4172edo
| ← 4171edo | 4172edo | 4173edo → |
4172 equal divisions of the octave (abbreviated 4172edo or 4172ed2), also called 4172-tone equal temperament (4172tet) or 4172 equal temperament (4172et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 4172 equal parts of about 0.288 ¢ each. Each step represents a frequency ratio of 21/4172, or the 4172nd root of 2.
Theory
The first 8 prime harmonics below 25% in 4172edo are 2, 5, 13, 17, 31, 37, 53, 61. Therefore, 4172edo can be thought of as a 2.5.13.17.31.37.53.61 subgroup temperament.
Subsets and supersets
4172's divisors are 1, 2, 4, 7, 14, 28, 149, 298, 596, 1043, 2086. Notable member of the group is 149edo, which is the smallest edo uniquely consistent in the 17-odd limit, although its approximations have long been diluted by edo of this size.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | -0.133 | -0.024 | -0.082 | +0.021 | +0.072 | -0.067 | +0.130 | +0.030 | -0.102 | +0.072 | -0.086 |
| relative (%) | -46 | -8 | -28 | +7 | +25 | -23 | +45 | +11 | -35 | +25 | -30 | |
| Steps (reduced) |
6612 (2440) |
9687 (1343) |
11712 (3368) |
13225 (709) |
14433 (1917) |
15438 (2922) |
16300 (3784) |
17053 (365) |
17722 (1034) |
18325 (1637) |
18872 (2184) | |