4edf

← 3edf

4edf

5edf →

Prime factorization	22
Step size	175.489¢
Octave	7\4edf (1228.42¢) (semiconvergent)
Twelfth	11\4edf (1930.38¢) (semiconvergent)
Consistency limit	6
Distinct consistency limit	2
Special properties	highly composite

4EDF is the equal division of the just perfect fifth into four parts of 175.489 cents each, corresponding to 6.8380 edo. It is related to the tetracot temperament, which tempers out 20000/19683 in the 5-limit.

Intervals

degree	cents value	octave-reduced cents value	Tetratonic notation
0			C
1	175.489		D
2	350.978		E
3	526.466		F
4	701.955		C
5	877.444		D
6	1052.933		E
second octave
7	1228.421	28.421	F
8	1403.910	203.910	C
nonet
9	1579.399	379.399	D
10	1754.888	554.888	E
11	1930.376	730.376	F
12	2105.865	905.865	C
13	2281.354	1081.354	D
third octave
14	2456.843	56.843	E

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: 171.4286 cents (4\7/4 = 1\7) to 180 cents (3\5/4 = 3\20)

Fifth							Cents	Comments
4\7							171.429
						27\47	172.340
					23\40		172.500
						42\73	172.603
				19\33			172.72
						53\92	172.826
					34\59		172.881
						49\85	172.941
			15\26				173.076
						56\97	173.196
					41\71		173.239
						67\116	173.276
				26\45			173.3	Flattone is in this region
						63\109	173.3945
					37\64		173.4375
						48\83	173.494
		11\19					173.684
						51\88	173.863
					40\69		173.913
						69\119	173.950
				29\50			174.000
						76\131	174.046	Golden meantone (696.2145¢)
					47\81		174.074
						65\112	174.107
			18\31				174.193	Meantone is in this region
						61\105	174.286
					43\74		174.324
						68\117	174.359
				25\43			174.419
						57\98	174.490
					32\55		174.54
						39\67	174.627
	7\12						175.000
						38\65	175.385
					31\53		175.472	The fifth closest to a just 3/2 for EDOs less than 200
						55\94	175.532	Garibaldi / Cassandra
				24\41			175.610
						65\111	175.675
					41\70		175.714
						58\99	175.75
			17\29				175.862
						61\104	175.9615
					44\75		176.000
						71\121	176.033	Golden neogothic (704.0956¢)
				27\46			176.087	Neogothic is in this region
						64\109	176.147
					37\63		176.1905
						47\80	176.250
		10\17					176.471
						43\73	176.712
					33\56		176.786
						56\95	176.842
				23\39			176.923
						59\100	177.000
					36\61		177.049
						49\83	177.108
			13\22				177.27	Archy is in this region
						42\71	177.648
					29\49		177.551
						45\76	177.532
				16\27			177.7
						35\59	177.966
					19\32		178.125
						22\37	178.378
3\5							180.000

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

• Repetitions 1 Repetitions 1 video by Carlo Serafini