Diaschismic rank three 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 '''diaschismic rank-3 family''' of [[temperament]]s [[tempering out|tempers out]] the diaschisma, [[2048/2025]]. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-12-12 14:40:06 UTC</tt>.<br>
: The original revision id was <tt>187507343</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">These are the rank three temperaments tempering out the diaschisma, 2048/2025. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.


==Julius==
== Rank-3 diaschismic ==
Commas: 176/175, 896/891
[[Subgroup]]: 2.3.5.7


Minimax tuning
[[Comma list]]: [[2048/2025]]
[|1 0 0 0 0&gt;, |11/8 1/2 -1/4 0 0&gt;, |11/4 -1 1/2 0 0&gt;,
|-1/8 3/2 -5/4 0 1&gt;, |11/8 -1/2 -1/4 0 1&gt;]
Eigenmonzos: 2, 12/11, 10/9


Lattice basis: 3/2, 9/7
{{Mapping|legend=1| 2 0 11 0 | 0 1 -2 0 | 0 0 0 1 }}
Map to lattice: [&lt;0 1 -2 2 -2|, &lt;0 0 0 -1 -1|]
Angle(3/2, 9/7) = 82.289 degrees


Map: [&lt;2 0 11 0 14|, &lt;0 1 -2 0 -4|, &lt;0 0 0 1 1 |]
: mapping generators: ~45/32, ~3, ~7
EDOs: 58, 68, 80, 126


Julius[22] [[Hobbits|hobbit]] [[transversal]]:
[[Optimal tuning]] ([[POTE]]): ~3/2 = 704.8982, ~7/4 = 972.1330
28/27, 16/15, 11/10, 9/8, 32/27, 6/5, 5/4, 9/7, 4/3, 15/11, 45/32, 22/15, 3/2, 14/9, 8/5, 5/3, 27/16, 16/9, 20/11, 15/8, 27/14, 2


==Augustus==
{{Optimal ET sequence|legend=1| 10, 12, 22, 34d, 46, 58, 68, 80, 126, 206cd, 332bccd }}
Commas: 385/384, 1232/1215


Minimax tuning
[[Badness]]: 0.442 × 10<sup>-3</sup>
[|1 0 0 0 0&gt;, |11/8 1/2 -1/4 0 0&gt;, |11/4 -1 1/2 0 0&gt;,
|45/16 3/4 -3/8 1/2 -1/2&gt;, |45/16 3/4 -3/8 -1/2 1/2&gt;]
Eigenmonzos: 2, 10/9, 14/11


Map: [&lt;2 0 11 0 3|, &lt;0 1 -2 0 3|, &lt;0 0 0 1 -1|]
== Julius aka varda ==
EDOs: 46, 68, 114, 124
[[Subgroup]]: 2.3.5.7.11


==Tiberius==
[[Comma list]]: 176/175, 896/891
Commas: 2048/2025, 441/440


Minimax tuning
{{Mapping|legend=1| 2 0 11 0 14 | 0 1 -2 0 -4 | 0 0 0 1 1 }}
[|1 0 0 0 0&gt;, |11/8 1/2 -1/4 0 0&gt;, |11/4 -1 1/2 0 0&gt;,
|0 0 0 1 0&gt;, |-3 2 -1 2 0&gt;]
Eigenmonzos: 2, 9/5, 7


Map: [&lt;2 0 11 0 -17|, &lt;0 1 -2 0 4|, &lt;0 0 0 1 2|]
[[Mapping to lattice]]: [{{val| 0 1 -2 2 -2 }}, {{val| 0 0 0 -1 -1 }}]
EDOs: 58, 148, 160


==Claudius==
Lattice basis:
Commas: 2048/2025, 540/539
: 3/2, 9/7
: Angle (3/2, 9/7) = 82.289 degrees


Minimax tuning
[[Minimax tuning]]:
[|1 0 0 0 0&gt;, |9/7 0 -3/7 2/7 1/7&gt;, |41/14 0 6/7 -4/7 -2/7&gt;,
* [[11-odd-limit]]
|41/14 0 -1/7 3/7 -2/7&gt;, |41/14 0 -1/7 -4/7 5/7&gt;]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 11/8 1/2 -1/4 0 0 }}, {{monzo| 11/4 -1 1/2 0 0 }}, {{monzo| -1/8 3/2 -5/4 0 1 }}, {{monzo| 11/8 -1/2 -1/4 0 1 }}]
Eigenmonzos: 2, 7/5, 11/5
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/3


Map: [&lt;2 0 11 0 15|, &lt;0 1 -2 0 1|, &lt;0 0 0 1 -2|]
{{Optimal ET sequence|legend=1| 12, 22, 34d, 46, 58, 68, 80, 126, 184c, 206cd, 264bccde }}
EDOs: 58, 80</pre></div>
 
<h4>Original HTML content:</h4>
[[Badness]]: 0.728 × 10<sup>-3</sup>
<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;Diaschismic rank three family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;These are the rank three temperaments tempering out the diaschisma, 2048/2025. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.&lt;br /&gt;
 
&lt;br /&gt;
Julius[22] [[hobbit]] [[transversal]]:
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Julius"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Julius&lt;/h2&gt;
: 28/27, 16/15, 11/10, 9/8, 32/27, 6/5, 5/4, 9/7, 4/3, 15/11, 45/32, 22/15, 3/2, 14/9, 8/5, 5/3, 27/16, 16/9, 20/11, 15/8, 27/14, 2
Commas: 176/175, 896/891&lt;br /&gt;
 
&lt;br /&gt;
Julius[24] hobbit transversal
Minimax tuning&lt;br /&gt;
: 28/27, 16/15, 11/10, 9/8, 7/6, 6/5, 40/33, 5/4, 9/7, 4/3, 15/11, 45/32, 16/11, 3/2, 14/9, 8/5, 33/20, 27/16, 7/4, 16/9, 20/11, 15/8, 27/14, 2
[|1 0 0 0 0&amp;gt;, |11/8 1/2 -1/4 0 0&amp;gt;, |11/4 -1 1/2 0 0&amp;gt;, &lt;br /&gt;
 
|-1/8 3/2 -5/4 0 1&amp;gt;, |11/8 -1/2 -1/4 0 1&amp;gt;]&lt;br /&gt;
=== Varda ===
Eigenmonzos: 2, 12/11, 10/9&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
&lt;br /&gt;
 
Lattice basis: 3/2, 9/7&lt;br /&gt;
Comma list: 176/175, 351/350, 364/363
Map to lattice: [&amp;lt;0 1 -2 2 -2|, &amp;lt;0 0 0 -1 -1|]&lt;br /&gt;
 
Angle(3/2, 9/7) = 82.289 degrees&lt;br /&gt;
Mapping: {{mapping| 2 0 11 0 14 24 | 0 1 -2 0 -4 -7 | 0 0 0 1 1 1 }}
&lt;br /&gt;
 
Map: [&amp;lt;2 0 11 0 14|, &amp;lt;0 1 -2 0 -4|, &amp;lt;0 0 0 1 1 |]&lt;br /&gt;
{{Optimal ET sequence|legend=1| 12f, 22, 34d, 46, 58, 80, 138cde, 242cde, 322bcde }}
EDOs: 58, 68, 80, 126&lt;br /&gt;
 
&lt;br /&gt;
Badness: 0.970 × 10<sup>-3</sup>
Julius[22] &lt;a class="wiki_link" href="/Hobbits"&gt;hobbit&lt;/a&gt; &lt;a class="wiki_link" href="/transversal"&gt;transversal&lt;/a&gt;:&lt;br /&gt;
 
28/27, 16/15, 11/10, 9/8, 32/27, 6/5, 5/4, 9/7, 4/3, 15/11, 45/32, 22/15, 3/2, 14/9, 8/5, 5/3, 27/16, 16/9, 20/11, 15/8, 27/14, 2&lt;br /&gt;
===== 17-limit =====
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-Augustus"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Augustus&lt;/h2&gt;
 
Commas: 385/384, 1232/1215&lt;br /&gt;
Comma list: 136/135, 176/175, 256/255, 351/350
&lt;br /&gt;
 
Minimax tuning&lt;br /&gt;
Mapping: {{mapping| 2 0 11 0 14 24 5 | 0 1 -2 0 -4 -7 1 | 0 0 0 1 1 1 0 }}
[|1 0 0 0 0&amp;gt;, |11/8 1/2 -1/4 0 0&amp;gt;, |11/4 -1 1/2 0 0&amp;gt;, &lt;br /&gt;
 
|45/16 3/4 -3/8 1/2 -1/2&amp;gt;, |45/16 3/4 -3/8 -1/2 1/2&amp;gt;]&lt;br /&gt;
{{Optimal ET sequence|legend=1| 12f, 22, 34d, 46, 58, 80, 126, 138cde }}
Eigenmonzos: 2, 10/9, 14/11&lt;br /&gt;
 
&lt;br /&gt;
=== Julius ===
Map: [&amp;lt;2 0 11 0 3|, &amp;lt;0 1 -2 0 3|, &amp;lt;0 0 0 1 -1|]&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
EDOs: 46, 68, 114, 124&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 176/175, 325/324, 896/891
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x-Tiberius"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Tiberius&lt;/h2&gt;
 
Commas: 2048/2025, 441/440&lt;br /&gt;
Mapping: {{mapping| 2 0 11 0 14 -18 | 0 1 -2 0 -4 8 | 0 0 0 1 1 0 }}
&lt;br /&gt;
 
Minimax tuning&lt;br /&gt;
{{Optimal ET sequence|legend=1| 12, 22f, 34d, 46, 68, 80, 148d, 228bcd }}
[|1 0 0 0 0&amp;gt;, |11/8 1/2 -1/4 0 0&amp;gt;, |11/4 -1 1/2 0 0&amp;gt;, &lt;br /&gt;
 
|0 0 0 1 0&amp;gt;, |-3 2 -1 2 0&amp;gt;]&lt;br /&gt;
Badness: 1.78 × 10<sup>-3</sup>
Eigenmonzos: 2, 9/5, 7&lt;br /&gt;
 
&lt;br /&gt;
===== 17-limit =====
Map: [&amp;lt;2 0 11 0 -17|, &amp;lt;0 1 -2 0 4|, &amp;lt;0 0 0 1 2|]&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17
EDOs: 58, 148, 160&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 136/135, 176/175, 256/255, 325/324
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="x-Claudius"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Claudius&lt;/h2&gt;
 
Commas: 2048/2025, 540/539&lt;br /&gt;
Mapping: {{mapping| 2 0 11 0 14 -18 5 | 0 1 -2 0 -4 8 1 | 0 0 0 1 1 0 0 }}
&lt;br /&gt;
 
Minimax tuning&lt;br /&gt;
{{Optimal ET sequence|legend=1| 12, 22f, 34d, 46, 68, 80, 148d, 228bcd }}
[|1 0 0 0 0&amp;gt;, |9/7 0 -3/7 2/7 1/7&amp;gt;, |41/14 0 6/7 -4/7 -2/7&amp;gt;,&lt;br /&gt;
 
|41/14 0 -1/7 3/7 -2/7&amp;gt;, |41/14 0 -1/7 -4/7 5/7&amp;gt;]&lt;br /&gt;
===== 19-limit =====
Eigenmonzos: 2, 7/5, 11/5&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17.19
&lt;br /&gt;
 
Map: [&amp;lt;2 0 11 0 15|, &amp;lt;0 1 -2 0 1|, &amp;lt;0 0 0 1 -2|]&lt;br /&gt;
Comma list: 136/135, 176/175, 190/189, 256/255, 286/285
EDOs: 58, 80&lt;/body&gt;&lt;/html&gt;</pre></div>
 
Mapping: {{mapping| 2 0 11 0 14 -18 5 -13 | 0 1 -2 0 -4 8 1 5 | 0 0 0 1 1 0 0 1 }}
 
{{Optimal ET sequence|legend=1| 12, 22fh, 34dh, 46, 68, 80, 148d, 228bcd }}
 
== Augustus ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 385/384, 1232/1215
 
{{Mapping|legend=1| 2 0 11 0 3 | 0 1 -2 0 3 | 0 0 0 1 -1 }}
 
[[Minimax tuning]]:
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 11/8 1/2 -1/4 0 0 }}, {{monzo| 11/4 -1 1/2 0 0 }}, {{monzo| 45/16 3/4 -3/8 1/2 -1/2 }}, {{monzo| 45/16 3/4 -3/8 -1/2 1/2 }}]
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.11/7
 
{{Optimal ET sequence|legend=1| 10, 12e, 22, 46, 68, 114, 170b }}
 
[[Badness]]: 1.58 × 10<sup>-3</sup>
 
== Tiberius ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 441/440, 2048/2025
 
{{Mapping|legend=1| 2 0 11 0 -17 | 0 1 -2 0 4 | 0 0 0 1 2 }}
 
[[Minimax tuning]]:
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 11/8 1/2 -1/4 0 0 }}, {{monzo| 11/4 -1 1/2 0 0 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| -3 2 -1 2 0 }}]
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5.7
 
{{Optimal ET sequence|legend=1| 46, 58, 104c, 160, 162ce, 172ce, 218ce }}
 
[[Badness]]: 1.71 × 10<sup>-3</sup>
 
== Claudius ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 540/539, 2048/2025
 
{{Mapping|legend=1| 2 0 11 0 15 | 0 1 -2 0 1 | 0 0 0 1 -2 }}
 
[[Minimax tuning]]:
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 9/7 0 -3/7 2/7 1/7 }}, {{monzo| 41/14 0 6/7 -4/7 -2/7 }}, {{monzo| 41/14 0 -1/7 3/7 -2/7 }}, {{monzo| 41/14 0 -1/7 -4/7 5/7 }}]
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/5
 
{{Optimal ET sequence|legend=1| 22, 46e, 48c, 58, 80, 138cde, 218cde, 286bcddeee, 366bccdddeee }}
 
[[Badness]]: 1.97 × 10<sup>-3</sup>
 
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Diaschismic rank-3 family| ]] <!-- main article -->
[[Category:Diaschismic| ]] <!-- key article -->
[[Category:Rank 3]]

Latest revision as of 00:35, 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 diaschismic rank-3 family of temperaments tempers out the diaschisma, 2048/2025. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.

Rank-3 diaschismic

Subgroup: 2.3.5.7

Comma list: 2048/2025

Mapping: [⟨2 0 11 0], ⟨0 1 -2 0], ⟨0 0 0 1]]

mapping generators: ~45/32, ~3, ~7

Optimal tuning (POTE): ~3/2 = 704.8982, ~7/4 = 972.1330

Optimal ET sequence: 10, 12, 22, 34d, 46, 58, 68, 80, 126, 206cd, 332bccd

Badness: 0.442 × 10-3

Julius aka varda

Subgroup: 2.3.5.7.11

Comma list: 176/175, 896/891

Mapping: [⟨2 0 11 0 14], ⟨0 1 -2 0 -4], ⟨0 0 0 1 1]]

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

Lattice basis:

3/2, 9/7
Angle (3/2, 9/7) = 82.289 degrees

Minimax tuning:

[[1 0 0 0 0⟩, [11/8 1/2 -1/4 0 0⟩, [11/4 -1 1/2 0 0⟩, [-1/8 3/2 -5/4 0 1⟩, [11/8 -1/2 -1/4 0 1⟩]
unchanged-interval (eigenmonzo) basis: 2.9/5.11/3

Optimal ET sequence: 12, 22, 34d, 46, 58, 68, 80, 126, 184c, 206cd, 264bccde

Badness: 0.728 × 10-3

Julius[22] hobbit transversal:

28/27, 16/15, 11/10, 9/8, 32/27, 6/5, 5/4, 9/7, 4/3, 15/11, 45/32, 22/15, 3/2, 14/9, 8/5, 5/3, 27/16, 16/9, 20/11, 15/8, 27/14, 2

Julius[24] hobbit transversal

28/27, 16/15, 11/10, 9/8, 7/6, 6/5, 40/33, 5/4, 9/7, 4/3, 15/11, 45/32, 16/11, 3/2, 14/9, 8/5, 33/20, 27/16, 7/4, 16/9, 20/11, 15/8, 27/14, 2

Varda

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 364/363

Mapping: [⟨2 0 11 0 14 24], ⟨0 1 -2 0 -4 -7], ⟨0 0 0 1 1 1]]

Optimal ET sequence: 12f, 22, 34d, 46, 58, 80, 138cde, 242cde, 322bcde

Badness: 0.970 × 10-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 136/135, 176/175, 256/255, 351/350

Mapping: [⟨2 0 11 0 14 24 5], ⟨0 1 -2 0 -4 -7 1], ⟨0 0 0 1 1 1 0]]

Optimal ET sequence: 12f, 22, 34d, 46, 58, 80, 126, 138cde

Julius

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 325/324, 896/891

Mapping: [⟨2 0 11 0 14 -18], ⟨0 1 -2 0 -4 8], ⟨0 0 0 1 1 0]]

Optimal ET sequence: 12, 22f, 34d, 46, 68, 80, 148d, 228bcd

Badness: 1.78 × 10-3

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 136/135, 176/175, 256/255, 325/324

Mapping: [⟨2 0 11 0 14 -18 5], ⟨0 1 -2 0 -4 8 1], ⟨0 0 0 1 1 0 0]]

Optimal ET sequence: 12, 22f, 34d, 46, 68, 80, 148d, 228bcd

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 136/135, 176/175, 190/189, 256/255, 286/285

Mapping: [⟨2 0 11 0 14 -18 5 -13], ⟨0 1 -2 0 -4 8 1 5], ⟨0 0 0 1 1 0 0 1]]

Optimal ET sequence: 12, 22fh, 34dh, 46, 68, 80, 148d, 228bcd

Augustus

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1232/1215

Mapping: [⟨2 0 11 0 3], ⟨0 1 -2 0 3], ⟨0 0 0 1 -1]]

Minimax tuning:

[[1 0 0 0 0⟩, [11/8 1/2 -1/4 0 0⟩, [11/4 -1 1/2 0 0⟩, [45/16 3/4 -3/8 1/2 -1/2⟩, [45/16 3/4 -3/8 -1/2 1/2⟩]
unchanged-interval (eigenmonzo) basis: 2.9/5.11/7

Optimal ET sequence: 10, 12e, 22, 46, 68, 114, 170b

Badness: 1.58 × 10-3

Tiberius

Subgroup: 2.3.5.7.11

Comma list: 441/440, 2048/2025

Mapping: [⟨2 0 11 0 -17], ⟨0 1 -2 0 4], ⟨0 0 0 1 2]]

Minimax tuning:

[[1 0 0 0 0⟩, [11/8 1/2 -1/4 0 0⟩, [11/4 -1 1/2 0 0⟩, [0 0 0 1 0⟩, [-3 2 -1 2 0⟩]
unchanged-interval (eigenmonzo) basis: 2.9/5.7

Optimal ET sequence: 46, 58, 104c, 160, 162ce, 172ce, 218ce

Badness: 1.71 × 10-3

Claudius

Subgroup: 2.3.5.7.11

Comma list: 540/539, 2048/2025

Mapping: [⟨2 0 11 0 15], ⟨0 1 -2 0 1], ⟨0 0 0 1 -2]]

Minimax tuning:

[[1 0 0 0 0⟩, [9/7 0 -3/7 2/7 1/7⟩, [41/14 0 6/7 -4/7 -2/7⟩, [41/14 0 -1/7 3/7 -2/7⟩, [41/14 0 -1/7 -4/7 5/7⟩]
unchanged-interval (eigenmonzo) basis: 2.7/5.11/5

Optimal ET sequence: 22, 46e, 48c, 58, 80, 138cde, 218cde, 286bcddeee, 366bccdddeee

Badness: 1.97 × 10-3

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