87ed7
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| ← 86ed7 | 87ed7 | 88ed7 → |
(semiconvergent)
87 equal divisions of the 7th harmonic (abbreviated 87ed7) is a nonoctave tuning system that divides the interval of 7/1 into 87 equal parts of about 38.7 ¢ each. Each step represents a frequency ratio of 71/87, or the 87th root of 7.
Theory
87ed7 is related to 31edo, but with the 7/1 rather than the 2/1 being just. The octave is slightly stretched (about 0.3862 ¢). Like 31edo, 87ed7 is consistent through the 12-integer-limit.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.4 | -4.6 | +0.8 | +1.7 | -4.2 | +0.0 | +1.2 | -9.1 | +2.1 | -8.0 | -3.8 |
| Relative (%) | +1.0 | -11.8 | +2.0 | +4.3 | -10.8 | +0.0 | +3.0 | -23.6 | +5.3 | -20.8 | -9.8 | |
| Steps (reduced) |
31 (31) |
49 (49) |
62 (62) |
72 (72) |
80 (80) |
87 (0) |
93 (6) |
98 (11) |
103 (16) |
107 (20) |
111 (24) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +12.5 | +0.4 | -2.9 | +1.5 | +12.8 | -8.8 | +13.8 | +2.5 | -4.6 | -7.7 | -7.2 | -3.4 |
| Relative (%) | +32.3 | +1.0 | -7.5 | +4.0 | +32.9 | -22.6 | +35.7 | +6.3 | -11.8 | -19.8 | -18.5 | -8.8 | |
| Steps (reduced) |
115 (28) |
118 (31) |
121 (34) |
124 (37) |
127 (40) |
129 (42) |
132 (45) |
134 (47) |
136 (49) |
138 (51) |
140 (53) |
142 (55) | |
Subsets and supersets
Since 87 factors into primes as 3 × 29, 87ed7 contains 3ed7 and 29ed7 as subset ed7's.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 38.7 | |
| 2 | 77.4 | 23/22 |
| 3 | 116.2 | 31/29 |
| 4 | 154.9 | 23/21, 35/32 |
| 5 | 193.6 | 19/17, 28/25 |
| 6 | 232.3 | 8/7 |
| 7 | 271.1 | 41/35 |
| 8 | 309.8 | |
| 9 | 348.5 | 11/9 |
| 10 | 387.2 | 5/4 |
| 11 | 425.9 | 23/18, 32/25 |
| 12 | 464.7 | 17/13 |
| 13 | 503.4 | |
| 14 | 542.1 | 26/19, 41/30 |
| 15 | 580.8 | 7/5 |
| 16 | 619.6 | 10/7 |
| 17 | 658.3 | 19/13, 41/28 |
| 18 | 697 | |
| 19 | 735.7 | 26/17 |
| 20 | 774.4 | 25/16, 36/23 |
| 21 | 813.2 | 8/5 |
| 22 | 851.9 | 18/11 |
| 23 | 890.6 | |
| 24 | 929.3 | 41/24 |
| 25 | 968.1 | 7/4 |
| 26 | 1006.8 | 34/19 |
| 27 | 1045.5 | |
| 28 | 1084.2 | |
| 29 | 1122.9 | |
| 30 | 1161.7 | |
| 31 | 1200.4 | 2/1 |
| 32 | 1239.1 | |
| 33 | 1277.8 | 23/11 |
| 34 | 1316.6 | 15/7 |
| 35 | 1355.3 | 35/16 |
| 36 | 1394 | 38/17 |
| 37 | 1432.7 | 16/7 |
| 38 | 1471.4 | |
| 39 | 1510.2 | |
| 40 | 1548.9 | 22/9 |
| 41 | 1587.6 | 5/2 |
| 42 | 1626.3 | 23/9, 41/16 |
| 43 | 1665.1 | 34/13 |
| 44 | 1703.8 | |
| 45 | 1742.5 | 41/15 |
| 46 | 1781.2 | 14/5 |
| 47 | 1819.9 | 20/7 |
| 48 | 1858.7 | 38/13, 41/14 |
| 49 | 1897.4 | |
| 50 | 1936.1 | |
| 51 | 1974.8 | 25/8 |
| 52 | 2013.6 | 16/5 |
| 53 | 2052.3 | 36/11 |
| 54 | 2091 | |
| 55 | 2129.7 | 41/12 |
| 56 | 2168.4 | 7/2 |
| 57 | 2207.2 | |
| 58 | 2245.9 | |
| 59 | 2284.6 | |
| 60 | 2323.3 | |
| 61 | 2362.1 | |
| 62 | 2400.8 | 4/1 |
| 63 | 2439.5 | |
| 64 | 2478.2 | |
| 65 | 2516.9 | 30/7 |
| 66 | 2555.7 | 35/8 |
| 67 | 2594.4 | |
| 68 | 2633.1 | 32/7 |
| 69 | 2671.8 | |
| 70 | 2710.5 | |
| 71 | 2749.3 | |
| 72 | 2788 | 5/1 |
| 73 | 2826.7 | 41/8 |
| 74 | 2865.4 | |
| 75 | 2904.2 | |
| 76 | 2942.9 | |
| 77 | 2981.6 | 28/5 |
| 78 | 3020.3 | 40/7 |
| 79 | 3059 | 41/7 |
| 80 | 3097.8 | |
| 81 | 3136.5 | |
| 82 | 3175.2 | 25/4 |
| 83 | 3213.9 | 32/5 |
| 84 | 3252.7 | |
| 85 | 3291.4 | |
| 86 | 3330.1 | |
| 87 | 3368.8 | 7/1 |