19edf

← 18edf

19edf

20edf →

Prime factorization	19 (prime)
Step size	36.945¢
Octave	32\19edf (1182.24¢)
Semitones (A1:m2)	5:1 (184.7¢ : 36.95¢)
Consistency limit	2
Distinct consistency limit	2

19EDF is the equal division of the just perfect fifth into 19 parts of 36.945 cents each, corresponding to 32.4807 edo (similar to every second step of 65edo). It tempers out the same commas as 65edo with the addition of |-103/19 65/19> (1.425 cents) resulting from its inexact 4/1.

Intervals

degree	cents value	corresponding JI intervals	comments
0		exact 1/1
1	36.945
2	73.89	24/23
3	110.835	16/15
4	147.78	12/11
5	184.725	10/9
6	221.67	25/22
7	258.615	36/31
8	295.56	19/16
9	332.505	63/52, 40/33
10	369.45	26/21
11	406.395	24/19, 19/15
12	443.34	31/24
13	480.285	33/25
14	517.23	27/20
15	554.175	11/8
16	591.12	45/32
17	628.065	23/16
18	665.01	22/15
19	701.955	exact 3/2	just perfect fifth
20	738.9
21	775.845
22	812.79	8/5
23	849.735	18/11
24	886.68	5/3
25	923.625
26	960.57
27	997.515	16/9
28	1034.46	20/11
29	1071.405	13/7
30	1108.35	36/19
31	1145.295	31/16
32	1182.24
33	1219.185
34	1256.13
35	1293.075
36	1330.02
37	1366.965
38	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: 36.09023 cents (4\7/19 = 4\133) to 37.89474 cents (3\5/19 = 3\95)

Fifth							Cents	Comments
4\7							36.0902
						27\47	36.2822
					23\40		36.3158
						42\73	36.3374
				19\33			36.36
						53\92	36.3844
					34\59		36.3961
						49\85	36.4087
			15\26				36.43725
						56\97	36.4633
					41\71		36.4175
						67\116	36.4791
				26\45			36.4912	Flattone is in this region
						63\109	36.5041
					37\64		36.5132
						48\83	36.52505
		11\19					36.5651
						51\88	36.6029
					40\69		36.6133
						69\119	36.6210
				29\50			36.6316
						76\131	36.6412	Golden meantone (696.2145¢)
					47\81		36.6472
						65\112	36.6541
			18\31				36.6723	Meantone is in this region
						61\105	36.6917
					43\74		36.6999
						68\117	36.70175
				25\43			36.7197
						57\98	36.7347
					32\55		36.7464
						39\67	36.76355
	7\12						36.8421
						38\65	36.9231
					31\53		36.9414	The fifth closest to a just 3/2 for EDOs less than 200
						55\94	36.9541	Garibaldi / Cassandra
				24\41			36.9705
						65\111	36.98435
					41\70		36.9925
						58\99	37.0016
			17\29				37.0236
						61\104	37.0445
					44\75		37.0526
						71\121	37.0596	Golden neogothic (704.0956¢)
				27\46			37.0709	Neogothic is in this region
						64\109	37.0835
					37\63		37.0927
						47\80	37.1053
		10\17					37.1517
						43\73	37.2026
					33\56		37.21805
						56\95	37.2299
				23\39			37.2470
						59\100	37.2632
					36\61		37.2735
						49\83	37.2388
			13\22				37.3206	Archy is in this region
						42\71	37.3610
					29\49		37.3792
						45\76	37.3961
				16\27			37.4269
						35\59	37.46655
					19\32		37.5000
						22\37	37.5533
3\5							37.8947

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.