475edo
| ← 474edo | 475edo | 476edo → |
475 equal divisions of the octave (abbreviated 475edo or 475ed2), also called 475-tone equal temperament (475tet) or 475 equal temperament (475et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 475 equal parts of about 2.53 ¢ each. Each step represents a frequency ratio of 21/475, or the 475th root of 2.
Theory
475edo is only consistent to the 5-odd-limit. The equal temperament tempers out [-14 -19 19⟩ (enneadeca) and [47 -15 -10⟩ (quintosec comma) in the 5-limit. In the 7-limit, the 475d val supports enneadecal and the patent val supports cotoneum.
It can be considered for the 2.3.5.11.13.19.23 subgroup, tempering out 2376/2375, 3250/3249, 11132/11115, 11979/11960, 14300/14283 and 42757/42750.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.36 | +0.21 | -1.25 | -0.58 | +0.74 | +1.15 | +0.59 | +0.78 | +1.16 | -0.61 |
| Relative (%) | +0.0 | +14.3 | +8.4 | -49.4 | -23.0 | +29.1 | +45.5 | +23.4 | +30.8 | +45.9 | -24.3 | |
| Steps (reduced) |
475 (0) |
753 (278) |
1103 (153) |
1333 (383) |
1643 (218) |
1758 (333) |
1942 (42) |
2018 (118) |
2149 (249) |
2308 (408) |
2353 (453) | |
Subsets and supersets
Since 475 factors into 52 × 19, 475edo has subset edos 5, 19, 25, and 95. 950edo, 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 | [753 -475⟩ | [⟨475 753]] | −0.1138 | 0.1138 | 4.50 |
| 2.3.5 | [-14 -19 19⟩, [47 -15 -10⟩ | [⟨475 753 1103]] | −0.1064 | 0.0935 | 3.70 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 157\475 | 396.63 | 98304/78125 | Squarschmidt |
| 5 | 329\475 (44\475) |
831.16 (111.16) |
160/99 (16/15) |
Quintosec |
| 19 | 197\475 (3\475) |
497.68 (7.58) |
4/3 (225/224) |
Enneadecal (475d) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct