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| Line 204: |
Line 204: |
| | | | | |
| |} | | |} |
| ==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.
| |
|
| |
| 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]]
| |
| 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
|
19EDF is the equal division of the just perfect fifth into 19 parts of 36.945 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
| degree
|
cents value
|
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
|
|