4ed5
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Prime factorization
22
Step size
696.578¢
Octave
2\4ed5 (1393.16¢) (→1\2ed5)
Twelfth
3\4ed5 (2089.74¢)
(semiconvergent)
Consistency limit
3
Distinct consistency limit
1
Special properties
| ← 3ed5 | 4ed5 | 5ed5 → |
(semiconvergent)
4 equal divisions of the 5th harmonic (abbreviated 4ed5) is a nonoctave tuning system that divides the interval of 5/1 into 4 equal parts of about 697 ¢ each. Each step represents a frequency ratio of 51/4, or the 4th root of 5.
Theory
 |
Todo: clarify |
By definition, one step of this tuning is equal to quarter-comma meantone, therefore it's just a circle of meantone fifths.
Interval table
| Steps | Cents | Approximate Ratios |
|---|---|---|
| 0 | 0 | 1/1, 21/20 |
| 1 | 696.578 | 3/2, 10/7, 11/7, 14/9, 17/11, 19/13 |
| 2 | 1393.157 | 7/3, 9/4, 11/5, 15/7, 20/9 |
| 3 | 2089.735 | 7/2, 10/3, 17/5 |
| 4 | 2786.314 | 5/1, 21/4 |
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +193 | +188 | -310 | +0 | -316 | +114 | -117 | -321 | +193 | +28 | -122 |
| Relative (%) | +27.7 | +27.0 | -44.5 | +0.0 | -45.3 | +16.4 | -16.8 | -46.1 | +27.7 | +4.0 | -17.6 | |
| Steps (reduced) |
2 (2) |
3 (3) |
3 (3) |
4 (0) |
4 (0) |
5 (1) |
5 (1) |
5 (1) |
6 (2) |
6 (2) |
6 (2) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -261 | +307 | +188 | +76 | -29 | -128 | -221 | -310 | +302 | +221 | +144 |
| Relative (%) | -37.5 | +44.1 | +27.0 | +10.9 | -4.1 | -18.4 | -31.8 | -44.5 | +43.3 | +31.8 | +20.7 | |
| Steps (reduced) |
6 (2) |
7 (3) |
7 (3) |
7 (3) |
7 (3) |
7 (3) |
7 (3) |
7 (3) |
8 (0) |
8 (0) |
8 (0) | |