460edo
| ← 459edo | 460edo | 461edo → |
460 equal divisions of the octave (abbreviated 460edo or 460ed2), also called 460-tone equal temperament (460tet) or 460 equal temperament (460et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 460 equal parts of about 2.61 ¢ each. Each step represents a frequency ratio of 21/460, or the 460th root of 2.
Theory
460edo is a very strong 19-limit system and is distinctly consistent to the 21-odd-limit, with harmonics of 3 to 19 all tuned flat.
The equal temperament tempers out the schisma, 32805/32768, in the 5-limit and 4375/4374 and 65536/65625 in the 7-limit, so that it supports pontiac, the 171 & 289 temperament. In the 11-limit it tempers of 3025/3024 and 9801/9800, and 43923/43904; in the 13-limit 1001/1000, 4225/4224 and 10648/10647, so that it supports deca, the 190 & 270 temperament; in the 17-limit 833/832, 1089/1088, 1225/1224, 1701/1700, 2058/2057, 2431/2430, 2601/2600 and 4914/4913; and in the 19-limit 1331/1330, 1445/1444, 1521/1520, 1540/1539, 1729/1728, 2376/2375, 2926/2925, 3136/3135, 3250/3249 and 4200/4199. It serves as the optimal patent val for various temperaments such as the rank-5 temperament tempering out 833/832 and 1001/1000.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.22 | -0.23 | -1.00 | -0.88 | -0.53 | -0.61 | -0.12 | +0.42 | +0.86 | +0.18 |
| Relative (%) | +0.0 | -8.3 | -8.7 | -38.3 | -33.9 | -20.2 | -23.3 | -4.7 | +16.2 | +32.9 | +7.0 | |
| Steps (reduced) |
460 (0) |
729 (269) |
1068 (148) |
1291 (371) |
1591 (211) |
1702 (322) |
1880 (40) |
1954 (114) |
2081 (241) |
2235 (395) |
2279 (439) | |
Subsets and supersets
Since 460 factors into 22 × 5 × 23, 460edo has subset edos 2, 4, 5, 10, 20, 23, 46, 92, 115, and 230.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-729 460⟩ | [⟨460 729]] | +0.0681 | 0.0681 | 2.61 |
| 2.3.5 | 32805/32768, [6 68 -49⟩ | [⟨460 729 1068]] | +0.0780 | 0.0573 | 2.20 |
| 2.3.5.7 | 4375/4374, 32805/32768, [-4 -2 -9 10⟩ | [⟨460 729 1068 1291]] | +0.1475 | 0.1303 | 4.99 |
| 2.3.5.7.11 | 3025/3024, 4375/4374, 32805/32768, 184877/184320 | [⟨460 729 1068 1291 1591]] | +0.1691 | 0.1243 | 4.76 |
| 2.3.5.7.11.13 | 1001/1000, 3025/3024, 4225/4224, 4375/4374, 26411/26364 | [⟨460 729 1068 1291 1591 1702]] | +0.1647 | 0.1139 | 4.36 |
| 2.3.5.7.11.13.17 | 833/832, 1001/1000, 1089/1088, 1225/1224, 1701/1700, 4225/4224 | [⟨460 729 1068 1291 1591 1702 1880]] | +0.1624 | 0.1056 | 4.05 |
| 2.3.5.7.11.13.17.19 | 833/832, 1001/1000, 1089/1088, 1225/1224, 1331/1330, 1445/1444, 1617/1615 | [⟨460 729 1068 1291 1591 1702 1880 1954]] | +0.1457 | 0.1082 | 4.15 |
- 460et has lower absolute errors in the 17- and 19-limit than any previous equal temperaments. It beats 422 in either subgroup, and is bettered by 494 in the 17-limit, and 525 in the 19-limit.
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 9\460 | 23.48 | 531441/524288 | Commatose |
| 1 | 121\460 | 315.65 | 6/5 | Egads |
| 1 | 191\460 | 498.26 | 4/3 | Pontiac |
| 10 | 121\460 (17\460) |
315.65 (44.35) |
6/5 (40/39) |
Deca |
| 20 | 66\460 (20\460) |
172.173 (52.173) |
169/153 (?) |
Calcium |
| 20 | 217\460 (10\460) |
566.086 (26.086) |
238/165 (?) |
Soviet ferris wheel |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct