Tuning systems for qanun

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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.

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)]]. All systems use 8 mandals, allowing a subdivision of the apotome into 7 parts.

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.

=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= 
==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.

Interval table (cents):
|| 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== 
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>

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

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

===Symmetrical model=== 
J.J. Weiss
Advantage: marked contrast between Segah of Ushaq and Segah of arabic Rast.
Equal 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>

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

===Super-symmetrical model=== 
J.J. Weiss
Equal division of 65/54 (320.98 cents) and its inverse. (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>

Interval table (cents): same as in previous model.

Interval table of just intervals:
XXX

===Unequal division of 65/54=== 
J.J. Weiss
Unequal division of 65/54 (320.98 cents)

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>

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=== 
J.J. Weiss
54/49: 168.2 cents
XXX

Mandal positions (cents): <span style="color: #00000a; font-family: Tahoma;">22|14|17|13|13|13|22</span>

Interval table 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>

Interval table XXX

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

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

Interval table XXX

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

==System Jacques Dudon (2006)== 

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

Interval table 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>


Interval table 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:38:&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:38 --><!-- ws:start:WikiTextTocRule:39: --><div style="margin-left: 1em;"><a href="#Explanation of the tuning tables">Explanation of the tuning tables</a></div>
<!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --><div style="margin-left: 1em;"><a href="#Older systems">Older systems</a></div>
<!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --><div style="margin-left: 2em;"><a href="#Older systems-First System J.J.Weiss">First System J.J.Weiss</a></div>
<!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --><div style="margin-left: 2em;"><a href="#Older systems-Variant with 128/119">Variant with 128/119</a></div>
<!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --><div style="margin-left: 2em;"><a href="#Older systems-Variant with 128/119 ascending/descending">Variant with 128/119 ascending/descending</a></div>
<!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --><div style="margin-left: 2em;"><a href="#Older systems-Variant with 43/40 ascending/descending">Variant with 43/40 ascending/descending</a></div>
<!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --><div style="margin-left: 1em;"><a href="#Newer systems">Newer systems</a></div>
<!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: --><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:46 --><!-- ws:start:WikiTextTocRule:47: --><div style="margin-left: 2em;"><a href="#Newer systems-Symmetrical and super-symmetrical models">Symmetrical and super-symmetrical models</a></div>
<!-- ws:end:WikiTextTocRule:47 --><!-- ws:start:WikiTextTocRule:48: --><div style="margin-left: 3em;"><a href="#Newer systems-Symmetrical and super-symmetrical models-Symmetrical model">Symmetrical model</a></div>
<!-- ws:end:WikiTextTocRule:48 --><!-- ws:start:WikiTextTocRule:49: --><div style="margin-left: 3em;"><a href="#Newer systems-Symmetrical and super-symmetrical models-Super-symmetrical model">Super-symmetrical model</a></div>
<!-- ws:end:WikiTextTocRule:49 --><!-- ws:start:WikiTextTocRule:50: --><div style="margin-left: 3em;"><a href="#Newer systems-Symmetrical and super-symmetrical models-Unequal division of 65/54">Unequal division of 65/54</a></div>
<!-- ws:end:WikiTextTocRule:50 --><!-- ws:start:WikiTextTocRule:51: --><div style="margin-left: 3em;"><a href="#Newer systems-Symmetrical and super-symmetrical models-Equal division of the Zarlinian semitone">Equal division of the Zarlinian semitone</a></div>
<!-- ws:end:WikiTextTocRule:51 --><!-- ws:start:WikiTextTocRule:52: --><div style="margin-left: 3em;"><a href="#Newer systems-Symmetrical and super-symmetrical models-Ascending/descending with 54/49">Ascending/descending with 54/49</a></div>
<!-- ws:end:WikiTextTocRule:52 --><!-- ws:start:WikiTextTocRule:53: --><div style="margin-left: 3em;"><a href="#Newer systems-Symmetrical and super-symmetrical models-Ascending/descending with 14/13">Ascending/descending with 14/13</a></div>
<!-- ws:end:WikiTextTocRule:53 --><!-- ws:start:WikiTextTocRule:54: --><div style="margin-left: 3em;"><a href="#Newer systems-Symmetrical and super-symmetrical models-Ascending/descending with 11/10">Ascending/descending with 11/10</a></div>
<!-- ws:end:WikiTextTocRule:54 --><!-- ws:start:WikiTextTocRule:55: --><div style="margin-left: 3em;"><a href="#Newer systems-Symmetrical and super-symmetrical models-Ascending/descending with 35/32">Ascending/descending with 35/32</a></div>
<!-- ws:end:WikiTextTocRule:55 --><!-- ws:start:WikiTextTocRule:56: --><div style="margin-left: 2em;"><a href="#Newer systems-System Jacques Dudon (2006)">System Jacques Dudon (2006)</a></div>
<!-- ws:end:WikiTextTocRule:56 --><!-- ws:start:WikiTextTocRule:57: --><div style="margin-left: 3em;"><a href="#Newer systems-System Jacques Dudon (2006)-Arithmetic system">Arithmetic system</a></div>
<!-- ws:end:WikiTextTocRule:57 --><!-- ws:start:WikiTextTocRule:58: --></div>
<!-- ws:end:WikiTextTocRule:58 -->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<br />
Practice of the Ensemble “Al-Kindi” and in Contemporary Western Music. PhD Thesis, MIAM/Istanbul Technical<br />
University.<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>. All systems use 8 mandals, allowing a subdivision of the apotome into 7 parts.<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="Explanation of the tuning tables"></a><!-- ws:end:WikiTextHeadingRule:0 -->Explanation of the tuning tables</h1>
 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.<br />
<br />
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 <strong>bold</strong>.<br />
<br />
Any given configuration of mandal positions, resulting in a certain set of pitches that can be played at a given time<br />
(base for a maqam tetrachord) is represented by a choice of one cell in each row.<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 />
Interval table (cents):<br />


<table class="wiki_table">
    <tr>
        <td>0<br />
</td>
        <td>22<br />
</td>
        <td><strong>35</strong><br />
</td>
        <td><strong>48</strong><br />
</td>
        <td><strong>60</strong><br />
</td>
        <td><strong>76</strong><br />
</td>
        <td>92<br />
</td>
        <td>114<br />
</td>
    </tr>
    <tr>
        <td>90<br />
</td>
        <td>112<br />
</td>
        <td><strong>125</strong><br />
</td>
        <td><strong>138</strong><br />
</td>
        <td><strong>150</strong><br />
</td>
        <td><strong>166</strong><br />
</td>
        <td>182<br />
</td>
        <td>204<br />
</td>
    </tr>
    <tr>
        <td>294<br />
</td>
        <td>316<br />
</td>
        <td><strong>330</strong><br />
</td>
        <td><strong>342</strong><br />
</td>
        <td><strong>354</strong><br />
</td>
        <td><strong>370</strong><br />
</td>
        <td>386<br />
</td>
        <td>408<br />
</td>
    </tr>
    <tr>
        <td>384<br />
</td>
        <td>406<br />
</td>
        <td><strong>420</strong><br />
</td>
        <td><strong>432</strong><br />
</td>
        <td><strong>444</strong><br />
</td>
        <td><strong>460</strong><br />
</td>
        <td>476<br />
</td>
        <td>498<br />
</td>
    </tr>
    <tr>
        <td>498<br />
</td>
        <td>520<br />
</td>
        <td><strong>533</strong><br />
</td>
        <td><strong>546</strong><br />
</td>
        <td><strong>558</strong><br />
</td>
        <td><strong>574</strong><br />
</td>
        <td>590<br />
</td>
        <td>612<br />
</td>
    </tr>
</table>

<br />
Interval table of just intervals (ascending, descending):<br />


<table class="wiki_table">
    <tr>
        <td>1/1<br />
</td>
        <td>81/80<br />
</td>
        <td>49/48<br />
</td>
        <td>1053/1024<br />
</td>
        <td>729/704<br />
</td>
        <td>2673/2560<br />
</td>
        <td>135/128<br />
</td>
        <td>2187/2048<br />
</td>
    </tr>
    <tr>
        <td>256/243<br />
</td>
        <td>16/15<br />
</td>
        <td>784/729, 128/119, 43/40 (asc.)<br />
320/297 (desc.)<br />
</td>
        <td>13/12 (asc.)<br />
88/81 (desc.)<br />
</td>
        <td>12/11 (asc.)<br />
128/117 (desc.)<br />
</td>
        <td>11/10, 208/189 (asc.)<br />
54/49 (desc.)<br />
</td>
        <td>10/9<br />
</td>
        <td>9/8<br />
</td>
    </tr>
    <tr>
        <td>32/27<br />
</td>
        <td>6/5<br />
</td>
        <td>98/81 (asc.)<br />
40/33 (desc.)<br />
</td>
        <td>39/32 (asc.)<br />
11/9 (desc.)<br />
</td>
        <td>27/22 (asc.)<br />
16/13 (desc.)<br />
</td>
        <td>99/80, 26/21 (asc.)<br />
243/196, XXX (desc.)<br />
</td>
        <td>5/4<br />
</td>
        <td>81/64<br />
</td>
    </tr>
    <tr>
        <td>8192/6581<br />
</td>
        <td>XXX<br />
</td>
        <td>25088/19683<br />
</td>
        <td>104/81<br />
</td>
        <td>XXX<br />
</td>
        <td>176/135<br />
</td>
        <td>320/243<br />
</td>
        <td>4/3<br />
</td>
    </tr>
    <tr>
        <td>XXX<br />
</td>
        <td>27/20<br />
</td>
        <td>351/258<br />
</td>
        <td>XXX<br />
</td>
        <td>243/176<br />
</td>
        <td>891/640<br />
</td>
        <td>45/32<br />
</td>
        <td>729/512<br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Older systems-Variant with 128/119"></a><!-- ws:end:WikiTextHeadingRule:6 -->Variant with 128/119</h2>
 128/119: 126.2 cents<br />
XXX<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Older systems-Variant with 128/119 ascending/descending"></a><!-- ws:end:WikiTextHeadingRule:8 -->Variant with 128/119 ascending/descending</h2>
 XXX<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Older systems-Variant with 43/40 ascending/descending"></a><!-- ws:end:WikiTextHeadingRule:10 -->Variant with 43/40 ascending/descending</h2>
 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 />
Interval table (cents):<br />


<table class="wiki_table">
    <tr>
        <td>0<br />
</td>
        <td>22<br />
</td>
        <td><strong>38</strong><br />
</td>
        <td><strong>54</strong><br />
</td>
        <td><strong>66</strong><br />
</td>
        <td><strong>78</strong><br />
</td>
        <td>92<br />
</td>
        <td>114<br />
</td>
    </tr>
    <tr>
        <td>90<br />
</td>
        <td>112<br />
</td>
        <td><strong>128</strong><br />
</td>
        <td><strong>143</strong><br />
</td>
        <td><strong>156</strong><br />
</td>
        <td><strong>168</strong><br />
</td>
        <td>182<br />
</td>
        <td>204<br />
</td>
    </tr>
    <tr>
        <td>294<br />
</td>
        <td>316<br />
</td>
        <td><strong>332</strong><br />
</td>
        <td><strong>348</strong><br />
</td>
        <td><strong>360</strong><br />
</td>
        <td><strong>372</strong><br />
</td>
        <td>386<br />
</td>
        <td>408<br />
</td>
    </tr>
</table>

<br />
Interval table of just intervals:<br />
XXX<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:&lt;h2&gt; --><h2 id="toc8"><a name="Newer systems-Symmetrical and super-symmetrical models"></a><!-- ws:end:WikiTextHeadingRule:16 -->Symmetrical and super-symmetrical models</h2>
 XXX<br />
Characteristics of super-symmetric systems: no difference between ascending and descending ratios.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:18:&lt;h3&gt; --><h3 id="toc9"><a name="Newer systems-Symmetrical and super-symmetrical models-Symmetrical model"></a><!-- ws:end:WikiTextHeadingRule:18 -->Symmetrical model</h3>
 J.J. Weiss<br />
Advantage: marked contrast between Segah of Ushaq and Segah of arabic Rast.<br />
Equal 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 />
Interval table (cents):<br />


<table class="wiki_table">
    <tr>
        <td>0<br />
</td>
        <td>22<br />
</td>
        <td><strong>35</strong><br />
</td>
        <td><strong>48</strong><br />
</td>
        <td><strong>66</strong><br />
</td>
        <td><strong>79</strong><br />
</td>
        <td>92<br />
</td>
        <td>114<br />
</td>
    </tr>
    <tr>
        <td>90<br />
</td>
        <td>112<br />
</td>
        <td><strong>125</strong><br />
</td>
        <td><strong>138</strong><br />
</td>
        <td><strong>156</strong><br />
</td>
        <td><strong>169</strong><br />
</td>
        <td>182<br />
</td>
        <td>204<br />
</td>
    </tr>
    <tr>
        <td>294<br />
</td>
        <td>316<br />
</td>
        <td><strong>329</strong><br />
</td>
        <td><strong>342</strong><br />
</td>
        <td><strong>360</strong><br />
</td>
        <td><strong>373</strong><br />
</td>
        <td>386<br />
</td>
        <td>408<br />
</td>
    </tr>
</table>

<br />
Interval table of just intervals:<br />
XXX<br />
<br />
<!-- ws:start:WikiTextHeadingRule:20:&lt;h3&gt; --><h3 id="toc10"><a name="Newer systems-Symmetrical and super-symmetrical models-Super-symmetrical model"></a><!-- ws:end:WikiTextHeadingRule:20 -->Super-symmetrical model</h3>
 J.J. Weiss<br />
Equal division of 65/54 (320.98 cents) and its inverse. (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 />
Interval table (cents): same as in previous model.<br />
<br />
Interval table of just intervals:<br />
XXX<br />
<br />
<!-- ws:start:WikiTextHeadingRule:22:&lt;h3&gt; --><h3 id="toc11"><a name="Newer systems-Symmetrical and super-symmetrical models-Unequal division of 65/54"></a><!-- ws:end:WikiTextHeadingRule:22 -->Unequal division of 65/54</h3>
 J.J. Weiss<br />
Unequal division of 65/54 (320.98 cents)<br />
<br />
XXX<br />
<br />
<!-- ws:start:WikiTextHeadingRule:24:&lt;h3&gt; --><h3 id="toc12"><a name="Newer systems-Symmetrical and 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 />
Interval table (cents):<br />


<table class="wiki_table">
    <tr>
        <td>0<br />
</td>
        <td>22<br />
</td>
        <td><strong>36</strong><br />
</td>
        <td><strong>50</strong><br />
</td>
        <td><strong>64</strong><br />
</td>
        <td><strong>76</strong><br />
</td>
        <td>92<br />
</td>
        <td>114<br />
</td>
    </tr>
    <tr>
        <td>90<br />
</td>
        <td>112<br />
</td>
        <td><strong>126</strong><br />
</td>
        <td><strong>140</strong><br />
</td>
        <td><strong>154</strong><br />
</td>
        <td><strong>168</strong><br />
</td>
        <td>182<br />
</td>
        <td>204<br />
</td>
    </tr>
    <tr>
        <td>294<br />
</td>
        <td>316<br />
</td>
        <td><strong>330</strong><br />
</td>
        <td><strong>344</strong><br />
</td>
        <td><strong>358</strong><br />
</td>
        <td><strong>372</strong><br />
</td>
        <td>386<br />
</td>
        <td>408<br />
</td>
    </tr>
</table>

<br />
Interval table of just intervals (ascending, descending):<br />


<table class="wiki_table">
    <tr>
        <td>1/1<br />
</td>
        <td>81/80<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>135/128<br />
</td>
        <td>2187/2048<br />
</td>
    </tr>
    <tr>
        <td>256/243<br />
</td>
        <td>16/15<br />
</td>
        <td>100/93 (asc.)<br />
242/225 (desc.)<br />
</td>
        <td>400/369 (asc.)<br />
248/225 (desc.)<br />
</td>
        <td>200/183 (asc.)<br />
82/75 (desc.)<br />
</td>
        <td>400/363 (asc.)<br />
248/225 (desc.)<br />
</td>
        <td>10/9<br />
</td>
        <td>9/8<br />
</td>
    </tr>
    <tr>
        <td>32/27<br />
</td>
        <td>6/5<br />
</td>
        <td>75/62 (asc.)<br />
121/100 (desc.)<br />
</td>
        <td>50/41 (asc.)<br />
61/50 (desc.)<br />
</td>
        <td>75/61 (asc.)<br />
49/40 (desc.)<br />
</td>
        <td>150/121 (asc.)<br />
31/25 (desc.)<br />
</td>
        <td>5/4<br />
</td>
        <td>81/64<br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:26:&lt;h3&gt; --><h3 id="toc13"><a name="Newer systems-Symmetrical and super-symmetrical models-Ascending/descending with 54/49"></a><!-- ws:end:WikiTextHeadingRule:26 -->Ascending/descending with 54/49</h3>
 J.J. Weiss<br />
54/49: 168.2 cents<br />
XXX<br />
<br />
Mandal positions (cents): <span style="color: #00000a; font-family: Tahoma;">22|14|17|13|13|13|22</span><br />
<br />
Interval table XXX<br />
<br />
<!-- ws:start:WikiTextHeadingRule:28:&lt;h3&gt; --><h3 id="toc14"><a name="Newer systems-Symmetrical and 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 />
Interval table XXX<br />
<br />
<!-- ws:start:WikiTextHeadingRule:30:&lt;h3&gt; --><h3 id="toc15"><a name="Newer systems-Symmetrical and 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 />
XXX<br />
<br />
Mandal positions (cents): <span style="color: #00000a; font-family: Tahoma;">22|16|15|13|13|13|22</span><br />
<br />
Interval table XXX<br />
<br />
<!-- ws:start:WikiTextHeadingRule:32:&lt;h3&gt; --><h3 id="toc16"><a name="Newer systems-Symmetrical and 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 />
XXX<br />
<br />
<!-- ws:start:WikiTextHeadingRule:34:&lt;h2&gt; --><h2 id="toc17"><a name="Newer systems-System Jacques Dudon (2006)"></a><!-- ws:end:WikiTextHeadingRule:34 -->System Jacques Dudon (2006)</h2>
 <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 />
Interval table XXX<br />
<br />
<!-- ws:start:WikiTextHeadingRule:36:&lt;h3&gt; --><h3 id="toc18"><a name="Newer systems-System Jacques Dudon (2006)-Arithmetic system"></a><!-- ws:end:WikiTextHeadingRule:36 -->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 />
<br />
Interval table XXX</body></html>
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