13edf

← 12edf

13edf

14edf →

Prime factorization	13 (prime)
Step size	53.9965¢
Octave	22\13edf (1187.92¢)
Semitones (A1:m2)	3:1 (162¢ : 54¢)
Consistency limit	4
Distinct consistency limit	4

13EDF is the equal division of the just perfect fifth into 13 parts of 53.9965 cents each, corresponding to 22.2236 edo. It is nearly identical to every ninth step of 200edo.

Intervals

degree	cents value	corresponding JI intervals	comments
0		exact 1/1
1	53.9965	33/32	pseudo-25/24
2	107.9931	17/16, 117/110, 16/15
3	161.9896	11/10
4	215.9862	17/15
5	269.9827	7/6
6	323.9792	77/64	pseudo-6/5
7	377.9758	56/45	pseudo-5/4
8	431.9723	9/7
9	485.9688	45/34	pseudo-4/3
10	539.9654	15/11
11	593.9619	55/39, 24/17
12	647.9585	16/11
13	701.9550	exact 3/2	just perfect fifth
14	755.9515	99/64
15	809.9481	51/32, 8/5
16	863.9446	33/20
17	917.9412	17/10
18	971.9377	7/4
19	1025.9342	29/16	pseudo-9/5
20	1079.9308	28/15	pseudo-15/8
21	1133.9273	52/27, 27/14
22	1187.9238	135/68	pseudo-octave
23	1241.9204	45/22
24	1295.9169	19/9, 36/17
25	1349.9135	24/11
26	1403.9100	exact 9/4	pythagorean major ninth

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: 52.74725 cents (4\7/13 = 4\91) to 55.384615 cents (3\5/13 = 3\65)

Fifth							Cents	Comments
4\7							52.74725
						27\47	53.0278
					23\40		53.0769
						42\73	53.1085
				19\33			53.14685
						53\92	53.1773
					34\59		53.1943
						49\85	53.2123
			15\26				53.2544
						56\97	53.2910
					41\71		53.3044
						67\116	53.31565
				26\45			53.3	Flattone is in this region
						63\109	53.35215
					37\64		53.3654
						48\83	53.3828
		11\19					53.4413
						51\88	53.4965
					40\69		53.5117
						69\119	53.52295
				29\50			53.5385
						76\131	53.55255	Golden meantone (696.2145¢)
					47\81		53.56125
						65\112	53.5714
			18\31				53.5980	Meantone is in this region
						61\105	53.6264
					43\74		53.63825
						68\117	53.6489
				25\43			53.6673
						57\98	53.6892
					32\55		53.7063
						39\67	53.73135
	7\12						53.84615
						38\65	53.9645
					31\53		53.9913	The fifth closest to a just 3/2 for EDOs less than 200
						55\94	54.0098	Garibaldi / Cassandra
				24\41			54.0338
						65\111	54.054
					41\70		54.0659
						58\99	54.07925
			17\29				54.1114
						61\104	54.1420
					44\75		54.15385
						71\121	54.1640	Golden neogothic (704.0956¢)
				27\46			54.1806	Neogothic is in this region
						64\109	54.1990
					37\63		54.21245
						47\80	54.2308
		10\17					54.2986
						43\73	54.3730
					33\56		54.3956
						56\95	54.4130
				23\39			54.4734
						59\100	54.4615
					36\61		54.4767
						49\83	54.4949
			13\22				54.54	Archy is in this region
						42\71	54.60455
					29\49		54.6311
						45\76	54.6558
				16\27			54.70085
						35\59	54.7588
					19\32		54.8077
						22\37	54.88565
3\5							55.3846

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.