21edf
| ← 20edf | 21edf | 22edf → |
21 equal divisions of the perfect fifth (abbreviated 21edf or 21ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 21 equal parts of about 33.4 ¢ each. Each step represents a frequency ratio of (3/2)1/21, or the 21st root of 3/2.
21EDF is related to 36edo, but with the 3/2 rather than the 2/1 being jus, which stretches the octave by about 3.3514 cents. Unlike 36edo, it is only consistent up to the 4-integer-limit, with discrepancy for the 5th harmonic.
Lookalikes: 36edo, 57edt, 93ed6, 101ed7, 129ed12
Approximations
Harmonics
| 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) | |
| 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 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 | |