1609edo
| ← 1608edo | 1609edo | 1610edo → |
1609 equal divisions of the octave (abbreviated 1609edo or 1609ed2), also called 1609-tone equal temperament (1609tet) or 1609 equal temperament (1609et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1609 equal parts of about 0.746 ¢ each. Each step represents a frequency ratio of 21/1609, or the 1609th root of 2.
Theory
1609edo is consistent to the 15-odd-limit, tempering out 67392/67375, 160083/160000, 50193/50176, 91125/91091 and 557568/557375 in the 13-limit. It is a strong 2.5.7.13.19 subgroup tuning, tempering out 401408/401375, 739375/739328, 479199083125/479174066176 and 32005131723019/32000000000000. Using the 2.5.7.11.13.19.43 subgroup it tempers out 5720/5719. The equal temperament supports exodia.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.153 | +0.013 | -0.025 | -0.168 | -0.006 | +0.203 | +0.063 | -0.307 | -0.367 | -0.225 |
| Relative (%) | +0.0 | -20.5 | +1.8 | -3.4 | -22.5 | -0.8 | +27.2 | +8.5 | -41.1 | -49.1 | -30.2 | |
| Steps (reduced) |
1609 (0) |
2550 (941) |
3736 (518) |
4517 (1299) |
5566 (739) |
5954 (1127) |
6577 (141) |
6835 (399) |
7278 (842) |
7816 (1380) |
7971 (1535) | |
Subsets and supersets
1609edo is the 254th prime edo.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-2550 1609⟩ | [⟨1609 2550]] | 0.0482 | 0.0482 | 6.46 |
| 2.3.5 | [-68 18 17⟩, [-38 65 -28⟩ | [⟨1609 2550 3736]] | 0.0302 | 0.0468 | 6.28 |
| 2.3.5.7 | 2460375/2458624, 1640558367/1638400000, [29 14 -16 -5⟩ | [⟨1609 2550 3736 4517]] | 0.0249 | 0.0416 | 5.58 |
| 2.3.5.7.11 | 160083/160000, 1240029/1239040, 151263/151250, 262766592/262609375 | [⟨1609 2550 3736 4517 5566]] | 0.0297 | 0.0384 | 5.15 |
| 2.3.5.7.11.13 | 67392/67375, 160083/160000, 50193/50176, 91125/91091, 557568/557375 | [⟨1609 2550 3736 4517 5566 5954]] | 0.0250 | 0.0365 | 4.89 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 150\1609 | 111.871 | 16/15 | Vavoom |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct