7edf

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7edf

8edf →

Prime factorization	7 (prime)
Step size	100.279¢
Octave	12\7edf (1203.35¢) (convergent)
Twelfth	19\7edf (1905.31¢) (convergent)
Consistency limit	10
Distinct consistency limit	4

Division of the just perfect fifth into 7 equal parts (7EDF) is related to 12 edo, but with the 3/2 rather than the 2/1 being just. The octave is about 3.3514 cents stretched and the step size is about 100.2793 cents. The patent val has a generally sharp tendency for harmonics up to 21, with the exception for 11 and 13.

Lookalikes: 12edo, 19ed3, 31ed6

Intervals

#	Cents	12edo notation
1	100.2793	C#, Db
2	200.5586	D
3	300.8379	D#, Eb
4	401.1171	E
5	501.3964	F
6	601.6757	F#, Gb
7	701.955	G
8	802.2343	G#, Ab
9	902.5136	A
10	1002.7929	A#, Bb
11	1103.0721	B
12	1203.3514	C
13	1303.6307	C#, Db
14	1403.91	D

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: 97.9592 cents (4\7/7 = 4\49) to 102.8571 cents (3\5/7 = 3\35)

Fifth							Cents	Comments
4\7							97.959
						27\47	98.480
					23\40		98.571
						42\73	98.630
				19\33			98.701
						53\92	98.758
					34\59		98.789
						49\85	98.8235
			15\26				98.901
						56\97	98.969	The generator closest to a just 18/17 for EDOs less than 1400
					41\71		98.994
						67\116	99.015
				26\45			99.048	Flattone is in this region
						63\109	99.083
					37\64		99.107
						48\83	99.139
		11\19					99.248
						51\88	99.351
					40\69		99.379
						69\119	99.400
				29\50			99.429
						76\131	99.455	Golden meantone (696.2145¢)
					47\81		99.471
						65\112	99.490
			18\31				99.539	Meantone is in this region
						61\105	99.592
					43\74		99.613	The generator closest to a just 16/9 for EDOs less than 1400
						68\117	99.634
				25\43			99.668
						57\98	99.7085
					32\55		99.740
						39\67	99.787
	7\12						100.000
						38\65	100.220
					31\53		100.2695	The fifth closest to a just 3/2 for EDOs less than 200
						55\94	100.304	Garibaldi / Cassandra
				24\41			100.348
						65\111	100.361
					41\70		100.408
						58\99	100.433
			17\29				100.493
						61\104	100.5495
					44\75		100.571
						71\121	100.590	Golden neogothic (704.0956¢)
				27\46			100.621	Neogothic is in this region
						64\109	100.655
					37\63		100.680
						47\80	100.714
		10\17					100.840
						43\73	100.9785
					33\56		101.020
						56\95	101.053
				23\39			101.099
						59\100	101.143
					36\61		101.171
						49\83	101.205
			13\22				101.299	Archy is in this region
						42\71	101.4085
					29\49		101.458
						45\76	101.504
				16\27			101.587
						35\59	101.695
					19\32		101.786	The generator closest to a just 9/5 for EDOs less than 1400
						22\37	101.9305
3\5							102.857

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.