638edo: Difference between revisions

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**Imported revision 601729698 - Original comment: **
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox ET}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{ED intro}}
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-12-08 14:10:26 UTC</tt>.<br>
 
: The original revision id was <tt>601729698</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
The equal temperament [[tempering out|tempers out]] the minortone comma, {{monzo| -16 35 -17 }}, in the 5-limit, [[4375/4374]] in the 7-limit, [[3025/3024]], [[9801/9800]], and 43923/43904, in the 11-limit; and [[625/624]], [[729/728]], [[1575/1573]], [[2200/2197]] and [[4225/4224]] in the 13-limit. It supplies the [[optimal patent val]] for [[quatracot]], the {{nowrap|224 &amp; 414}} temperament.  
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
 
<h4>Original Wikitext content:</h4>
=== Odd harmonics ===
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">The 638 equal temperaments divide the octave into 638 equal parts of 1.881 cents each. It tempers out the minortone comma, [[tel/50031545098999707|50031545098999707]]/50000000000000000, in the 5-limit, 4375/4374 in the 7-limit, 43923/43904, 3025/3024, 9801/9800 in the 11-limit, and 625/624, 729/728, 1575/1573, 2200/2197 and 4225/4224 in the 13-limit. It supplies the optimal patent val for [[Ragismic microtemperaments#Quatracot|quatracot temperament]]. 638 factors as 2*11*29.</pre></div>
{{Harmonics in equal|638}}
<h4>Original HTML content:</h4>
 
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;638edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 638 equal temperaments divide the octave into 638 equal parts of 1.881 cents each. It tempers out the minortone comma, &lt;a class="wiki_link" href="http://tel.wikispaces.com/50031545098999707"&gt;50031545098999707&lt;/a&gt;/50000000000000000, in the 5-limit, 4375/4374 in the 7-limit, 43923/43904, 3025/3024, 9801/9800 in the 11-limit, and 625/624, 729/728, 1575/1573, 2200/2197 and 4225/4224 in the 13-limit. It supplies the optimal patent val for &lt;a class="wiki_link" href="/Ragismic%20microtemperaments#Quatracot"&gt;quatracot temperament&lt;/a&gt;. 638 factors as 2*11*29.&lt;/body&gt;&lt;/html&gt;</pre></div>
=== Subsets and supersets ===
Since 638 factors as {{factorisation|638}}, 638edo has subset edos {{EDOs| 2, 11, 22, 29, 58, and 319 }}.
 
== Regular temperament properties ==
{| 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
| {{monzo| -1011 638 }}
| {{mapping| 638 1011 }}
| +0.1223
| 0.1223
| 6.50
|-
| 2.3.5
| {{monzo| -51 19 9 }}, {{monzo| -16 35 -17 }}
| {{mapping| 638 1011 1481 }}
| +0.1869
| 0.1353
| 7.19
|-
| 2.3.5.7
| 4375/4374, 2100875/2097152, {{monzo| -11 5 11 -8 }}
| {{mapping| 638 1011 1481 1791 }}
| +0.1556
| 0.1291
| 6.86
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 825000/823543, 1265625/1261568
| {{mapping| 638 1011 1481 1791 2207 }}
| +0.1373
| 0.1212
| 6.44
|-
| 2.3.5.7.11.13
| 625/624, 729/728, 1575/1573, 2200/2197, 823680/823543
| {{mapping| 638 1011 1481 1791 2207 2361 }}
| +0.1043
| 0.1330
| 7.07
|}
 
=== Rank-2 temperaments ===
{| 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
| 97\638
| 182.45
| 10/9
| [[Mitonic]]
|-
| 1
| 313\638
| 588.71
| 7/5
| [[Untriton]]
|-
| 2
| 94\638
| 176.80
| 448/405
| [[Quatracot]]
|}
<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
 
[[Category:Quatracot]]

Latest revision as of 13:32, 13 March 2026

← 637edo 638edo 639edo →
Prime factorization 2 × 11 × 29
Step size 1.88088 ¢ 
Fifth 373\638 (701.567 ¢)
Semitones (A1:m2) 59:49 (111 ¢ : 92.16 ¢)
Consistency limit 11
Distinct consistency limit 11

638 equal divisions of the octave (abbreviated 638edo or 638ed2), also called 638-tone equal temperament (638tet) or 638 equal temperament (638et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 638 equal parts of about 1.88 ¢ each. Each step represents a frequency ratio of 21/638, or the 638th root of 2.

Theory

The equal temperament tempers out the minortone comma, [-16 35 -17⟩, in the 5-limit, 4375/4374 in the 7-limit, 3025/3024, 9801/9800, and 43923/43904, in the 11-limit; and 625/624, 729/728, 1575/1573, 2200/2197 and 4225/4224 in the 13-limit. It supplies the optimal patent val for quatracot, the 224 & 414 temperament.

Odd harmonics

Approximation of odd harmonics in 638edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -0.388 -0.734 -0.174 -0.775 -0.221 +0.225 +0.759 +0.374 -0.334 -0.561 -0.061
Relative (%) -20.6 -39.0 -9.2 -41.2 -11.7 +11.9 +40.4 +19.9 -17.8 -29.9 -3.3
Steps
(reduced) 		1011
(373) 		1481
(205) 		1791
(515) 		2022
(108) 		2207
(293) 		2361
(447) 		2493
(579) 		2608
(56) 		2710
(158) 		2802
(250) 		2886
(334)

Subsets and supersets

Since 638 factors as 2 × 11 × 29, 638edo has subset edos 2, 11, 22, 29, 58, and 319.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-1011 638⟩ [⟨638 1011]] +0.1223 0.1223 6.50
2.3.5 [-51 19 9⟩, [-16 35 -17⟩ [⟨638 1011 1481]] +0.1869 0.1353 7.19
2.3.5.7 4375/4374, 2100875/2097152, [-11 5 11 -8⟩ [⟨638 1011 1481 1791]] +0.1556 0.1291 6.86
2.3.5.7.11 3025/3024, 4375/4374, 825000/823543, 1265625/1261568 [⟨638 1011 1481 1791 2207]] +0.1373 0.1212 6.44
2.3.5.7.11.13 625/624, 729/728, 1575/1573, 2200/2197, 823680/823543 [⟨638 1011 1481 1791 2207 2361]] +0.1043 0.1330 7.07

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 97\638 182.45 10/9 Mitonic
1 313\638 588.71 7/5 Untriton
2 94\638 176.80 448/405 Quatracot

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

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