470edo
| ← 469edo | 470edo | 471edo → |
470 equal divisions of the octave (abbreviated 470edo or 470ed2), also called 470-tone equal temperament (470tet) or 470 equal temperament (470et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 470 equal parts of about 2.55 ¢ each. Each step represents a frequency ratio of 21/470, or the 470th root of 2.
Theory
470 = 5 × 94, and 470edo shares the fifth with 94edo. Unlike 94edo, however, 470edo is only consistent to the 5-odd-limit. Using the patent val, the equal temperament tempers out 703125/702464, 823543/820125, and 1500625/1492992 in the 7-limit; 3025/3024, 4000/3993, 6250/6237, 19712/19683, and 117649/117128 in the 11-limit. It supports uniwiz and decimetra.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.17 | -0.78 | -1.17 | +0.17 | -0.53 | -0.27 | +1.21 | -0.19 | -0.64 | -1.21 |
| Relative (%) | +0.0 | +6.8 | -30.6 | -45.7 | +6.7 | -20.7 | -10.8 | +47.4 | -7.4 | -25.1 | -47.2 | |
| Steps (reduced) |
470 (0) |
745 (275) |
1091 (151) |
1319 (379) |
1626 (216) |
1739 (329) |
1921 (41) |
1997 (117) |
2126 (246) |
2283 (403) |
2328 (448) | |
Subsets and supersets
Since 470 factors into 2 × 5 × 47, 470edo has subset edos 2, 5, 10, 47, 94, and 235.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | 1600000/1594323, [-77 -10 40⟩ | [⟨470 745 1091]] | +0.0759 | 0.1897 | 7.43 |
| 2.3.5.7 | 703125/702464, 1500625/1492992, 1600000/1594323 | [⟨470 745 1091 1319]] | +0.1608 | 0.2205 | 8.64 |
| 2.3.5.7.11 | 3025/3024, 4000/3993, 19712/19683, 117649/117128 | [⟨470 745 1091 1319 1626]] | +0.1187 | 0.2144 | 8.40 |
| 2.3.5.7.11.13 | 625/624, 1575/1573, 2080/2079, 13720/13689, 15379/15360 | [⟨470 745 1091 1319 1626 1739]] | +0.1227 | 0.1959 | 7.67 |
| 2.3.5.7.11.13.17 | 595/594, 625/624, 833/832, 1575/1573, 3185/3179, 8624/8619 | [⟨470 745 1091 1319 1626 1739 1921]] | +0.1148 | 0.1824 | 7.14 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 133\470 | 339.57 | 243/200 | Amity (5-limit) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct