | Intervals of 19edf
|-
|-
| | 1
! Degree
| | 36.945
! [[Cent]]s
! Corresponding<br />JI intervals
! comments
|-
|-
| | 2
! colspan="2" | 0
| | 73.89
| '''exact [[1/1]]'''
|  
|-
|-
| | 3
| 1
| | 110.835
| 36.945
|  
|  
|-
|-
| | 4
| 2
| | 147.78
| 73.89
| [[24/23]]
|  
|-
|-
| | 5
| 3
| | 184.725
| 110.835
| [[16/15]]
|  
|-
|-
| | 6
| 4
| | 221.67
| 147.78
| [[12/11]]
|  
|-
|-
| | 7
| 5
| | 258.615
| 184.725
| [[10/9]]
|  
|-
|-
| | 8
| 6
| | 295.56
| 221.67
| [[25/22]]
|  
|-
|-
| | 9
| 7
| | 332.505
| 258.615
| 36/31
|  
|-
|-
| | 10
| 8
| | 369.45
| 295.56
| [[19/16]]
|  
|-
|-
| | 11
| 9
| | 406.395
| 332.505
| 63/52, 40/33
|  
|-
|-
| | 12
| 10
| | 443.34
| 369.45
| [[26/21]]
|  
|-
|-
| | 13
| 11
| | 480.285
| 406.395
| [[24/19]], [[19/15]]
|  
|-
|-
| | 14
| 12
| | 517.23
| 443.34
| 31/24
|  
|-
|-
| | 15
| 13
| | 554.175
| 480.285
| 33/25
|  
|-
|-
| | 16
| 14
| | 591.12
| 517.23
| [[27/20]]
|  
|-
|-
| | 17
| 15
| | 628.065
| 554.175
| [[11/8]]
|  
|-
|-
| | 18
| 16
| | 665.01
| 591.12
| [[45/32]]
|  
|-
|-
| | 19
| 17
| | 701.955
| 628.065
| [[23/16]]
|  
|-
|-
| | 20
| 18
| | 738.9
| 665.01
| [[22/15]]
|  
|-
|-
| | 21
| 19
| | 775.845
| 701.955
| '''exact [[3/2]]'''
| just perfect fifth
|-
|-
| | 22
| 20
| | 812.79
| 738.9
|  
|  
|-
|-
| | 23
| 21
| | 849.735
| 775.845
|  
|  
|-
|-
| | 24
| 22
| | 886.68
| 812.79
| [[8/5]]
|  
|-
|-
| | 25
| 23
| | 923.625
| 849.735
| [[18/11]]
|  
|-
|-
| | 26
| 24
| | 960.57
| 886.68
| [[5/3]]
|  
|-
|-
| | 27
| 25
| | 997.515
| 923.625
|  
|  
|-
|-
| | 28
| 26
| | 1034.46
| 960.57
|  
|  
|-
|-
| | 29
| 27
| | 1071.405
| 997.515
| [[16/9]]
|  
|-
|-
| | 30
| 28
| | 1108.35
| 1034.46
| [[20/11]]
|  
|-
|-
| | 31
| 29
| | 1145.295
| 1071.405
| [[13/7]]
|  
|-
|-
| | 32
| 30
| | 1182.24
| 1108.35
| [[36/19]]
|  
|-
|-
| | 33
| 31
| | 1219.185
| 1145.295
| 31/16
|
|-
| 32
| 1182.24
|
|
|-
| 33
| 1219.185
|
|
|-
| 34
| 1256.13
|
|
|-
| 35
| 1293.075
|
|
|-
| 36
| 1330.02
|
|
|-
| 37
| 1366.965
|
|
|-
| 38
| 1403.91
| '''exact''' 9/4
|
|}
|}
{{todo|expand}}

Latest revision as of 14:52, 8 July 2025

← 18edf 19edf 20edf →
Prime factorization 19 (prime)
Step size 36.945 ¢ 
Octave 32\19edf (1182.24 ¢)
Twelfth 51\19edf (1884.2 ¢)
Consistency limit 3
Distinct consistency limit 3

Theory

19edf corresponds to 32.4807 edo (similar to every second step of 65edo). It tempers out the same commas as 65edo with the addition of [-103/19 65/19⟩ (1.425 ¢) resulting from its inexact 4/1. It is not as similar to 32edo as 13edf and 16edf are to 22edo and 27edo.

Harmonics

Approximation of harmonics in 19edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -17.8 -17.8 +1.4 -15.4 +1.4 -6.8 -16.3 +1.4 +3.7 -13.5 -16.3
Relative (%) -48.1 -48.1 +3.9 -41.8 +3.9 -18.5 -44.2 +3.9 +10.1 -36.5 -44.2
Steps
(reduced) 		32
(13) 		51
(13) 		65
(8) 		75
(18) 		84
(8) 		91
(15) 		97
(2) 		103
(8) 		108
(13) 		112
(17) 		116
(2)
Approximation of harmonics in 19edf
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) -7.1 +12.4 +3.7 +2.9 +8.7 -16.3 +0.9 -14.0 +12.4 +5.7 +2.6
Relative (%) -19.3 +33.4 +10.1 +7.7 +23.6 -44.2 +2.4 -37.9 +33.4 +15.4 +7.1
Steps
(reduced) 		120
(6) 		124
(10) 		127
(13) 		130
(16) 		133
(0) 		135
(2) 		138
(5) 		140
(7) 		143
(10) 		145
(12) 		147
(14)

Intervals

Intervals of 19edf
Degree Cents Corresponding
JI intervals 		comments
0 exact 1/1
1 36.945
2 73.89 24/23
3 110.835 16/15
4 147.78 12/11
5 184.725 10/9
6 221.67 25/22
7 258.615 36/31
8 295.56 19/16
9 332.505 63/52, 40/33
10 369.45 26/21
11 406.395 24/19, 19/15
12 443.34 31/24
13 480.285 33/25
14 517.23 27/20
15 554.175 11/8
16 591.12 45/32
17 628.065 23/16
18 665.01 22/15
19 701.955 exact 3/2 just perfect fifth
20 738.9
21 775.845
22 812.79 8/5
23 849.735 18/11
24 886.68 5/3
25 923.625
26 960.57
27 997.515 16/9
28 1034.46 20/11
29 1071.405 13/7
30 1108.35 36/19
31 1145.295 31/16
32 1182.24
33 1219.185
34 1256.13
35 1293.075
36 1330.02
37 1366.965
38 1403.91 exact 9/4
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