35ed4
| ← 33ed4 | 35ed4 | 37ed4 → |
35 equal divisions of the 4th harmonic (abbreviated 35ed4) is a nonoctave tuning system that divides the interval of 4/1 into 35 equal parts of about 68.6 ¢ each. Each step represents a frequency ratio of 41/35, or the 35th root of 4. Željko Stojanović recommends the use of 35ed4.
Intervals
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 68.6 | 26/25, 27/26, 28/27 |
| 2 | 137.1 | 13/12 |
| 3 | 205.7 | 9/8 |
| 4 | 274.3 | 7/6 |
| 5 | 342.9 | 11/9 |
| 6 | 411.4 | 14/11 |
| 7 | 480 | 4/3 |
| 8 | 548.6 | 11/8 |
| 9 | 617.1 | 10/7 |
| 10 | 685.7 | 3/2 |
| 11 | 754.3 | 14/9 |
| 12 | 822.9 | 8/5 |
| 13 | 891.4 | 5/3 |
| 14 | 960 | 7/4 |
| 15 | 1028.6 | 9/5, 11/6 |
| 16 | 1097.1 | 15/8, 17/9 |
| 17 | 1165.7 | 35/18 |
| 18 | 1234.3 | |
| 19 | 1302.9 | 17/8 |
| 20 | 1371.4 | 11/5 |
| 21 | 1440 | 16/7, 7/3 |
| 22 | 1508.6 | 12/5 |
| 23 | 1577.1 | 5/2 |
| 24 | 1645.7 | 13/5 |
| 25 | 1714.3 | 8/3 |
| 26 | 1782.9 | 14/5 |
| 27 | 1851.4 | 35/12 |
| 28 | 1920 | 3/1 |
| 29 | 1988.6 | 28/9 |
| 30 | 2057.1 | 23/7 |
| 31 | 2125.7 | 17/5 |
| 32 | 2194.3 | 13/4, 10/3 |
| 33 | 2262.9 | 11/3, 15/4 |
| 34 | 2331.4 | |
| 35 | 2400 | 4/1 |
Ratios above were based on direct approximation, not subgroup interpretation.
Harmonics
35ed4 can be approached as a 4.3.10.7 subgroup tuning. Alternatively it can be approached as a 2.3.5.11.13.17 tuning because the error on all those primes is in the same direction and by similar amounts, so intervals built from them turn out better than one would expect from the raw numbers.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +34.3 | +18.0 | +0.0 | +25.1 | -16.2 | -8.8 | +34.3 | -32.5 | -9.2 | +31.5 | +18.0 |
| Relative (%) | +50.0 | +26.3 | +0.0 | +36.6 | -23.7 | -12.9 | +50.0 | -47.4 | -13.4 | +46.0 | +26.3 | |
| Steps (reduced) |
18 (18) |
28 (28) |
35 (0) |
41 (6) |
45 (10) |
49 (14) |
53 (18) |
55 (20) |
58 (23) |
61 (26) |
63 (28) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +16.6 | +25.5 | -25.4 | +0.0 | +32.2 | +1.8 | -23.2 | +25.1 | +9.2 | -2.7 | -11.1 |
| Relative (%) | +24.2 | +37.1 | -37.1 | +0.0 | +46.9 | +2.6 | -33.9 | +36.6 | +13.4 | -4.0 | -16.2 | |
| Steps (reduced) |
65 (30) |
67 (32) |
68 (33) |
70 (0) |
72 (2) |
73 (3) |
74 (4) |
76 (6) |
77 (7) |
78 (8) |
79 (9) | |