1600edo
| ← 1599edo | 1600edo | 1601edo → |
1600 equal divisions of the octave (abbreviated 1600edo or 1600ed2), also called 1600-tone equal temperament (1600tet) or 1600 equal temperament (1600et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1600 equal parts of exactly 0.75 ¢ each. Each step represents a frequency ratio of 21/1600, or the 1600th root of 2.
Theory
1600edo is a very strong 37-limit system, being distinctly consistent in the 37-odd-limit with a smaller relative error than anything else with this property until 4501. It is also the first division past 311 with a lower 43-limit relative error.
In the 5-limit, it supports kwazy. In the 11-limit, it supports the rank-3 temperament thor. In higher limits, it tempers out 12376/12375 in the 17-limit and due to being consistent higher than 33-odd-limit it enables the essentially tempered flashmic chords.
Odd harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.000 | +0.045 | -0.064 | +0.174 | -0.068 | +0.222 | +0.045 | +0.237 | +0.226 | +0.173 | +0.214 |
| relative (%) | +0 | +6 | -8 | +23 | -9 | +30 | +6 | +32 | +30 | +23 | +29 | |
| Steps (reduced) |
1600 (0) |
2536 (936) |
3715 (515) |
4492 (1292) |
5535 (735) |
5921 (1121) |
6540 (140) |
6797 (397) |
7238 (838) |
7773 (1373) |
7927 (1527) | |
Subsets and supersets
Since 1600 factors into 26 × 52, 1600edo has subset edos 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 160, 200, 320, 400, and 800.
One step of it is the relative cent for 16. Its high divisibility, high consistency limit, and compatibility with the decimal system make it a candidate for interval size measure. One step of 1600edo is already used as a measure called śata in the context of 16edo Armodue theory.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | [-53 10 16⟩, [26 -75 40⟩ | [⟨1600 2536 3715]] | -0.0003 | 0.0228 | 3.04 |
| 2.3.5.7 | 4375/4374, [36 -5 0 -10⟩, [-17 5 16 -10⟩ | [⟨1600 2536 3715 4492]] | -0.0157 | 0.0332 | 4.43 |
| 2.3.5.7.11 | 3025/3024, 4375/4374, [24 -1 -5 0 1⟩, [15 1 7 -8 -3⟩ | [⟨1600 2536 3715 4492 5535]] | -0.0172 | 0.0329 | 4.39 |
| 2.3.5.7.11.13 | 3025/3024, 4096/4095, 4375/4374, 78125/78078, 823875/823543 | [⟨1600 2536 3715 4492 5535 5921]] | -0.0087 | 0.0356 | 4.75 |
| 2.3.5.7.11.13.17 | 2500/2499, 3025/3024, 4096/4095, 4375/4374, 14875/14872, 63888/63869 | [⟨1600 2536 3715 4492 5535 5921 6540]] | -0.0163 | 0.0331 | 4.41 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 2 | 217\1600 | 162.75 | 1125/1024 | Kwazy |
| 32 | 23\1600 | 17.25 | ? | Dam / dike / polder |
| 32 | 121\1600 (21/1600) |
90.75 (15.75) |
48828125/46294416 (?) |
Windrose |
| 32 | 357\1600 (7\1600) |
267.75 (5.25) |
245/143 (?) |
Germanium |
| 80 | 629\1600 (9\1600) |
471.75 (6.75) |
130/99 (?) |
Tetraicosic |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct