Tuning systems for qanun: Difference between revisions

Wikispaces>hstraub
**Imported revision 254681902 - Original comment: **
Wikispaces>hstraub
**Imported revision 254750304 - Original comment: **
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2011-09-16 07:06:01 UTC</tt>.<br>
: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2011-09-16 10:48:16 UTC</tt>.<br>
: The original revision id was <tt>254681902</tt>.<br>
: The original revision id was <tt>254750304</tt>.<br>
: The revision comment was: <tt></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>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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J.J. Weiss
J.J. Weiss
Advantage: marked contrast between Segah of Ushaq and Segah of arabic Rast.
Advantage: marked contrast between Segah of Ushaq and Segah of arabic Rast.
Equal division of 65/54 (320.98 cents)
Aliquot division of 65/54 (320.98 cents)


Mandal positions (cents): &lt;span style="font-family: Tahoma;"&gt;22|13|13|1&lt;/span&gt;&lt;span style="color: #00000a; font-family: Tahoma;"&gt;8&lt;/span&gt;&lt;span style="font-family: Tahoma;"&gt;|13|13|22&lt;/span&gt;
Mandal positions (cents): &lt;span style="font-family: Tahoma;"&gt;22|13|13|1&lt;/span&gt;&lt;span style="color: #00000a; font-family: Tahoma;"&gt;8&lt;/span&gt;&lt;span style="font-family: Tahoma;"&gt;|13|13|22&lt;/span&gt;
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Characteristics of super-symmetric systems: no difference between ascending and descending ratios.
Characteristics of super-symmetric systems: no difference between ascending and descending ratios.


===Equal division of 65/54 and its inverse===  
===Aliquot division of 65/54 and its inverse===  
J.J. Weiss
J.J. Weiss
(XXX Is that formulation correct?)
(XXX Is that formulation correct?)
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XXX
XXX


===Unequal division of 65/54===  
===Non-aliquot division of 65/54===  
J.J. Weiss
J.J. Weiss


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XXX
XXX


==System Jacques Dudon (2006)==  
==Systems by Jacques Dudon==
===Aliquot system (2006)===  


Mandal positions (cents): &lt;span style="color: #00000a; font-family: Tahoma;"&gt;21,5 | 14,5 | 14,5 | 14,5 | 15 | 12 | 21,5&lt;/span&gt;
Mandal positions (cents): &lt;span style="color: #00000a; font-family: Tahoma;"&gt;21,5 | 14,5 | 14,5 | 14,5 | 15 | 12 | 21,5&lt;/span&gt;
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<h4>Original HTML content:</h4>
<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;tuning systems for qanun&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;span style="font-size: 150%;"&gt;&lt;strong&gt;Tuning systems for the qanun&lt;/strong&gt;&lt;/span&gt;&lt;br /&gt;
<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;tuning systems for qanun&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;span style="font-size: 150%;"&gt;&lt;strong&gt;Tuning systems for the qanun&lt;/strong&gt;&lt;/span&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextTocRule:40:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:40 --&gt;&lt;!-- ws:start:WikiTextTocRule:41: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Notation"&gt;Notation&lt;/a&gt;&lt;/div&gt;
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&lt;!-- ws:end:WikiTextTocRule:41 --&gt;&lt;!-- ws:start:WikiTextTocRule:42: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Explanation of the tuning tables"&gt;Explanation of the tuning tables&lt;/a&gt;&lt;/div&gt;
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&lt;!-- ws:end:WikiTextTocRule:42 --&gt;&lt;!-- ws:start:WikiTextTocRule:43: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Older systems"&gt;Older systems&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:44 --&gt;&lt;!-- ws:start:WikiTextTocRule:45: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Older systems"&gt;Older systems&lt;/a&gt;&lt;/div&gt;
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&lt;!-- ws:end:WikiTextTocRule:44 --&gt;&lt;!-- ws:start:WikiTextTocRule:45: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Older systems-First System J.J.Weiss-Variant with 128/119"&gt;Variant with 128/119&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:46 --&gt;&lt;!-- ws:start:WikiTextTocRule:47: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Older systems-First System J.J.Weiss-Variant with 128/119"&gt;Variant with 128/119&lt;/a&gt;&lt;/div&gt;
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&lt;!-- ws:end:WikiTextTocRule:47 --&gt;&lt;!-- ws:start:WikiTextTocRule:48: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Older systems-First System J.J.Weiss-Variant with 128/119 ascending/descending"&gt;Variant with 128/119 ascending/descending&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:46 --&gt;&lt;!-- ws:start:WikiTextTocRule:47: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Older systems-First System J.J.Weiss-Variant with 43/40 ascending/descending"&gt;Variant with 43/40 ascending/descending&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:48 --&gt;&lt;!-- ws:start:WikiTextTocRule:49: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Older systems-First System J.J.Weiss-Variant with 43/40 ascending/descending"&gt;Variant with 43/40 ascending/descending&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:47 --&gt;&lt;!-- ws:start:WikiTextTocRule:48: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Newer systems"&gt;Newer systems&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:49 --&gt;&lt;!-- ws:start:WikiTextTocRule:50: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Newer systems"&gt;Newer systems&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:48 --&gt;&lt;!-- ws:start:WikiTextTocRule:49: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Newer systems-System 2, better suited for ottoman maqams"&gt;System 2, better suited for ottoman maqams&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:50 --&gt;&lt;!-- ws:start:WikiTextTocRule:51: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Newer systems-System 2, better suited for ottoman maqams"&gt;System 2, better suited for ottoman maqams&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:49 --&gt;&lt;!-- ws:start:WikiTextTocRule:50: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Newer systems-Symmetrical model"&gt;Symmetrical model&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:51 --&gt;&lt;!-- ws:start:WikiTextTocRule:52: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Newer systems-Symmetrical model"&gt;Symmetrical model&lt;/a&gt;&lt;/div&gt;
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&lt;!-- ws:end:WikiTextTocRule:51 --&gt;&lt;!-- ws:start:WikiTextTocRule:52: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Newer systems-Super-symmetrical models-Equal division of 65/54 and its inverse"&gt;Equal division of 65/54 and its inverse&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:53 --&gt;&lt;!-- ws:start:WikiTextTocRule:54: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Newer systems-Super-symmetrical models-Aliquot division of 65/54 and its inverse"&gt;Aliquot division of 65/54 and its inverse&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:52 --&gt;&lt;!-- ws:start:WikiTextTocRule:53: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Newer systems-Super-symmetrical models-Unequal division of 65/54"&gt;Unequal division of 65/54&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:54 --&gt;&lt;!-- ws:start:WikiTextTocRule:55: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Newer systems-Super-symmetrical models-Non-aliquot division of 65/54"&gt;Non-aliquot division of 65/54&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:53 --&gt;&lt;!-- ws:start:WikiTextTocRule:54: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Newer systems-Super-symmetrical models-Equal division of the Zarlinian semitone"&gt;Equal division of the Zarlinian semitone&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:55 --&gt;&lt;!-- ws:start:WikiTextTocRule:56: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Newer systems-Super-symmetrical models-Equal division of the Zarlinian semitone"&gt;Equal division of the Zarlinian semitone&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:54 --&gt;&lt;!-- ws:start:WikiTextTocRule:55: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Newer systems-Super-symmetrical models-Ascending/descending with 54/49"&gt;Ascending/descending with 54/49&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:56 --&gt;&lt;!-- ws:start:WikiTextTocRule:57: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Newer systems-Super-symmetrical models-Ascending/descending with 54/49"&gt;Ascending/descending with 54/49&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:55 --&gt;&lt;!-- ws:start:WikiTextTocRule:56: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Newer systems-Super-symmetrical models-Ascending/descending with 14/13"&gt;Ascending/descending with 14/13&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:57 --&gt;&lt;!-- ws:start:WikiTextTocRule:58: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Newer systems-Super-symmetrical models-Ascending/descending with 14/13"&gt;Ascending/descending with 14/13&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:56 --&gt;&lt;!-- ws:start:WikiTextTocRule:57: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Newer systems-Super-symmetrical models-Ascending/descending with 11/10"&gt;Ascending/descending with 11/10&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:58 --&gt;&lt;!-- ws:start:WikiTextTocRule:59: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Newer systems-Super-symmetrical models-Ascending/descending with 11/10"&gt;Ascending/descending with 11/10&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:57 --&gt;&lt;!-- ws:start:WikiTextTocRule:58: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Newer systems-Super-symmetrical models-Ascending/descending with 35/32"&gt;Ascending/descending with 35/32&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:59 --&gt;&lt;!-- ws:start:WikiTextTocRule:60: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Newer systems-Super-symmetrical models-Ascending/descending with 35/32"&gt;Ascending/descending with 35/32&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:58 --&gt;&lt;!-- ws:start:WikiTextTocRule:59: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Newer systems-System Jacques Dudon (2006)"&gt;System Jacques Dudon (2006)&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:60 --&gt;&lt;!-- ws:start:WikiTextTocRule:61: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Newer systems-Systems by Jacques Dudon"&gt;Systems by Jacques Dudon&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:59 --&gt;&lt;!-- ws:start:WikiTextTocRule:60: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Newer systems-System Jacques Dudon (2006)-Arithmetic system"&gt;Arithmetic system&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:61 --&gt;&lt;!-- ws:start:WikiTextTocRule:62: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Newer systems-Systems by Jacques Dudon-Aliquot system (2006)"&gt;Aliquot system (2006)&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:60 --&gt;&lt;!-- ws:start:WikiTextTocRule:61: --&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:62 --&gt;&lt;!-- ws:start:WikiTextTocRule:63: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Newer systems-Systems by Jacques Dudon-Arithmetic system"&gt;Arithmetic system&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:61 --&gt;Julien Jalaleddine Weiss, used with permission.&lt;br /&gt;
&lt;!-- ws:end:WikiTextTocRule:63 --&gt;&lt;!-- ws:start:WikiTextTocRule:64: --&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:64 --&gt;Julien Jalaleddine Weiss, used with permission.&lt;br /&gt;
Reference: Pohlit, Stefan. 2011. Julien Jalâl Ed-Dine Weiss – A New Qānūn System: Its Application in the Performance Practice of the Ensemble “Al-Kindi” and in Contemporary Western Music. PhD Thesis, MIAM/Istanbul Technical University.&lt;br /&gt;
Reference: Pohlit, Stefan. 2011. Julien Jalâl Ed-Dine Weiss – A New Qānūn System: Its Application in the Performance Practice of the Ensemble “Al-Kindi” and in Contemporary Western Music. PhD Thesis, MIAM/Istanbul Technical University.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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For the steps in between, additional symbols are used - altogether 7 symbols for raising pitches and 7 for lowering pitches.&lt;br /&gt;
For the steps in between, additional symbols are used - altogether 7 symbols for raising pitches and 7 for lowering pitches.&lt;br /&gt;
This gives 15 potential different pitches per base note. Seven base notes (C, D, E, F, G, A, B or Do, Re, Mi, Fa, Sol, La, Si) lead to a notation system of 7*15=105 pitches. See the following document.&lt;br /&gt;
This gives 15 potential different pitches per base note. Seven base notes (C, D, E, F, G, A, B or Do, Re, Mi, Fa, Sol, La, Si) lead to a notation system of 7*15=105 pitches. See the following document.&lt;br /&gt;
&lt;!-- ws:start:WikiTextFileRule:806:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/Tableaux%20JJW%20VIII-2011.pdf?h=52&amp;amp;w=320&amp;quot; class=&amp;quot;WikiFile&amp;quot; id=&amp;quot;wikitext@@file@@Tableaux JJW VIII-2011.pdf&amp;quot; title=&amp;quot;File: Tableaux JJW VIII-2011.pdf&amp;quot; width=&amp;quot;320&amp;quot; height=&amp;quot;52&amp;quot; /&amp;gt; --&gt;&lt;div class="objectEmbed"&gt;&lt;a href="/file/view/Tableaux%20JJW%20VIII-2011.pdf/253043932/Tableaux%20JJW%20VIII-2011.pdf" onclick="ws.common.trackFileLink('/file/view/Tableaux%20JJW%20VIII-2011.pdf/253043932/Tableaux%20JJW%20VIII-2011.pdf');"&gt;&lt;img src="http://www.wikispaces.com/i/mime/32/application/pdf.png" height="32" width="32" alt="Tableaux JJW VIII-2011.pdf" /&gt;&lt;/a&gt;&lt;div&gt;&lt;a href="/file/view/Tableaux%20JJW%20VIII-2011.pdf/253043932/Tableaux%20JJW%20VIII-2011.pdf" onclick="ws.common.trackFileLink('/file/view/Tableaux%20JJW%20VIII-2011.pdf/253043932/Tableaux%20JJW%20VIII-2011.pdf');" class="filename" title="Tableaux JJW VIII-2011.pdf"&gt;Tableaux JJW VIII-2011.pdf&lt;/a&gt;&lt;br /&gt;&lt;ul&gt;&lt;li&gt;&lt;a href="/file/detail/Tableaux%20JJW%20VIII-2011.pdf"&gt;Details&lt;/a&gt;&lt;/li&gt;&lt;li&gt;&lt;a href="/file/view/Tableaux%20JJW%20VIII-2011.pdf/253043932/Tableaux%20JJW%20VIII-2011.pdf"&gt;Download&lt;/a&gt;&lt;/li&gt;&lt;li style="color: #666"&gt;130 KB&lt;/li&gt;&lt;/ul&gt;&lt;/div&gt;&lt;/div&gt;&lt;!-- ws:end:WikiTextFileRule:806 --&gt;&lt;br /&gt;
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(used with permission J. J. Weiss/S. Pohlit)&lt;br /&gt;
(used with permission J. J. Weiss/S. Pohlit)&lt;br /&gt;
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  J.J. Weiss&lt;br /&gt;
  J.J. Weiss&lt;br /&gt;
Advantage: marked contrast between Segah of Ushaq and Segah of arabic Rast.&lt;br /&gt;
Advantage: marked contrast between Segah of Ushaq and Segah of arabic Rast.&lt;br /&gt;
Equal division of 65/54 (320.98 cents)&lt;br /&gt;
Aliquot division of 65/54 (320.98 cents)&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Mandal positions (cents): &lt;span style="font-family: Tahoma;"&gt;22|13|13|1&lt;/span&gt;&lt;span style="color: #00000a; font-family: Tahoma;"&gt;8&lt;/span&gt;&lt;span style="font-family: Tahoma;"&gt;|13|13|22&lt;/span&gt;&lt;br /&gt;
Mandal positions (cents): &lt;span style="font-family: Tahoma;"&gt;22|13|13|1&lt;/span&gt;&lt;span style="color: #00000a; font-family: Tahoma;"&gt;8&lt;/span&gt;&lt;span style="font-family: Tahoma;"&gt;|13|13|22&lt;/span&gt;&lt;br /&gt;
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  Characteristics of super-symmetric systems: no difference between ascending and descending ratios.&lt;br /&gt;
  Characteristics of super-symmetric systems: no difference between ascending and descending ratios.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc11"&gt;&lt;a name="Newer systems-Super-symmetrical models-Equal division of 65/54 and its inverse"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;Equal division of 65/54 and its inverse&lt;/h3&gt;
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc11"&gt;&lt;a name="Newer systems-Super-symmetrical models-Aliquot division of 65/54 and its inverse"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;Aliquot division of 65/54 and its inverse&lt;/h3&gt;
  J.J. Weiss&lt;br /&gt;
  J.J. Weiss&lt;br /&gt;
(XXX Is that formulation correct?)&lt;br /&gt;
(XXX Is that formulation correct?)&lt;br /&gt;
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XXX&lt;br /&gt;
XXX&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc12"&gt;&lt;a name="Newer systems-Super-symmetrical models-Unequal division of 65/54"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;Unequal division of 65/54&lt;/h3&gt;
&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc12"&gt;&lt;a name="Newer systems-Super-symmetrical models-Non-aliquot division of 65/54"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;Non-aliquot division of 65/54&lt;/h3&gt;
  J.J. Weiss&lt;br /&gt;
  J.J. Weiss&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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XXX&lt;br /&gt;
XXX&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;a name="Newer systems-System Jacques Dudon (2006)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;System Jacques Dudon (2006)&lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;a name="Newer systems-Systems by Jacques Dudon"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;Systems by Jacques Dudon&lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:38:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc19"&gt;&lt;a name="Newer systems-Systems by Jacques Dudon-Aliquot system (2006)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:38 --&gt;Aliquot system (2006)&lt;/h3&gt;
  &lt;br /&gt;
  &lt;br /&gt;
Mandal positions (cents): &lt;span style="color: #00000a; font-family: Tahoma;"&gt;21,5 | 14,5 | 14,5 | 14,5 | 15 | 12 | 21,5&lt;/span&gt;&lt;br /&gt;
Mandal positions (cents): &lt;span style="color: #00000a; font-family: Tahoma;"&gt;21,5 | 14,5 | 14,5 | 14,5 | 15 | 12 | 21,5&lt;/span&gt;&lt;br /&gt;
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Interval table of just intervals XXX&lt;br /&gt;
Interval table of just intervals XXX&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:38:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc19"&gt;&lt;a name="Newer systems-System Jacques Dudon (2006)-Arithmetic system"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:38 --&gt;Arithmetic system&lt;/h3&gt;
&lt;!-- ws:start:WikiTextHeadingRule:40:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc20"&gt;&lt;a name="Newer systems-Systems by Jacques Dudon-Arithmetic system"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:40 --&gt;Arithmetic system&lt;/h3&gt;
  &lt;br /&gt;
  &lt;br /&gt;
Mandal positions (cents): &lt;span style="color: #00000a; font-family: Tahoma;"&gt;21,5 | 12 | 15 | 14,5 | 14,5 | 14,5 | 21,5 &lt;/span&gt;&lt;br /&gt;
Mandal positions (cents): &lt;span style="color: #00000a; font-family: Tahoma;"&gt;21,5 | 12 | 15 | 14,5 | 14,5 | 14,5 | 21,5 &lt;/span&gt;&lt;br /&gt;