581edo: Difference between revisions
Cleanup; +prime error table; +categories |
+infobox; +RTT table and rank-2 temperaments |
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{{Infobox ET | |||
| Prime factorization = 7 × 73 | |||
| Step size = 2.06540¢ | |||
| Fifth = 340\581 (702.24¢) | |||
| Semitones = 56:43 (115.66¢ : 88.81¢) | |||
| Consistency = 25 | |||
}} | |||
{{EDO intro|581}} | {{EDO intro|581}} | ||
== Theory == | |||
581edo is a very strong 19- and 23-limit system, distinctly [[consistent]] to the [[25-odd-limit]]. It tempers out [[2401/2400]] in the 7-limit, [[3025/3024]], [[19712/19683]], 151263/151250 in the 11-limit, and [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]] in the 13-limit. It [[support]]s and gives a good tuning for [[newt]], the 41 & 229 microtemperament, which has a neutral thirds generator. It is the first division after 270 with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]], and the first past 311 with a lower 23-limit relative error, and not until [[1578edo|1578]] do we reach a lower 23-limit relative error. | 581edo is a very strong 19- and 23-limit system, distinctly [[consistent]] to the [[25-odd-limit]]. It tempers out [[2401/2400]] in the 7-limit, [[3025/3024]], [[19712/19683]], 151263/151250 in the 11-limit, and [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]] in the 13-limit. It [[support]]s and gives a good tuning for [[newt]], the 41 & 229 microtemperament, which has a neutral thirds generator. It is the first division after 270 with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]], and the first past 311 with a lower 23-limit relative error, and not until [[1578edo|1578]] do we reach a lower 23-limit relative error. | ||
=== Prime harmonics === | |||
{{Harmonics in equal|581|columns=11}} | {{Harmonics in equal|581|columns=11}} | ||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | |||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list|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| 921 -581 }} | |||
| [{{val| 581 921 }}] | |||
| -0.0891 | |||
| 0.0891 | |||
| 4.32 | |||
|- | |||
| 2.3.5 | |||
| {{monzo| -29 -11 20 }}, {{monzo| 33 -34 9 }} | |||
| [{{val| 581 921 1349 }}] | |||
| -0.0475 | |||
| 0.0936 | |||
| 4.53 | |||
|- | |||
| 2.3.5.7 | |||
| 2401/2400, 33554432/33480783, 48828125/48771072 | |||
| [{{val| 581 921 1349 1631 }}] | |||
| -0.0222 | |||
| 0.0922 | |||
| 4.46 | |||
|- | |||
| 2.3.5.7.11 | |||
| 2401/2400, 3025/3024, 19712/19683, 234375/234256 | |||
| [{{val| 581 921 1349 1631 2010 }}] | |||
| -0.0261 | |||
| 0.0828 | |||
| 4.01 | |||
|- | |||
| 2.3.5.7.11.13 | |||
| 2080/2079, 2401/2400, 3025/3024, 4096/4095, 78125/78078 | |||
| [{{val| 581 921 1349 1631 2010 2150 }}] | |||
| -0.0259 | |||
| 0.0756 | |||
| 3.66 | |||
|- | |||
| 2.3.5.7.11.13.17 | |||
| 1225/1224, 2058/2057, 2080/2079, 2401/2400, 4096/4095, 13013/13005 | |||
| [{{val| 581 921 1349 1631 2010 2150 2375 }}] | |||
| -0.0355 | |||
| 0.0738 | |||
| 3.58 | |||
|- | |||
| 2.3.5.7.11.13.17.19 | |||
| 1216/1215, 1225/1224, 1540/1539, 1729/1728, 2058/2057, 2080/2079, 10985/10982 | |||
| [{{val| 581 921 1349 1631 2010 2150 2375 2468 }}] | |||
| -0.0283 | |||
| 0.0717 | |||
| 3.47 | |||
|} | |||
=== 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 | |||
| 17\581 | |||
| 35.11 | |||
| 1990656/1953125 | |||
| [[Gammic]] (5-limit) | |||
|- | |||
| 1 | |||
| 64\581 | |||
| 132.19 | |||
| {{monzo| -38 5 13 }} | |||
| [[Astro]] | |||
|- | |||
| 1 | |||
| 170\581 | |||
| 351.12 | |||
| 49/40 | |||
| [[Newt]] | |||
|- | |||
| 1 | |||
| 282\581 | |||
| 582.44 | |||
| 7/5 | |||
| [[Neptune]] | |||
|} | |||
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | [[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | ||
[[Category:Newt]] | [[Category:Newt]] | ||