634edo: Difference between revisions

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== Theory ==
== Theory ==
634edo is a good 13-limit and no-17 higher-limit system. It tempers out the {{monzo| -53 10 16 }} ([[kwazy comma]]); 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. 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.  


=== Prime harmonics ===
=== Prime harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
634edo has subset edos [[2edo]] and 317edo.
Since 634 factors into 2 × 317, 634edo has [[2edo]] and [[317edo]] as its subsets.


== Regular temperament properties ==
== Regular temperament properties ==
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| 2.3
| 2.3
| {{monzo| 1005 -634 }}
| {{monzo| 1005 -634 }}
| [{{val| 634 1005 }}]
| {{mapping| 634 1005 }}
| -0.0799
| -0.0799
| 0.0799
| 0.0799
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| 2.3.5
| 2.3.5
| {{monzo| -53 10 16 }}, {{monzo| 33 -34 9 }}
| {{monzo| -53 10 16 }}, {{monzo| 33 -34 9 }}
| [{{val| 634 1005 1472 }}]
| {{mapping| 634 1005 1472 }}
| -0.0254
| -0.0254
| 0.1009
| 0.1009
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| 2.3.5.7
| 2.3.5.7
| 420175/419904, 703125/702464, 33554432/33480783
| 420175/419904, 703125/702464, 33554432/33480783
| [{{val| 634 1005 1472 1780 }}]
| {{mapping| 634 1005 1472 1780 }}
| -0.0422
| -0.0422
| 0.0921
| 0.0921
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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
| [{{val| 634 1005 1472 1780 2193 }}]
| {{mapping| 634 1005 1472 1780 2193 }}
| -0.0031
| -0.0031
| 0.1135
| 0.1135
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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
| [{{val| 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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{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods <br>per 8ve
! Periods<br>per 8ve
! Generator <br>(Reduced)
! Generator*
! Cents <br>(Reduced)
! Cents*
! Associated<br>Ratio
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
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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

Revision as of 07:47, 25 October 2023

← 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

Template:EDO intro

Theory

634edo is a good 13-limit and no-17 higher-limit system. The equal temperament 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
1 241\634 456.15 125/96 Qak
1 263\634 497.79 4/3 Gary
1 311\634 588.64 [-14 15 -4⟩ Countritonic (5-limit)
2 86\634 162.78 1125/1024 Kwazy

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

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