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| Line 214: |
Line 214: |
| | pythagorean ninth | | | pythagorean ninth |
| |} | | |} |
|
| |
| == 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.
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|
| |
| Generator range: 42.85714 cents (4\7/16 = 1\28) to 45 cents (3\5/16 = 3\80)
| |
| {| class="wikitable center-all"
| |
| ! colspan="7" |Fifth
| |
| !Cents
| |
| !Comments
| |
| |-
| |
| |4\7|| || || || || || ||42.8571||
| |
| |-
| |
| | || || || || || || 27\47||43.0851||
| |
| |-
| |
| | || || || || ||23\40|| ||43.1250||
| |
| |-
| |
| | || || || || || ||42\73||43.1507||
| |
| |-
| |
| | || || || ||19\33|| || ||43.{{Overline|18}}||
| |
| |-
| |
| | || || || || || ||53\92 ||43.2065||
| |
| |-
| |
| | || || || || ||34\59|| ||43.2203||
| |
| |-
| |
| | || || || || || || 49\85||43.2353||
| |
| |-
| |
| | || || ||15\26|| || || ||43.2692||
| |
| |-
| |
| | || || || || || ||56\97||43.2990||
| |
| |-
| |
| | || || || || ||41\71|| ||43.3099||
| |
| |-
| |
| | || || || || || ||67\116|| 43.3190||
| |
| |-
| |
| | || || || || 26\45|| || ||43.{{Overline|3}}||[[Flattone]] is in this region
| |
| |-
| |
| | || || || || || ||63\109||43.3486||
| |
| |-
| |
| | || || || || ||37\64 || ||43.3594 ||
| |
| |-
| |
| | || || || || || ||48\83 ||43.3735||
| |
| |-
| |
| | || ||11\19|| || || || ||43.42105||
| |
| |-
| |
| | || || || || || || 51\88||43.465{{Overline|90}}||
| |
| |-
| |
| | || || || || ||40\69|| ||43.4783||
| |
| |-
| |
| | || || || || || ||69\119||43.4874||
| |
| |-
| |
| | || || || ||29\50|| || ||43.5000||
| |
| |-
| |
| | || || || || || ||76\131||43.51145||[[Golden meantone]] (696.2145¢)
| |
| The generator closest to a just [[9/7]] for EDOs less than 800
| |
| |-
| |
| | || || || || ||47\81|| ||43.{{Overline|518}}||
| |
| |-
| |
| | || || || || || ||65\112||43.5268||
| |
| |-
| |
| | || || ||18\31|| || || ||43.5484||[[Meantone]] is in this region
| |
| |-
| |
| | || || || || || ||61\105||43.5714||
| |
| |-
| |
| | || || || || ||43\74|| ||43.5{{Overline|810}}||
| |
| |-
| |
| | || || || || || ||68\117||43.5897||
| |
| |-
| |
| | || || || ||25\43|| || ||43.60465||
| |
| |-
| |
| | || || || || || ||57\98||43.62245||
| |
| |-
| |
| | || || || || ||32\55|| ||43.{{Overline|63}}||
| |
| |-
| |
| | || || || || || ||39\67||43.6567||
| |
| |-
| |
| | ||7\12|| || || || || ||43.7500||
| |
| |-
| |
| | || || || || || ||38\65||43.84615||
| |
| |-
| |
| | || || || || ||31\53|| ||43.8679||The fifth closest to a just [[3/2]] for EDOs less than 200
| |
| |-
| |
| | || || || || || ||55\94||43.8830||[[Garibaldi]] / [[Cassandra]]
| |
| |-
| |
| | || || || ||24\41|| || ||43.9024||
| |
| |-
| |
| | || || || || || ||65\111||43.{{Overline|1=918}}||
| |
| |-
| |
| | || || || || ||41\70|| ||43.9286||
| |
| |-
| |
| | || || || || || ||58\99||43.{{Overline|1=93}}||
| |
| |-
| |
| | || || ||17\29|| || || ||43.9655||
| |
| |-
| |
| | || || || || || ||61\104||43.9904||
| |
| |-
| |
| | || || || || ||44\75|| ||44.0000||
| |
| |-
| |
| | || || || || || ||71\121||44.0083||Golden neogothic (704.0956¢)
| |
| |-
| |
| | || || || ||27\46|| || ||44.0217||[[Neogothic]] is in this region
| |
| |-
| |
| | || || || || || ||64\109||44.0367||
| |
| |-
| |
| | || || || || ||37\63|| ||44.0476||
| |
| |-
| |
| | || || || || || ||47\80||44.0625||
| |
| |-
| |
| | || ||10\17|| || || || ||44.11765||
| |
| |-
| |
| | || || || || || ||43\73||44.1781||
| |
| |-
| |
| | || || || || ||33\56|| ||44.1964||
| |
| |-
| |
| | || || || || || ||56\95||44.2105||
| |
| |-
| |
| | || || || ||23\39|| || ||44.3208||
| |
| |-
| |
| | || || || || || ||59\100||43.2500||
| |
| |-
| |
| | || || || || ||36\61|| ||44.2623||
| |
| |-
| |
| | || || || || || ||49\83||44.2771||
| |
| |-
| |
| | || || ||13\22|| || || ||44.3{{Overline|18}}||[[Archy]] is in this region
| |
| |-
| |
| | || || || || || ||42\71||44.3662||
| |
| |-
| |
| | || || || || ||29\49|| ||44.3878||
| |
| |-
| |
| | || || || || || ||45\76||44.4079||
| |
| |-
| |
| | || || || ||16\27|| || ||44.{{Overline|4}}||
| |
| |-
| |
| | || || || || || ||35\59||44.4915||
| |
| |-
| |
| | || || || || ||19\32|| ||44.53125||
| |
| |-
| |
| | || || || || || ||22\37||44.{{Overline|594}}||
| |
| |-
| |
| |3\5|| || || || || || ||45.0000||
| |
| |}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.
| |
|
| |
| Tunings below 7\12 on this chart are called "positive tunings" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth tunings such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 13\22 through 16\27, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.
| |
|
| |
|
| == Music == | | == Music == |
| Prime factorization
|
24
|
| Step size
|
43.8722 ¢
|
| Octave
|
27\16edf (1184.55 ¢)
|
| Twelfth
|
43\16edf (1886.5 ¢)
|
| Consistency limit
|
3
|
| Distinct consistency limit
|
3
|
16EDF is the equal division of the just perfect fifth into 16 parts of 43.8722 cents each, corresponding to 27.3522 edo (similar to every third step of 82edo). 16edf contains good approximations of the 7th and 13th harmonics.
It serves as a good approximation to halftone temperament, containing the ~7/5 generator at 13 steps.
Lookalikes: 27edo, 43edt
Intervals
| degree
|
cents value
|
corresponding
JI intervals
|
Halftone[6] notation (using ups and downs)
|
comments
|
| 0
|
0.0000
|
1/1
|
C
|
|
| 1
|
43.8722
|
40/39, 39/38
|
^C
|
|
| 2
|
87.7444
|
20/19
|
Db
|
|
| 3
|
131.6166
|
55/51, (27/25)
|
vD
|
|
| 4
|
175.4888
|
(21/19)
|
D
|
|
| 5
|
219.3609
|
|
vE
|
|
| 6
|
263.2331
|
(7/6)
|
E
|
|
| 7
|
307.1053
|
|
Fb
|
|
| 8
|
350.9775
|
60/49, 49/40
|
vF
|
|
| 9
|
394.8497
|
(44/35)
|
F
|
|
| 10
|
438.7219
|
(9/7)
|
Ab
|
|
| 11
|
482.5941
|
|
vA
|
|
| 12
|
526.4663
|
(19/14)
|
A
|
|
| 13
|
570.3384
|
(25/18), 153/110, 112/81
|
B
|
|
| 14
|
614.2106
|
(10/7)
|
Cb
|
|
| 15
|
658.0828
|
19/13
|
vC
|
|
| 16
|
701.9550
|
3/2 (exact)
|
C
|
just perfect fifth
|
| 17
|
745.8272
|
20/13
|
|
|
| 18
|
789.6994
|
30/19
|
|
|
| 19
|
833.5716
|
55/34
|
|
|
| 20
|
877.4438
|
|
|
|
| 21
|
921.3159
|
|
|
|
| 22
|
965.1881
|
|
7/4
|
|
| 23
|
1009.0603
|
|
|
|
| 24
|
1052.9325
|
90/49, (11/6)
|
|
|
| 25
|
1096.8047
|
(66/35)
|
|
|
| 26
|
1140.6769
|
|
|
|
| 27
|
1184.5491
|
|
|
|
| 28
|
1228.4213
|
128/63
|
|
|
| 29
|
1272.2934
|
25/12
|
|
|
| 30
|
1316.1656
|
15/7
|
|
|
| 31
|
1360.0378
|
57/26
|
|
|
| 32
|
1403.9100
|
9/4 (exact)
|
|
pythagorean ninth
|
Music
schizophrenic lullaby fugue by nationalsolipsism
Neptune by Nae Ayy