20edf

← 19edf

20edf

21edf →

Prime factorization	22 × 5
Step size	35.0978¢
Octave	34\20edf (1193.32¢) (→17\10edf)
Twelfth	54\20edf (1895.28¢) (→27\10edf)
Consistency limit	7
Distinct consistency limit	6

Intervals

The first steps up to two just perfect fifths should give a feeling of the granularity of this system…

Degrees
1	35.1
2	70.2
3	105.29
4	140.39
5	175.49
6	210.59
7	245.68
8	280.78
9	315.88
10	350.98
11	386.075
12	421.17
13	456.27
14	491.37
15	526.47
16	561.56
17	596.66
18	631.76
19	666.86
20	701.955
21	737.05
22	772.15
23	807.25
24	842.35
25	877.44
26	912.54
27	947.64
28	982.74
29	1017.835
30	1052.93
31	1088.03
32	1123.13
33	1158.23
34	1193.32
35	1228.42
36	1263.52
37	1298.62
38	1333.715
39	1368.81
40	1403.91

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 "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 EDOs pop up in our continuum.

Generator range: 34.28571 cents (4\7/20 = 1\35) to 36 cents (3\5/20 = 3/100)

Fifth							Cents	Comments
4\7							34.2857
						27\47	34.4681
					23\40		34.5000
						42\73	34.52055
				19\33			34.54
						53\92	34.5652
					34\59		34.5763
						49\85	34.5882
			15\26				34.6153
						56\97	34.6392
					41\71		34.6479
						67\116	34.6551
				26\45			34.6	Flattone is in this region
						63\109	34.6789
					37\64		34.6875
						48\83	34.6988
		11\19					34.7368	The generator closest to a just 11/9 for EDOs less than 2000
						51\88	34.772
					40\69		34.7826
						69\119	34.7899
				29\50			34.8000
						76\131	34.8092	Golden meantone (696.2145¢)
					47\81		34.814
						65\112	34.8214
			18\31				34.8387	Meantone is in this region
						61\105	34.8571
					43\74		34.864
						68\117	34.8718
				25\43			34.8837
						57\98	34.8980
					32\55		34.90
						39\67	34.9254
	7\12						35.0000
						38\65	35.0769
					31\53		35.0843	The fifth closest to a just 3/2 for EDOs less than 200
						55\94	35.1064	Garibaldi / Cassandra
				24\41			35.12195
						65\111	35.135
					41\70		35.1429
						58\99	35.15
			17\29				35.1724
						61\104	35.1923
					44\75		35.2000
						71\121	35.2066	Golden neogothic (704.0956¢)
				27\46			35.2174	Neogothic is in this region
						64\109	35.2294
					37\63		35.2381
						47\80	35.2500
		10\17					35.2941
						43\73	35.3425
					33\56		35.3571
						56\95	35.3684
				23\39			35.3846
						59\100	35.4000
					36\61		35.4098
						49\83	35.4217
			13\22				35.45	Archy is in this region
						42\71	35.4930
					29\49		35.5102
						45\76	35.5263
				16\27			35.5
						35\59	35.5932
					19\32		35.6250
						22\37	35.675
3\5							36.0000

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.