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Octagar family: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Technical data page}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
The '''octagar family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the octagar comma, [[4000/3969]].  
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-10-25 00:17:34 UTC</tt>.<br>
: The original revision id was <tt>173261849</tt>.<br>
: The revision comment was: <tt></tt><br>
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>
<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">Octagar temperament is the 7-limit [[planar temperament]] tempering out 4000/3969. While most 7-limit planar temperaments exhibit a tendency towards tuning flat, many people seem to prefer a slight sharp tendency instead. Octagar provides this; for instance the 7 odd limit minimax tuning has fifths and 7s 2.245 cents sharp, with just major thirds.


===Vital statistics===
For [[rank-2 temperament]]s that temper out the octagar comma, see [[Octagar temperaments]].
[[Comma]] c = 4000/3969


7-limit minimax: 3 and 7 1/6c sharp, 5 just
== Octagar ==
[|1 0 0 0&gt;, |5/6 1/3 1/2 -1/3&gt;,  
Octagar is generated by a [[3/2|perfect fifth]] and a minor sixth of [[~]][[63/40]], two of which make ~[[5/2]], and three make an interval short of two octaves by a [[64/63|septimal comma]].
|0 0 1 0&gt;, |5/6 -2/3 1/2 2/3&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 7/6, 5/4


9-limit minimax: 3 1/8c sharp, 5 just, 7 1/4c sharp
While many 7-limit rank-3 temperaments exhibit a tendency towards tuning flat, a slight sharp tendency is often preferred instead. Octagar provides this; for instance the [[7-odd-limit]] [[minimax tuning]] has [[3/1|3]]'s and [[7/1|7]]'s 2.245 cents sharp, with just [[5/1|5]]'s.
[|1 0 0 0&gt;, |5/8 1/2 3/8 -1/4&gt;,
|0 0 1 0&gt;, |5/4 -1 3/4 1/2&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 5/4, 9/7


Lattice basis: 63/50 length 0.8966, 21/20 length 1.0605
This temperament is catalogued as ''octagari'' in [[Graham Breed]]'s [https://x31eq.com/temper/ Temperament Finder].  
Angle(63/50, 21/20) = 97.743 cents
Map to lattice: [&lt;0 -1 -2 -1|, &lt;0 -1 0 2|]
[[edo|EDOs]]: [[162edo|162]], [[189edo|189]], [[215edo|215]], [[230edo|230]]


===[[Hobbits|Hobbit bases]]===
[[Subgroup]]: 2.3.5.7
{2, 3, 7/5} subgroup


12: 50/49, 256000/250047
[[Comma list]]: [[4000/3969]]
15: 256000/250047, 1029/1000
 
23: 12800000/12252303, 107163/102400</pre></div>
{{Mapping|legend=1| 1 0 1 4 | 0 1 0 -2 | 0 0 2 3 }}
<h4>Original HTML content:</h4>
: Mapping generators: ~2, ~3, ~63/40
<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;Octagar family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Octagar temperament is the 7-limit &lt;a class="wiki_link" href="/planar%20temperament"&gt;planar temperament&lt;/a&gt; tempering out 4000/3969. While most 7-limit planar temperaments exhibit a tendency towards tuning flat, many people seem to prefer a slight sharp tendency instead. Octagar provides this; for instance the 7 odd limit minimax tuning has fifths and 7s 2.245 cents sharp, with just major thirds.&lt;br /&gt;
 
&lt;br /&gt;
[[Mapping to lattice]]: [{{val| 0 -1 -2 -1 }}, {{val| 0 -1 0 2 }}]
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc0"&gt;&lt;a name="x--Vital statistics"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Vital statistics&lt;/h3&gt;
 
&lt;a class="wiki_link" href="/Comma"&gt;Comma&lt;/a&gt; c = 4000/3969&lt;br /&gt;
Lattice basis:  
&lt;br /&gt;
: 63/50 length = 0.8966, 21/20 length = 1.0605
7-limit minimax: 3 and 7 1/6c sharp, 5 just&lt;br /&gt;
: Angle (63/50, 21/20) = 97.743 degrees
[|1 0 0 0&amp;gt;, |5/6 1/3 1/2 -1/3&amp;gt;, &lt;br /&gt;
 
|0 0 1 0&amp;gt;, |5/6 -2/3 1/2 2/3&amp;gt;]&lt;br /&gt;
[[Optimal tuning]]s:
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 7/6, 5/4&lt;br /&gt;
* [[WE]]: ~2 = 1199.5353{{c}}, ~3/2 = 703.3499{{c}}, ~63/40 = 792.6380{{c}}
&lt;br /&gt;
: [[error map]]: {{val| -0.465 +0.930 -1.502 +1.459 }}
9-limit minimax: 3 1/8c sharp, 5 just, 7 1/4c sharp&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.3764{{c}}, ~63/40 = 792.6299{{c}}
[|1 0 0 0&amp;gt;, |5/8 1/2 3/8 -1/4&amp;gt;, &lt;br /&gt;
: error map: {{val| 0.000 +1.421 -1.054 +2.311 }}
|0 0 1 0&amp;gt;, |5/4 -1 3/4 1/2&amp;gt;]&lt;br /&gt;
 
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 5/4, 9/7&lt;br /&gt;
[[Minimax tuning]]:
&lt;br /&gt;
* [[7-odd-limit]]
Lattice basis: 63/50 length 0.8966, 21/20 length 1.0605&lt;br /&gt;
: {{monzo list| 1 0 0 0 | 5/6 1/3 1/2 -1/3 | 0 0 1 0 | 5/6 -2/3 1/2 2/3 }}
Angle(63/50, 21/20) = 97.743 cents&lt;br /&gt;
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7/3
Map to lattice: [&amp;lt;0 -1 -2 -1|, &amp;lt;0 -1 0 2|]&lt;br /&gt;
* [[9-odd-limit]]
&lt;a class="wiki_link" href="/edo"&gt;EDOs&lt;/a&gt;: &lt;a class="wiki_link" href="/162edo"&gt;162&lt;/a&gt;, &lt;a class="wiki_link" href="/189edo"&gt;189&lt;/a&gt;, &lt;a class="wiki_link" href="/215edo"&gt;215&lt;/a&gt;, &lt;a class="wiki_link" href="/230edo"&gt;230&lt;/a&gt;&lt;br /&gt;
: {{monzo list| 1 0 0 0 | 5/8 1/2 3/8 -1/4 | 0 0 1 0 | 5/4 -1 3/4 1/2 }}
&lt;br /&gt;
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.9/7
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc1"&gt;&lt;a name="x--Hobbit bases"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;&lt;a class="wiki_link" href="/Hobbits"&gt;Hobbit bases&lt;/a&gt;&lt;/h3&gt;
 
{2, 3, 7/5} subgroup&lt;br /&gt;
{{Optimal ET sequence|legend=1| 12, 26, 27, 39d, 41, 53, 80, 94, 121, 174d, 215, 295d, 336d }}
&lt;br /&gt;
 
12: 50/49, 256000/250047&lt;br /&gt;
[[Badness]] (Sintel): 0.951
15: 256000/250047, 1029/1000&lt;br /&gt;
 
23: 12800000/12252303, 107163/102400&lt;/body&gt;&lt;/html&gt;</pre></div>
[[Projection pair]]s: <code>5 3969/800 7 27783/4000</code> to 2.3.7/5
 
{{Databox|[[Hobbit|Hobbit bases]]|
2.3.7/5 subgroup
* 12: 50/49, 256000/250047
* 15: 256000/250047, 1029/1000
* 23: 12800000/12252303, 107163/102400
}}
 
== Nakika ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 100/99, 245/242
 
{{Mapping|legend=1| 1 0 1 4 4 | 0 1 0 -2 -2 | 0 0 2 3 4 }}
 
Mapping to lattice: [{{val| 0 1 2 1 2 }}, {{val| 0 -1 0 2 2 }}]
 
Lattice basis:
: 11/7 length = 0.798, 22/21 length = 0.906
: Angle (11/7, 22/21) = 97.747 degrees
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6137{{c}}, ~3/2 = 703.6571{{c}}, ~11/7 = 791.5207{{c}}
: [[error map]]: {{val| -0.386 +1.316 -3.659 -2.351 +6.678 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.6783{{c}}, ~11/7 = 791.5168{{c}}
: error map: {{val| 0.000 +1.723 -3.280 -1.632 +7.393 }}
 
{{Optimal ET sequence|legend=1| 12, 15, 26, 27e, 41, 109e }}
 
[[Badness]] (Sintel): 0.647
 
[[Projection pair]]s: <code>5 242/49 7 21296/3087 11 234256/21609</code> to 2.3.11/7
 
[[Associated temperament]]: [[octacot]]
 
Scales: [[nakika12]]     
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 100/99, 105/104, 245/242
 
Mapping: {{mapping| 1 0 1 4 4 2 | 0 1 0 -2 -2 -1 | 0 0 2 3 4 5 }}
 
Optimal tunings:
* WE: ~2 = 1200.5397{{c}}, ~3/2 = 702.2038{{c}}, ~11/7 = 789.6574{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0977{{c}}, ~11/7 = 789.5686{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 14cf, 15, 26, 29, 41 }}
 
Badness (Sintel): 0.832
 
== Octasand ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 540/539, 2200/2187
 
{{Mapping|legend=1| 1 0 1 4 -5 | 0 1 0 -2 7 | 0 0 2 3 -4 }}
 
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.5255{{c}}, ~3/2 = 703.2719{{c}}, ~63/40 = 792.6321{{c}}
: [[error map]]: {{val| -0.475 +0.842 -1.524 +1.578 +0.108 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.2662{{c}}, ~63/40 = 792.6212{{c}}
: error map: {{val| 0.000 +1.311 -1.071 +2.505 +1.061 }}
 
{{Optimal ET sequence|legend=1| 27e, 39d, 41, 53, 80, 94, 121, 174d, 215, 295d }}
 
[[Badness]] (Sintel): 1.20
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 352/351, 540/539
 
Mapping: {{mapping| 1 0 1 4 -5 0 | 0 1 0 -2 7 4 | 0 0 2 3 -4 -4 }}
 
Optimal tunings:
* WE: ~2 = 1199.5112{{c}}, ~3/2 = 703.2823{{c}}, ~63/40 = 792.6491{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.3684{{c}}, ~63/40 = 792.7875{{c}}
 
{{Optimal ET sequence|legend=0| 27e, 41, 53, 80, 94, 121, 174d, 215, 295d, 336def }}
 
Badness (Sintel): 1.03
 
[[Category:Temperament families]]
[[Category:Octagar family| ]] <!-- main article -->
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Octagar_family"