5edf

← 4edf

5edf

6edf →

Prime factorization	5 (prime)
Step size	140.391¢
Octave	9\5edf (1263.52¢)
Twelfth	14\5edf (1965.47¢)
Consistency limit	2
Distinct consistency limit	2

5EDF is the equal division of the just perfect fifth into five parts of 140.391 cents each, corresponding to 8.5476 edo. It is close to the Bleu generator chain and every second step of 17edo. 4 steps of 5edf is a fraction of a cent away to the seventh harmonic (which is 112/81 instead of 7/4 since the equave is 3/2), which is an extremely accurate approximation for the size of this scale.

Intervals

degree	cents value	octave-reduced cents value	approximate ratios	Neptunian notation
0			1/1	perfect unison	C
1	140.391		13/12, 49/45	augmented unison, minor second	C#, Db
2	280.782		75/64, 20/17, 13/11	major second, minor third	D, Eb
3	421.173		14/11, 23/18	major third, diminished fourth	E, Fb
4	561.564		11/8, 18/13, 25/18	perfect fourth	F
5	701.955		3/2	perfect fifth	C
6	842.346		21/13, 13/8, 18/11	augmented fifth, minor sixth	C#, Db
7	982.737		7/4, 30/17	major sixth, minor seventh	D, Eb
8	1123.128			major seventh, minor octave	E, Fb
9	1263.519	63.519		major octave	F
10	1403.910	203.910			C
11	1544.301	344.301			C#, Db
12	1684.692	484.692			D, Eb
13	1825.083	625.083			E
14	1965.474	765.474			F
15	2105.865	905.865			C
16	2246.256	1046.256			C#, Db
17	2386.647	1186.647			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: 137.1429 cents (4\7/5 = 4\35) to 144 cents (3\5/5 = 3\25)

Fifth							Cents	Comments
4\7							137.143
						27\47	137.872
					23\40		138.000
						42\73	138.082
				19\33			138.18
						53\92	138.261
					34\59		138.305
						49\85	138.353
			15\26				138.4615
						56\97	138.557
					41\71		138.5915	The generator closest to a just 13/12 for EDOs less than 1000
						67\116	138.621
				26\45			138.6	Flattone is in this region
						63\109	138.716
					37\64		138.750
						48\83	138.795
		11\19					138.947
						51\88	139.09
					40\69		139.130
						69\119	139.160
				29\50			139.200
						76\131	139.237	Golden meantone (696.2145¢)
					47\81		139.259
						65\112	139.286
			18\31				139.355	Meantone is in this region
						61\105	139.429
					43\74		139.459
						68\117	139.487
				25\43			139.535
						57\98	139.592
					32\55		139.63
						39\67	139.7015
	7\12						140.000
						38\65	140.308
					31\53		140.377	The fifth closest to a just 3/2 for EDOs less than 200
						55\94	140.4255	Garibaldi / Cassandra
				24\41			140.488
						65\111	140.540
					41\70		140.571
						58\99	140.60
			17\29				140.690
						61\104	140.769
					44\75		140.800
						71\121	140.826	Golden neogothic (704.0956¢)
				27\46			140.870	Neogothic is in this region
						64\109	140.917
					37\63		140.952
						47\80	141.000
		10\17					141.1765
						43\73	141.370
					33\56		141.429
						56\95	141.474
				23\39			141.5385
						59\100	141.600
					36\61		141.639
						49\83	141.687
			13\22				141.81	Archy is in this region
						42\71	141.972
					29\49		142.041
						45\76	142.105
				16\27			142.2
						35\59	142.373
					19\32		142.500
						22\37	142.702
3\5							144.000

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.