1759edo
| ← 1758edo | 1759edo | 1760edo → |
1759 equal divisions of the octave (abbreviated 1759edo or 1759ed2), also called 1759-tone equal temperament (1759tet) or 1759 equal temperament (1759et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1759 equal parts of about 0.682 ¢ each. Each step represents a frequency ratio of 21/1759, or the 1759th root of 2.
Theory
1759edo is consistent to the 23-odd-limit, tempering out 3025/3024, 2500/2499, 5985/5984, 4225/4224, 6175/6174, 3520/3519, 14875/14872 and 256000/255879. It supports etampesic. Essentially tempered chords in 1759et include vicetertismic chords.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.035 | -0.185 | -0.094 | -0.096 | -0.050 | +0.104 | -0.071 | +0.037 | -0.129 | -0.294 |
| Relative (%) | +0.0 | +5.1 | -27.2 | -13.7 | -14.0 | -7.3 | +15.3 | -10.4 | +5.5 | -18.9 | -43.1 | |
| Steps (reduced) |
1759 (0) |
2788 (1029) |
4084 (566) |
4938 (1420) |
6085 (808) |
6509 (1232) |
7190 (154) |
7472 (436) |
7957 (921) |
8545 (1509) |
8714 (1678) | |
Subsets and supersets
1759edo is the 274th prime EDO.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [2788 -1759⟩ | [⟨1759 2788]] | −0.0110 | 0.0110 | 1.61 |
| 2.3.5 | [-68 18 17⟩, [-4 -59 42⟩ | [⟨1759 2788 4084]] | +0.0193 | 0.0437 | 6.41 |
| 2.3.5.7 | 420175/419904, 48828125/48771072, [44 -28 5 -4⟩ | [⟨1759 2788 4084 4938]] | +0.0228 | 0.0383 | 5.61 |
| 2.3.5.7.11 | 3025/3024, 420175/419904, 1953125/1951488, 645922816/645700815 | [⟨1759 2788 4084 4938 6085]] | +0.0238 | 0.0343 | 5.03 |
| 2.3.5.7.11.13 | 3025/3024, 4225/4224, 256000/255879, 4100625/4100096, 420175/419904 | [⟨1759 2788 4084 4938 6085 6509]] | +0.0221 | 0.0316 | 4.63 |
| 2.3.5.7.11.13.17 | 3025/3024, 2500/2499, 4225/4224, 56595/56576, 14875/14872, 256000/255879 | [⟨1759 2788 4084 4938 6085 6509 7190]] | +0.0153 | 0.0336 | 4.93 |
| 2.3.5.7.11.13.17.19 | 3025/3024, 2500/2499, 5985/5984, 4225/4224, 6175/6174, 14875/14872, 256000/255879 | [⟨1759 2788 4084 4938 6085 6509 7190 7472]] | +0.0155 | 0.0315 | 4.62 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 164\1759 | 111.882 | 16/15 | Vavoom |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct