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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 [[temperament]]s are [[rank-3 temperament]]s [[tempering out]] the octagar comma, [[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 7's 2.245 cents sharp, with just major thirds.
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-01-13 19:42:42 UTC</tt>.<br>
: The original revision id was <tt>193167794</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===
[[Linear temperament]]s that temper out the octagar comma can be found in [[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;,
This temperament is also known as ''octagari''.
|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
[[Subgroup]]: 2.3.5.7
[|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
[[Comma list]]: [[4000/3969]]
Angle(63/50, 21/20) = 97.743 cents
Map to lattice: [&lt;0 -1 -2 -1|, &lt;0 -1 0 2|]
[[edo|EDOs]]: [[12edo|12]], [[26edo|26]], [[27edo|27]], [[41edo|41]], [[53edo|53]], [[94edo|94]], [[121edo|121]], [[162edo|162]], [[189edo|189]], [[215edo|215]], [[230edo|230]]
Badness: 0.000216


===[[Hobbits|Hobbit bases]]===
{{Mapping|legend=1| 1 0 1 4 | 0 1 0 -2 | 0 0 2 3 }}
{2, 3, 7/5} subgroup


12: 50/49, 256000/250047
: Mapping generators: ~2, ~3, ~63/40
15: 256000/250047, 1029/1000
23: 12800000/12252303, 107163/102400


==Nakika==
[[Mapping to lattice]]: [{{val| 0 -1 -2 -1 }}, {{val| 0 -1 0 2 }}]
Commas: 100/99, 245/242
Associated linear temperament: [[Tetracot family|octacot]]
Lattice basis: 11/7 0.798 22/21 0.906
Angle(11/7, 22/21) = 97.747 degrees
Map to lattice: [&lt;0 1 2 1 2|, &lt;0 -1 0 2 2|]


Map: [&lt;1 0 1 4 4|, &lt;0 1 0 -2 -2|, &lt;0 0 2 3 4|]
Lattice basis:  
EDOs: 12, 15, 26, 29, 41
: 63/50 length = 0.8966, 21/20 length = 1.0605
Badness: 0.000539
: Angle (63/50, 21/20) = 97.743 degrees


Scales: [[nakika12]]</pre></div>
[[Optimal tuning]] (POTE): ~3/2 = 703.6224, ~21/20 = 89.3227
<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;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;
[[Minimax tuning]]:
&lt;br /&gt;
* [[7-odd-limit]]
&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;
: {{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 }}
&lt;a class="wiki_link" href="/Comma"&gt;Comma&lt;/a&gt; c = 4000/3969&lt;br /&gt;
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7/3
&lt;br /&gt;
* [[9-odd-limit]]
7-limit minimax: 3 and 7 1/6c sharp, 5 just&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 }}
[|1 0 0 0&amp;gt;, |5/6 1/3 1/2 -1/3&amp;gt;, &lt;br /&gt;
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.9/7
|0 0 1 0&amp;gt;, |5/6 -2/3 1/2 2/3&amp;gt;]&lt;br /&gt;
 
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 7/6, 5/4&lt;br /&gt;
{{Optimal ET sequence|legend=1| 12, 26, 27, 39d, 41, 53, 80, 94, 121, 174d, 215, 295d, 336d }}
&lt;br /&gt;
 
9-limit minimax: 3 1/8c sharp, 5 just, 7 1/4c sharp&lt;br /&gt;
[[Badness]]: 0.216 × 10<sup>-3</sup>
[|1 0 0 0&amp;gt;, |5/8 1/2 3/8 -1/4&amp;gt;, &lt;br /&gt;
 
|0 0 1 0&amp;gt;, |5/4 -1 3/4 1/2&amp;gt;]&lt;br /&gt;
[[Projection pair]]s: 5 - 3969/800, 7 - 27783/4000 to 2.3.7/5
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 5/4, 9/7&lt;br /&gt;
 
&lt;br /&gt;
{{Databox|[[Hobbit|Hobbit bases]]|
Lattice basis: 63/50 length 0.8966, 21/20 length 1.0605&lt;br /&gt;
2.3.7/5 subgroup
Angle(63/50, 21/20) = 97.743 cents&lt;br /&gt;
* 12: 50/49, 256000/250047
Map to lattice: [&amp;lt;0 -1 -2 -1|, &amp;lt;0 -1 0 2|]&lt;br /&gt;
* 15: 256000/250047, 1029/1000
&lt;a class="wiki_link" href="/edo"&gt;EDOs&lt;/a&gt;: &lt;a class="wiki_link" href="/12edo"&gt;12&lt;/a&gt;, &lt;a class="wiki_link" href="/26edo"&gt;26&lt;/a&gt;, &lt;a class="wiki_link" href="/27edo"&gt;27&lt;/a&gt;, &lt;a class="wiki_link" href="/41edo"&gt;41&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53&lt;/a&gt;, &lt;a class="wiki_link" href="/94edo"&gt;94&lt;/a&gt;, &lt;a class="wiki_link" href="/121edo"&gt;121&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;
* 23: 12800000/12252303, 107163/102400
Badness: 0.000216&lt;br /&gt;
}}
&lt;br /&gt;
 
&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;
== Nakika ==
{2, 3, 7/5} subgroup&lt;br /&gt;
[[Subgroup]]: 2.3.5.7.11
&lt;br /&gt;
 
12: 50/49, 256000/250047&lt;br /&gt;
[[Comma list]]: 100/99, 245/242
15: 256000/250047, 1029/1000&lt;br /&gt;
 
23: 12800000/12252303, 107163/102400&lt;br /&gt;
{{Mapping|legend=1| 1 0 1 4 4 | 0 1 0 -2 -2 | 0 0 2 3 4 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x-Nakika"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Nakika&lt;/h2&gt;
Mapping to lattice: [{{val| 0 1 2 1 2 }}, {{val| 0 -1 0 2 2 }}]
Commas: 100/99, 245/242&lt;br /&gt;
 
Associated linear temperament: &lt;a class="wiki_link" href="/Tetracot%20family"&gt;octacot&lt;/a&gt;&lt;br /&gt;
Lattice basis:
Lattice basis: 11/7 0.798 22/21 0.906&lt;br /&gt;
: 11/7 length = 0.798, 22/21 length = 0.906
Angle(11/7, 22/21) = 97.747 degrees&lt;br /&gt;
: Angle (11/7, 22/21) = 97.747 degrees
Map to lattice: [&amp;lt;0 1 2 1 2|, &amp;lt;0 -1 0 2 2|]&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]] (POTE): ~3/2 = 703.8837, ~21/20 = 87.8919
Map: [&amp;lt;1 0 1 4 4|, &amp;lt;0 1 0 -2 -2|, &amp;lt;0 0 2 3 4|]&lt;br /&gt;
 
EDOs: 12, 15, 26, 29, 41&lt;br /&gt;
{{Optimal ET sequence|legend=1| 12, 15, 26, 27e, 41, 109e }}
Badness: 0.000539&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]]: 0.539 × 10<sup>-3</sup>
Scales: &lt;a class="wiki_link" href="/nakika12"&gt;nakika12&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
 
[[Projection pair]]s: 5 - 242/49, 7 - 21296/3087, 11 - 234256/21609 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 tuning (POTE): ~3/2 = 701.8881, ~21/20 = 87.4143
 
Optimal ET sequence: {{Optimal ET sequence| 12f, 14cf, 15, 26, 29, 41 }}
 
== 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]] (POTE): ~3/2 = 703.5501, ~21/20 = 89.3956
 
{{Optimal ET sequence|legend=1| 12e, 14c, 27e, 41, 53, 80, 94, 121 }}
 
=== 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 tuning (POTE): ~3/2 = 703.5688, ~21/20 = 89.4032
 
Optimal ET sequence: {{Optimal ET sequence| 12e, 14c, 27e, 41, 53, 80, 94, 121 }}
 
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Octagar family| ]] <!-- main article -->
[[Category:Octagar| ]] <!-- key article -->
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Octagar_family"