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]]

Latest revision as of 00:31, 24 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The octagar family of temperaments are rank-3 temperaments 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.

Linear temperaments that temper out the octagar comma can be found in octagar temperaments.

Octagar

This temperament is also known as octagari.

Subgroup: 2.3.5.7

Comma list: 4000/3969

Mapping: [⟨1 0 1 4], ⟨0 1 0 -2], ⟨0 0 2 3]]

Mapping generators: ~2, ~3, ~63/40

Mapping to lattice: [⟨0 -1 -2 -1], ⟨0 -1 0 2]]

Lattice basis:

63/50 length = 0.8966, 21/20 length = 1.0605
Angle (63/50, 21/20) = 97.743 degrees

Optimal tuning (POTE): ~3/2 = 703.6224, ~21/20 = 89.3227

Minimax tuning:

[[1 0 0 0⟩, [5/6 1/3 1/2 -1/3⟩, [0 0 1 0⟩, [5/6 -2/3 1/2 2/3⟩]
unchanged-interval (eigenmonzo) basis: 2.5.7/3
[[1 0 0 0⟩, [5/8 1/2 3/8 -1/4⟩, [0 0 1 0⟩, [5/4 -1 3/4 1/2⟩]
unchanged-interval (eigenmonzo) basis: 2.5.9/7

Optimal ET sequence: 12, 26, 27, 39d, 41, 53, 80, 94, 121, 174d, 215, 295d, 336d

Badness: 0.216 × 10-3

Projection pairs: 5 - 3969/800, 7 - 27783/4000 to 2.3.7/5

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: [⟨1 0 1 4 4], ⟨0 1 0 -2 -2], ⟨0 0 2 3 4]]

Mapping to lattice: [⟨0 1 2 1 2], ⟨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 (POTE): ~3/2 = 703.8837, ~21/20 = 87.8919

Optimal ET sequence: 12, 15, 26, 27e, 41, 109e

Badness: 0.539 × 10-3

Projection pairs: 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: [⟨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: 12f, 14cf, 15, 26, 29, 41

Octasand

Subgroup: 2.3.5.7.11

Comma list: 540/539, 2200/2187

Mapping: [⟨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: 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: [⟨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: 12e, 14c, 27e, 41, 53, 80, 94, 121

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