12edf

← 11edf

12edf

13edf →

Prime factorization	22 × 3 (highly composite)
Step size	58.4963 ¢
Octave	21\12edf (1228.42 ¢) (→ 7\4edf)
Twelfth	33\12edf (1930.38 ¢) (→ 11\4edf)
Consistency limit	3
Distinct consistency limit	3

12EDF is the equal division of the just perfect fifth into 12 parts of 58.49625 cents each, corresponding to 20.5141 edo (similar to every second step of 41edo). It is an intersection of 3edf~5edo and 4edf~7edo relations, and could pass as both 20edo and 21edo, with both relations nearly breaking down by this point. It is related to the dodecacot temperament, which tempers out 3087/3125 and 10976/10935 in the 7-limit.

Intervals

degree	cents value	corresponding JI intervals	comments
0		exact 1/1
1	58.49625	91/88, 88/85
2	116.9925	15/14
3	175.48875	21/19
4	233.9850	8/7
5	292.48125	45/38
6	350.9775	11/9, 27/22
7	409.47375	19/15
8	467.9700	21/16
9	526.46625	19/14
10	584.9625	7/5
11	643.4588	13/9
12	701.9550	exact 3/2	just perfect fifth
13	760.45125	273/176, 132/85
14	818.9475	8/5
15	877.44375	63/38
16	935.94	12/7
17	994.43625	135/76
18	1052.9325	11/6, 81/44
19	1111.42875	19/10
20	1169.925	63/32
21	1228.42125	57/28
22	1286.9175	21/10
23	1345.41375	13/6
24	1403.91	exact 9/4

Scale tree

EDFs can be approximated by subdividing diatonic fifths between 4\7 and 3\5.

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: 57.14286 cents (4\7/12 = 1\21) to 60 cents (3\5/12 = 1\20)

Fifth							Cents	Comments
4\7							57.1429
						27\47	57.4468
					23\40		57.5000
						42\73	57.53425
				19\33			57.57
						53\92	57.6087
					34\59		57.6271
						49\85	57.6471
			15\26				57.6923
						56\97	57.7320
					41\71		57.7465
						67\116	57.7586
				26\45			57.7	Flattone is in this region
						63\109	57.7982
					37\64		57.8125
						48\83	57.8313
		11\19					57.8947
						51\88	57.954
					40\69		57.9710
						69\119	57.9832
				29\50			58.000
						76\131	58.0153	Golden meantone (696.2145¢)
					47\81		58.0247
						65\112	58.0357
			18\31				58.0645	Meantone is in this region
						61\105	58.0952
					43\74		58.108
						68\117	58.1967
				25\43			58.1935
						57\98	58.1633
					32\55		58.18
						39\67	58.2090	The generator closest to a just 7/5 for EDOs less than 2400
	7\12						58.3
						38\65	58.4615
					31\53		58.4906	The fifth closest to a just 3/2 for EDOs less than 200
						55\94	58,5106	Garibaldi / Cassandra
				24\41			58.5366
						65\111	58.558
					41\70		58.5714
						58\99	58.58
			17\29				58.6207
						61\104	58.65385
					44\75		58.6
						71\121	58.6777	Golden neogothic (704.0956¢)
				27\46			58.69565	Neogothic is in this region
						64\109	58.7155
					37\63		58.7302
						47\80	58.7500
		10\17					58.8235
						43\73	58.9041
					33\56		58.9286
						56\95	58.9474
				23\39			58.9744
						59\100	59.0000
					36\61		59.0164
						49\83	59.0361
			13\22				59.09	Archy is in this region
						42\71	59.1549
					29\49		59.1837
						45\76	59.2105
				16\27			59.259
						35\59	59.3220
					19\32		59.3750
						22\37	59.459
3\5							60.0000

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.