465edo
| ← 464edo | 465edo | 466edo → |
465 equal divisions of the octave (abbreviated 465edo or 465ed2), also called 465-tone equal temperament (465tet) or 465 equal temperament (465et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 465 equal parts of about 2.58 ¢ each. Each step represents a frequency ratio of 21/465, or the 465th root of 2.
Theory
465edo is only consistent to the 5-odd-limit, and the errors of harmonics beyond 3 tend to be quite large. It can be considered for the 2.3.5.11.13.17 subgroup, tempering out 936/935, 1377/1375, 71874/71825, 131648/131625 and 225000/224939. It supports counterschismic in the 5-limit, and birds and belobog in the 7-limit using the patent val.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.02 | +0.78 | -1.08 | +0.94 | +0.76 | +0.85 | -0.74 | -1.18 | +0.10 | +0.77 |
| Relative (%) | +0.0 | -0.8 | +30.3 | -42.0 | +36.4 | +29.6 | +33.0 | -28.6 | -45.6 | +3.9 | +29.9 | |
| Steps (reduced) |
465 (0) |
737 (272) |
1080 (150) |
1305 (375) |
1609 (214) |
1721 (326) |
1901 (41) |
1975 (115) |
2103 (243) |
2259 (399) |
2304 (444) | |
Subsets and supersets
Since 465 factors into 3 × 5 × 31, 465edo has subset edos 3, 5, 15, 31, 93, and 155. 930edo, which doubles it, gives a good correction to the harmonic 7.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-737 465⟩ | [⟨465 737]] | +0.0062 | 0.0062 | 0.24 |
| 2.3.5 | [25 15 -21⟩, [-22 30 -11⟩ | [⟨465 737 1080]] | -0.1083 | 0.1619 | 6.27 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 193\465 | 498.06 | 4/3 | Counterschismic |
| 5 | 322\465 (43\465) |
830.97 (110.97) |
80/49 (15/14) |
Qintosec (465) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct