69ed4
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69 equal divisions of the 4th harmonic (abbreviated 69ed4) is a nonoctave tuning system that divides the interval of 4/1 into 69 equal parts of about 34.8 ¢ each. Each step represents a frequency ratio of 41/69, or the 69th root of 4.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 34.8 | |
| 2 | 69.6 | |
| 3 | 104.3 | 35/33 |
| 4 | 139.1 | |
| 5 | 173.9 | 21/19 |
| 6 | 208.7 | 35/31 |
| 7 | 243.5 | |
| 8 | 278.3 | 34/29 |
| 9 | 313 | |
| 10 | 347.8 | |
| 11 | 382.6 | |
| 12 | 417.4 | |
| 13 | 452.2 | 13/10 |
| 14 | 487 | |
| 15 | 521.7 | 23/17, 27/20 |
| 16 | 556.5 | 29/21 |
| 17 | 591.3 | 31/22 |
| 18 | 626.1 | 33/23 |
| 19 | 660.9 | 22/15 |
| 20 | 695.7 | |
| 21 | 730.4 | 29/19, 35/23 |
| 22 | 765.2 | 14/9 |
| 23 | 800 | 35/22 |
| 24 | 834.8 | 34/21 |
| 25 | 869.6 | |
| 26 | 904.3 | |
| 27 | 939.1 | |
| 28 | 973.9 | |
| 29 | 1008.7 | 34/19 |
| 30 | 1043.5 | 31/17 |
| 31 | 1078.3 | |
| 32 | 1113 | 19/10 |
| 33 | 1147.8 | 33/17 |
| 34 | 1182.6 | |
| 35 | 1217.4 | |
| 36 | 1252.2 | 35/17 |
| 37 | 1287 | 21/10 |
| 38 | 1321.7 | 15/7 |
| 39 | 1356.5 | |
| 40 | 1391.3 | 29/13 |
| 41 | 1426.1 | |
| 42 | 1460.9 | |
| 43 | 1495.7 | |
| 44 | 1530.4 | |
| 45 | 1565.2 | 37/15 |
| 46 | 1600 | |
| 47 | 1634.8 | |
| 48 | 1669.6 | |
| 49 | 1704.3 | |
| 50 | 1739.1 | |
| 51 | 1773.9 | |
| 52 | 1808.7 | 37/13 |
| 53 | 1843.5 | 29/10 |
| 54 | 1878.3 | |
| 55 | 1913 | |
| 56 | 1947.8 | |
| 57 | 1982.6 | 22/7 |
| 58 | 2017.4 | |
| 59 | 2052.2 | |
| 60 | 2087 | 10/3 |
| 61 | 2121.7 | 17/5 |
| 62 | 2156.5 | |
| 63 | 2191.3 | |
| 64 | 2226.1 | |
| 65 | 2260.9 | |
| 66 | 2295.7 | |
| 67 | 2330.4 | |
| 68 | 2365.2 | |
| 69 | 2400 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +17.4 | +11.1 | +0.0 | -3.7 | -6.3 | +5.1 | +17.4 | -12.6 | +13.7 | -12.2 | +11.1 |
| Relative (%) | +50.0 | +31.9 | +0.0 | -10.7 | -18.1 | +14.6 | +50.0 | -36.2 | +39.3 | -35.0 | +31.9 | |
| Steps (reduced) |
35 (35) |
55 (55) |
69 (0) |
80 (11) |
89 (20) |
97 (28) |
104 (35) |
109 (40) |
115 (46) |
119 (50) |
124 (55) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +11.6 | -12.3 | +7.4 | +0.0 | -0.6 | +4.8 | +15.5 | -3.7 | +16.2 | +5.2 | -2.2 |
| Relative (%) | +33.5 | -35.4 | +21.2 | +0.0 | -1.7 | +13.8 | +44.7 | -10.7 | +46.5 | +15.0 | -6.3 | |
| Steps (reduced) |
128 (59) |
131 (62) |
135 (66) |
138 (0) |
141 (3) |
144 (6) |
147 (9) |
149 (11) |
152 (14) |
154 (16) |
156 (18) | |