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{{Infobox ET}}
{{Infobox ET}}
'''19EDF''' is the [[EDF|equal division of the just perfect fifth]] into 19 parts of 36.945 [[cent|cents]] each, corresponding 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 cents) resulting from its inexact 4/1.


==Intervals==
== Theory ==
{| class="wikitable"
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]].
 
== 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
|-
|-
! | degree
! Degree
! | cents value
! [[Cent]]s
! | corresponding <br>JI intervals
! Corresponding<br />JI intervals
! | comments
! comments
|-
|-
! colspan="2" | 0
! colspan="2" | 0
| | '''exact [[1/1]]'''
| '''exact [[1/1]]'''
| |  
|  
|-
|-
| | 1
| 1
| | 36.945
| 36.945
| |  
|  
| |  
|  
|-
|-
| | 2
| 2
| | 73.89
| 73.89
| | [[24/23]]
| [[24/23]]
| |  
|  
|-
|-
| | 3
| 3
| | 110.835
| 110.835
| | [[16/15]]
| [[16/15]]
| |  
|  
|-
|-
| | 4
| 4
| | 147.78
| 147.78
| | [[12/11]]
| [[12/11]]
| |  
|  
|-
|-
| | 5
| 5
| | 184.725
| 184.725
| | [[10/9]]
| [[10/9]]
| |  
|  
|-
|-
| | 6
| 6
| | 221.67
| 221.67
| | [[25/22]]
| [[25/22]]
| |  
|  
|-
|-
| | 7
| 7
| | 258.615
| 258.615
| | 36/31
| 36/31
| |  
|  
|-
|-
| | 8
| 8
| | 295.56
| 295.56
| | [[19/16]]
| [[19/16]]
| |  
|  
|-
|-
| | 9
| 9
| | 332.505
| 332.505
| | 63/52, 40/33
| 63/52, 40/33
| |  
|  
|-
|-
| | 10
| 10
| | 369.45
| 369.45
| | [[26/21]]
| [[26/21]]
| |  
|  
|-
|-
| | 11
| 11
| | 406.395
| 406.395
| | [[24/19]], [[19/15]]
| [[24/19]], [[19/15]]
| |  
|  
|-
|-
| | 12
| 12
| | 443.34
| 443.34
| | 31/24
| 31/24
| |  
|  
|-
|-
| | 13
| 13
| | 480.285
| 480.285
| | 33/25
| 33/25
| |  
|  
|-
|-
| | 14
| 14
| | 517.23
| 517.23
| | [[27/20]]
| [[27/20]]
| |  
|  
|-
|-
| | 15
| 15
| | 554.175
| 554.175
| | [[11/8]]
| [[11/8]]
| |  
|  
|-
|-
| | 16
| 16
| | 591.12
| 591.12
| | [[45/32]]
| [[45/32]]
| |  
|  
|-
|-
| | 17
| 17
| | 628.065
| 628.065
| | [[23/16]]
| [[23/16]]
| |  
|  
|-
|-
| | 18
| 18
| | 665.01
| 665.01
| | [[22/15]]
| [[22/15]]
| |  
|  
|-
|-
| | 19
| 19
| | 701.955
| 701.955
| | '''exact [[3/2]]'''
| '''exact [[3/2]]'''
| | just perfect fifth
| just perfect fifth
|-
|-
| | 20
| 20
| | 738.9
| 738.9
| |  
|  
| |  
|  
|-
|-
| | 21
| 21
| | 775.845
| 775.845
| |  
|  
| |  
|  
|-
|-
| | 22
| 22
| | 812.79
| 812.79
| | [[8/5]]
| [[8/5]]
| |  
|  
|-
|-
| | 23
| 23
| | 849.735
| 849.735
| | [[18/11]]
| [[18/11]]
| |  
|  
|-
|-
| | 24
| 24
| | 886.68
| 886.68
| | [[5/3]]
| [[5/3]]
| |  
|  
|-
|-
| | 25
| 25
| | 923.625
| 923.625
| |  
|  
| |  
|  
|-
|-
| | 26
| 26
| | 960.57
| 960.57
| |  
|  
| |  
|  
|-
|-
| | 27
| 27
| | 997.515
| 997.515
| | [[16/9]]
| [[16/9]]
| |  
|  
|-
|-
| | 28
| 28
| | 1034.46
| 1034.46
| | [[20/11]]
| [[20/11]]
| |  
|  
|-
|-
| | 29
| 29
| | 1071.405
| 1071.405
| | [[13/7]]
| [[13/7]]
| |  
|  
|-
|-
| | 30
| 30
| | 1108.35
| 1108.35
| | [[36/19]]
| [[36/19]]
| |  
|  
|-
|-
| | 31
| 31
| | 1145.295
| 1145.295
| | 31/16
| 31/16
| |  
|  
|-
|-
| | 32
| 32
| | 1182.24
| 1182.24
| |  
|  
| |  
|  
|-
|-
| | 33
| 33
| | 1219.185
| 1219.185
| |  
|  
| |  
|  
|-
|-
|34
| 34
|1256.13
| 1256.13
|
|  
|
|  
|-
|-
|35
| 35
|1293.075
| 1293.075
|
|  
|
|  
|-
|-
|36
| 36
|1330.02
| 1330.02
|
|  
|
|  
|-
|-
|37
| 37
|1366.965
| 1366.965
|
|  
|
|  
|-
|-
|38
| 38
|1403.91
| 1403.91
|'''exact''' 9/4
| '''exact''' 9/4
|
|  
|}
|}
==Scale tree==
If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking [[Mediant|"freshman sums"]] of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO.


If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum.
{{todo|expand}}
 
Generator range: 36.09023 cents (4\7/19 = 4\133) to 37.89474 cents (3\5/19 = 3\95)
{| class="wikitable center-all"
! colspan="7" | Fifth
!Cents
! Comments
|-
| 4\7|| || || || || || || 36.0902||
|-
| || || || || ||  ||27\47||36.2822||
|-
|  || || || || ||23\40||  ||36.3158 ||
|-
| || || || || || ||42\73 ||36.3374 ||
|-
| ||  || || || 19\33||  || ||36.{{Overline|36}}||
|-
| || || || || || ||53\92||36.3844||
|-
| || || ||  || ||34\59|| ||36.3961||
|-
| || ||  || || || ||49\85||36.4087 ||
|-
|  || || ||15\26|| ||  || ||36.43725||
|-
| || ||  || || || ||56\97||36.4633||
|-
| || ||  || || || 41\71|| ||36.4175||
|-
|  || || || || || ||67\116|| 36.4791||
|-
| || ||  || || 26\45||  || ||36.4912||[[Flattone]] is in this region
|-
| ||  || || || || ||63\109||36.5041||
|-
| || || ||  || ||37\64||  ||36.5132||
|-
| ||  || || || || ||48\83 || 36.52505||
|-
| ||  ||11\19|| || ||  || || 36.5651||
|-
| || || || || || || 51\88||36.6029||
|-
| || || ||  || ||40\69 || ||36.6133||
|-
|  || || || || || ||69\119|| 36.6210||
|-
|  || || || || 29\50|| ||  ||36.6316||
|-
| ||  || ||  || || || 76\131||36.6412||[[Golden meantone]] (696.2145¢)
|-
| || || || || ||47\81|| ||36.6472||
|-
| || || || || || ||65\112||36.6541||
|-
| || || ||18\31|| || || ||36.6723||[[Meantone]] is in this region
|-
| || || || || || ||61\105||36.6917||
|-
| || || || || ||43\74|| ||36.6999||
|-
| || || || || || ||68\117||36.70175||
|-
| || || || ||25\43|| || ||36.7197||
|-
| || || || || || ||57\98||36.7347||
|-
| || || || || ||32\55|| ||36.7464||
|-
| || || || || || ||39\67||36.76355||
|-
| ||7\12|| || || || || ||36.8421||
|-
| || || || || || ||38\65||36.9231||
|-
| || || || || ||31\53|| ||36.9414||The fifth closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || ||55\94||36.9541||[[Garibaldi]] / [[Cassandra]]
|-
| || || || ||24\41|| || ||36.9705||
|-
| || || || || || ||65\111||36.98435||
|-
| || || || || ||41\70|| ||36.9925||
|-
| || || || || || ||58\99||37.0016||
|-
| || || ||17\29|| || || ||37.0236||
|-
| || || || || || ||61\104||37.0445||
|-
| || || || || ||44\75|| ||37.0526||
|-
| || || || || || ||71\121||37.0596||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||37.0709||[[Neogothic]] is in this region
|-
| || || || || || ||64\109||37.0835||
|-
| || || || || ||37\63|| ||37.0927||
|-
| || || || || || ||47\80||37.1053||
|-
| || ||10\17|| || || || ||37.1517||
|-
| || || || || || ||43\73||37.2026||
|-
| || || || || ||33\56|| ||37.21805||
|-
| || || || || || ||56\95||37.2299||
|-
| || || || ||23\39|| || ||37.2470||
|-
| || || || || || ||59\100||37.2632||
|-
| || || || || ||36\61|| ||37.2735||
|-
| || || || || || ||49\83||37.2388||
|-
| || || ||13\22|| || || ||37.3206||[[Archy]] is in this region
|-
| || || || || || ||42\71||37.3610||
|-
| || || || || ||29\49|| ||37.3792||
|-
| || || || || || ||45\76||37.3961||
|-
| || || || ||16\27|| || ||37.4269||
|-
| || || || || || ||35\59||37.46655||
|-
| || || || || ||19\32|| ||37.5000||
|-
| || || || || || ||22\37||37.5533||
|-
|3\5|| || || || || || ||37.8947||
|}Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.
[[Category:Edf]]
[[Category:Edonoi]]

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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