User:Francium/1987edo
| ← 1986edo | 1987edo | 1988edo → |
1987 equal divisions of the octave (abbreviated 1987edo or 1987ed2), also called 1987-tone equal temperament (1987tet) or 1987 equal temperament (1987et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1987 equal parts of about 0.604 ¢ each. Each step represents a frequency ratio of 21/1987, or the 1987th root of 2.
Theory
1987edo is only consistent to the 3-limit, having a quite large error in all of its lower harmonics. Its harmonic 15 is an exception with a relative error of 0.8 percent.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.194 | +0.199 | -0.129 | +0.217 | +0.066 | +0.137 | +0.005 | +0.128 | +0.222 | +0.281 | -0.192 |
| Relative (%) | -32.0 | +32.9 | -21.4 | +35.9 | +10.9 | +22.6 | +0.8 | +21.1 | +36.8 | +46.5 | -31.8 | |
| Steps (reduced) |
3149 (1162) |
4614 (640) |
5578 (1604) |
6299 (338) |
6874 (913) |
7353 (1392) |
7763 (1802) |
8122 (174) |
8441 (493) |
8728 (780) |
8988 (1040) | |
Subsets and supersets
1987edo is the 300th prime edo. 5961edo, which triples it, gives a good correction to its harmonic 5.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-3149 1987⟩ | [⟨1987 3149]] | 0.0611 | 0.0611 | 10.12 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 522\1987 | 315.249 | 6/5 | Parakleismic |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct