116ed5
| ← 115ed5 | 116ed5 | 117ed5 → |
116 equal divisions of the 5th harmonic (abbreviated 116ed5) is a nonoctave tuning system that divides the interval of 5/1 into 116 equal parts of about 24 ¢ each. Each step represents a frequency ratio of 51/116, or the 116th root of 5.
It is almost exactly identical to 50edo, having the same vals for all of the first 43 prime harmonics. The relative error of its approximations is almost identical to 50edo too. It supports all the same rank-2 temperaments as 50edo, most notably meantone. Compared to 50edo, the octave is stretched by about 1 cent.
All of that could make 116ed5 useful as a piano tuning to be used alongside other instruments in 50edo. Because pianos usually use stretched octaves, 116ed5 accommodates this, while also being so close to 50edo that it still preserves all 50edo’s properties.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.0 | -4.4 | +2.0 | +0.0 | -3.4 | -6.0 | +3.0 | -8.8 | +1.0 | +4.1 | -2.4 |
| Relative (%) | +4.2 | -18.2 | +8.3 | +0.0 | -14.1 | -25.1 | +12.5 | -36.5 | +4.2 | +17.2 | -9.9 | |
| Steps (reduced) |
50 (50) |
79 (79) |
100 (100) |
116 (0) |
129 (13) |
140 (24) |
150 (34) |
158 (42) |
166 (50) |
173 (57) |
179 (63) | |