83ed4
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83 equal divisions of the 4th harmonic (abbreviated 83ed4) is a nonoctave tuning system that divides the interval of 4/1 into 83 equal parts of about 28.9 ¢ each. Each step represents a frequency ratio of 41/83, or the 83rd root of 4.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 28.9 | |
| 2 | 57.8 | 30/29, 31/30 |
| 3 | 86.7 | 21/20 |
| 4 | 115.7 | 31/29 |
| 5 | 144.6 | 38/35 |
| 6 | 173.5 | |
| 7 | 202.4 | |
| 8 | 231.3 | |
| 9 | 260.2 | |
| 10 | 289.2 | 13/11 |
| 11 | 318.1 | |
| 12 | 347 | 11/9 |
| 13 | 375.9 | |
| 14 | 404.8 | |
| 15 | 433.7 | 9/7 |
| 16 | 462.7 | 17/13, 30/23 |
| 17 | 491.6 | |
| 18 | 520.5 | 23/17, 27/20 |
| 19 | 549.4 | |
| 20 | 578.3 | |
| 21 | 607.2 | |
| 22 | 636.1 | 13/9 |
| 23 | 665.1 | |
| 24 | 694 | |
| 25 | 722.9 | |
| 26 | 751.8 | 17/11 |
| 27 | 780.7 | 11/7 |
| 28 | 809.6 | |
| 29 | 838.6 | |
| 30 | 867.5 | 33/20, 38/23 |
| 31 | 896.4 | |
| 32 | 925.3 | 29/17 |
| 33 | 954.2 | |
| 34 | 983.1 | 30/17 |
| 35 | 1012 | |
| 36 | 1041 | 31/17 |
| 37 | 1069.9 | 13/7 |
| 38 | 1098.8 | 17/9 |
| 39 | 1127.7 | |
| 40 | 1156.6 | 37/19, 39/20 |
| 41 | 1185.5 | |
| 42 | 1214.5 | |
| 43 | 1243.4 | |
| 44 | 1272.3 | |
| 45 | 1301.2 | |
| 46 | 1330.1 | |
| 47 | 1359 | |
| 48 | 1388 | 29/13 |
| 49 | 1416.9 | |
| 50 | 1445.8 | 30/13 |
| 51 | 1474.7 | |
| 52 | 1503.6 | 31/13 |
| 53 | 1532.5 | |
| 54 | 1561.4 | 37/15 |
| 55 | 1590.4 | |
| 56 | 1619.3 | |
| 57 | 1648.2 | |
| 58 | 1677.1 | 29/11 |
| 59 | 1706 | |
| 60 | 1734.9 | 30/11 |
| 61 | 1763.9 | |
| 62 | 1792.8 | 31/11 |
| 63 | 1821.7 | |
| 64 | 1850.6 | |
| 65 | 1879.5 | |
| 66 | 1908.4 | |
| 67 | 1937.3 | |
| 68 | 1966.3 | |
| 69 | 1995.2 | |
| 70 | 2024.1 | 29/9 |
| 71 | 2053 | |
| 72 | 2081.9 | 10/3 |
| 73 | 2110.8 | |
| 74 | 2139.8 | 31/9 |
| 75 | 2168.7 | 7/2 |
| 76 | 2197.6 | |
| 77 | 2226.5 | |
| 78 | 2255.4 | |
| 79 | 2284.3 | |
| 80 | 2313.3 | 19/5 |
| 81 | 2342.2 | |
| 82 | 2371.1 | |
| 83 | 2400 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +14.5 | +6.5 | +0.0 | -10.4 | -8.0 | +14.3 | +14.5 | +13.0 | +4.0 | +12.5 | +6.5 |
| Relative (%) | +50.0 | +22.4 | +0.0 | -36.0 | -27.6 | +49.5 | +50.0 | +44.8 | +14.0 | +43.4 | +22.4 | |
| Steps (reduced) |
42 (42) |
66 (66) |
83 (0) |
96 (13) |
107 (24) |
117 (34) |
125 (42) |
132 (49) |
138 (55) |
144 (61) |
149 (66) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +12.5 | -0.2 | -3.9 | +0.0 | +10.7 | -1.5 | -8.4 | -10.4 | -8.1 | -1.9 | +7.9 |
| Relative (%) | +43.2 | -0.5 | -13.6 | +0.0 | +37.0 | -5.2 | -28.9 | -36.0 | -28.1 | -6.6 | +27.2 | |
| Steps (reduced) |
154 (71) |
158 (75) |
162 (79) |
166 (0) |
170 (4) |
173 (7) |
176 (10) |
179 (13) |
182 (16) |
185 (19) |
188 (22) | |