1559edo
| ← 1558edo | 1559edo | 1560edo → |
1559 equal divisions of the octave (abbreviated 1559edo or 1559ed2), also called 1559-tone equal temperament (1559tet) or 1559 equal temperament (1559et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1559 equal parts of about 0.77 ¢ each. Each step represents a frequency ratio of 21/1559, or the 1559th root of 2.
Theory
1559edo is consistent to the 9-odd-limit. Using the patent val, it tempers out [38 -2 -15⟩ (luna comma) and [71 -99 37⟩ (raider comma) in the 5-limit; 43046721/43025920, 78125000/78121827 and [32 1 -6 -7⟩ in the 7-limit, 3025/3024, 180224/180075, 50014503/50000000 and 56953125/56942116 in the 11-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.033 | +0.088 | +0.257 | -0.195 | +0.011 | -0.273 | +0.370 | -0.179 | +0.314 | +0.314 |
| Relative (%) | +0.0 | +4.3 | +11.4 | +33.4 | -25.4 | +1.4 | -35.5 | +48.1 | -23.3 | +40.8 | +40.8 | |
| Steps (reduced) |
1559 (0) |
2471 (912) |
3620 (502) |
4377 (1259) |
5393 (716) |
5769 (1092) |
6372 (136) |
6623 (387) |
7052 (816) |
7574 (1338) |
7724 (1488) | |
Subsets and supersets
1559edo is the 246th prime edo.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [2471 -1559⟩ | [⟨1559 2471]] | −0.0106 | 0.0106 | 1.38 |
| 2.3.5 | [38 -2 -15⟩, [71 -99 37⟩ | [⟨1559 2471 3620]] | −0.0196 | 0.0155 | 2.01 |
| 2.3.5.7 | 43046721/43025920, 78125000/78121827, 12884901888/12867859375 | [⟨1559 2471 3620 4377]] | −0.0376 | 0.0339 | 4.40 |
| 2.3.5.7.11 | 3025/3024, 180224/180075, 50014503/50000000, 56953125/56942116 | [⟨1559 2471 3620 4377 5393]] | −0.0188 | 0.0483 | 6.27 |
| 2.3.5.7.11.13 | 1716/1715, 3025/3024, 4096/4095, 492128/492075, 5282739/5281250 | [⟨1559 2471 3620 4377 5393 5769]] | −0.0162 | 0.0445 | 5.78 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 251\1559 | 193.201 | 262144/234375 | Luna |
| 1 | 446\1559 | 343.297 | 8000/6561 | Raider |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct