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764edo: Difference between revisions

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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>2011-07-13 19:06:48 UTC</tt>.<br>
 
: The original revision id was <tt>241243741</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
764edo is a very strong 17-limit system, [[consistent]] to the [[17-odd-limit]] or the no-19 no-29 [[41-odd-limit]]. It is the fourteenth [[zeta integral edo]]. In the 5-limit it [[tempering out|tempers out]] the hemithirds comma, {{monzo| 38 -2 -15 }}; in the 7-limit [[4375/4374]]; in the 11-limit [[3025/3024]] and [[9801/9800]]; in the 13-limit [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]], [[6656/6655]] and [[10648/10647]]; and in the 17-limit [[2431/2430]], [[2500/2499]], [[4914/4913]] and [[5832/5831]]. It provides the [[optimal patent val]] for the [[abigail]] temperament in the 11-limit.
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>
In higher limits, it is a strong no-19 and no-29 37-limit tuning, and an exceptional 2.11.23.31.37 subgroup system, with errors less than 2%.
<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 //764 equal division// divides the octave into 764 equal parts of 1.571 cents each. It a a strong 17-limit system and is distinctly consistent to the 17-limit. In the 5-limit it tempers out the hemithirds comma, |38 -2 -15&gt;; in the 7-limit 4375/4374; in the 11-limit 3025/3024 and 9801/9800; in the 13-limit 1716/1715, 2080/2079, 4096/4095, 4225/4224, 6656/6655 and 10648/10647; and in the 17-limit 2431/2430, 2500/2499, 4914/4913 and 5832/5831. It provides the [[optimal patent val]] for [[Ragismic microtemperaments#Abigail|abigail temperament]] in the 11-limit.</pre></div>
 
<h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;764edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;764 equal division&lt;/em&gt; divides the octave into 764 equal parts of 1.571 cents each. It a a strong 17-limit system and is distinctly consistent to the 17-limit. In the 5-limit it tempers out the hemithirds comma, |38 -2 -15&amp;gt;; in the 7-limit  4375/4374; in the 11-limit 3025/3024 and 9801/9800; in the 13-limit 1716/1715, 2080/2079, 4096/4095, 4225/4224, 6656/6655 and 10648/10647; and in the 17-limit 2431/2430, 2500/2499, 4914/4913 and 5832/5831. It provides the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt; for &lt;a class="wiki_link" href="/Ragismic%20microtemperaments#Abigail"&gt;abigail temperament&lt;/a&gt; in the 11-limit.&lt;/body&gt;&lt;/html&gt;</pre></div>
{{Harmonics in equal|764|columns=15}}
 
=== Subsets and supersets ===
Since 764 factors into {{factorization|764}}, 764edo has subset edos 2, 4, 191, and 382. In addition, one step of 764edo is exactly 22 [[jinn]]s ([[16808edo|22\16808]]).
 
== 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| 1211 -764 }}
| {{mapping| 764 1211 }}
| −0.0439
| 0.0439
| 2.80
|-
| 2.3.5
| {{monzo| 38 -2 -15 }}, {{monzo| 25 -48 22 }}
| {{mapping| 764 1211 1774 }}
| −0.0399
| 0.0363
| 2.31
|-
| 2.3.5.7
| 4375/4374, 52734375/52706752, {{monzo| 31 -6 -2 -6 }}
| {{mapping| 764 1211 1774 2145 }}
| −0.0552
| 0.0412
| 2.62
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 131072/130977, 35156250/35153041
| {{mapping| 764 1211 1774 2145 2643 }}
| −0.0436
| 0.0435
| 2.77
|-
| 2.3.5.7.11.13
| 1716/1715, 2080/2079, 3025/3024, 4096/4095, 10549994/10546875
| {{mapping| 764 1211 1774 2145 2643 2827 }}
| −0.0267
| 0.0548
| 3.49
|-
| 2.3.5.7.11.13.17
| 1716/1715, 2080/2079, 2431/2430, 2500/2499, 4096/4095, 4914/4913
| {{mapping| 764 1211 1774 2145 2643 2827 3123 }}
| −0.0327
| 0.0528
| 3.36
|-
| 2.3.5.7.11.13.17.23
| 1716/1715, 2080/2079, 2024/2023, 2431/2430, 2500/2499, 3520/3519, 4096/4095
| {{mapping| 764 1211 1774 2145 2643 2827 3123 3456 }}
| −0.0286
| 0.0506
| 3.22
|}
* 764et has lower absolute errors than any previous equal temperaments in the 13- and 17-limit. In the 13-limit it beats [[684edo|684]] and is only bettered by [[935edo|935]]. In the 17-limit it beats [[742edo|742]] and is only bettered by [[814edo|814]].  
* It is best at the no-19 23-limit, where it has a lower relative error than any previous equal temperaments, past [[494edo|494]] and before [[1578edo|1578]].
 
=== 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
| 123\764
| 193.19
| 262144/234375
| [[Lunatic]] (7-limit)
|-
| 1
| 277\764
| 435.08
| 9/7
| [[Supermajor]]
|-
| 2
| 133\764
| 208.90
| 44/39
| [[Abigail]]
|-
| 2
| 277\764<br />(105\764)
| 435.08<br />(164.92)
| 9/7<br />(11/10)
| [[Semisupermajor]]
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
[[Category:Abigail]]
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