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>2010-09-04 14:19:17 UTC</tt>.<br>
: The original revision id was <tt>160395237</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 ==
9-limit minimax: 3 1/c sharp, 5 just, 7 1/4c sharp
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]].
Lattice basis: 63/50 length 0.8966, 21/20 length 1.0605
 
Angle(63/50, 21/20) = 97.743 cents
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.
Map to lattice: [&lt;0 -1 -2 -1|, &lt;0 -1 0 2|]
 
EDOs: 162, 189, 215, 230</pre></div>
This temperament is catalogued as ''octagari'' in [[Graham Breed]]'s [https://x31eq.com/temper/ Temperament Finder].
<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 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.&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
&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;
[[Comma list]]: [[4000/3969]]
Comma c = 4000/3969&lt;br /&gt;
 
7-limit minimax: 3 and 7 1/6c sharp, 5 just&lt;br /&gt;
{{Mapping|legend=1| 1 0 1 4 | 0 1 0 -2 | 0 0 2 3 }}
9-limit minimax: 3 1/c sharp, 5 just, 7 1/4c sharp&lt;br /&gt;
: Mapping generators: ~2, ~3, ~63/40
Lattice basis: 63/50 length 0.8966, 21/20 length 1.0605&lt;br /&gt;
 
Angle(63/50, 21/20) = 97.743 cents&lt;br /&gt;
[[Mapping to lattice]]: [{{val| 0 -1 -2 -1 }}, {{val| 0 -1 0 2 }}]
Map to lattice: [&amp;lt;0 -1 -2 -1|, &amp;lt;0 -1 0 2|]&lt;br /&gt;
 
EDOs: 162, 189, 215, 230&lt;/body&gt;&lt;/html&gt;</pre></div>
Lattice basis:
: 63/50 length = 0.8966, 21/20 length = 1.0605
: Angle (63/50, 21/20) = 97.743 degrees
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5353{{c}}, ~3/2 = 703.3499{{c}}, ~63/40 = 792.6380{{c}}
: [[error map]]: {{val| -0.465 +0.930 -1.502 +1.459 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.3764{{c}}, ~63/40 = 792.6299{{c}}
: error map: {{val| 0.000 +1.421 -1.054 +2.311 }}
 
[[Minimax tuning]]:
* [[7-odd-limit]]
: {{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 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7/3
* [[9-odd-limit]]
: {{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 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.9/7
 
{{Optimal ET sequence|legend=1| 12, 26, 27, 39d, 41, 53, 80, 94, 121, 174d, 215, 295d, 336d }}
 
[[Badness]] (Sintel): 0.951
 
[[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]]

Latest revision as of 16:28, 13 March 2026

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 rank-3 temperaments tempers out the octagar comma, 4000/3969.

For rank-2 temperaments that temper out the octagar comma, see Octagar temperaments.

Octagar

Octagar is generated by a 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 septimal comma.

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's and 7's 2.245 cents sharp, with just 5's.

This temperament is catalogued as octagari in Graham Breed's Temperament Finder.

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 tunings:

  • WE: ~2 = 1199.5353 ¢, ~3/2 = 703.3499 ¢, ~63/40 = 792.6380 ¢
error map: ⟨-0.465 +0.930 -1.502 +1.459]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.3764 ¢, ~63/40 = 792.6299 ¢
error map: ⟨0.000 +1.421 -1.054 +2.311]

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 (Sintel): 0.951

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 tunings:

  • WE: ~2 = 1199.6137 ¢, ~3/2 = 703.6571 ¢, ~11/7 = 791.5207 ¢
error map: ⟨-0.386 +1.316 -3.659 -2.351 +6.678]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.6783 ¢, ~11/7 = 791.5168 ¢
error map: ⟨0.000 +1.723 -3.280 -1.632 +7.393]

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

Badness (Sintel): 0.647

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 tunings:

  • WE: ~2 = 1200.5397 ¢, ~3/2 = 702.2038 ¢, ~11/7 = 789.6574 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0977 ¢, ~11/7 = 789.5686 ¢

Optimal ET sequence: 12f, 14cf, 15, 26, 29, 41

Badness (Sintel): 0.832

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 tunings:

  • WE: ~2 = 1199.5255 ¢, ~3/2 = 703.2719 ¢, ~63/40 = 792.6321 ¢
error map: ⟨-0.475 +0.842 -1.524 +1.578 +0.108]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.2662 ¢, ~63/40 = 792.6212 ¢
error map: ⟨0.000 +1.311 -1.071 +2.505 +1.061]

Optimal ET sequence: 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: [⟨1 0 1 4 -5 0], ⟨0 1 0 -2 7 4], ⟨0 0 2 3 -4 -4]]

Optimal tunings:

  • WE: ~2 = 1199.5112 ¢, ~3/2 = 703.2823 ¢, ~63/40 = 792.6491 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.3684 ¢, ~63/40 = 792.7875 ¢

Optimal ET sequence: 27e, 41, 53, 80, 94, 121, 174d, 215, 295d, 336def

Badness (Sintel): 1.03

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