15edf

← 14edf

15edf

16edf →

Prime factorization	3 × 5
Step size	46.797¢
Octave	26\15edf (1216.72¢)
Twelfth	41\15edf (1918.68¢)
Consistency limit	2
Distinct consistency limit	2

15EDF is the equal division of the just perfect fifth into 15 parts of 46.797 cents each, corresponding to 25.6427 edo (similar to every third step of 77edo). The 3edf~5edo correspondence has completely collapsed by this point, with this EDF being closer to 26edo than 25edo.

Lookalikes: 26edo, 41edt

Intervals

degree	cents value	corresponding JI intervals	comments
0		exact 1/1
1	46.797
2	93.594	19/18
3	140.391	13/12
4	187.188	10/9
5	233.985	8/7
6	280.782	20/17
7	327.579
8	374.376
9	421.173	51/40
10	467.97	21/16
11	514.767	27/20
12	561.564	18/13
13	608.361	27/19
14	655.158
15	701.955	exact 3/2	just perfect fifth
16	748.752
17	795.549	19/12
18	842.346	13/8
19	889.143	5/3
20	935.94	12/7
21	982.737	30/17
22	1029.534
23	1076.331
24	1123.128	153/80
25	1168.925	63/32
26	1216.722	81/40
27	1263.519	27/13
28	1310.316	81/38
29	1357.113
30	1403.91	exact 9/4

Scale tree

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: 45.71429 cents (4\7/15 = 4\105) to 48 cents (3\5/15 = 1\25)

Fifth							Cents	Comments
4\7							45.7143
						27\47	45.95745
					23\40		46.0000
						42\73	46.0274
				19\33			46.06
						53\92	46.0870
					34\59		46.1017
						49\85	46.11765
			15\26				46.15385
						56\97	46.1856
					41\71		46.1972
						67\116	46.2069
				26\45			46.2	Flattone is in this region
						63\109	46.2385
					37\64		46.2500
						48\83	46.2651
		11\19					46.3158
						51\88	46.36
					40\69		46.3768
						69\119	46.38655
				29\50			46.4000
						76\131	46.4122	Golden meantone (696.2145¢)
					47\81		46.41975
						65\112	46.4286
			18\31				46.4516	Meantone is in this region
						61\105	46.4762
					43\74		46.486
						68\117	46.4957
				25\43			46.5116
						57\98	46.5306
					32\55		46.54
						39\67	46.5672
	7\12						46.6
						38\65	46.7692
					31\53		46.79245	The fifth closest to a just 3/2 for EDOs less than 200
						55\94	46.8085	Garibaldi / Cassandra
				24\41			46.8293
						65\111	46.843
					41\70		46.8571
						58\99	46.86
			17\29				46.89655
						61\104	46.9231
					44\75		46.93
						71\121	46.94215	Golden neogothic (704.0956¢)
				27\46			46.9565	Neogothic is in this region
						64\109	46.9725
					37\63		46.9841
						47\80	47.0000
		10\17					47.0588	The generator closest to a just 21/16 for EDOs less than 200
						43\73	47.1233
					33\56		47.1429
						56\95	47.1579
				23\39			47.1795
						59\100	47.2000
					36\61		47.2131
						49\83	47.2289
			13\22				47.27	Archy is in this region
						42\71	47.3239
					29\49		47.3469
						45\76	47.3864
				16\27			47.407
						35\59	47.4576
					19\32		47.5000
						22\37	47.567
3\5							48.0000

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.