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| Line 226: |
Line 226: |
|
| |
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| EDOs: 31, 369, 400, 431, 462 | | EDOs: 31, 369, 400, 431, 462 |
| ==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: 28.09524 cents (4\7/18 = 2\63) to 40 cents (3\5/18 = 1\30)
| |
| {| class="wikitable center-all"
| |
| ! colspan="7" |Fifth
| |
| !Cents
| |
| !Comments
| |
| |-
| |
| | 4\7 || || || || || || || 38.0952||
| |
| |-
| |
| | || || || || || ||27\47||38.2979||
| |
| |-
| |
| | || || || || ||23\40|| ||38.{{Overline|3}}||
| |
| |-
| |
| | || || || || || ||42\73 ||38.3562||
| |
| |-
| |
| | || || || ||19\33 || || ||38.{{Overline|38}}||
| |
| |-
| |
| | || || || || || || 53\92||38.4058 ||
| |
| |-
| |
| | || || || || ||34\59|| ||38.4181||
| |
| |-
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| | || || || || || || 49\85||38.4314 ||
| |
| |-
| |
| | || || ||15\26|| || || || 38.4615||
| |
| |-
| |
| | || || || || || ||56\97 ||38.4880||
| |
| |-
| |
| | || || || || ||41\71|| || 38.49765||
| |
| |-
| |
| | || || || || || ||67\116|| 38.50575||
| |
| |-
| |
| | || || || ||26\45|| || ||38.{{Overline|518}}||[[Flattone]] is in this region
| |
| |-
| |
| | || || || || || ||63\109||38.5321||
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| |-
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| | || || || || ||37\64|| ||38.5417||
| |
| |-
| |
| | || || || || || ||48\83||38.5542||
| |
| |-
| |
| | || ||11\19|| || || || ||38.5965||
| |
| |-
| |
| | || || || || || ||51\88||38.{{Overline|63}}||
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| |-
| |
| | || || || || ||40\69|| ||38.6473||
| |
| |-
| |
| | || || || || || ||69\119||38.6555||
| |
| |-
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| | || || || ||29\50|| || ||38.{{Overline|6}}||
| |
| |-
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| | || || || || || ||76\131||38.6768||[[Golden meantone]] (696.2145¢)
| |
| |-
| |
| | || || || || || 47\81|| ||38.6831||
| |
| |-
| |
| | || || || || || || 65\112||38.6905||
| |
| |-
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| | || || ||18\31|| || || ||38.7097||[[Meantone]] is in this region
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| |-
| |
| | || || || || || || 61\105||38.7302||
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| |-
| |
| | || || || || ||43\74|| ||38.{{Overline|738}}||
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| |-
| |
| | || || || || || ||68\117||38.7464||
| |
| |-
| |
| | || || || || 25\43|| || ||38.7597||
| |
| |-
| |
| | || || || || || ||57\98||38.7755||
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| |-
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| | || || || || ||32\55|| ||38.{{Overline|78}}||
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| |-
| |
| | || || || || || ||39\67||38.8060||
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| |-
| |
| | ||7\12|| || || || || || 38.{{Overline|8}}||
| |
| |-
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| | || || || || || || 38\65|| 38.9743||
| |
| |-
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| | || || || || ||31\53|| || 38.9937||The fifth closest to a just [[3/2]] for EDOs less than 200
| |
| |-
| |
| | || || || || || ||55\94||39.0071||[[Garibaldi]] / [[Cassandra]]
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| |-
| |
| | || || || ||24\41|| || ||39.0244||
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| |-
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| | || || || || || ||65\111 ||39.{{Overline|039}} ||
| |
| |-
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| | || || || || ||41\70|| ||39.0476||
| |
| |-
| |
| | || || || || || ||58\99||39.0572||
| |
| |-
| |
| | || || ||17\29|| || || ||39.0805||
| |
| |-
| |
| | || || || || || ||61\104||39.1026||
| |
| |-
| |
| | || || || || ||44\75|| ||39.{{Overline|1}}||
| |
| |-
| |
| | || || || || || ||71\121||39.1185||Golden neogothic (704.0956¢)
| |
| |-
| |
| | || || || ||27\46|| || ||39.1304||[[Neogothic]] is in this region
| |
| |-
| |
| | || || || || || ||64\109||39.1437||
| |
| |-
| |
| | || || || || ||37\63|| ||39.1534||
| |
| |-
| |
| | || || || || || ||47\80||39.1{{Overline|6}}||
| |
| |-
| |
| | || ||10\17|| || || || ||39.2157||
| |
| |-
| |
| | || || || || || ||43\73||39.2694||
| |
| |-
| |
| | || || || || ||33\56|| ||39.2857||
| |
| |-
| |
| | || || || || || ||56\95||39.29825||
| |
| |-
| |
| | || || || ||23\39|| || ||39.3162||
| |
| |-
| |
| | || || || || || ||59\100||39.{{Overline|3}}||
| |
| |-
| |
| | || || || || ||36\61|| ||39.3443||
| |
| |-
| |
| | || || || || || ||49\83||39.3574||
| |
| |-
| |
| | || || ||13\22|| || || ||39.{{Overline|39}}||[[Archy]] is in this region
| |
| |-
| |
| | || || || || || ||42\71||39.4366||
| |
| |-
| |
| | || || || || ||29\49|| ||39.4558||
| |
| |-
| |
| | || || || || || ||45\76||39.4737||
| |
| |-
| |
| | || || || ||16\27|| || ||39.5062||
| |
| |-
| |
| | || || || || || ||35\59||39.5480||
| |
| |-
| |
| | || || || || ||19\32|| ||39.58{{Overline|3}}||
| |
| |-
| |
| | || || || || || ||22\37||39.{{Overline|639}}||
| |
| |-
| |
| |3\5|| || || || || || ||40.0000||
| |
| |}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
|
2 × 32
|
| Step size
|
38.9975 ¢
|
| Octave
|
31\18edf (1208.92 ¢)
|
| Twelfth
|
49\18edf (1910.88 ¢)
|
| Consistency limit
|
4
|
| Distinct consistency limit
|
4
|
18EDF is the equal division of the just perfect fifth into 18 parts of 38.9975 cents each, corresponding to 30.7712 edo. It is related to the regular temperament which tempers out 2401/2400 and 8589934592/8544921875 in the 7-limit; 5632/5625, 46656/46585, and 166698/166375 in the 11-limit, which is supported by 31edo, 369edo, 400edo, 431edo, and 462edo.
Lookalikes: 31edo, 49edt
Intervals
| 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