7ed4
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7 equal divisions of the 4th harmonic (abbreviated 7ed4) is a nonoctave tuning system that divides the interval of 4/1 into 7 equal parts of about 343 ¢ each. Each step represents a frequency ratio of 41/7, or the 7th root of 4.
Theory
7ed4 is the result of taking every other step of 7edo. As the step size of 7ed4 is less than 5c flat of the undecimal neutral third 11/9, 7ed4 also roughly matches a stack of neutral thirds.
Intervals
| Step | Interval (¢) | JI approximated | Simplified ratio |
|---|---|---|---|
| 1 | 342.86 | 78/64 | 39/32 |
| 2 | 685.71 | 6/4 | 3/2 |
| 3 | 1028.57 | 29/16 | |
| 4 | 1371.43 | 64/29 | |
| 5 | 1714.29 | 16/6, 35/13 | 8/3 |
| 6 | 2057.14 | 13/4 | |
| 7 | 2400.00 | 4/1 |
The subgroup interpretation used is 4.6.13.29.35. Other interpretations are possible.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +171 | +155 | +0 | -43 | -16 | +60 | +171 | -32 | +128 | -37 | +155 |
| Relative (%) | +50.0 | +45.3 | +0.0 | -12.7 | -4.7 | +17.4 | +50.0 | -9.5 | +37.3 | -10.8 | +45.3 | |
| Steps (reduced) |
4 (4) |
6 (6) |
7 (0) |
8 (1) |
9 (2) |
10 (3) |
11 (4) |
11 (4) |
12 (5) |
12 (5) |
13 (6) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +17 | -112 | +112 | +0 | -105 | +139 | +45 | -43 | -128 | +134 | +57 |
| Relative (%) | +4.8 | -32.6 | +32.6 | +0.0 | -30.6 | +40.5 | +13.2 | -12.7 | -37.3 | +39.2 | +16.8 | |
| Steps (reduced) |
13 (6) |
13 (6) |
14 (0) |
14 (0) |
14 (0) |
15 (1) |
15 (1) |
15 (1) |
15 (1) |
16 (2) |
16 (2) | |
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