Tuning systems for qanun: Difference between revisions

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<span style="font-size: 150%;">**Tuning systems for the qanun**</span>
[[toc]]
Julien Jalaleddine Weiss, used with permission.
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.

Online version of Stefan Pohlit's dissertation: see [[http://stefanpohlit.com/dissertation.engl..htm]]

The tuning tables on this page are specifically designed for the tuning system of the [[qanun]] (see the link for details on the system of tuning and playing a qanun with mandals/orabs).

The possible pitches of a string obtained via raising/lowering the mandals lie within one [[2187_2048|apotome (2187/2048, 113.7 cents)]], in both directions. All systems divde the apotome into 7 parts, which requires 14 mandals per string (one apotome up and down).

The first rough subdivision of the apotome is always into one [[81_80|syntonic comma (81/80, 21.5 cents)]], one [[25_24|Zarlinian semitone (25/24, 70.7 cents)]] and another syntonic comma. The middle part (25/24, Zarlinian semitone) is then further subdivided into 5 (unequal or equal) parts. The various systems differ mainly in the division of the middle part.

=Notation= 
Raising a pitch by an apotome is notated with "#", lowering a pitch by the same amount is notated with "b".
For the steps in between, additional symbols are used - altogether 7 symbols for raising pitches and 7 for lowering pitches.
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.
[[file:Tableaux JJW VIII-2011.pdf]]

(used with permission J. J. Weiss/S. Pohlit)

The base notes without accidentals form a pythagorean diatonic scale, with a major third of [[81_80|81/80]], a major sixth of [[27_16|27/16]] and a major seventh of [[243_128|243/128]].

A first characteristic of this system is that sharps are higher than flats: C# is one apotome (114 cents) above C, while Db is 9/8 (214 cents) minus one apotome = 90 cents. Sharps being higher than flats is a property of pythagorean systems (unlike [[meantone]] systems).

A second characteristic is that one mandal position below C# (i.e. one syntonic comma down) gives 114 - 22 = 92 cents, differenig from Db by one [[32805_32768|schisma (2 cents)]].
The interval of the schisma is present and can be played on a qanun in any of the tuning systems described here.

=Older systems= 
==First System J.J.Weiss== 
Luthier: Ejder Gulec.
Subdivision of 25/24 into 65/64 (26 cents), 144/143 (12 cents) and 55/54 (32 cents).
65/64 and 55/54 are each split into two.

This gives the following interval positions of the mandals: 22, 13, 13, 12, 16, 16, 22 cents.

XXX

===Variant with 128/119=== 
128/119: 126.2 cents
XXX

===Variant with 128/119 ascending/descending=== 
XXX

===Variant with 43/40 ascending/descending=== 
43/40: 125.2 cents
XXX

=Newer systems= 
==System 2, better suited for ottoman maqams== 
J.J. Weiss, Qanun no. 9. Luthier: Kenan Ozten.

Mandal positions (cents): <span style="color: #00000a; font-family: Tahoma;">22|16|15|12|13|14|22</span>

XXX

==Symmetrical model== 
J.J. Weiss
Advantage: marked contrast between Segah of Ushaq and Segah of arabic Rast.
Aliquot division of 65/54 (320.98 cents)

Mandal positions (cents): <span style="font-family: Tahoma;">22|13|13|1</span><span style="color: #00000a; font-family: Tahoma;">8</span><span style="font-family: Tahoma;">|13|13|22</span>

XXX

==Super-symmetrical models== 
Characteristics of super-symmetric systems: no difference between ascending and descending ratios.

===Aliquot division of 65/54 and its inverse=== 
J.J. Weiss
(XXX Is that formulation correct?)
Ascending/descending with 43/40 (125.2 cents).

Mandal positions (cents): <span style="font-family: Tahoma;">22|13|13|1</span><span style="color: #00000a; font-family: Tahoma;">8</span><span style="font-family: Tahoma;">|13|13</span><span style="color: #00000a; font-family: Tahoma;">|</span><span style="font-family: Tahoma;">22</span>

XXX

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

XXX

===Equal division of the Zarlinian semitone=== 
J.J. Weiss
This is the simplest variant for luthiers...

Mandal positions (cents): <span style="color: #00000a; font-family: Tahoma;">22|14|14|14|14|14|22</span>

===Ascending/descending with 54/49=== 
J.J. Weiss
[[54_49|54/49]]: 168.2 cents, Zalzal's mujannab (Al Farabi)

Mandal positions (cents): <span style="color: #00000a; font-family: Tahoma;">22|14|17|13|13|13|22</span>
<span style="color: #00000a; font-family: Tahoma;">XXX</span> There seem to be typos here...

XXX

===Ascending/descending with 14/13=== 
J.J. Weiss
<span style="color: #00000a; font-family: Tahoma;">14/13: 128.3 cents</span>
<span style="color: #00000a; font-family: Tahoma;">XXX</span>

Mandal positions (cents): <span style="color: #00000a; font-family: Tahoma;">22, 16, 13, 12, 13, 16, 22</span>

XXX

===Ascending/descending with 11/10=== 
J.J. Weiss
11/10: 165 cents

Mandal positions (cents): <span style="color: #00000a; font-family: Tahoma;">22|16|15|13|13|13|22</span>

XXX

===Ascending/descending with 35/32=== 
J.J. Weiss
35/32: 155.14 cents

XXX

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

Mandal positions (cents): <span style="color: #00000a; font-family: Tahoma;">21,5 | 14,5 | 14,5 | 14,5 | 15 | 12 | 21,5</span>

XXX

===Arithmetic system=== 

Mandal positions (cents): <span style="color: #00000a; font-family: Tahoma;">21,5 | 12 | 15 | 14,5 | 14,5 | 14,5 | 21,5 </span>

XXX

Original HTML content:

<html><head><title>tuning systems for qanun</title></head><body><span style="font-size: 150%;"><strong>Tuning systems for the qanun</strong></span><br />
<!-- ws:start:WikiTextTocRule:40:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --><div style="margin-left: 1em;"><a href="#Notation">Notation</a></div>
<!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --><div style="margin-left: 1em;"><a href="#Older systems">Older systems</a></div>
<!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --><div style="margin-left: 2em;"><a href="#Older systems-First System J.J.Weiss">First System J.J.Weiss</a></div>
<!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --><div style="margin-left: 3em;"><a href="#Older systems-First System J.J.Weiss-Variant with 128/119">Variant with 128/119</a></div>
<!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --><div style="margin-left: 3em;"><a href="#Older systems-First System J.J.Weiss-Variant with 128/119 ascending/descending">Variant with 128/119 ascending/descending</a></div>
<!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --><div style="margin-left: 3em;"><a href="#Older systems-First System J.J.Weiss-Variant with 43/40 ascending/descending">Variant with 43/40 ascending/descending</a></div>
<!-- ws:end:WikiTextTocRule:46 --><!-- ws:start:WikiTextTocRule:47: --><div style="margin-left: 1em;"><a href="#Newer systems">Newer systems</a></div>
<!-- ws:end:WikiTextTocRule:47 --><!-- ws:start:WikiTextTocRule:48: --><div style="margin-left: 2em;"><a href="#Newer systems-System 2, better suited for ottoman maqams">System 2, better suited for ottoman maqams</a></div>
<!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --><div style="margin-left: 2em;"><a href="#Newer systems-Symmetrical model">Symmetrical model</a></div>
<!-- ws:end:WikiTextTocRule:49 --><!-- ws:start:WikiTextTocRule:50: --><div style="margin-left: 2em;"><a href="#Newer systems-Super-symmetrical models">Super-symmetrical models</a></div>
<!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --><div style="margin-left: 3em;"><a href="#Newer systems-Super-symmetrical models-Aliquot division of 65/54 and its inverse">Aliquot division of 65/54 and its inverse</a></div>
<!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --><div style="margin-left: 3em;"><a href="#Newer systems-Super-symmetrical models-Non-aliquot division of 65/54">Non-aliquot division of 65/54</a></div>
<!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextTocRule:53: --><div style="margin-left: 3em;"><a href="#Newer systems-Super-symmetrical models-Equal division of the Zarlinian semitone">Equal division of the Zarlinian semitone</a></div>
<!-- ws:end:WikiTextTocRule:53 --><!-- ws:start:WikiTextTocRule:54: --><div style="margin-left: 3em;"><a href="#Newer systems-Super-symmetrical models-Ascending/descending with 54/49">Ascending/descending with 54/49</a></div>
<!-- ws:end:WikiTextTocRule:54 --><!-- ws:start:WikiTextTocRule:55: --><div style="margin-left: 3em;"><a href="#Newer systems-Super-symmetrical models-Ascending/descending with 14/13">Ascending/descending with 14/13</a></div>
<!-- ws:end:WikiTextTocRule:55 --><!-- ws:start:WikiTextTocRule:56: --><div style="margin-left: 3em;"><a href="#Newer systems-Super-symmetrical models-Ascending/descending with 11/10">Ascending/descending with 11/10</a></div>
<!-- ws:end:WikiTextTocRule:56 --><!-- ws:start:WikiTextTocRule:57: --><div style="margin-left: 3em;"><a href="#Newer systems-Super-symmetrical models-Ascending/descending with 35/32">Ascending/descending with 35/32</a></div>
<!-- ws:end:WikiTextTocRule:57 --><!-- ws:start:WikiTextTocRule:58: --><div style="margin-left: 2em;"><a href="#Newer systems-Systems by Jacques Dudon">Systems by Jacques Dudon</a></div>
<!-- ws:end:WikiTextTocRule:58 --><!-- ws:start:WikiTextTocRule:59: --><div style="margin-left: 3em;"><a href="#Newer systems-Systems by Jacques Dudon-Aliquot system (2006)">Aliquot system (2006)</a></div>
<!-- ws:end:WikiTextTocRule:59 --><!-- ws:start:WikiTextTocRule:60: --><div style="margin-left: 3em;"><a href="#Newer systems-Systems by Jacques Dudon-Arithmetic system">Arithmetic system</a></div>
<!-- ws:end:WikiTextTocRule:60 --><!-- ws:start:WikiTextTocRule:61: --></div>
<!-- ws:end:WikiTextTocRule:61 -->Julien Jalaleddine Weiss, used with permission.<br />
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.<br />
<br />
Online version of Stefan Pohlit's dissertation: see <a class="wiki_link_ext" href="http://stefanpohlit.com/dissertation.engl..htm" rel="nofollow">http://stefanpohlit.com/dissertation.engl..htm</a><br />
<br />
The tuning tables on this page are specifically designed for the tuning system of the <a class="wiki_link" href="/qanun">qanun</a> (see the link for details on the system of tuning and playing a qanun with mandals/orabs).<br />
<br />
The possible pitches of a string obtained via raising/lowering the mandals lie within one <a class="wiki_link" href="/2187_2048">apotome (2187/2048, 113.7 cents)</a>, in both directions. All systems divde the apotome into 7 parts, which requires 14 mandals per string (one apotome up and down).<br />
<br />
The first rough subdivision of the apotome is always into one <a class="wiki_link" href="/81_80">syntonic comma (81/80, 21.5 cents)</a>, one <a class="wiki_link" href="/25_24">Zarlinian semitone (25/24, 70.7 cents)</a> and another syntonic comma. The middle part (25/24, Zarlinian semitone) is then further subdivided into 5 (unequal or equal) parts. The various systems differ mainly in the division of the middle part.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Notation"></a><!-- ws:end:WikiTextHeadingRule:0 -->Notation</h1>
 Raising a pitch by an apotome is notated with &quot;#&quot;, lowering a pitch by the same amount is notated with &quot;b&quot;.<br />
For the steps in between, additional symbols are used - altogether 7 symbols for raising pitches and 7 for lowering pitches.<br />
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.<br />
<!-- ws:start:WikiTextFileRule:62:&lt;img src=&quot;http://www.wikispaces.com/site/embedthumbnail/file/Tableaux%20JJW%20VIII-2011.pdf?h=52&amp;w=320&quot; class=&quot;WikiFile&quot; id=&quot;wikitext@@file@@Tableaux JJW VIII-2011.pdf&quot; title=&quot;File: Tableaux JJW VIII-2011.pdf&quot; width=&quot;320&quot; height=&quot;52&quot; /&gt; --><div class="objectEmbed"><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');"><img src="http://www.wikispaces.com/i/mime/32/application/pdf.png" height="32" width="32" alt="Tableaux JJW VIII-2011.pdf" /></a><div><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">Tableaux JJW VIII-2011.pdf</a><br /><ul><li><a href="/file/detail/Tableaux%20JJW%20VIII-2011.pdf">Details</a></li><li><a href="/file/view/Tableaux%20JJW%20VIII-2011.pdf/253043932/Tableaux%20JJW%20VIII-2011.pdf">Download</a></li><li style="color: #666">130 KB</li></ul></div></div><!-- ws:end:WikiTextFileRule:62 --><br />
<br />
(used with permission J. J. Weiss/S. Pohlit)<br />
<br />
The base notes without accidentals form a pythagorean diatonic scale, with a major third of <a class="wiki_link" href="/81_80">81/80</a>, a major sixth of <a class="wiki_link" href="/27_16">27/16</a> and a major seventh of <a class="wiki_link" href="/243_128">243/128</a>.<br />
<br />
A first characteristic of this system is that sharps are higher than flats: C# is one apotome (114 cents) above C, while Db is 9/8 (214 cents) minus one apotome = 90 cents. Sharps being higher than flats is a property of pythagorean systems (unlike <a class="wiki_link" href="/meantone">meantone</a> systems).<br />
<br />
A second characteristic is that one mandal position below C# (i.e. one syntonic comma down) gives 114 - 22 = 92 cents, differenig from Db by one <a class="wiki_link" href="/32805_32768">schisma (2 cents)</a>.<br />
The interval of the schisma is present and can be played on a qanun in any of the tuning systems described here.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Older systems"></a><!-- ws:end:WikiTextHeadingRule:2 -->Older systems</h1>
 <!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="Older systems-First System J.J.Weiss"></a><!-- ws:end:WikiTextHeadingRule:4 -->First System J.J.Weiss</h2>
 Luthier: Ejder Gulec.<br />
Subdivision of 25/24 into 65/64 (26 cents), 144/143 (12 cents) and 55/54 (32 cents).<br />
65/64 and 55/54 are each split into two.<br />
<br />
This gives the following interval positions of the mandals: 22, 13, 13, 12, 16, 16, 22 cents.<br />
<br />
XXX<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="Older systems-First System J.J.Weiss-Variant with 128/119"></a><!-- ws:end:WikiTextHeadingRule:6 -->Variant with 128/119</h3>
 128/119: 126.2 cents<br />
XXX<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="Older systems-First System J.J.Weiss-Variant with 128/119 ascending/descending"></a><!-- ws:end:WikiTextHeadingRule:8 -->Variant with 128/119 ascending/descending</h3>
 XXX<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="Older systems-First System J.J.Weiss-Variant with 43/40 ascending/descending"></a><!-- ws:end:WikiTextHeadingRule:10 -->Variant with 43/40 ascending/descending</h3>
 43/40: 125.2 cents<br />
XXX<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h1&gt; --><h1 id="toc6"><a name="Newer systems"></a><!-- ws:end:WikiTextHeadingRule:12 -->Newer systems</h1>
 <!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Newer systems-System 2, better suited for ottoman maqams"></a><!-- ws:end:WikiTextHeadingRule:14 -->System 2, better suited for ottoman maqams</h2>
 J.J. Weiss, Qanun no. 9. Luthier: Kenan Ozten.<br />
<br />
Mandal positions (cents): <span style="color: #00000a; font-family: Tahoma;">22|16|15|12|13|14|22</span><br />
<br />
XXX<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name="Newer systems-Symmetrical model"></a><!-- ws:end:WikiTextHeadingRule:16 -->Symmetrical model</h2>
 J.J. Weiss<br />
Advantage: marked contrast between Segah of Ushaq and Segah of arabic Rast.<br />
Aliquot division of 65/54 (320.98 cents)<br />
<br />
Mandal positions (cents): <span style="font-family: Tahoma;">22|13|13|1</span><span style="color: #00000a; font-family: Tahoma;">8</span><span style="font-family: Tahoma;">|13|13|22</span><br />
<br />
XXX<br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Newer systems-Super-symmetrical models"></a><!-- ws:end:WikiTextHeadingRule:18 -->Super-symmetrical models</h2>
 Characteristics of super-symmetric systems: no difference between ascending and descending ratios.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:20:&lt;h3&gt; --><h3 id="toc10"><a name="Newer systems-Super-symmetrical models-Aliquot division of 65/54 and its inverse"></a><!-- ws:end:WikiTextHeadingRule:20 -->Aliquot division of 65/54 and its inverse</h3>
 J.J. Weiss<br />
(XXX Is that formulation correct?)<br />
Ascending/descending with 43/40 (125.2 cents).<br />
<br />
Mandal positions (cents): <span style="font-family: Tahoma;">22|13|13|1</span><span style="color: #00000a; font-family: Tahoma;">8</span><span style="font-family: Tahoma;">|13|13</span><span style="color: #00000a; font-family: Tahoma;">|</span><span style="font-family: Tahoma;">22</span><br />
<br />
XXX<br />
<br />
<!-- ws:start:WikiTextHeadingRule:22:&lt;h3&gt; --><h3 id="toc11"><a name="Newer systems-Super-symmetrical models-Non-aliquot division of 65/54"></a><!-- ws:end:WikiTextHeadingRule:22 -->Non-aliquot division of 65/54</h3>
 J.J. Weiss<br />
<br />
XXX<br />
<br />
<!-- ws:start:WikiTextHeadingRule:24:&lt;h3&gt; --><h3 id="toc12"><a name="Newer systems-Super-symmetrical models-Equal division of the Zarlinian semitone"></a><!-- ws:end:WikiTextHeadingRule:24 -->Equal division of the Zarlinian semitone</h3>
 J.J. Weiss<br />
This is the simplest variant for luthiers...<br />
<br />
Mandal positions (cents): <span style="color: #00000a; font-family: Tahoma;">22|14|14|14|14|14|22</span><br />
<br />
<!-- ws:start:WikiTextHeadingRule:26:&lt;h3&gt; --><h3 id="toc13"><a name="Newer systems-Super-symmetrical models-Ascending/descending with 54/49"></a><!-- ws:end:WikiTextHeadingRule:26 -->Ascending/descending with 54/49</h3>
 J.J. Weiss<br />
<a class="wiki_link" href="/54_49">54/49</a>: 168.2 cents, Zalzal's mujannab (Al Farabi)<br />
<br />
Mandal positions (cents): <span style="color: #00000a; font-family: Tahoma;">22|14|17|13|13|13|22</span><br />
<span style="color: #00000a; font-family: Tahoma;">XXX</span> There seem to be typos here...<br />
<br />
XXX<br />
<br />
<!-- ws:start:WikiTextHeadingRule:28:&lt;h3&gt; --><h3 id="toc14"><a name="Newer systems-Super-symmetrical models-Ascending/descending with 14/13"></a><!-- ws:end:WikiTextHeadingRule:28 -->Ascending/descending with 14/13</h3>
 J.J. Weiss<br />
<span style="color: #00000a; font-family: Tahoma;">14/13: 128.3 cents</span><br />
<span style="color: #00000a; font-family: Tahoma;">XXX</span><br />
<br />
Mandal positions (cents): <span style="color: #00000a; font-family: Tahoma;">22, 16, 13, 12, 13, 16, 22</span><br />
<br />
XXX<br />
<br />
<!-- ws:start:WikiTextHeadingRule:30:&lt;h3&gt; --><h3 id="toc15"><a name="Newer systems-Super-symmetrical models-Ascending/descending with 11/10"></a><!-- ws:end:WikiTextHeadingRule:30 -->Ascending/descending with 11/10</h3>
 J.J. Weiss<br />
11/10: 165 cents<br />
<br />
Mandal positions (cents): <span style="color: #00000a; font-family: Tahoma;">22|16|15|13|13|13|22</span><br />
<br />
XXX<br />
<br />
<!-- ws:start:WikiTextHeadingRule:32:&lt;h3&gt; --><h3 id="toc16"><a name="Newer systems-Super-symmetrical models-Ascending/descending with 35/32"></a><!-- ws:end:WikiTextHeadingRule:32 -->Ascending/descending with 35/32</h3>
 J.J. Weiss<br />
35/32: 155.14 cents<br />
<br />
XXX<br />
<br />
<!-- ws:start:WikiTextHeadingRule:34:&lt;h2&gt; --><h2 id="toc17"><a name="Newer systems-Systems by Jacques Dudon"></a><!-- ws:end:WikiTextHeadingRule:34 -->Systems by Jacques Dudon</h2>
 <!-- ws:start:WikiTextHeadingRule:36:&lt;h3&gt; --><h3 id="toc18"><a name="Newer systems-Systems by Jacques Dudon-Aliquot system (2006)"></a><!-- ws:end:WikiTextHeadingRule:36 -->Aliquot system (2006)</h3>
 <br />
Mandal positions (cents): <span style="color: #00000a; font-family: Tahoma;">21,5 | 14,5 | 14,5 | 14,5 | 15 | 12 | 21,5</span><br />
<br />
XXX<br />
<br />
<!-- ws:start:WikiTextHeadingRule:38:&lt;h3&gt; --><h3 id="toc19"><a name="Newer systems-Systems by Jacques Dudon-Arithmetic system"></a><!-- ws:end:WikiTextHeadingRule:38 -->Arithmetic system</h3>
 <br />
Mandal positions (cents): <span style="color: #00000a; font-family: Tahoma;">21,5 | 12 | 15 | 14,5 | 14,5 | 14,5 | 21,5 </span><br />
<br />
XXX</body></html>

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 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2011-09-26 07:37:38 UTC</tt>.<br>
+: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2011-09-27 03:34:14 UTC</tt>.<br>
-: The original revision id was <tt>258183286</tt>.<br>
+: The original revision id was <tt>258553166</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>
@@ Line 15: / Line 15: @@
 The tuning tables on this page are specifically designed for the tuning system of the [[qanun]] (see the link for details on the system of tuning and playing a qanun with mandals/orabs).
-The possible pitches of a string obtained via raising/lowering the mandals lie within one [[2187_2048|apotome (2187/2048, 113.7 cents)]], in both directions. All systems divde the apotome into 7 parts, which requires 15 mandals per string (one apotome up and down).
+The possible pitches of a string obtained via raising/lowering the mandals lie within one [[2187_2048|apotome (2187/2048, 113.7 cents)]], in both directions. All systems divde the apotome into 7 parts, which requires 14 mandals per string (one apotome up and down).
 The first rough subdivision of the apotome is always into one [[81_80|syntonic comma (81/80, 21.5 cents)]], one [[25_24|Zarlinian semitone (25/24, 70.7 cents)]] and another syntonic comma. The middle part (25/24, Zarlinian semitone) is then further subdivided into 5 (unequal or equal) parts. The various systems differ mainly in the division of the middle part.
@@ Line 27: / Line 27: @@
 (used with permission J. J. Weiss/S. Pohlit)
+The base notes without accidentals form a pythagorean diatonic scale, with a major third of [[81_80|81/80]], a major sixth of [[27_16|27/16]] and a major seventh of [[243_128|243/128]].
+A first characteristic of this system is that sharps are higher than flats: C# is one apotome (114 cents) above C, while Db is 9/8 (214 cents) minus one apotome = 90 cents. Sharps being higher than flats is a property of pythagorean systems (unlike [[meantone]] systems).
+A second characteristic is that one mandal position below C# (i.e. one syntonic comma down) gives 114 - 22 = 92 cents, differenig from Db by one [[32805_32768|schisma (2 cents)]].
-XXX THE FOLLOWING PARTS OF THIS PAGE NEED MAJOR CORRECTIONS XXX
+The interval of the schisma is present and can be played on a qanun in any of the tuning systems described here.
-=Explanation of the tuning tables=
-The whole table covers roughly the range of a fourth (the range where the ajnas - maqam tetrachords - reside). Each row corresponds to one string of the qanun. In the first column stands the basic (relative) tuning of a string while the following columns indicate possible intervals that can be reached via raising/lowering the mandals.
-The first table contains the cent values and the second the just intervals, sometimes differing between ascending and descending ratios. The middle part (which is the one that is mainly varying among the systems) is marked in **bold**.
-Any given configuration of mandal positions, resulting in a certain set of pitches that can be played at a given time
-(base for a maqam tetrachord) is represented by a choice of one cell in each row.
 =Older systems=
@@ Line 48: / Line 42: @@
 This gives the following interval positions of the mandals: 22, 13, 13, 12, 16, 16, 22 cents.
-Interval table (cents):
+XXX
-|| 0 || 22 || **35** || **48** || **60** || **76** || 92 || 114 ||
-|| 90 || 112 || **125** || **138** || **150** || **166** || 182 || 204 ||
-|| 294 || 316 || **330** || **342** || **354** || **370** || 386 || 408 ||
-|| 384 || 406 || **420** || **432** || **444** || **460** || 476 || 498 ||
-|| 498 || 520 || **533** || **546** || **558** || **574** || 590 || 612 ||
-Interval table of just intervals (ascending, descending):
-|| 1/1 || 81/80 || **49/48** || **1053/1024** || **729/704** || **2673/2560** || 135/128 || 2187/2048 ||
-|| 256/243 || 16/15 || **784/729, 128/119, 43/40 (asc.)**
-**320/297 (desc.)** || **13/12 (asc.)**
-**88/81 (desc.)** || **12/11 (asc.)**
-**128/117 (desc.)** || **11/10, 208/189 (asc.)**
-**54/49 (desc.)** || 10/9 || 9/8 ||
-|| 32/27 || 6/5 || **98/81 (asc.)**
-**40/33 (desc.)** || **39/32 (asc.)**
-**11/9 (desc.)** || **27/22 (asc.)**
-**16/13 (desc.)** || **99/80, 26/21 (asc.)**
-**243/196, XXX (desc.)** || 5/4 || 81/64 ||
-|| 8192/6581 || XXX || **25088/19683** || **104/81** || **XXX** || **176/135** || 320/243 || 4/3 ||
-|| XXX || 27/20 || **351/258** || **XXX** || **243/176** || **891/640** || 45/32 || 729/512 ||
 ===Variant with 128/119===
@@ Line 87: / Line 61: @@
 Mandal positions (cents): &lt;span style="color: #00000a; font-family: Tahoma;"&gt;22|16|15|12|13|14|22&lt;/span&gt;
-Interval table (cents):
-|| 0 || 22 || **38** || **54** || **66** || **78** || 92 || 114 ||
-|| 90 || 112 || **128** || **143** || **156** || **168** || 182 || 204 ||
-|| 294 || 316 || **332** || **348** || **360** || **372** || 386 || 408 ||
-Interval table of just intervals:
 XXX
@@ Line 102: / Line 70: @@
 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;
-Interval table (cents):
-|| 0 || 22 || **35** || **48** || **66** || **79** || 92 || 114 ||
-|| 90 || 112 || **125** || **138** || **156** || **169** || 182 || 204 ||
-|| 294 || 316 || **329** || **342** || **360** || **373** || 386 || 408 ||
-Interval table of just intervals:
 XXX
@@ Line 120: / Line 82: @@
 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&lt;/span&gt;&lt;span style="color: #00000a; font-family: Tahoma;"&gt;|&lt;/span&gt;&lt;span style="font-family: Tahoma;"&gt;22&lt;/span&gt;
-Interval table (cents): same as in previous model.
-Interval table of just intervals:
 XXX
@@ Line 128: / Line 87: @@
 J.J. Weiss
-Interval table (cents):
-|| 0 || 22 ||   ||   ||   ||   ||   ||   ||
-|| 90 || 112 || **126** || **138** || **156** || **168** || 182 || 204 ||
-|| 294 || 316 || **330** || **342** || **360** || **372** || 386 || 408 ||
-Interval table of just intervals:
 XXX
@@ Line 141: / Line 94: @@
 Mandal positions (cents): &lt;span style="color: #00000a; font-family: Tahoma;"&gt;22|14|14|14|14|14|22&lt;/span&gt;
-Interval table (cents):
-|| 0 || 22 || **36** || **50** || **64** || **76** || 92 || 114 ||
-|| 90 || 112 || **126** || **140** || **154** || **168** || 182 || 204 ||
-|| 294 || 316 || **330** || **344** || **358** || **372** || 386 || 408 ||
-Interval table of just intervals (ascending, descending):
-|| 1/1 || 81/80 ||   ||   ||   ||   || 135/128 || 2187/2048 ||
-|| 256/243 || 16/15 || **100/93 (asc.)**
-**242/225 (desc.)** || **400/369 (asc.)**
-**248/225 (desc.)** || **200/183 (asc.)**
-**82/75 (desc.)** || **400/363 (asc.)**
-**248/225 (desc.)** || 10/9 || 9/8 ||
-|| 32/27 || 6/5 || **75/62 (asc.)**
-**121/100 (desc.)** || **50/41 (asc.)**
-**61/50 (desc.)** || **75/61 (asc.)**
-**49/40 (desc.)** || **150/121 (asc.)**
-**31/25 (desc.)** || 5/4 || 81/64 ||
 ===Ascending/descending with 54/49===
@@ Line 167: / Line 102: @@
 &lt;span style="color: #00000a; font-family: Tahoma;"&gt;XXX&lt;/span&gt; There seem to be typos here...
-Interval table (cents):
+XXX
-|| 0 || 22 || **36** || **48** || **60** || **76** || 92 || 114 ||
-|| 90 || 112 || **126** || **138** || **156** || **168** || 182 || 204 ||
-|| 294 || 316 || **330** ||   || **360** || **374** || 386 || 408 ||
-Interval table of just intervals XXX
 ===Ascending/descending with 14/13===
@@ Line 181: / Line 111: @@
 Mandal positions (cents): &lt;span style="color: #00000a; font-family: Tahoma;"&gt;22, 16, 13, 12, 13, 16, 22&lt;/span&gt;
-Interval table (cents):
-||   ||   ||   ||   ||   ||   ||   ||   ||
-|| 90 || 112 || **128** || **141** || **153** || **166** || 182 || 204 ||
-|| 294 || 316 || **332** || **345** || **357** || **370** || 386 || 408 ||
-Interval table of just intervals:
 XXX
@@ Line 195: / Line 119: @@
 Mandal positions (cents): &lt;span style="color: #00000a; font-family: Tahoma;"&gt;22|16|15|13|13|13|22&lt;/span&gt;
-Interval table (cents):
-XXX
-Interval table of just intervals:
 XXX
@@ Line 205: / Line 125: @@
 /32: 155.14 cents
-Interval table (cents):
-||   ||   ||   ||   ||   ||   ||   ||   ||
-|| 90 || 112 || **126** || **138.99** || **155.13** || **168** || 182 || 204 ||
-|| 294 || 316 || **330** || **342.90** || **359** || **372** || 386 || 408 ||
-Interval table of just intervals:
 XXX
@@ Line 218: / Line 132: @@
 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;
-Interval table (cents):
+XXX
-|| 0 || 21.5 || **36** || **50.5** || **65** || **80** || 92 || 113.7 ||
-|| 90 || 112 || **126** || **141** || **156** || **170** ||   || 204 ||
-|| 294 || 315.5 || **330** || **345** || **359.5** || **374.3** ||   || 408 ||
-Interval table of just intervals XXX
 ===Arithmetic system===
@@ Line 229: / Line 138: @@
 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;
-Interval table (cents):
+XXX</pre></div>
-|| 0 || 21.5 || **33.5** || **48.5** || **63** || **77.5** || 92 || 113.7 ||
-||   ||   || **124** || **138** || **153** || **169** || 182 || 204 ||
-|| 294 || 315.5 || **327.5** || **342.5** || **357** || **372** || 386 || 408 ||
-XXX Are there typos?
-Interval table of just intervals XXX</pre></div>
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+&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 35/32"&gt;Ascending/descending with 35/32&lt;/a&gt;&lt;/div&gt;
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+&lt;!-- ws:end:WikiTextTocRule:59 --&gt;&lt;!-- ws:start:WikiTextTocRule:60: --&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: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:60 --&gt;&lt;!-- ws:start:WikiTextTocRule:61: --&gt;&lt;/div&gt;
-&lt;!-- ws:end:WikiTextTocRule:63 --&gt;&lt;!-- ws:start:WikiTextTocRule:64: --&gt;&lt;/div&gt;
+&lt;!-- ws:end:WikiTextTocRule:61 --&gt;Julien Jalaleddine Weiss, used with permission.&lt;br /&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;
 &lt;br /&gt;
@@ Line 267: / Line 169: @@
 The tuning tables on this page are specifically designed for the tuning system of the &lt;a class="wiki_link" href="/qanun"&gt;qanun&lt;/a&gt; (see the link for details on the system of tuning and playing a qanun with mandals/orabs).&lt;br /&gt;
 &lt;br /&gt;
-The possible pitches of a string obtained via raising/lowering the mandals lie within one &lt;a class="wiki_link" href="/2187_2048"&gt;apotome (2187/2048, 113.7 cents)&lt;/a&gt;, in both directions. All systems divde the apotome into 7 parts, which requires 15 mandals per string (one apotome up and down).&lt;br /&gt;
+The possible pitches of a string obtained via raising/lowering the mandals lie within one &lt;a class="wiki_link" href="/2187_2048"&gt;apotome (2187/2048, 113.7 cents)&lt;/a&gt;, in both directions. All systems divde the apotome into 7 parts, which requires 14 mandals per string (one apotome up and down).&lt;br /&gt;
 &lt;br /&gt;
 The first rough subdivision of the apotome is always into one &lt;a class="wiki_link" href="/81_80"&gt;syntonic comma (81/80, 21.5 cents)&lt;/a&gt;, one &lt;a class="wiki_link" href="/25_24"&gt;Zarlinian semitone (25/24, 70.7 cents)&lt;/a&gt; and another syntonic comma. The middle part (25/24, Zarlinian semitone) is then further subdivided into 5 (unequal or equal) parts. The various systems differ mainly in the division of the middle part.&lt;br /&gt;
@@ Line 275: / Line 177: @@
 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;
-&lt;!-- ws:start:WikiTextFileRule:809:&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:809 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextFileRule:62:&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:62 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 (used with permission J. J. Weiss/S. Pohlit)&lt;br /&gt;
 &lt;br /&gt;
+The base notes without accidentals form a pythagorean diatonic scale, with a major third of &lt;a class="wiki_link" href="/81_80"&gt;81/80&lt;/a&gt;, a major sixth of &lt;a class="wiki_link" href="/27_16"&gt;27/16&lt;/a&gt; and a major seventh of &lt;a class="wiki_link" href="/243_128"&gt;243/128&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
+A first characteristic of this system is that sharps are higher than flats: C# is one apotome (114 cents) above C, while Db is 9/8 (214 cents) minus one apotome = 90 cents. Sharps being higher than flats is a property of pythagorean systems (unlike &lt;a class="wiki_link" href="/meantone"&gt;meantone&lt;/a&gt; systems).&lt;br /&gt;
 &lt;br /&gt;
+A second characteristic is that one mandal position below C# (i.e. one syntonic comma down) gives 114 - 22 = 92 cents, differenig from Db by one &lt;a class="wiki_link" href="/32805_32768"&gt;schisma (2 cents)&lt;/a&gt;.&lt;br /&gt;
+The interval of the schisma is present and can be played on a qanun in any of the tuning systems described here.&lt;br /&gt;
 &lt;br /&gt;
-XXX THE FOLLOWING PARTS OF THIS PAGE NEED MAJOR CORRECTIONS XXX&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Older systems"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Older systems&lt;/h1&gt;
-&lt;br /&gt;
+  &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Older systems-First System J.J.Weiss"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;First System J.J.Weiss&lt;/h2&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Explanation of the tuning tables"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Explanation of the tuning tables&lt;/h1&gt;
- The whole table covers roughly the range of a fourth (the range where the ajnas - maqam tetrachords - reside). Each row corresponds to one string of the qanun. In the first column stands the basic (relative) tuning of a string while the following columns indicate possible intervals that can be reached via raising/lowering the mandals.&lt;br /&gt;
-&lt;br /&gt;
-The first table contains the cent values and the second the just intervals, sometimes differing between ascending and descending ratios. The middle part (which is the one that is mainly varying among the systems) is marked in &lt;strong&gt;bold&lt;/strong&gt;.&lt;br /&gt;
-&lt;br /&gt;
-Any given configuration of mandal positions, resulting in a certain set of pitches that can be played at a given time&lt;br /&gt;
-(base for a maqam tetrachord) is represented by a choice of one cell in each row.&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="Older systems"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Older systems&lt;/h1&gt;
-  &lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Older systems-First System J.J.Weiss"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;First System J.J.Weiss&lt;/h2&gt;
   Luthier: Ejder Gulec.&lt;br /&gt;
 Subdivision of 25/24 into 65/64 (26 cents), 144/143 (12 cents) and 55/54 (32 cents).&lt;br /&gt;
@@ Line 300: / Line 196: @@
 This gives the following interval positions of the mandals: 22, 13, 13, 12, 16, 16, 22 cents.&lt;br /&gt;
 &lt;br /&gt;
-Interval table (cents):&lt;br /&gt;
+XXX&lt;br /&gt;
-&lt;table class="wiki_table"&gt;
-    &lt;tr&gt;
-        &lt;td&gt;0&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;22&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;35&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;48&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;60&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;76&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;92&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;114&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;90&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;112&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;125&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;138&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;150&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;166&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;182&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;204&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;294&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;316&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;330&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;342&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;354&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;370&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;386&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;408&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;384&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;406&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;420&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;432&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;444&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;460&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;476&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;498&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;498&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;520&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;533&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;546&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;558&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;574&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;590&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;612&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-&lt;/table&gt;
 &lt;br /&gt;
-Interval table of just intervals (ascending, descending):&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="Older systems-First System J.J.Weiss-Variant with 128/119"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Variant with 128/119&lt;/h3&gt;
-&lt;table class="wiki_table"&gt;
-    &lt;tr&gt;
-        &lt;td&gt;1/1&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;81/80&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;49/48&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;1053/1024&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;729/704&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;2673/2560&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;135/128&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;2187/2048&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;256/243&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;16/15&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;784/729, 128/119, 43/40 (asc.)&lt;/strong&gt;&lt;br /&gt;
-&lt;strong&gt;320/297 (desc.)&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;13/12 (asc.)&lt;/strong&gt;&lt;br /&gt;
-&lt;strong&gt;88/81 (desc.)&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;12/11 (asc.)&lt;/strong&gt;&lt;br /&gt;
-&lt;strong&gt;128/117 (desc.)&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;11/10, 208/189 (asc.)&lt;/strong&gt;&lt;br /&gt;
-&lt;strong&gt;54/49 (desc.)&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;10/9&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;9/8&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;32/27&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;6/5&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;98/81 (asc.)&lt;/strong&gt;&lt;br /&gt;
-&lt;strong&gt;40/33 (desc.)&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;39/32 (asc.)&lt;/strong&gt;&lt;br /&gt;
-&lt;strong&gt;11/9 (desc.)&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;27/22 (asc.)&lt;/strong&gt;&lt;br /&gt;
-&lt;strong&gt;16/13 (desc.)&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;99/80, 26/21 (asc.)&lt;/strong&gt;&lt;br /&gt;
-&lt;strong&gt;243/196, XXX (desc.)&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;5/4&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;81/64&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;8192/6581&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;XXX&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;25088/19683&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;104/81&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;XXX&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;176/135&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;320/243&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;4/3&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;XXX&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;27/20&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;351/258&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;XXX&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;243/176&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;891/640&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;45/32&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;729/512&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-&lt;/table&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc4"&gt;&lt;a name="Older systems-First System J.J.Weiss-Variant with 128/119"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Variant with 128/119&lt;/h3&gt;
 /119: 126.2 cents&lt;br /&gt;
 XXX&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="Older systems-First System J.J.Weiss-Variant with 128/119 ascending/descending"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Variant with 128/119 ascending/descending&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc4"&gt;&lt;a name="Older systems-First System J.J.Weiss-Variant with 128/119 ascending/descending"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Variant with 128/119 ascending/descending&lt;/h3&gt;
   XXX&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc6"&gt;&lt;a name="Older systems-First System J.J.Weiss-Variant with 43/40 ascending/descending"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Variant with 43/40 ascending/descending&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="Older systems-First System J.J.Weiss-Variant with 43/40 ascending/descending"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Variant with 43/40 ascending/descending&lt;/h3&gt;
 /40: 125.2 cents&lt;br /&gt;
 XXX&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Newer systems"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Newer systems&lt;/h1&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc6"&gt;&lt;a name="Newer systems"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Newer systems&lt;/h1&gt;
-  &lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Newer systems-System 2, better suited for ottoman maqams"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;System 2, better suited for ottoman maqams&lt;/h2&gt;
+  &lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Newer systems-System 2, better suited for ottoman maqams"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;System 2, better suited for ottoman maqams&lt;/h2&gt;
   J.J. Weiss, Qanun no. 9. Luthier: Kenan Ozten.&lt;br /&gt;
 &lt;br /&gt;
 Mandal positions (cents): &lt;span style="color: #00000a; font-family: Tahoma;"&gt;22|16|15|12|13|14|22&lt;/span&gt;&lt;br /&gt;
 &lt;br /&gt;
-Interval table (cents):&lt;br /&gt;
-&lt;table class="wiki_table"&gt;
-    &lt;tr&gt;
-        &lt;td&gt;0&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;22&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;38&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;54&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;66&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;78&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;92&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;114&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;90&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;112&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;128&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;143&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;156&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;168&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;182&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;204&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;294&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;316&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;332&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;348&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;360&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;372&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;386&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;408&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-&lt;/table&gt;
-&lt;br /&gt;
-Interval table of just intervals:&lt;br /&gt;
 XXX&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="Newer systems-Symmetrical model"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Symmetrical model&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Newer systems-Symmetrical model"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Symmetrical model&lt;/h2&gt;
   J.J. Weiss&lt;br /&gt;
 Advantage: marked contrast between Segah of Ushaq and Segah of arabic Rast.&lt;br /&gt;
@@ Line 590: / Line 224: @@
 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;
 &lt;br /&gt;
-Interval table (cents):&lt;br /&gt;
-&lt;table class="wiki_table"&gt;
-    &lt;tr&gt;
-        &lt;td&gt;0&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;22&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;35&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;48&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;66&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;79&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;92&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;114&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;90&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;112&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;125&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;138&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;156&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;169&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;182&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;204&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;294&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;316&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;329&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;342&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;360&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;373&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;386&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;408&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-&lt;/table&gt;
-&lt;br /&gt;
-Interval table of just intervals:&lt;br /&gt;
 XXX&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="Newer systems-Super-symmetrical models"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Super-symmetrical models&lt;/h2&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Newer systems-Super-symmetrical models"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Super-symmetrical models&lt;/h2&gt;
   Characteristics of super-symmetric systems: no difference between ascending and descending ratios.&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-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;
+&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc10"&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:20 --&gt;Aliquot division of 65/54 and its inverse&lt;/h3&gt;
   J.J. Weiss&lt;br /&gt;
 (XXX Is that formulation correct?)&lt;br /&gt;
@@ Line 664: / Line 236: @@
 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&lt;/span&gt;&lt;span style="color: #00000a; font-family: Tahoma;"&gt;|&lt;/span&gt;&lt;span style="font-family: Tahoma;"&gt;22&lt;/span&gt;&lt;br /&gt;
 &lt;br /&gt;
-Interval table (cents): same as in previous model.&lt;br /&gt;
-&lt;br /&gt;
-Interval table of just intervals:&lt;br /&gt;
 XXX&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-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;
+&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc11"&gt;&lt;a name="Newer systems-Super-symmetrical models-Non-aliquot division of 65/54"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;Non-aliquot division of 65/54&lt;/h3&gt;
   J.J. Weiss&lt;br /&gt;
 &lt;br /&gt;
-Interval table (cents):&lt;br /&gt;
-&lt;table class="wiki_table"&gt;
-    &lt;tr&gt;
-        &lt;td&gt;0&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;22&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;90&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;112&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;126&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;138&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;156&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;168&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;182&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;204&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;294&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;316&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;330&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;342&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;360&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;372&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;386&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;408&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-&lt;/table&gt;
-&lt;br /&gt;
-Interval table of just intervals:&lt;br /&gt;
 XXX&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc13"&gt;&lt;a name="Newer systems-Super-symmetrical models-Equal division of the Zarlinian semitone"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;Equal division of the Zarlinian semitone&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-Equal division of the Zarlinian semitone"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;Equal division of the Zarlinian semitone&lt;/h3&gt;
   J.J. Weiss&lt;br /&gt;
 This is the simplest variant for luthiers...&lt;br /&gt;
@@ Line 742: / Line 249: @@
 Mandal positions (cents): &lt;span style="color: #00000a; font-family: Tahoma;"&gt;22|14|14|14|14|14|22&lt;/span&gt;&lt;br /&gt;
 &lt;br /&gt;
-Interval table (cents):&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc13"&gt;&lt;a name="Newer systems-Super-symmetrical models-Ascending/descending with 54/49"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;Ascending/descending with 54/49&lt;/h3&gt;
-&lt;table class="wiki_table"&gt;
-    &lt;tr&gt;
-        &lt;td&gt;0&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;22&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;36&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;50&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;64&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;76&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;92&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;114&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;90&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;112&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;126&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;140&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;154&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;168&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;182&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;204&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;294&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;316&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;330&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;344&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;358&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;372&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;386&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;408&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-&lt;/table&gt;
-&lt;br /&gt;
-Interval table of just intervals (ascending, descending):&lt;br /&gt;
-&lt;table class="wiki_table"&gt;
-    &lt;tr&gt;
-        &lt;td&gt;1/1&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;81/80&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;135/128&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;2187/2048&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;256/243&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;16/15&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;100/93 (asc.)&lt;/strong&gt;&lt;br /&gt;
-&lt;strong&gt;242/225 (desc.)&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;400/369 (asc.)&lt;/strong&gt;&lt;br /&gt;
-&lt;strong&gt;248/225 (desc.)&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;200/183 (asc.)&lt;/strong&gt;&lt;br /&gt;
-&lt;strong&gt;82/75 (desc.)&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;400/363 (asc.)&lt;/strong&gt;&lt;br /&gt;
-&lt;strong&gt;248/225 (desc.)&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;10/9&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;9/8&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;32/27&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;6/5&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;75/62 (asc.)&lt;/strong&gt;&lt;br /&gt;
-&lt;strong&gt;121/100 (desc.)&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;50/41 (asc.)&lt;/strong&gt;&lt;br /&gt;
-&lt;strong&gt;61/50 (desc.)&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;75/61 (asc.)&lt;/strong&gt;&lt;br /&gt;
-&lt;strong&gt;49/40 (desc.)&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;150/121 (asc.)&lt;/strong&gt;&lt;br /&gt;
-&lt;strong&gt;31/25 (desc.)&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;5/4&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;81/64&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-&lt;/table&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc14"&gt;&lt;a name="Newer systems-Super-symmetrical models-Ascending/descending with 54/49"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;Ascending/descending with 54/49&lt;/h3&gt;
   J.J. Weiss&lt;br /&gt;
 &lt;a class="wiki_link" href="/54_49"&gt;54/49&lt;/a&gt;: 168.2 cents, Zalzal's mujannab (Al Farabi)&lt;br /&gt;
@@ Line 879: / Line 256: @@
 &lt;span style="color: #00000a; font-family: Tahoma;"&gt;XXX&lt;/span&gt; There seem to be typos here...&lt;br /&gt;
 &lt;br /&gt;
-Interval table (cents):&lt;br /&gt;
+XXX&lt;br /&gt;
-&lt;table class="wiki_table"&gt;
-    &lt;tr&gt;
-        &lt;td&gt;0&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;22&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;36&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;48&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;60&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;76&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;92&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;114&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;90&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;112&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;126&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;138&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;156&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;168&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;182&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;204&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;294&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;316&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;330&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;360&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;374&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;386&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;408&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-&lt;/table&gt;
 &lt;br /&gt;
-Interval table of just intervals XXX&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc14"&gt;&lt;a name="Newer systems-Super-symmetrical models-Ascending/descending with 14/13"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;Ascending/descending with 14/13&lt;/h3&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc15"&gt;&lt;a name="Newer systems-Super-symmetrical models-Ascending/descending with 14/13"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;Ascending/descending with 14/13&lt;/h3&gt;
   J.J. Weiss&lt;br /&gt;
 &lt;span style="color: #00000a; font-family: Tahoma;"&gt;14/13: 128.3 cents&lt;/span&gt;&lt;br /&gt;
@@ Line 949: / Line 265: @@
 Mandal positions (cents): &lt;span style="color: #00000a; font-family: Tahoma;"&gt;22, 16, 13, 12, 13, 16, 22&lt;/span&gt;&lt;br /&gt;
 &lt;br /&gt;
-Interval table (cents):&lt;br /&gt;
-&lt;table class="wiki_table"&gt;
-    &lt;tr&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;90&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;112&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;128&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;141&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;153&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;166&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;182&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;204&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;294&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;316&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;332&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;345&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;357&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;370&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;386&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;408&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-&lt;/table&gt;
-&lt;br /&gt;
-Interval table of just intervals:&lt;br /&gt;
 XXX&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc16"&gt;&lt;a name="Newer systems-Super-symmetrical models-Ascending/descending with 11/10"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;Ascending/descending with 11/10&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc15"&gt;&lt;a name="Newer systems-Super-symmetrical models-Ascending/descending with 11/10"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;Ascending/descending with 11/10&lt;/h3&gt;
   J.J. Weiss&lt;br /&gt;
 /10: 165 cents&lt;br /&gt;
@@ Line 1,019: / Line 273: @@
 Mandal positions (cents): &lt;span style="color: #00000a; font-family: Tahoma;"&gt;22|16|15|13|13|13|22&lt;/span&gt;&lt;br /&gt;
 &lt;br /&gt;
-Interval table (cents):&lt;br /&gt;
-XXX&lt;br /&gt;
-&lt;br /&gt;
-Interval table of just intervals:&lt;br /&gt;
 XXX&lt;br /&gt;
 &lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc17"&gt;&lt;a name="Newer systems-Super-symmetrical models-Ascending/descending with 35/32"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&gt;Ascending/descending with 35/32&lt;/h3&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc16"&gt;&lt;a name="Newer systems-Super-symmetrical models-Ascending/descending with 35/32"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;Ascending/descending with 35/32&lt;/h3&gt;
   J.J. Weiss&lt;br /&gt;
 /32: 155.14 cents&lt;br /&gt;
 &lt;br /&gt;
-Interval table (cents):&lt;br /&gt;
-&lt;table class="wiki_table"&gt;
-    &lt;tr&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;90&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;112&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;126&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;138.99&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;155.13&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;168&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;182&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;204&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;294&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;316&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;330&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;342.90&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;359&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;372&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;386&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;408&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-&lt;/table&gt;
-&lt;br /&gt;
-Interval table of just intervals:&lt;br /&gt;
 XXX&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-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:34:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc17"&gt;&lt;a name="Newer systems-Systems by Jacques Dudon"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&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;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc18"&gt;&lt;a name="Newer systems-Systems by Jacques Dudon-Aliquot system (2006)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;Aliquot system (2006)&lt;/h3&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;
 &lt;br /&gt;
-Interval table (cents):&lt;br /&gt;
+XXX&lt;br /&gt;
-&lt;table class="wiki_table"&gt;
-    &lt;tr&gt;
-        &lt;td&gt;0&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;21.5&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;36&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;50.5&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;65&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;80&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;92&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;113.7&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;90&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;112&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;126&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;141&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;156&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;170&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;204&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;294&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;315.5&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;330&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;345&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;359.5&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;374.3&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;408&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-&lt;/table&gt;
 &lt;br /&gt;
-Interval table of just intervals XXX&lt;br /&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-Arithmetic system"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:38 --&gt;Arithmetic system&lt;/h3&gt;
-&lt;br /&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;
 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;
 &lt;br /&gt;
-Interval table (cents):&lt;br /&gt;
+XXX&lt;/body&gt;&lt;/html&gt;</pre></div>
-&lt;table class="wiki_table"&gt;
-    &lt;tr&gt;
-        &lt;td&gt;0&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;21.5&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;33.5&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;48.5&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;63&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;77.5&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;92&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;113.7&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;124&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;138&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;153&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;169&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;182&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;204&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-    &lt;tr&gt;
-        &lt;td&gt;294&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;315.5&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;327.5&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;342.5&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;357&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;&lt;strong&gt;372&lt;/strong&gt;&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;386&lt;br /&gt;
-&lt;/td&gt;
-        &lt;td&gt;408&lt;br /&gt;
-&lt;/td&gt;
-    &lt;/tr&gt;
-&lt;/table&gt;
-XXX Are there typos?&lt;br /&gt;
-&lt;br /&gt;
-Interval table of just intervals XXX&lt;/body&gt;&lt;/html&gt;</pre></div>

Tuning systems for qanun: Difference between revisions

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