460edo: Difference between revisions
+prime error table; +links; +category |
+RTT table and rank-2 temperaments |
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The '''460 equal divisions of the octave''' divides the octave into 460 equal parts of 2.609 cents each. | The '''460 equal divisions of the octave''' divides the octave into 460 equal parts of 2.609 cents each. | ||
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 [[support]]s [[ | 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 [[support]]s [[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 five temperament tempering out 833/832 and 1001/1000. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|460}} | {{Harmonics in equal|460}} | ||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | |||
! rowspan="2" | Subgroup | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br>8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3 | |||
| {{monzo| -729 460 }} | |||
| [{{val| 460 729 }}] | |||
| +0.0681 | |||
| 0.0681 | |||
| 2.61 | |||
|- | |||
| 2.3.5 | |||
| 32805/32768, {{monzo| 6 68 -49 }} | |||
| [{{val| 460 729 1068 }}] | |||
| +0.0780 | |||
| 0.0573 | |||
| 2.20 | |||
|- | |||
| 2.3.5.7 | |||
| 4375/4374, 32805/32768, {{monzo| -4 -2 -9 10 }} | |||
| [{{val| 460 729 1068 1291 }}] | |||
| +0.1475 | |||
| 0.1303 | |||
| 4.99 | |||
|- | |||
| 2.3.5.7.11 | |||
| 3025/3024, 4375/4374, 32805/32768, 184877/184320 | |||
| [{{val| 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 | |||
| [{{val| 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 | |||
| [{{val| 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 | |||
| [{{val| 460 729 1068 1291 1591 1702 1880 1954 }}] | |||
| +0.1457 | |||
| 0.1082 | |||
| 4.15 | |||
|} | |||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+Table of rank-2 temperaments by generator | |||
! Periods<br>per octave | |||
! Generator<br>(reduced) | |||
! Cents<br>(reduced) | |||
! Associated<br>ratio | |||
! Temperaments | |||
|- | |||
| 1 | |||
| 121\460 | |||
| 315.65 | |||
| 6/5 | |||
| [[Egads]] | |||
|- | |||
| 1 | |||
| 191\460 | |||
| 498.26 | |||
| 4/3 | |||
| [[Helmholtz]] / [[pontiac]] | |||
|- | |||
| 10 | |||
| 121\460<br>(17\460) | |||
| 315.65<br>(44.35) | |||
| 6/5<br>(40/39) | |||
| [[Deca]] | |||
|} | |||
[[Category:Equal divisions of the octave]] | [[Category:Equal divisions of the octave]] | ||