3edf

← 2edf

3edf

4edf →

Prime factorization	3 (prime)
Step size	233.985¢
Octave	5\3edf (1169.93¢) (convergent)
Twelfth	8\3edf (1871.88¢) (convergent)
Consistency limit	10
Distinct consistency limit	2

3edf, if the attempt is made to use it as an actual scale, would divide the just perfect fifth into three equal parts, each of size 233.985 cents, which is to say (3/2)^1/3 as a frequency ratio. It corresponds to 5.1285 edo. If we want to consider it to be a temperament, it tempers out 16/15, 21/20, 28/27, 81/80, and 256/243 as well as 5edo.

Factoids about 3edf

3edf's step size is close to the slendric temperament, which tempers out 1029/1024 in the 2.3.7 subgroup.

Intervals

#	Cents
1	233.99
2	467.97
3	701.96

Scale tree

EDF scales can be approximated in EDOs by subdividing diatonic fifths. 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: 228.5714 cents (4\7/3 = 4\21) to 240 cents (3\5/3 = 1\5)

Fifth							Cents	Comments
4\7							228.571
						27\47	229.787
					23\40		230.000
						42\73	230.137
				19\33			230.30
						53\92	230.435
					34\59		230.5085
						49\85	230.588
			15\26				230.769
						56\97	230.928
					41\71		230.986
						67\116	231.0345
				26\45			231.1	Flattone is in this region
						63\109	231.193	The generator closest to a just 8/7 for EDOs less than 600
					37\64		231.25
						48\83	231.325
		11\19					231.579
						51\88	231.81
					40\69		231.884
						69\119	231.933
				29\50			232.000
						76\131	232.061	Golden meantone (696.2145¢)
					47\81		232.099
						65\112	232.143
			18\31				232.258	Meantone is in this region
						61\105	232.381
					43\74		232.432
						68\117	232.479
				25\43			232.558
						57\98	232.653
					32\55		232.72
						39\67	232.836
	7\12						233.3
						38\65	233.846
					31\53		233.962	The fifth closest to a just 3/2 for EDOs less than 200
						55\94	234.043	Garibaldi / Cassandra
				24\41			234.146
						65\111	234.234
					41\70		234.286
						58\99	234.34
			17\29				234.483
						61\104	234.615
					44\75		234.6
						71\121	234.711	Golden neogothic (704.0956¢)
				27\46			234.783	Neogothic is in this region
						64\109	234.862
					37\63		234.921
						47\80	235.000
		10\17					235.294
						43\73	235.616
					33\56		235.714
						56\95	235.7895
				23\39			235.897
						59\100	236.000
					36\61		236.066
						49\83	236.145
			13\22				236.36	Archy is in this region
						42\71	236.620
					29\49		236.735
						45\76	236.842
				16\27			237.037
						35\59	237.288
					19\32		237.500
						22\37	237.837
3\5							240.000

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.

Music

• Sequences & Chaos by Bazil Müzik