Prime factorization

3 × 7

Step size

33.4264¢

Octave

36\21edf (1203.35¢) (→12\7edf)

Twelfth

57\21edf (1905.31¢) (→19\7edf)

Consistency limit

4

Distinct consistency limit

4

Division of the just perfect fifth into 21 equal parts (21EDF) is related to 36 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 33.4264 cents. Unlike 36edo, it is only consistent up to the 4-integer-limit, with discrepancy for the 5th harmonic.

Lookalikes: 36edo, 57edt

Approximations

3-limit (Pythagorean) approximations (same as 7edf):

2/1 = 1200 cents; 36 degrees of 21edf = 1203.3514... cents.

4/3 = 498.045... cents; 15 degrees of 21edf = 501.9634... cents.

9/8 = 203.910... cents; 6 degrees of 21edf = 200.5585... cents.

16/9 = 996.090... cents; 30 degrees of 21edf = 1002.7928... cents.

27/16 = 905.865... cents; 27 degrees of 21edf = 902.5135... cents.

32/27 = 294.135... cents; 9 degrees of 21edf = 300.8379... cents.

81/64 = 407.820... cents; 12 degrees of 21edf = 401.1171... cents.

128/81 = 792.180... cents; 24 degrees of 21edf = 802.2342... cents.

7-limit approximations:

7 only:

7/4 = 968.826... cents; 29 degrees of 21edf = 969.3664... cents.

8/7 = 231.174... cents; 7 degrees of 21edf = 233.985... cents.

49/32 = 737.652... cents; 22 degrees of 21edf = 733.333... cents.

64/49 = 462.348... cents; 14 degrees of 21edf = 467.97... cents.

3 and 7:

7/6 = 266.871... cents; 8 degrees of 21edf = 267.4114... cents.

12/7 = 933.129... cents; 28 degrees of 21edf = 935.94... cents.

9/7 = 435.084... cents; 13 degrees of 21edf = 434.5435... cents.

14/9 = 764.916... cents; 23 degrees of 21edf = 768.8078... cents.

28/27 = 62.961... cents; 2 degrees of 21edf = 66.8528... cents.

27/14 = 1137.039... cents; 34 degrees of 21edf = 1136.4985... cents.

21/16 = 470.781... cents; 14 degrees of 21edf = 467.97... cents.

32/21 = 729.219... cents; 22 degrees of 21edf = 735.3814... cents.

49/48 = 35.697... cents; 1 degree of 21edf = 33.4264... cents.

96/49 = 1164.303... cents; 35 degrees of 21edf = 1169.925... cents.

49/36 = 533.742... cents; 16 degrees of 21edf = 534.8228... cents.

72/49 = 666.258... cents; 20 degrees of 21edf = 668.5285... cents.

64/63 = 27.264... cents; 1 degree of 21edf = 33.4264... cents.

63/32 = 1172.736... cents; 35 degrees of 21edf = 1169.925... cents.

The following table gives an overview of all degrees of 36edo.

Degree	Size in cents	Approximate ratios of 2.3.7	Additional ratios of 2.3.7.13.17
0		1/1
1	33.4264	64/63, 49/48
2	66.8529	28/27
3	100.2793	256/243	17/16, 18/17
4	133.7057	243/224	14/13, 13/12
5	167.1321	54/49
6	200.5586	9/8
7	233.985	8/7
8	267.4114	7/6
9	300.8379	32/27
10	334.2643	98/81	17/14
11	367.6907	243/196	16/13, 26/21, 21/17
12	401.1171	81/64
13	434.5436	9/7
14	467.97	64/49, 21/16	17/13
15	501.3964	4/3
16	534.8229	49/36
17	568.2493		18/13
18	601.6757
19	635.1021		13/9
20	668.5286	72/49
21	701.955	3/2
22	735.3814	49/32, 32/21	26/17
23	768.8079	14/9
24	802.2343	128/81
25	835.6607	392/243	13/8, 21/13, 34/21
26	869.0871	81/49	28/17
27	902.5136	27/16
28	935.94	12/7
29	969.3664	7/4
30	1002.7929	16/9
31	1036.2193	49/27
32	1069.6457	448/243	13/7, 24/13
33	1103.0721	243/128	32/17, 17/9
34	1136.4986	27/14
35	1169.925	63/32, 96/49
36	1203.3514	2/1
37	1236.7779	128/63, 49/24
38	1270.2043	56/27
39	1303.6307	512/243	17/8, 36/17
40	1337.05715	243/112	28/13, 13/6
41	1370.4836	108/49
42	1403.91	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: 32.65306 cents (4\7/21 = 4\147) to 34.28571 cents (3\5/21 = 1\35)

Fifth							Cents	Comments
4\7							32.6531
						27\47	32.82675
					23\40		32.8571
						42\73	32.8767
				19\33			32.9004
						53\92	32.91925
					34\59		32.9298
						49\85	32.9412
			15\26				33.9670
						56\97	32.9897
					41\71		32.9980
						67\116	33.0049
				26\45			33.0159	Flattone is in this region
						63\109	33.0275
					37\64		33.0357
						48\83	33.0465
		11\19					33.0827
						51\88	33.1167
					40\69		33.1263
						69\119	33.13325
				29\50			33.1429
						76\131	33.1516	Golden meantone (696.2145¢)
					47\81		33.1570
						65\112	33.1633
			18\31				33.1797	Meantone is in this region
						61\105	33.1973
					43\74		33.2046
						68\117	33.2112
				25\43			33.2226
						57\98	33.23615
					32\55		33.24675
						39\67	33.2623
	7\12						33.3
						38\65	33.4066
					31\53		33.4232	The fifth closest to a just 3/2 for EDOs less than 200
						55\94	33.43465	Garibaldi / Cassandra
				24\41			33.4495
						65\111	33.4620
					41\70		33.4694
						58\99	33.4776
			17\29				33.4975
						61\104	33.5165
					44\75		33.5328
						71\121	33.5301	Golden neogothic (704.0956¢)
				27\46			33.5404	Neogothic is in this region
						64\109	33.5518
					37\63		33.5601
						47\80	33.5714
		10\17					33.61345	The generator closest to a just 17/14 for EDOs less than 4200
						43\73	33.6595
					33\56		33.6735
						56\95	33.6842
				23\39			33.6996
						59\100	33.7143
					36\61		33.72365
						49\83	33.7349
			13\22				33.7662	Archy is in this region
						42\71	33.8028
					29\49		33.8192
						45\76	33.8346
				16\27			33.8624
						35\59	33.8983
					19\32		33.9286
						22\37	33.9768
3\5							34.2857

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.

21edf

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