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 ===
Line 9: Line 9:


=== 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 ==
{{comma basis begin}}
{| 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
| 2.3
| {{monzo| 1005 -634 }}
| {{Monzo| 1005 -634 }}
| {{mapping| 634 1005 }}
| {{Mapping| 634 1005 }}
| &minus;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 }}
| &minus;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 }}
| &minus;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 }}
| &minus;0.0031
| −0.0031
| 0.1135
| 0.1135
| 6.00
| 6.00
Line 44: 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 }}
| {{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 ===
{{rank-2 begin}}
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>ratio*
! 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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| 1125/1024
| 1125/1024
| [[Kwazy]]
| [[Kwazy]]
{{rank-2 end}}
|}
{{orf}}
<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

Latest revision as of 12:08, 20 May 2026

← 633edo 634edo 635edo →
Prime factorization 2 × 317
Step size 1.89274 ¢ 
Fifth 371\634 (702.208 ¢)
Semitones (A1:m2) 61:47 (115.5 ¢ : 88.96 ¢)
Consistency limit 9
Distinct consistency limit 9

634 equal divisions of the octave (abbreviated 634edo or 634ed2), also called 634-tone equal temperament (634tet) or 634 equal temperament (634et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 634 equal parts of about 1.89 ¢ each. Each step represents a frequency ratio of 21/634, or the 634th root of 2.

Theory

634edo is a good 13-limit and no-17 higher-limit system. As an equal temperament, it tempers out [-53 10 16⟩ (kwazy comma) and [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

Approximation of prime harmonics in 634edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.253 -0.194 +0.259 -0.529 -0.149 -0.854 -0.352 +0.117 +0.076 +0.075
Relative (%) +0.0 +13.4 -10.2 +13.7 -28.0 -7.9 -45.1 -18.6 +6.2 +4.0 +4.0
Steps
(reduced) 		634
(0) 		1005
(371) 		1472
(204) 		1780
(512) 		2193
(291) 		2346
(444) 		2591
(55) 		2693
(157) 		2868
(332) 		3080
(544) 		3141
(605)

Subsets and supersets

Since 634 factors into 2 × 317, 634edo has 2edo and 317edo as its subsets.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [1005 -634⟩ [⟨634 1005]] −0.0799 0.0799 4.22
2.3.5 [-53 10 16⟩, [33 -34 9⟩ [⟨634 1005 1472]] −0.0254 0.1009 5.33
2.3.5.7 420175/419904, 703125/702464, 33554432/33480783 [⟨634 1005 1472 1780]] −0.0422 0.0921 4.86
2.3.5.7.11 9801/9800, 19712/19683, 41503/41472, 703125/702464 [⟨634 1005 1472 1780 2193]] −0.0031 0.1135 6.00
2.3.5.7.11.13 1716/1715, 2080/2079, 4096/4095, 14641/14625, 31250/31213 [⟨634 1005 1472 1780 2193 2346]] +0.0041 0.1048 5.54

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 53\634 100.32 675/637 Heptacot
1 241\634 456.15 125/96 Qak
1 263\634 497.79 4/3 Gary
1 311\634 588.64 351/250 Garitritonic
2 263\634
(54\634) 		497.79
(102.21) 		4/3
(35/33) 		Gariwizmic
2 86\634 162.78 1125/1024 Kwazy

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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