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634edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|634}}
{{ED intro}}


== Theory ==
== Theory ==
634edo is a good 13-limit and no-17 higher-limit system. The equal temperament [[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.  
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 ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
Since 634 factors into {{factorization|634}}, 634edo has [[2edo]] and [[317edo]] as its subsets.
Since 634 factors into 2 × 317, 634edo has [[2edo]] and [[317edo]] as its subsets.


== 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|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" | Tuning error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
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|-
|-
| 2.3
| 2.3
| {{monzo| 1005 -634 }}
| {{Monzo| 1005 -634 }}
| {{mapping| 634 1005 }}
| {{Mapping| 634 1005 }}
| -0.0799
| −0.0799
| 0.0799
| 0.0799
| 4.22
| 4.22
|-
|-
| 2.3.5
| 2.3.5
| {{monzo| -53 10 16 }}, {{monzo| 33 -34 9 }}
| {{Monzo| -53 10 16 }}, {{monzo| 33 -34 9 }}
| {{mapping| 634 1005 1472 }}
| {{Mapping| 634 1005 1472 }}
| -0.0254
| −0.0254
| 0.1009
| 0.1009
| 5.33
| 5.33
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| 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 }}
| {{Mapping| 634 1005 1472 1780 }}
| -0.0422
| −0.0422
| 0.0921
| 0.0921
| 4.86
| 4.86
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| 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 }}
| {{Mapping| 634 1005 1472 1780 2193 }}
| -0.0031
| −0.0031
| 0.1135
| 0.1135
| 6.00
| 6.00
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| 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 }}
| {{Mapping| 634 1005 1472 1780 2193 2346 }}
| +0.0041
| +0.0041
| 0.1048
| 0.1048
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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
| 1
| 53\634
| 100.32
| 675/637
| [[Heptacot]]
|-
|-
| 1
| 1
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| 311\634
| 311\634
| 588.64
| 588.64
| {{monzo| -14 15 -4 }}
| 351/250
| [[Countritonic]] (5-limit)
| [[Garitritonic]]
|-
| 2
| 263\634<br>(54\634)
| 497.79<br>(102.21)
| 4/3<br>(35/33)
| [[Gariwizmic]]
|-
|-
| 2
| 2
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| [[Kwazy]]
| [[Kwazy]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<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
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