460edo: Difference between revisions
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== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" | Subgroup | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning Error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
| Line 76: | Line 76: | ||
|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator<br>( | ! Generator<br>(Reduced) | ||
! Cents<br>( | ! Cents<br>(Reduced) | ||
! Associated<br> | ! Associated<br>Ratio | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|9\460 | | 9\460 | ||
|23.48 | | 23.48 | ||
|531441/524288 | | 531441/524288 | ||
|[[Commatose]] | | [[Commatose]] | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 97: | Line 97: | ||
| 498.26 | | 498.26 | ||
| 4/3 | | 4/3 | ||
| [[ | | [[Pontiac]] | ||
|- | |- | ||
| 10 | | 10 | ||
| Line 105: | Line 105: | ||
| [[Deca]] | | [[Deca]] | ||
|- | |- | ||
|20 | | 20 | ||
|66\460<br>(20\460) | | 66\460<br>(20\460) | ||
|172.173<br>(52.173) | | 172.173<br>(52.173) | ||
|169/153<br>(?) | | 169/153<br>(?) | ||
|[[Calcium]] | | [[Calcium]] | ||
|- | |- | ||
|20 | | 20 | ||
|217\460<br>(10\460) | | 217\460<br>(10\460) | ||
|566.086<br>(26.086) | | 566.086<br>(26.086) | ||
|238/165<br>(?) | | 238/165<br>(?) | ||
|[[Soviet Ferris wheel]] | | [[Soviet Ferris wheel]] | ||
|} | |} | ||
Revision as of 13:51, 8 March 2023
| ← 459edo | 460edo | 461edo → |
The 460 equal divisions of the octave (460edo), or the 460(-tone) equal temperament (460tet, 460et) when viewed from a regular temperament perspective, divides the octave into 460 equal parts of about 2.61 cents each.
Theory
460edo is a very strong 19-limit system and is uniquely consistent to the 21-odd-limit, with harmonics of 3 to 19 all tuned flat. It 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. In the 11-limit it tempers of 43923/43904, 3025/3024 and 9801/9800; in the 13-limit 1001/1000, 4225/4224 and 10648/10647; 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. 460edo supports the 460 & 1789 temperament in the 2.9.5.7.11.13 subgroup called commatose.
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 |
Rank-2 temperaments
| Periods per 8ve |
Generator (Reduced) |
Cents (Reduced) |
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 |