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643edo: Difference between revisions

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**Imported revision 241390639 - Original comment: **
 
m Text replacement - "[[Helmholtz temperament|" to "[[Helmholtz (temperament)|"
 
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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:genewardsmith|genewardsmith]] and made on <tt>2011-07-14 16:47:55 UTC</tt>.<br>
 
: The original revision id was <tt>241390639</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
643edo is [[consistency|distinctly consistent]] to the [[21-odd-limit]], with a generally flat tendency, but the [[5/1|5th harmonic]] is only 0.000439 cents sharp as the denominator of a convergent to log<sub>2</sub>5, after [[146edo|146]] and before [[4004edo|4004]]. As an equal temperament, it [[tempering out|tempers out]] [[32805/32768]] in the 5-limit and [[2401/2400]] in the 7-limit, so that it [[support]]s the [[sesquiquartififths]] temperament. In the 11-limit it tempers out [[3025/3024]] and 151263/151250; in the 13-limit [[1001/1000]], [[1716/1715]] and [[4225/4224]]; in the 17-limit [[1089/1088]], [[1701/1700]], [[2431/2430]] and [[2601/2600]]; and in the 19-limit 1331/1330, [[1521/1520]], [[1729/1728]], 2376/2375 and 2926/2925. It provides the [[optimal patent val]] for the rank-3 13-limit [[vili]] 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>
=== Prime 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 //643 equal division// divides the octave into 643 equal parts of 1.866 cents each. It is uniquely [[consistent]] to the 21-limit, with a generally flat tendency. It tempers out 32805/32768 in the 5-limit and 2401/2400 in the 7-limit, so that it supports [[Schismatic family#Sesquiquartififths|sesquiquartififths temperament]]. In the 11-limit it tempers out 3025/3024 and 151263/151250; in the 13-limit 1001/1000, 1716/1715 and 4225/4224; in the 17-limit 1089/1088, 1701/1700, 2431/2430 and 2601/2600; and in the 19-limit 1331/1330, 1521/1520, 1729/1728, 2376/2375 and 2926/2925. It provides the [[optimal patent val]] for the rank three 13-limit temperament [[Breed family#Vili|vili temperament]].</pre></div>
{{Harmonics in equal|643}}
<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;643edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;643 equal division&lt;/em&gt; divides the octave into 643 equal parts of 1.866 cents each. It is uniquely &lt;a class="wiki_link" href="/consistent"&gt;consistent&lt;/a&gt; to the 21-limit, with a generally flat tendency. It tempers out 32805/32768 in the 5-limit and 2401/2400 in the 7-limit, so that it supports &lt;a class="wiki_link" href="/Schismatic%20family#Sesquiquartififths"&gt;sesquiquartififths temperament&lt;/a&gt;. In the 11-limit it tempers out 3025/3024 and 151263/151250; in the 13-limit 1001/1000, 1716/1715 and 4225/4224; in the 17-limit 1089/1088, 1701/1700, 2431/2430 and 2601/2600; and in the 19-limit 1331/1330, 1521/1520, 1729/1728, 2376/2375 and 2926/2925. It provides the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt; for the rank three 13-limit temperament &lt;a class="wiki_link" href="/Breed%20family#Vili"&gt;vili temperament&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>
=== Subsets and supersets ===
643edo is the 117th [[prime edo]].
 
== 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| -1019 643 }}
| {{mapping| 643 1019 }}
| +0.0771
| 0.0771
| 4.13
|-
| 2.3.5
| 32805/32768, {{monzo| 1 99 -68 }}
| {{mapping| 643 1019 1493 }}
| +0.0513
| 0.7270
| 3.90
|-
| 2.3.5.7
| 2401/2400, 32805/32768, {{monzo| 9 21 -17 -1 }}
| {{mapping| 643 1019 1493 1805 }}
| +0.0600
| 0.0647
| 3.47
|-
| 2.3.5.7.11
| 2401/2400, 3025/3024, 32805/32768, 391314/390625
| {{mapping| 643 1019 1493 1805 2224 }}
| +0.0927
| 0.0874
| 4.68
|-
| 2.3.5.7.11.13
| 1001/1000, 1716/1715, 3025/3024, 4225/4224, 32805/32768
| {{mapping| 643 1019 1493 1805 2224 2379 }}
| +0.1094
| 0.0881
| 4.72
|-
| 2.3.5.7.11.13.17
| 1001/1000, 1089/1088, 1701/1700, 1716/1715, 2601/2600, 4225/4224
|{{mapping| 643 1019 1493 1805 2224 2379 2628 }}
| +0.1094
| 0.0816
| 4.37
|-
| 2.3.5.7.11.13.17.19
| 1001/1000, 1089/1088, 1521/1520, 1701/1700, 1716/1715, 1729/1728, 2601/2600
| {{mapping| 643 1019 1493 1805 2224 2379 2628 2731 }}
| +0.1186
| 0.0801
| 4.29
|}
 
=== 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
| 94\643
| 175.43
| 448/405
| [[Sesquiquartififths]]
|-
| 1
| 267\643
| 498.29
| 4/3
| [[Helmholtz (temperament)|Helmholtz]]
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
== Music ==
; [[Francium]]
* "Bobson Dugnutt" from ''Don't Give Your Kids These Names!'' (2025) − [https://open.spotify.com/track/1ROUQlzxJR7pDpM8GLujol Spotify] | [https://francium223.bandcamp.com/track/bobson-dugnutt Bandcamp] | [https://www.youtube.com/watch?v=Bg2w1__AW4k YouTube] − in Botolphic, 643edo tuning
 
[[Category:Sesquiquartififths]]
[[Category:Vili]]
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