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
Error	Absolute (¢)	+3.35	+3.35	+6.70	-11.92	+6.70	+7.24	+10.05	+6.70
	Relative (%)	+10.0	+10.0	+20.1	-35.7	+20.1	+21.7	+30.1	+20.1
Steps (reduced)		36 (15)	57 (15)	72 (9)	83 (20)	93 (9)	101 (17)	108 (3)	114 (9)

contd.

Harmonic		10	11	12	13	14	15	16	17
Error	Absolute (¢)	-8.57	-6.44	+10.05	+5.19	+10.59	-8.57	+13.41	+8.73
	Relative (%)	-25.6	-19.3	+30.1	+15.5	+31.7	-25.6	+40.1	+26.1
Steps (reduced)		119 (14)	124 (19)	129 (3)	133 (7)	137 (11)	140 (14)	144 (18)	147 (0)

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.

Intervals

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

Intervals of 21edf

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