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|
| Line 152: |
Line 152: |
| | 9/4 | | | 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.
| |
|
| |
| Generator range: 40.33613 cents (4\7/17 = 4\119) to 42.35294 cents (3\5/17 = 3\85)
| |
| {| class="wikitable center-all"
| |
| ! colspan="7" | Fifth
| |
| !Cents
| |
| !Comments
| |
| |-
| |
| |4\7|| || || || || || ||40.3361||
| |
| |-
| |
| | || || || || || ||27\47 ||40.5507||
| |
| |-
| |
| | || || || || || 23\40|| ||40.5882||
| |
| |-
| |
| | || || || || || || 42\73||40.6124 ||
| |
| |-
| |
| | || || || ||19\33|| || || 40.6417||
| |
| |-
| |
| | || || || || || || 53\92||40.6650||
| |
| |-
| |
| | || || || || ||34\59|| ||40.6780||
| |
| |-
| |
| | || || || || || ||49\85||40.6920||
| |
| |-
| |
| | || || ||15\26|| || || ||40.7240||
| |
| |-
| |
| | || || || || || ||56\97|| 40.7520||
| |
| |-
| |
| | || || || || ||41\71|| ||40.7622||
| |
| |-
| |
| | || || || || || ||67\116||40.7708 ||
| |
| |-
| |
| | || || || ||26\45|| || ||40.7843||[[Flattone]] is in this region
| |
| |-
| |
| | || || || || || || 63\109||40.7897||
| |
| |-
| |
| | || || || || ||37\64|| || 40.8088||
| |
| |-
| |
| | || || || || || ||48\83||40.8221||
| |
| |-
| |
| | || || 11\19|| || || || ||40.8669||
| |
| |-
| |
| | || || || || || ||51\88||40.{{Overline|90}}||
| |
| |-
| |
| | || || || || ||40\69|| ||40.9207||
| |
| |-
| |
| | || || || || || ||69\119||40.9293||
| |
| |-
| |
| | || || || ||29\50|| || ||40.9412||
| |
| |-
| |
| | || || || || || ||76\131||40.95195||[[Golden meantone]] (696.2145¢)
| |
| |-
| |
| | || || || || ||47\81|| ||40.9586||
| |
| |-
| |
| | || || || || || ||65\112||40.6994||
| |
| |-
| |
| | || || ||18\31|| || || ||40.9867||[[Meantone]] is in this region
| |
| |-
| |
| | || || || || || ||61\105||41.0084||
| |
| |-
| |
| | || || || || ||43\74|| ||41.1075||
| |
| |-
| |
| | || || || || || ||68\117||41.0256||
| |
| |-
| |
| | || || || ||25\43|| || ||41.0397||
| |
| |-
| |
| | || || || || || ||57\98||41.0564||
| |
| |-
| |
| | || || || || ||32\55|| ||41.0695||
| |
| |-
| |
| | || || || || || ||39\67||41.0887||
| |
| |-
| |
| | ||7\12|| || || || || ||41.1765||
| |
| |-
| |
| | || || || || || ||38\65||41.2670||
| |
| |-
| |
| | || || || || ||31\53|| ||41.2875||The fifth closest to a just [[3/2]] for EDOs less than 200
| |
| |-
| |
| | || || || || || ||55\94||41.3016||[[Garibaldi]] / [[Cassandra]]
| |
| |-
| |
| | || || || ||24\41|| || ||41.3199||
| |
| |-
| |
| | || || || || || ||65\111||41.33545||
| |
| |-
| |
| | || || || || ||41\70|| ||41.3445||
| |
| |-
| |
| | || || || || || ||58\99||41.3547||
| |
| |-
| |
| | || || ||17\29|| || || ||41.3793||
| |
| |-
| |
| | || || || || || ||61\104||41.4027||
| |
| |-
| |
| | || || || || ||44\75|| ||41.4118||
| |
| |-
| |
| | || || || || || ||71\121||41.4195||Golden neogothic (704.0956¢)
| |
| |-
| |
| | || || || ||27\46|| || ||41.4322||[[Neogothic]] is in this region
| |
| |-
| |
| | || || || || || ||64\109||41.4463||
| |
| |-
| |
| | || || || || ||37\63|| ||41.4566||
| |
| |-
| |
| | || || || || || ||47\80||41.4706||
| |
| |-
| |
| | || ||10\17|| || || || ||41.5225||
| |
| |-
| |
| | || || || || || ||43\73||41.5764||
| |
| |-
| |
| | || || || || ||33\56|| ||41.5966||
| |
| |-
| |
| | || || || || || ||56\95||41.6099||
| |
| |-
| |
| | || || || ||23\39|| || ||41.6290||
| |
| |-
| |
| | || || || || || ||59\100||41.6471||
| |
| |-
| |
| | || || || || ||36\61|| ||41.6586||
| |
| |-
| |
| | || || || || || ||49\83||41.6726||
| |
| |-
| |
| | || || ||13\22|| || || ||41.7112||[[Archy]] is in this region
| |
| |-
| |
| | || || || || || ||42\71||41.7564||The generator closest to a just 14/11 for EDOs less than 3400
| |
| |-
| |
| | || || || || ||29\49|| ||41.7768||
| |
| |-
| |
| | || || || || || ||45\76||41.7957||
| |
| |-
| |
| | || || || ||16\27|| || ||41.8301||
| |
| |-
| |
| | || || || || || ||35\59||41.8744||
| |
| |-
| |
| | || || || || ||19\32|| ||41.9118||
| |
| |-
| |
| | || || || || || ||22\37||41.9714||
| |
| |-
| |
| |3\5|| || || || || || ||42.3529||
| |
| |}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]]
| |
| [[Category:todo:improve synopsis]]
| |
| Prime factorization
|
17 (prime)
|
| Step size
|
41.2915 ¢
|
| Octave
|
29\17edf (1197.45 ¢)
(semiconvergent)
|
| Twelfth
|
46\17edf (1899.41 ¢)
(semiconvergent)
|
| Consistency limit
|
6
|
| Distinct consistency limit
|
6
|
17EDF is the Division of the just perfect fifth into 17 equal parts. It is related to 29 edo, but with the 3/2 rather than the 2/1 being just. The octave is about 2.5474 cents compressed and the step size is about 41.2915 cents. Unlike 29edo, it is only consistent up to the 6-integer-limit, with discrepancy for the 7th harmonic.
Lookalikes: 29edo, 46edt
Intervals
| Degree
|
Cents
|
Approx. ratios of the 15-odd-limit
|
| 0
|
0.0000
|
1/1
|
| 1
|
41.2915
|
25/24~33/32~56/55~81/80
|
| 2
|
82.5829
|
21/20
|
| 3
|
123.8744
|
16/15, 15/14, 14/13, 13/12
|
| 4
|
165.1659
|
12/11, 11/10
|
| 5
|
206.4574
|
9/8
|
| 6
|
248.7488
|
8/7, 7/6, 15/13
|
| 7·
|
289.0403
|
13/11
|
| 8
|
330.3318
|
6/5, 11/9
|
| 9
|
371.6232
|
5/4, 16/13
|
| 10
|
412.9147
|
14/11
|
| 11
|
455.2062
|
9/7, 13/10
|
| 12·
|
495.4976
|
4/3
|
| 13
|
536.7891
|
11/8, 15/11
|
| 14
|
578.0806
|
7/5, 18/13
|
| 15
|
619.3721
|
10/7, 13/9
|
| 16
|
660.6635
|
16/11, 22/15
|
| 17·
|
701.9550
|
3/2
|
| 18
|
743.2465
|
14/9, 20/13
|
| 19
|
784.5379
|
11/7
|
| 20
|
825.8294
|
8/5, 13/8
|
| 21
|
867.1209
|
5/3, 18/11
|
| 22·
|
908.4124
|
22/13
|
| 23
|
949.7038
|
7/4, 12/7, 26/15
|
| 24
|
990.9952
|
16/9
|
| 25
|
1032.3287
|
11/6, 20/11
|
| 26
|
1073.5782
|
15/8, 28/15, 13/7, 24/13
|
| 27
|
1114.8697
|
40/21
|
| 28
|
1156.1612
|
48/25~64/33~55/28 ~160/81
|
| 29
|
1197.4526
|
2/1
|
| 30
|
1238.7441
|
25/12~33/16~112/55~81/40
|
| 31
|
1280.0356
|
21/10
|
| 32
|
1321.3271
|
32/15, 15/7, 28/13, 13/6
|
| 33
|
1362.6185
|
24/11, 11/5
|
| 34
|
1403.9100
|
9/4
|