21edf

← 20edf

21edf

22edf →

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

Harmonics

Approximation of harmonics in 21edf

Harmonic		2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Error	Absolute (¢)	+3.35	+3.35	+6.70	-11.92	+6.70	+7.24	+10.05	+6.70	-8.57	-6.44	+10.05	+5.19	+10.59	-8.57	+13.41
	Relative (%)	+10.0	+10.0	+20.1	-35.7	+20.1	+21.7	+30.1	+20.1	-25.6	-19.3	+30.1	+15.5	+31.7	-25.6	+40.1
Steps (reduced)		36 (15)	57 (15)	72 (9)	83 (20)	93 (9)	101 (17)	108 (3)	114 (9)	119 (14)	124 (19)	129 (3)	133 (7)	137 (11)	140 (14)	144 (18)

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