487edo
| ← 486edo | 487edo | 488edo → |
487 equal divisions of the octave (abbreviated 487edo or 487ed2), also called 487-tone equal temperament (487tet) or 487 equal temperament (487et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 487 equal parts of about 2.46 ¢ each. Each step represents a frequency ratio of 21/487, or the 487th root of 2.
Theory
487edo is distinctly consistent to the 13-odd-limit. The equal temperament tempers out [24 -21 4⟩ (vulture comma) and [55 -1 -23⟩ (counterwürschmidt comma) in the 5-limit, 4375/4374 (ragisma), 235298/234375 (triwellisma), and 33554432/33480783 (garischisma) in the 7-limit, 5632/5625, 12005/11979, 19712/19683, 41503/41472 in the 11-limit, 676/675, 1001/1000, 2080/2079, 4096/4095, and 4225/4224 in the 13-limit. It supports semidimfourth, seniority, and vulture.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.30 | +0.54 | -0.45 | +0.63 | -0.28 | +1.00 | +0.64 | +0.06 | +0.40 | +0.75 |
| Relative (%) | +0.0 | +12.3 | +22.1 | -18.2 | +25.7 | -11.4 | +40.6 | +25.9 | +2.5 | +16.3 | +30.6 | |
| Steps (reduced) |
487 (0) |
772 (285) |
1131 (157) |
1367 (393) |
1685 (224) |
1802 (341) |
1991 (43) |
2069 (121) |
2203 (255) |
2366 (418) |
2413 (465) | |
Subsets and supersets
487edo is the 93rd prime edo.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [772 -487⟩ | [⟨487 772]] | −0.0958 | 0.0958 | 3.89 |
| 2.3.5 | [24 -21 4⟩, [55 -1 -23⟩ | [⟨487 772 1131]] | −0.1421 | 0.1020 | 4.14 |
| 2.3.5.7 | 4375/4374, 235298/234375, 33554432/33480783 | [⟨487 772 1131 1367]] | −0.0667 | 0.1577 | 6.40 |
| 2.3.5.7.11 | 4375/4374, 5632/5625, 12005/11979, 41503/41472 | [⟨487 772 1131 1367 1685]] | −0.0899 | 0.1485 | 6.03 |
| 2.3.5.7.11.13 | 676/675, 1001/1000, 4096/4095, 4375/4374, 12005/11979 | [⟨487 772 1131 1367 1685 1802]] | −0.0623 | 0.1490 | 6.05 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 131\487 | 322.79 | 3087/2560 | Seniority |
| 1 | 157\487 | 386.86 | 5/4 | Counterwürschmidt |
| 1 | 182\487 | 448.46 | 35/27 | Semidimfourth |
| 1 | 193\487 | 475.56 | 320/243 | Vulture |
| 1 | 202\487 | 497.74 | 4/3 | Gary |
| 1 | 227\487 | 559.34 | 864/625 | Tritriple (5-limit) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct