22edf

← 21edf

22edf

23edf →

Prime factorization	2 × 11
Step size	31.907¢
Octave	38\22edf (1212.47¢) (→19\11edf)
Twelfth	60\22edf (1914.42¢) (→30\11edf)
Consistency limit	2
Distinct consistency limit	2

22EDF is the equal division of the just perfect fifth into 22 parts of 31.907 cents each, corresponding to 37.6092 edo (similar to every fifth step of 188edo).

Intervals

degree	cents value	corresponding JI intervals	comments
0		exact 1/1
1	31.907	55/54
2	63.8141	(28/27), (27/26)
3	95.7211
4	127.6282	14/13
5	159.5352	57/52
6	191.4423
7	223.3493	8/7
8	255.2564
9	287.1634	13/11
10	319.0705	6/5
11	350.9775	60/49, 49/40
12	382.8845	5/4
13	414.7916	14/11
14	446.6986
15	478.6057
16	510.5127
17	542.4198	26/19
18	574.3268	39/28
19	606.2339	64/45
20	638.1409	(13/9)
21	670.048	81/55
22	701.955	exact 3/2	just perfect fifth
23	733.862	55/36
24	765.7691	14/9, 81/52
25	797.6761
26	828.5732	21/13
27	861.4902	171/104
28	893.3973
29	925.3043	12/7
30	956.2114
31	988.1184	39/22
32	1020.0255	9/5
33	1052.9235	90/49, 147/80
34	1084.8395	15/8
35	1116.7466	21/11
36	1148.6536
37	1180.5607
38	1211.4677
39	1244.3748	39/19
40	1276.2816	117/56
41	1308.1889	32/15
42	1340.0959	13/6
43	1372.003	243/110
44	1403.91	exact 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 "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: 31.16883 cents (4\7/22 = 2\77) to 32.72 cents (3\5/22 = 3\110)

Fifth							Cents	Comments
4\7							31.1688
						27\47	31.3346
					23\40		31.36
						42\73	31.3823
				19\33			31.4045
						53\92	31.4229
					34\59		31.4330
						49\85	31.44385
			15\26				31.4685
						56\97	31.4902
					41\71		31.4981
						67\116	31.5047
				26\45			31.51	Flattone is in this region
						63\109	31.5263
					37\64		31.53409
						48\83	31.5444
		11\19					31.57895
						51\88	31.6116
					40\69		31.62055
						69\119	31.6272
				29\50			31.63
						76\131	31.6447	Golden meantone (696.2145¢)
					47\81		31.6498
						65\112	31.6558
			18\31				31.67155	Meantone is in this region
						61\105	31.6883
					43\74		31.6953
						68\117	31.7016
				25\43			31.7125
						57\98	31.7254
					32\55		31.7355
						39\67	31.7503
	7\12						31.81
						38\65	31.8881
					31\53		31.90395	The fifth closest to a just 3/2 for EDOs less than 200
						55\94	31.9149	Garibaldi / Cassandra
				24\41			31.92905
						65\111	31.9410
					41\70		31.94805
						58\99	31.9559
			17\29				31.9749
						61\104	31.9930
					44\75		32.0000
						71\121	32.0060	Golden neogothic (704.0956¢)
				27\46			32.0158	Neogothic is in this region
						64\109	32.0267
					37\63		32.0346
						47\80	32.054
		10\17					32.0856
						43\73	32.1295
					33\56		32.1429
						56\95	32.1531
				23\39			32.1768
						59\100	32.18
					36\61		32.1908
						49\83	32.2015
			13\22				32.2314	Archy is in this region
						42\71	32.2663
					29\49		32.2820
						45\76	32.29665
				16\27			32.32
						35\59	32.3575
					19\32		32.3863
						22\37	32.432
3\5							32.72

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.