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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Todo|inline=1| intro }}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-06-21 22:03:52 UTC</tt>.<br>
: The original revision id was <tt>238088361</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">[[toc|flat]]


=Marveltwin and Marvel=
== Marveltwin and marvel ==
The //marveltwin comma//, 325/324, bears a curiously close analogy to the marvel comma, 225/224. 325/324 can be added to the [[11-limit]] version of marvel, which tempers out 225/224 and 385/384, to get [[13-limit]] marvel, aka hecate. But it's also interesting to leave 11 out of it. From 225/224 we get that a 5-limit approximation for 7 is 225/224 * 7 = 225/32. Similarly from 325/324 we get a 5-limit approximation of 13 from 324/325 * 13 = 324/25. If we define the major/minor transformation of the 5-limit as the result of fixing 2 and 3 and replacing 5 by 24/5, then major/minor applied to 225/32 is 162/25, which is (324/25)/2. Similarly, major/minor applied to 324/25 is 225/16 = 2 * (225/32). 225/224 tells us that two 16/15 in a row are an approximate 8/7, and 325/324 tells us two 10/9 in a row are an approximate 16/13. Needless to say, major/minor applied to 16/15 is 10/9, and applied to 10/9 is 16/15.
The marveltwin comma, [[325/324]], bears a curiously close analogy to the marvel comma, [[225/224]]. 325/324 can be added to the [[11-limit]] version of marvel, which tempers out 225/224 and 385/384, to get [[13-limit]] marvel, aka hecate. But it's also interesting to leave 11 out of it. From 225/224 we get that a 5-limit approximation for 7 is 225/224 * 7 = 225/32. Similarly from 325/324 we get a 5-limit approximation of 13 from 324/325 * 13 = 324/25. If we define the major/minor transformation of the 5-limit as the result of fixing 2 and 3 and replacing 5 by 24/5, then major/minor applied to 225/32 is 162/25, which is (324/25)/2. Similarly, major/minor applied to 324/25 is 225/16 = 2 * (225/32). 225/224 tells us that two 16/15 in a row are an approximate 8/7, and 325/324 tells us two 10/9 in a row are an approximate 16/13. Needless to say, major/minor applied to 16/15 is 10/9, and applied to 10/9 is 16/15.


=Rank five=
== Rank-5 temperaments ==
Comma: 325/324
[[Subgroup]]: 2.3.5.7.11.13


13 and 15 limit minimax tuning
[[Comma list]]: 325/324
|| [1 0 0 0 0 0&gt; ||
|| [0 1 0 0 0 0&gt; ||
|| [2/3 4/3 1/3 0 0 -1/3&gt; ||
|| [2/3 4/3 -2/3 1 0 -1/3&gt; ||
|| [2/3 4/3 -2/3 0 1 -1/3&gt; ||
|| [2/3 4/3 -2/3 0 0 2/3&gt; ||


Fifths are pure; 5, 7, 11 and 13 are all flat by (325/324)^(1/3), which is 1.778 cents.
[[Mapping]]:
Eigenmonzo subgroup: 2.3.7/5.11/5.13/5
{| class="right-all"
|-
| {{lbrack}}⟨ || 1 || 0 || 0 || 0 || 0 || 2 || {{rbrack}},
|-
| ⟨ || 0 || 1 || 0 || 0 || 0 || 4 || {{rbrack}},
|-
| ⟨ || 0 || 0 || 1 || 0 || 0 || -2 || {{rbrack}},
|-
| ⟨ || 0 || 0 || 0 || 1 || 0 || 0 || {{rbrack}},
|-
| ⟨ || 0 || 0 || 0 || 0 || 1 || 0 || {{rbrack}}{{rbrack}}
|}


[[Spectrum of a temperament|Spectrum]]: 4/3, 6/5, 10/9, 5/4, 9/8, 8/7, 16/15, 18/13, 13/12, 7/6, 16/13, 7/5, 11/8, 9/7, 12/11, 15/14, 11/10, 11/9, 13/10, 14/13, 15/13, 14/11, 15/11, 13/11
[[Minimax tuning]]s:
* 13- and 15-odd-limit
: {| class="right-all"
|-
| {{lbrack}}{{lbrack}} || 1 || 0 || 0 || 0 || 0 || 0 || ⟩
|-
| {{lbrack}} || 0 || 1 || 0 || 0 || 0 || 0 || ⟩
|-
| {{lbrack}} || 2/3 || 4/3 || 1/3 || 0 || 0 || -1/3 || ⟩
|-
| {{lbrack}} || 2/3 || 4/3 || -2/3 || 1 || 0 || -1/3 || ⟩
|-
| {{lbrack}} || 2/3 || 4/3 || -2/3 || 0 || 1 || -1/3 || ⟩
|-
| {{lbrack}} || 2/3 || 4/3 || -2/3 || 0 || 0 || 2/3 || ⟩{{rbrack}}
|}
: 3 pure; 5, 7, 11 and 13 all flat by (325/324)<sup>1/3</sup>, which is 1.778 cents.
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7/5.11/5.13/5


Map:  
[[Complexity spectrum]]: 4/3, 6/5, 10/9, 5/4, 9/8, 8/7, 16/15, 18/13, 13/12, 7/6, 16/13, 7/5, 11/8, 9/7, 12/11, 15/14, 11/10, 11/9, 13/10, 14/13, 15/13, 14/11, 15/11, 13/11
|| &lt;1 0 0 0 0 2] ||
|| &lt;0 1 0 0 0 4] ||
|| &lt;0 0 1 0 0 -2] ||
|| &lt;0 0 0 1 0 0] ||
|| &lt;0 0 0 0 1 0] ||


Edos: 7, 12, 15, 19, 26, 34, 41, 46, 53, 72, 87, 121, 140, 159, 193, 212, 299, 333
{{Optimal ET sequence|legend=1| 7, 12, 15, 19, 26, 34, 41, 46, 53, 72, 87, 121, 140, 159, 193, 212, 299, 333 }}


=Rank four=
== Rank-4 temperaments ==
=== 225/224 ===
[[Subgroup]]: 2.3.5.7.11.13


==225/224==
[[Comma list]]: 225/224, 325/324


13-limit eigenmonzo subgroup: 2.7.11/5.13/5
{{Mapping|legend=1| 1 0 0 -5 0 2 | 0 1 0 2 0 4 | 0 0 1 2 0 -2 | 0 0 0 0 1 0 }}
15-limit eigenmonzo subgroup: 2.7.15/11.15/13


[[Spectrum of a temperament|Spectrum]]: 4/3, 5/4, 6/5, 16/15, 15/14, 9/8, 10/9, 7/5, 9/7, 7/6, 18/13, 8/7, 13/12, 16/13, 11/8, 12/11, 15/13, 11/10, 15/11, 11/9, 13/10, 14/13, 14/11, 13/11
[[Minimax tuning]]s:  
* 13-limit unchanged-interval (eigenmonzo) basis: 2.7.11/5.13/5
* 15-limit unchanged-interval (eigenmonzo) basis: 2.7.15/11.15/13


Map:
{{Optimal ET sequence|legend=1| 12, 19, 41, 53, 72, 166 }}
|| &lt;1 0 0 -5 0 2] ||
|| &lt;0 1 0 2 0 4] ||
|| &lt;0 0 1 2 0 -2] ||
|| &lt;0 0 0 0 1 0] ||
Edos: 12, 19, 41, 53, 72, 166
10^6 * Badness: 3.668


==385/384==
[[Complexity spectrum]]: 4/3, 5/4, 6/5, 16/15, 15/14, 9/8, 10/9, 7/5, 9/7, 7/6, 18/13, 8/7, 13/12, 16/13, 11/8, 12/11, 15/13, 11/10, 15/11, 11/9, 13/10, 14/13, 14/11, 13/11


Spectrum: 4/3, 6/5, 10/9, 5/4, 9/8, 8/7, 11/8, 7/6, 12/11, 16/15, 18/13, 13/12, 9/7, 7/5, 16/13, 11/9, 11/10, 13/11, 14/13, 15/14, 15/11, 13/10, 15/13, 14/11
[[Badness]]: 3.668 × 10<sup>-6</sup>


Edos: 7, 15, 19, 26, 34, 41, 46, 53, 72, 87, 140, 159, 212, 299
=== 385/384 ===
10^6 * Badness: 2.206
See [[Keenanismic family #Martwin]].  


==364/363==
=== 364/363 ===
[[Subgroup]]: 2.3.5.7.11.13


Spectrum: 4/3, 6/5, 10/9, 11/8, 5/4, 12/11, 14/11, 9/8, 13/11, 11/9, 8/7, 13/12, 18/13, 16/15, 7/6, 11/10, 7/5, 16/13, 9/7, 15/11, 15/14, 13/10, 15/13, 14/13
[[Comma list]]: 325/324, 364/363


Edos: 15, 26, 41, 46, 72, 87, 121, 159, 193, 239, 280
[[Complexity spectrum]]: 4/3, 6/5, 10/9, 11/8, 5/4, 12/11, 14/11, 9/8, 13/11, 11/9, 8/7, 13/12, 18/13, 16/15, 7/6, 11/10, 7/5, 16/13, 9/7, 15/11, 15/14, 13/10, 15/13, 14/13
10^6 * Badness: 3.011


==441/440==
{{Optimal ET sequence|legend=1| 15, 26, 41, 46, 72, 87, 121, 159, 193, 239, 280 }}


Spectrum: 4/3, 6/5, 8/7, 10/9, 14/11, 5/4, 7/6, 9/8, 7/5, 16/15, 13/12, 18/13, 9/7, 11/8, 12/11, 15/14, 16/13, 11/9, 11/10, 13/11, 15/11, 14/13, 13/10, 15/13
[[Badness]]: 3.011 × 10<sup>-6</sup>


Edos: 12, 15, 26, 41, 46, 72, 87, 159
=== 441/440 ===
10^6 * Badness: 3.037
[[Subgroup]]: 2.3.5.7.11.13


==169/168==
[[Comma list]]: 325/324, 441/440


Spectrum: 10/9, 6/5, 4/3, 14/13, 13/12, 16/13, 5/4, 18/13, 9/8, 13/10, 8/7, 7/6, 15/13, 16/15, 9/7, 7/5, 11/8, 12/11, 13/11, 11/10, 15/14, 11/9, 14/11, 15/11
[[Complexity spectrum]]: 4/3, 6/5, 8/7, 10/9, 14/11, 5/4, 7/6, 9/8, 7/5, 16/15, 13/12, 18/13, 9/7, 11/8, 12/11, 15/14, 16/13, 11/9, 11/10, 13/11, 15/11, 14/13, 13/10, 15/13


Edos: 7, 19, 26, 46, 53, 72, 152
{{Optimal ET sequence|legend=1| 12, 15, 26, 41, 46, 72, 87, 159 }}
10^6 * Badness: 2.975


==540/539==
[[Badness]]: 3.037 × 10<sup>-6</sup>


Spectrum: 4/3, 6/5, 7/6, 9/7, 10/9, 5/4, 7/5, 9/8, 15/14, 8/7, 16/15, 18/13, 13/12, 12/11, 11/9, 11/10, 11/8, 15/11, 16/13, 14/13, 15/13, 13/10, 13/11, 14/11
=== 169/168 ===
[[Subgroup]]: 2.3.5.7.11.13


Edos: 19, 41, 53, 72, 121, 166, 193
[[Comma list]]: 169/168, 325/324
10^6 * Badness: 3.281


==352/351==
[[Complexity spectrum]]: 10/9, 6/5, 4/3, 14/13, 13/12, 16/13, 5/4, 18/13, 9/8, 13/10, 8/7, 7/6, 15/13, 16/15, 9/7, 7/5, 11/8, 12/11, 13/11, 11/10, 15/14, 11/9, 14/11, 15/11


Spectrum: 4/3, 10/9, 6/5, 9/8, 5/4, 13/11, 16/13, 11/9, 13/12, 12/11, 16/15, 18/13, 8/7, 7/6, 11/8, 9/7, 15/11, 11/10, 7/5, 14/13, 13/10, 15/14, 14/11, 15/13
{{Optimal ET sequence|legend=1| 7, 19, 26, 46, 53, 72, 152 }}


Edos: 7, 34, 41, 46, 53, 80, 87, 121, 140, 261, 358, 401
[[Badness]]: 2.975 × 10<sup>-6</sup>
10^6 * Badness: 3.434


==625/624==
=== 540/539 ===
[[Subgroup]]: 2.3.5.7.11.13


Spectrum: 6/5, 18/13, 15/13, 5/4, 4/3, 10/9, 13/12, 13/10, 16/15, 9/8, 16/13, 8/7, 7/5, 7/6, 15/14, 11/8, 9/7, 11/10, 12/11, 14/13, 15/11, 11/9, 13/11, 14/11
[[Comma list]]: 325/324, 540/539


Edos: 15, 19, 34, 53, 72, 87, 121, 140, 159, 193, 212, 299, 333
[[Complexity spectrum]]: 4/3, 6/5, 7/6, 9/7, 10/9, 5/4, 7/5, 9/8, 15/14, 8/7, 16/15, 18/13, 13/12, 12/11, 11/9, 11/10, 11/8, 15/11, 16/13, 14/13, 15/13, 13/10, 13/11, 14/11
10^6 * Badness: 3.563


=Rank three=
{{Optimal ET sequence|legend=1| 19, 41, 53, 72, 121, 166, 193 }}


==Portending==
[[Badness]]: 3.281 × 10<sup>-6</sup>
Commas: 325/324, 364/363, 441/440


Spectrum: 8/7, 4/3, 11/8, 6/5, 14/11, 7/6, 10/9, 12/11, 5/4, 13/11, 9/8, 7/5, 11/9, 9/7, 18/13, 13/12, 16/15, 11/10, 15/14, 16/13, 14/13, 15/11, 13/10, 15/13
=== 352/351 ===
[[Subgroup]]: 2.3.5.7.11.13


Edos: 15, 26, 41, 46, 72, 87, 159
[[Comma list]]: 325/324, 352/351
10^5 * Badness: 62.715


==Marvel (Hecate)==
[[Complexity spectrum]]: 4/3, 10/9, 6/5, 9/8, 5/4, 13/11, 16/13, 11/9, 13/12, 12/11, 16/15, 18/13, 8/7, 7/6, 11/8, 9/7, 15/11, 11/10, 7/5, 14/13, 13/10, 15/14, 14/11, 15/13
Commas: 225/224, 325/324, 385/384


Spectrum: 4/3, 5/4, 16/15, 15/14, 6/5, 9/8, 7/5, 9/7, 7/6, 10/9, 8/7, 18/13, 11/8, 12/11, 13/12, 11/9, 11/10, 15/13, 15/11, 16/13, 13/11, 14/13, 13/10, 14/11
{{Optimal ET sequence|legend=1| 7, 34, 41, 46, 53, 80, 87, 121, 140, 261, 358, 401 }}


Edos: 19, 41, 53, 72, 166
[[Badness]]: 3.434 × 10<sup>-6</sup>
10^5 * Badness: 72.113


==Sumatra==
=== 625/624 ===
Commas: 325/324, 385/384, 625/624
[[Subgroup]]: 2.3.5.7.11.13


Spectrum: 6/5, 18/13, 15/13, 5/4, 4/3, 10/9, 13/12, 13/10, 16/15, 9/8, 8/7, 11/8, 11/10, 7/5, 12/11, 7/6, 16/13, 15/11, 15/14, 11/9, 9/7, 13/11, 14/13, 14/11
[[Comma list]]: 325/324, 625/624


Edos: 15, 19, 34, 53, 72, 87, 140, 159, 212, 299
[[Complexity spectrum]]: 6/5, 18/13, 15/13, 5/4, 4/3, 10/9, 13/12, 13/10, 16/15, 9/8, 16/13, 8/7, 7/5, 7/6, 15/14, 11/8, 9/7, 11/10, 12/11, 14/13, 15/11, 11/9, 13/11, 14/11
Optimal patent val: [[299edo]]
10^5 * Badness: 68.005


</pre></div>
{{Optimal ET sequence|legend=1| 15, 19, 34, 53, 72, 87, 121, 140, 159, 193, 212, 299, 333 }}
<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;Marveltwin&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:30:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt;&lt;a href="#Marveltwin and Marvel"&gt;Marveltwin and Marvel&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;!-- ws:start:WikiTextTocRule:32: --&gt; | &lt;a href="#Rank five"&gt;Rank five&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&gt; | &lt;a href="#Rank four"&gt;Rank four&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:33 --&gt;&lt;!-- ws:start:WikiTextTocRule:34: --&gt;&lt;!-- ws:end:WikiTextTocRule:34 --&gt;&lt;!-- ws:start:WikiTextTocRule:35: --&gt;&lt;!-- ws:end:WikiTextTocRule:35 --&gt;&lt;!-- ws:start:WikiTextTocRule:36: --&gt;&lt;!-- ws:end:WikiTextTocRule:36 --&gt;&lt;!-- ws:start:WikiTextTocRule:37: --&gt;&lt;!-- ws:end:WikiTextTocRule:37 --&gt;&lt;!-- ws:start:WikiTextTocRule:38: --&gt;&lt;!-- ws:end:WikiTextTocRule:38 --&gt;&lt;!-- ws:start:WikiTextTocRule:39: --&gt;&lt;!-- ws:end:WikiTextTocRule:39 --&gt;&lt;!-- ws:start:WikiTextTocRule:40: --&gt;&lt;!-- ws:end:WikiTextTocRule:40 --&gt;&lt;!-- ws:start:WikiTextTocRule:41: --&gt;&lt;!-- ws:end:WikiTextTocRule:41 --&gt;&lt;!-- ws:start:WikiTextTocRule:42: --&gt; | &lt;a href="#Rank three"&gt;Rank three&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:42 --&gt;&lt;!-- ws:start:WikiTextTocRule:43: --&gt;&lt;!-- ws:end:WikiTextTocRule:43 --&gt;&lt;!-- ws:start:WikiTextTocRule:44: --&gt;&lt;!-- ws:end:WikiTextTocRule:44 --&gt;&lt;!-- ws:start:WikiTextTocRule:45: --&gt;&lt;!-- ws:end:WikiTextTocRule:45 --&gt;&lt;!-- ws:start:WikiTextTocRule:46: --&gt;
&lt;!-- ws:end:WikiTextTocRule:46 --&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Marveltwin and Marvel"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Marveltwin and Marvel&lt;/h1&gt;
The &lt;em&gt;marveltwin comma&lt;/em&gt;, 325/324, bears a curiously close analogy to the marvel comma, 225/224. 325/324 can be added to the &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt; version of marvel, which tempers out 225/224 and 385/384, to get &lt;a class="wiki_link" href="/13-limit"&gt;13-limit&lt;/a&gt; marvel, aka hecate. But it's also interesting to leave 11 out of it. From 225/224 we get that a 5-limit approximation for 7 is 225/224 * 7 = 225/32. Similarly from 325/324 we get a 5-limit approximation of 13 from 324/325 * 13 = 324/25. If we define the major/minor transformation of the 5-limit as the result of fixing 2 and 3 and replacing 5 by 24/5, then major/minor applied to 225/32 is 162/25, which is (324/25)/2. Similarly, major/minor applied to 324/25 is 225/16 = 2 * (225/32). 225/224 tells us that two 16/15 in a row are an approximate 8/7, and 325/324 tells us two 10/9 in a row are an approximate 16/13. Needless to say, major/minor applied to 16/15 is 10/9, and applied to 10/9 is 16/15.&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Rank five"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Rank five&lt;/h1&gt;
Comma: 325/324&lt;br /&gt;
&lt;br /&gt;
13 and 15 limit minimax tuning&lt;br /&gt;


[[Badness]]: 3.563 × 10<sup>-6</sup>


&lt;table class="wiki_table"&gt;
== Rank-3 temperaments ==
    &lt;tr&gt;
Notable rank-3 temperaments of marveltwin include:
        &lt;td&gt;[1 0 0 0 0 0&amp;gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;[0 1 0 0 0 0&amp;gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;[2/3 4/3 1/3 0 0 -1/3&amp;gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;[2/3 4/3 -2/3 1 0 -1/3&amp;gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;[2/3 4/3 -2/3 0 1 -1/3&amp;gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;[2/3 4/3 -2/3 0 0 2/3&amp;gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
* [[Portent|Portending]] → [[Gamelismic family #Portending|Gamelismic family]]
Fifths are pure; 5, 7, 11 and 13 are all flat by (325/324)^(1/3), which is 1.778 cents. &lt;br /&gt;
: +385/384, 441/440
Eigenmonzo subgroup: 2.3.7/5.11/5.13/5&lt;br /&gt;
* [[Marvel|Marvel (hecate)]] → [[Marvel family #Hecate|Marvel family]]
&lt;br /&gt;
: +225/224, 385/384
&lt;a class="wiki_link" href="/Spectrum%20of%20a%20temperament"&gt;Spectrum&lt;/a&gt;: 4/3, 6/5, 10/9, 5/4, 9/8, 8/7, 16/15, 18/13, 13/12, 7/6, 16/13, 7/5, 11/8, 9/7, 12/11, 15/14, 11/10, 11/9, 13/10, 14/13, 15/13, 14/11, 15/11, 13/11&lt;br /&gt;
* [[Enlil|Enlil a.k.a. sumatra]] → [[Kleismic rank-3 family #Enlil|Kleismic rank-3 family]]
&lt;br /&gt;
: +385/384, 625/624
Map: &lt;br /&gt;


 
[[Category:Regular temperament theory]]
&lt;table class="wiki_table"&gt;
[[Category:Commatic realms]]
    &lt;tr&gt;
[[Category:Marveltwin]]
        &lt;td&gt;&amp;lt;1 0 0 0 0 2]&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&amp;lt;0 1 0 0 0 4]&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&amp;lt;0 0 1 0 0 -2]&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&amp;lt;0 0 0 1 0 0]&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&amp;lt;0 0 0 0 1 0]&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;br /&gt;
Edos: 7, 12, 15, 19, 26, 34, 41, 46, 53, 72, 87, 121, 140, 159, 193, 212, 299, 333&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Rank four"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Rank four&lt;/h1&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Rank four-225/224"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;225/224&lt;/h2&gt;
&lt;br /&gt;
13-limit eigenmonzo subgroup: 2.7.11/5.13/5&lt;br /&gt;
15-limit eigenmonzo subgroup: 2.7.15/11.15/13&lt;br /&gt;
&lt;br /&gt;
&lt;a class="wiki_link" href="/Spectrum%20of%20a%20temperament"&gt;Spectrum&lt;/a&gt;: 4/3, 5/4, 6/5, 16/15, 15/14, 9/8, 10/9, 7/5, 9/7, 7/6, 18/13, 8/7, 13/12, 16/13, 11/8, 12/11, 15/13, 11/10, 15/11, 11/9, 13/10, 14/13, 14/11, 13/11&lt;br /&gt;
&lt;br /&gt;
Map:&lt;br /&gt;
 
 
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;td&gt;&amp;lt;1 0 0 -5 0 2]&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&amp;lt;0 1 0 2 0 4]&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&amp;lt;0 0 1 2 0 -2]&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&amp;lt;0 0 0 0 1 0]&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
Edos: 12, 19, 41, 53, 72, 166&lt;br /&gt;
10^6 * Badness: 3.668&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Rank four-385/384"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;385/384&lt;/h2&gt;
&lt;br /&gt;
Spectrum: 4/3, 6/5, 10/9, 5/4, 9/8, 8/7, 11/8, 7/6, 12/11, 16/15, 18/13, 13/12, 9/7, 7/5, 16/13, 11/9, 11/10, 13/11, 14/13, 15/14, 15/11, 13/10, 15/13, 14/11&lt;br /&gt;
&lt;br /&gt;
Edos: 7, 15, 19, 26, 34, 41, 46, 53, 72, 87, 140, 159, 212, 299&lt;br /&gt;
10^6 * Badness: 2.206&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Rank four-364/363"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;364/363&lt;/h2&gt;
&lt;br /&gt;
Spectrum: 4/3, 6/5, 10/9, 11/8, 5/4, 12/11, 14/11, 9/8, 13/11, 11/9, 8/7, 13/12, 18/13, 16/15, 7/6, 11/10, 7/5, 16/13, 9/7, 15/11, 15/14, 13/10, 15/13, 14/13&lt;br /&gt;
&lt;br /&gt;
Edos: 15, 26, 41, 46, 72, 87, 121, 159, 193, 239, 280&lt;br /&gt;
10^6 * Badness: 3.011&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Rank four-441/440"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;441/440&lt;/h2&gt;
&lt;br /&gt;
Spectrum: 4/3, 6/5, 8/7, 10/9, 14/11, 5/4, 7/6, 9/8, 7/5, 16/15, 13/12, 18/13, 9/7, 11/8, 12/11, 15/14, 16/13, 11/9, 11/10, 13/11, 15/11, 14/13, 13/10, 15/13&lt;br /&gt;
&lt;br /&gt;
Edos: 12, 15, 26, 41, 46, 72, 87, 159&lt;br /&gt;
10^6 * Badness: 3.037&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Rank four-169/168"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;169/168&lt;/h2&gt;
&lt;br /&gt;
Spectrum: 10/9, 6/5, 4/3, 14/13, 13/12, 16/13, 5/4, 18/13, 9/8, 13/10, 8/7, 7/6, 15/13, 16/15, 9/7, 7/5, 11/8, 12/11, 13/11, 11/10, 15/14, 11/9, 14/11, 15/11&lt;br /&gt;
&lt;br /&gt;
Edos: 7, 19, 26, 46, 53, 72, 152&lt;br /&gt;
10^6 * Badness: 2.975&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Rank four-540/539"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;540/539&lt;/h2&gt;
&lt;br /&gt;
Spectrum: 4/3, 6/5, 7/6, 9/7, 10/9, 5/4, 7/5, 9/8, 15/14, 8/7, 16/15, 18/13, 13/12, 12/11, 11/9, 11/10, 11/8, 15/11, 16/13, 14/13, 15/13, 13/10, 13/11, 14/11&lt;br /&gt;
&lt;br /&gt;
Edos: 19, 41, 53, 72, 121, 166, 193&lt;br /&gt;
10^6 * Badness: 3.281&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Rank four-352/351"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;352/351&lt;/h2&gt;
&lt;br /&gt;
Spectrum: 4/3, 10/9, 6/5, 9/8, 5/4, 13/11, 16/13, 11/9, 13/12, 12/11, 16/15, 18/13, 8/7, 7/6, 11/8, 9/7, 15/11, 11/10, 7/5, 14/13, 13/10, 15/14, 14/11, 15/13&lt;br /&gt;
&lt;br /&gt;
Edos: 7, 34, 41, 46, 53, 80, 87, 121, 140, 261, 358, 401&lt;br /&gt;
10^6 * Badness: 3.434&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc10"&gt;&lt;a name="Rank four-625/624"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;625/624&lt;/h2&gt;
&lt;br /&gt;
Spectrum: 6/5, 18/13, 15/13, 5/4, 4/3, 10/9, 13/12, 13/10, 16/15, 9/8, 16/13, 8/7, 7/5, 7/6, 15/14, 11/8, 9/7, 11/10, 12/11, 14/13, 15/11, 11/9, 13/11, 14/11&lt;br /&gt;
&lt;br /&gt;
Edos: 15, 19, 34, 53, 72, 87, 121, 140, 159, 193, 212, 299, 333&lt;br /&gt;
10^6 * Badness: 3.563&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc11"&gt;&lt;a name="Rank three"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;Rank three&lt;/h1&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc12"&gt;&lt;a name="Rank three-Portending"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;Portending&lt;/h2&gt;
Commas: 325/324, 364/363, 441/440&lt;br /&gt;
&lt;br /&gt;
Spectrum: 8/7, 4/3, 11/8, 6/5, 14/11, 7/6, 10/9, 12/11, 5/4, 13/11, 9/8, 7/5, 11/9, 9/7, 18/13, 13/12, 16/15, 11/10, 15/14, 16/13, 14/13, 15/11, 13/10, 15/13&lt;br /&gt;
&lt;br /&gt;
Edos: 15, 26, 41, 46, 72, 87, 159&lt;br /&gt;
10^5 * Badness: 62.715&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc13"&gt;&lt;a name="Rank three-Marvel (Hecate)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;Marvel (Hecate)&lt;/h2&gt;
Commas: 225/224, 325/324, 385/384&lt;br /&gt;
&lt;br /&gt;
Spectrum: 4/3, 5/4, 16/15, 15/14, 6/5, 9/8, 7/5, 9/7, 7/6, 10/9, 8/7, 18/13, 11/8, 12/11, 13/12, 11/9, 11/10, 15/13, 15/11, 16/13, 13/11, 14/13, 13/10, 14/11&lt;br /&gt;
&lt;br /&gt;
Edos: 19, 41, 53, 72, 166&lt;br /&gt;
10^5 * Badness: 72.113&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc14"&gt;&lt;a name="Rank three-Sumatra"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;Sumatra&lt;/h2&gt;
Commas: 325/324, 385/384, 625/624&lt;br /&gt;
&lt;br /&gt;
Spectrum: 6/5, 18/13, 15/13, 5/4, 4/3, 10/9, 13/12, 13/10, 16/15, 9/8, 8/7, 11/8, 11/10, 7/5, 12/11, 7/6, 16/13, 15/11, 15/14, 11/9, 9/7, 13/11, 14/13, 14/11&lt;br /&gt;
&lt;br /&gt;
Edos: 15, 19, 34, 53, 72, 87, 140, 159, 212, 299&lt;br /&gt;
Optimal patent val: &lt;a class="wiki_link" href="/299edo"&gt;299edo&lt;/a&gt;&lt;br /&gt;
10^5 * Badness: 68.005&lt;/body&gt;&lt;/html&gt;</pre></div>

Latest revision as of 13:30, 27 August 2025

Todo: intro

Marveltwin and marvel

The marveltwin comma, 325/324, bears a curiously close analogy to the marvel comma, 225/224. 325/324 can be added to the 11-limit version of marvel, which tempers out 225/224 and 385/384, to get 13-limit marvel, aka hecate. But it's also interesting to leave 11 out of it. From 225/224 we get that a 5-limit approximation for 7 is 225/224 * 7 = 225/32. Similarly from 325/324 we get a 5-limit approximation of 13 from 324/325 * 13 = 324/25. If we define the major/minor transformation of the 5-limit as the result of fixing 2 and 3 and replacing 5 by 24/5, then major/minor applied to 225/32 is 162/25, which is (324/25)/2. Similarly, major/minor applied to 324/25 is 225/16 = 2 * (225/32). 225/224 tells us that two 16/15 in a row are an approximate 8/7, and 325/324 tells us two 10/9 in a row are an approximate 16/13. Needless to say, major/minor applied to 16/15 is 10/9, and applied to 10/9 is 16/15.

Rank-5 temperaments

Subgroup: 2.3.5.7.11.13

Comma list: 325/324

Mapping:

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

Minimax tunings:

  • 13- and 15-odd-limit
[[ 1 0 0 0 0 0 ⟩
[ 0 1 0 0 0 0 ⟩
[ 2/3 4/3 1/3 0 0 -1/3 ⟩
[ 2/3 4/3 -2/3 1 0 -1/3 ⟩
[ 2/3 4/3 -2/3 0 1 -1/3 ⟩
[ 2/3 4/3 -2/3 0 0 2/3 ⟩]
3 pure; 5, 7, 11 and 13 all flat by (325/324)1/3, which is 1.778 cents.
unchanged-interval (eigenmonzo) basis: 2.3.7/5.11/5.13/5

Complexity spectrum: 4/3, 6/5, 10/9, 5/4, 9/8, 8/7, 16/15, 18/13, 13/12, 7/6, 16/13, 7/5, 11/8, 9/7, 12/11, 15/14, 11/10, 11/9, 13/10, 14/13, 15/13, 14/11, 15/11, 13/11

Optimal ET sequence: 7, 12, 15, 19, 26, 34, 41, 46, 53, 72, 87, 121, 140, 159, 193, 212, 299, 333

Rank-4 temperaments

225/224

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 325/324

Mapping: [⟨1 0 0 -5 0 2], ⟨0 1 0 2 0 4], ⟨0 0 1 2 0 -2], ⟨0 0 0 0 1 0]]

Minimax tunings:

  • 13-limit unchanged-interval (eigenmonzo) basis: 2.7.11/5.13/5
  • 15-limit unchanged-interval (eigenmonzo) basis: 2.7.15/11.15/13

Optimal ET sequence: 12, 19, 41, 53, 72, 166

Complexity spectrum: 4/3, 5/4, 6/5, 16/15, 15/14, 9/8, 10/9, 7/5, 9/7, 7/6, 18/13, 8/7, 13/12, 16/13, 11/8, 12/11, 15/13, 11/10, 15/11, 11/9, 13/10, 14/13, 14/11, 13/11

Badness: 3.668 × 10-6

385/384

See Keenanismic family #Martwin.

364/363

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363

Complexity spectrum: 4/3, 6/5, 10/9, 11/8, 5/4, 12/11, 14/11, 9/8, 13/11, 11/9, 8/7, 13/12, 18/13, 16/15, 7/6, 11/10, 7/5, 16/13, 9/7, 15/11, 15/14, 13/10, 15/13, 14/13

Optimal ET sequence: 15, 26, 41, 46, 72, 87, 121, 159, 193, 239, 280

Badness: 3.011 × 10-6

441/440

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 441/440

Complexity spectrum: 4/3, 6/5, 8/7, 10/9, 14/11, 5/4, 7/6, 9/8, 7/5, 16/15, 13/12, 18/13, 9/7, 11/8, 12/11, 15/14, 16/13, 11/9, 11/10, 13/11, 15/11, 14/13, 13/10, 15/13

Optimal ET sequence: 12, 15, 26, 41, 46, 72, 87, 159

Badness: 3.037 × 10-6

169/168

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324

Complexity spectrum: 10/9, 6/5, 4/3, 14/13, 13/12, 16/13, 5/4, 18/13, 9/8, 13/10, 8/7, 7/6, 15/13, 16/15, 9/7, 7/5, 11/8, 12/11, 13/11, 11/10, 15/14, 11/9, 14/11, 15/11

Optimal ET sequence: 7, 19, 26, 46, 53, 72, 152

Badness: 2.975 × 10-6

540/539

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 540/539

Complexity spectrum: 4/3, 6/5, 7/6, 9/7, 10/9, 5/4, 7/5, 9/8, 15/14, 8/7, 16/15, 18/13, 13/12, 12/11, 11/9, 11/10, 11/8, 15/11, 16/13, 14/13, 15/13, 13/10, 13/11, 14/11

Optimal ET sequence: 19, 41, 53, 72, 121, 166, 193

Badness: 3.281 × 10-6

352/351

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351

Complexity spectrum: 4/3, 10/9, 6/5, 9/8, 5/4, 13/11, 16/13, 11/9, 13/12, 12/11, 16/15, 18/13, 8/7, 7/6, 11/8, 9/7, 15/11, 11/10, 7/5, 14/13, 13/10, 15/14, 14/11, 15/13

Optimal ET sequence: 7, 34, 41, 46, 53, 80, 87, 121, 140, 261, 358, 401

Badness: 3.434 × 10-6

625/624

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 625/624

Complexity spectrum: 6/5, 18/13, 15/13, 5/4, 4/3, 10/9, 13/12, 13/10, 16/15, 9/8, 16/13, 8/7, 7/5, 7/6, 15/14, 11/8, 9/7, 11/10, 12/11, 14/13, 15/11, 11/9, 13/11, 14/11

Optimal ET sequence: 15, 19, 34, 53, 72, 87, 121, 140, 159, 193, 212, 299, 333

Badness: 3.563 × 10-6

Rank-3 temperaments

Notable rank-3 temperaments of marveltwin include:

+385/384, 441/440
+225/224, 385/384
+385/384, 625/624
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