6edf

← 5edf

6edf

7edf →

Prime factorization	2 × 3
Step size	116.993¢
Octave	10\6edf (1169.93¢) (→5\3edf)
Semitones (A1:m2)	2:0 (234¢ : 0¢)
Consistency limit	2
Distinct consistency limit	2
Special properties	highly composite

6EDF is the equal division of the just perfect fifth into six parts of 116.9925 cents each, corresponding to 10.2571 edo. It is related to the miracle temperament, which tempers out 225/224 and 1029/1024 in the 7-limit.

Intervals

degrees	cents ~ cents octave-reduced	approximate ratios	Neptunian notation
0	0 (perfect unison, 1:1)	1/1	C
1	117	16/15, 15/14	C#
2	234	8/7	Db
3	351	11/9, 27/22	D
4	468	21/16	E
5	585	7/5, 45/32	F
6	702 (just perfect fifth, 3:2)	3/2	C
7	819	8/5, 21/13	C#
8	936	12/7, 55/32	Db
9	1053	11/6	D
10	1170	49/25, 160/81, 2/1	E
11	1287 ~ 87		F
12	1404 ~ 204 (just major whole tone/ninth, 9:4)		C
13	1521 ~ 321		C#
14	1638 ~ 438		Db
15	1755 ~ 555		D
16	1872 ~ 672		E
17	1988 ~ 788		F
18	2106 ~ 906 (Pythagorean major sixth, 27:8)		C
19	2223 ~ 1023		C#
20	2340 ~ 1140		Db
21	2457 ~ 57		D
22	2574 ~ 174		E
23	2691 ~ 291		F
24	2808 ~ 408 (Pythagorean major third, 81:16)		C

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: 114.2857 cents (4\7/6 = 2\21) to 120 cents (3\5/6 = 1\10)

Fifth							Cents	Comments
4\7							114.286
						27\47	114.894
					23\40		115.000
						42\73	115.0685
				19\33			115.15
						53\92	115.217
					34\59		115.254
						49\85	115.294
			15\26				115.385
						56\97	115.464
					41\71		115.493
						67\116	115.517
				26\45			115.5	Flattone is in this region
						63\109	115.596
					37\64		115.625
						48\83	115.663
		11\19					115.684
						51\88	115.90
					40\69		115.942
						69\119	115.966
				29\50			116.000
						76\131	116.0305	Golden meantone (696.2145¢)
					47\81		116.049
						65\112	116.071
			18\31				116.129	Meantone is in this region
						61\105	116.1905
					43\74		116.216
						68\117	116.239
				25\43			116.279
						57\98	116.3265
					32\55		116.36
						39\67	116.418
	7\12						116.6
						38\65	116.923
					31\53		116.981	The fifth closest to a just 3/2 for EDOs less than 200
						55\94	117.021	Garibaldi / Cassandra
				24\41			117.073
						65\111	117.117
					41\70		117.143
						58\99	117.17
			17\29				117.241
						61\104	117.308
					44\75		117.3
						71\121	117.355	Golden neogothic (704.0956¢)
				27\46			117.391	Neogothic is in this region
						64\109	117.431
					37\63		117.460
						47\80	117.500
		10\17					117.647
						43\73	117.808
					33\56		117.857
						56\95	117.895
				23\39			117.949
						59\100	118.000
					36\61		118.033
						49\83	118.072
			13\22				118.18	Archy is in this region
						42\71	118.310
					29\49		118.367
						45\76	118.421
				16\27			118.518
						35\59	118.644
					19\32		118.750
						22\37	118.918	The generator closest to a just 15/14 for EDOs less than 1200
3\5							120.000

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.

Compositions

• Metashakti by Carlo Serafini