634edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
634edo is a good 13-limit and no-17 higher-limit system. | 634edo is a good 13-limit and no-17 higher-limit system. As an equal temperament, it [[tempering out|tempers out]] {{monzo| -53 10 16 }} ([[kwazy comma]]) and {{monzo| 33 -34 9 }} (countritonic comma) in the 5-limit; 420175/419904 ([[wizma]]), 703125/702464 ([[meter]]), and 33554432/33480783 ([[garischisma]]) in the 7-limit; [[9801/9800]], [[19712/19683]], [[41503/41472]] in the 11-limit; [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]], 14641/14625, and 31250/31213 in the 13-limit. | ||
=== Prime harmonics === | === Prime harmonics === | ||
| Line 9: | Line 9: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
Since 634 factors into | Since 634 factors into 2 × 317, 634edo has [[2edo]] and [[317edo]] as its subsets. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
| Line 17: | Line 17: | ||
! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br | ! rowspan="2" | Optimal<br>8ve stretch (¢) | ||
! colspan="2" | Tuning error | ! colspan="2" | Tuning error | ||
|- | |- | ||
| Line 24: | Line 24: | ||
|- | |- | ||
| 2.3 | | 2.3 | ||
| {{ | | {{Monzo| 1005 -634 }} | ||
| {{ | | {{Mapping| 634 1005 }} | ||
| −0.0799 | | −0.0799 | ||
| 0.0799 | | 0.0799 | ||
| Line 31: | Line 31: | ||
|- | |- | ||
| 2.3.5 | | 2.3.5 | ||
| {{ | | {{Monzo| -53 10 16 }}, {{monzo| 33 -34 9 }} | ||
| {{ | | {{Mapping| 634 1005 1472 }} | ||
| −0.0254 | | −0.0254 | ||
| 0.1009 | | 0.1009 | ||
| Line 39: | Line 39: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 420175/419904, 703125/702464, 33554432/33480783 | | 420175/419904, 703125/702464, 33554432/33480783 | ||
| {{ | | {{Mapping| 634 1005 1472 1780 }} | ||
| −0.0422 | | −0.0422 | ||
| 0.0921 | | 0.0921 | ||
| Line 46: | Line 46: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 9801/9800, 19712/19683, 41503/41472, 703125/702464 | | 9801/9800, 19712/19683, 41503/41472, 703125/702464 | ||
| {{ | | {{Mapping| 634 1005 1472 1780 2193 }} | ||
| −0.0031 | | −0.0031 | ||
| 0.1135 | | 0.1135 | ||
| Line 53: | Line 53: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 1716/1715, 2080/2079, 4096/4095, 14641/14625, 31250/31213 | | 1716/1715, 2080/2079, 4096/4095, 14641/14625, 31250/31213 | ||
| {{ | | {{Mapping| 634 1005 1472 1780 2193 2346 }} | ||
| +0.0041 | | +0.0041 | ||
| 0.1048 | | 0.1048 | ||
| Line 63: | Line 63: | ||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
|- | |- | ||
! Periods<br | ! Periods<br>per 8ve | ||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br | ! Associated<br>ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |||
| 1 | |||
| 53\634 | |||
| 100.32 | |||
| 675/637 | |||
| [[Heptacot]] | |||
|- | |- | ||
| 1 | | 1 | ||
| Line 84: | Line 90: | ||
| 311\634 | | 311\634 | ||
| 588.64 | | 588.64 | ||
| | | 351/250 | ||
| [[ | | [[Garitritonic]] | ||
|- | |||
| 2 | |||
| 263\634<br>(54\634) | |||
| 497.79<br>(102.21) | |||
| 4/3<br>(35/33) | |||
| [[Gariwizmic]] | |||
|- | |- | ||
| 2 | | 2 | ||
| Line 93: | Line 105: | ||
| [[Kwazy]] | | [[Kwazy]] | ||
|} | |} | ||
<nowiki />* [[Normal | <nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | ||