53ed7
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| ← 52ed7 | 53ed7 | 54ed7 → |
53 equal divisions of the 7th harmonic (abbreviated 53ed7) is a nonoctave tuning system that divides the interval of 7/1 into 53 equal parts of about 63.6 ¢ each. Each step represents a frequency ratio of 71/53, or the 53rd root of 7.
Theory
53ed7 is related to 19edo, 30edt, and Carlos Beta, but with the 7/1 rather than the 2/1 being just. The octave is about 7.6923 cents stretched. Like 19edo, 53ed7 is consistent to the 10-integer-limit, but the patent val has a generally sharp tendency for harmonics up to 16, with exception for 11th harmonic.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +7.7 | +4.9 | +15.4 | +10.4 | +12.6 | +0.0 | +23.1 | +9.9 | +18.1 | -19.7 | +20.3 |
| Relative (%) | +12.1 | +7.8 | +24.2 | +16.4 | +19.9 | +0.0 | +36.3 | +15.5 | +28.5 | -31.1 | +32.0 | |
| Steps (reduced) |
19 (19) |
30 (30) |
38 (38) |
44 (44) |
49 (49) |
53 (0) |
57 (4) |
60 (7) |
63 (10) |
65 (12) |
68 (15) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +8.9 | +7.7 | +15.4 | +30.8 | -10.6 | +17.5 | -12.5 | +25.8 | +4.9 | -12.0 | -25.4 |
| Relative (%) | +13.9 | +12.1 | +24.2 | +48.4 | -16.7 | +27.6 | -19.7 | +40.6 | +7.8 | -19.0 | -40.0 | |
| Steps (reduced) |
70 (17) |
72 (19) |
74 (21) |
76 (23) |
77 (24) |
79 (26) |
80 (27) |
82 (29) |
83 (30) |
84 (31) |
85 (32) | |
Subsets and supersets
53ed7 is the 16th prime ed7. It does not contain any nontrivial subset ed7's.
Intervals
| # | Cents | Approximate ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 63.6 | 21/20, 25/24, 27/26, 28/27 |
| 2 | 127.1 | 13/12, 14/13, 15/14, 16/15 |
| 3 | 190.7 | 9/8, 10/9 |
| 4 | 254.3 | 7/6, 8/7 |
| 5 | 317.8 | 6/5 |
| 6 | 381.4 | 5/4 |
| 7 | 444.9 | 9/7 |
| 8 | 508.5 | 4/3 |
| 9 | 572.1 | 7/5, 18/13 |
| 10 | 635.6 | 10/7, 13/9 |
| 11 | 699.2 | 3/2 |
| 12 | 762.8 | 14/9 |
| 13 | 826.3 | 8/5, 13/8 |
| 14 | 889.9 | 5/3 |
| 15 | 953.4 | 7/4, 12/7 |
| 16 | 1017.0 | 9/5 |
| 17 | 1080.6 | 15/8 |
| 18 | 1144.1 | 27/14, 35/18 |
| 19 | 1207.7 | 2/1 |
| 20 | 1271.3 | 21/10, 25/12 |
| 21 | 1334.8 | 13/6 |
| 22 | 1398.4 | 9/4 |
| 23 | 1461.9 | 7/3 |
| 24 | 1525.5 | 12/5 |
| 25 | 1589.1 | 5/2 |
| 26 | 1652.6 | 13/5 |
| 27 | 1716.2 | 8/3 |
| 28 | 1779.8 | 14/5 |
| 29 | 1843.3 | 20/7, 26/9 |
| 30 | 1906.9 | 3/1 |
| 31 | 1970.4 | 25/8, 28/9 |
| 32 | 2034.0 | 13/4 |
| 33 | 2097.6 | 10/3 |
| 34 | 2161.1 | 7/2 |
| 35 | 2224.7 | 18/5 |
| 36 | 2288.3 | 15/4 |
| 37 | 2351.8 | 35/9 |
| 38 | 2415.4 | 4/1 |
| 39 | 2478.9 | 21/5, 25/6 |
| 40 | 2542.5 | 13/3 |
| 41 | 2606.1 | 9/2 |
| 42 | 2669.6 | 14/3 |
| 43 | 2733.2 | 24/5 |
| 44 | 2796.8 | 5/1 |
| 45 | 2860.3 | 21/4, 26/5 |
| 46 | 2923.9 | 16/3 |
| 47 | 2987.4 | 28/5 |
| 48 | 3051.0 | 35/6 |
| 49 | 3114.6 | 6/1 |
| 50 | 3178.1 | 50/8, 56/9 |
| 51 | 3241.7 | 13/2 |
| 52 | 3305.3 | 27/4 |
| 53 | 3368.8 | 7/1 |