Tuning systems for qanun: Difference between revisions

Wikispaces>hstraub
**Imported revision 248375203 - Original comment: **
Wikispaces>hstraub
**Imported revision 249050231 - Original comment: **
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2011-08-25 06:03:42 UTC</tt>.<br>
: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2011-08-29 04:24:11 UTC</tt>.<br>
: The original revision id was <tt>248375203</tt>.<br>
: The original revision id was <tt>249050231</tt>.<br>
: The revision comment was: <tt></tt><br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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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.
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 per octave.
XXX to do: include J.J. Weiss's notation table here.
XXX Does this notation system have a standard name?


=Explanation of the tuning tables=  
=Explanation of the tuning tables=  
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<h4>Original HTML content:</h4>
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<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;tuning systems for qanun&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;span style="font-size: 150%;"&gt;&lt;strong&gt;Tuning systems for the qanun&lt;/strong&gt;&lt;/span&gt;&lt;br /&gt;
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;tuning systems for qanun&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;span style="font-size: 150%;"&gt;&lt;strong&gt;Tuning systems for the qanun&lt;/strong&gt;&lt;/span&gt;&lt;br /&gt;
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&lt;!-- ws:end:WikiTextTocRule:58 --&gt;Julien Jalaleddine Weiss, used with permission.&lt;br /&gt;
&lt;!-- ws:end:WikiTextTocRule:60 --&gt;&lt;!-- ws:start:WikiTextTocRule:61: --&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:61 --&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&lt;br /&gt;
Reference: Pohlit, Stefan. 2011. Julien Jalâl Ed-Dine Weiss – A New Qānūn System: Its Application in the Performance&lt;br /&gt;
Practice of the Ensemble “Al-Kindi” and in Contemporary Western Music. PhD Thesis, MIAM/Istanbul Technical&lt;br /&gt;
Practice of the Ensemble “Al-Kindi” and in Contemporary Western Music. PhD Thesis, MIAM/Istanbul Technical&lt;br /&gt;
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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;
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;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Explanation of the tuning tables"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Explanation of the tuning tables&lt;/h1&gt;
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Notation&lt;/h1&gt;
Raising a pitch by an apotome is notated with &amp;quot;#&amp;quot;, lowering a pitch by the same amount is notated with &amp;quot;b&amp;quot;.&lt;br /&gt;
For the steps in between, additional symbols are used - altogether 7 symbols for raising pitches and 7 for lowering pitches.&lt;br /&gt;
This gives 15 potential different pitches per base note. Seven base notes (C, D, E, F, G, A, B or Do, Re, Mi, Fa, Sol, La, Si) lead to a notation system of 7*15=105 pitches per octave.&lt;br /&gt;
XXX to do: include J.J. Weiss's notation table here.&lt;br /&gt;
XXX Does this notation system have a standard name?&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="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;
  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;
&lt;br /&gt;
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(base for a maqam tetrachord) is represented by a choice of one cell in each row.&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;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;
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  &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: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;
  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;
Subdivision of 25/24 into 65/64 (26 cents), 144/143 (12 cents) and 55/54 (32 cents).&lt;br /&gt;
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&lt;br /&gt;
&lt;br /&gt;
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&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Older systems-Variant with 128/119"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Variant with 128/119&lt;/h2&gt;
  128/119: 126.2 cents&lt;br /&gt;
  128/119: 126.2 cents&lt;br /&gt;
XXX&lt;br /&gt;
XXX&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Older systems-Variant with 128/119 ascending/descending"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Variant with 128/119 ascending/descending&lt;/h2&gt;
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  XXX&lt;br /&gt;
  XXX&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Older systems-Variant with 43/40 ascending/descending"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Variant with 43/40 ascending/descending&lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Older systems-Variant with 43/40 ascending/descending"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Variant with 43/40 ascending/descending&lt;/h2&gt;
  43/40: 125.2 cents&lt;br /&gt;
  43/40: 125.2 cents&lt;br /&gt;
XXX&lt;br /&gt;
XXX&lt;br /&gt;
&lt;br /&gt;
&lt;br /&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: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: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;
  &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;
  J.J. Weiss, Qanun no. 9. Luthier: Kenan Ozten.&lt;br /&gt;
  J.J. Weiss, Qanun no. 9. Luthier: Kenan Ozten.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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XXX&lt;br /&gt;
XXX&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Newer systems-Symmetrical and super-symmetrical models"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Symmetrical and 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-Symmetrical and super-symmetrical models"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Symmetrical and super-symmetrical models&lt;/h2&gt;
  XXX&lt;br /&gt;
  XXX&lt;br /&gt;
Characteristics of super-symmetric systems: no difference between ascending and descending ratios.&lt;br /&gt;
Characteristics of super-symmetric systems: no difference between ascending and descending ratios.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc9"&gt;&lt;a name="Newer systems-Symmetrical and super-symmetrical models-Symmetrical model"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Symmetrical model&lt;/h3&gt;
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc10"&gt;&lt;a name="Newer systems-Symmetrical and super-symmetrical models-Symmetrical model"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Symmetrical model&lt;/h3&gt;
  J.J. Weiss&lt;br /&gt;
  J.J. Weiss&lt;br /&gt;
Advantage: marked contrast between Segah of Ushaq and Segah of arabic Rast.&lt;br /&gt;
Advantage: marked contrast between Segah of Ushaq and Segah of arabic Rast.&lt;br /&gt;
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XXX&lt;br /&gt;
XXX&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc10"&gt;&lt;a name="Newer systems-Symmetrical and super-symmetrical models-Super-symmetrical model"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Super-symmetrical model&lt;/h3&gt;
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc11"&gt;&lt;a name="Newer systems-Symmetrical and super-symmetrical models-Super-symmetrical model"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;Super-symmetrical model&lt;/h3&gt;
  J.J. Weiss&lt;br /&gt;
  J.J. Weiss&lt;br /&gt;
Equal division of 65/54 (320.98 cents) and its inverse. (XXX Is that formulation correct?)&lt;br /&gt;
Equal division of 65/54 (320.98 cents) and its inverse. (XXX Is that formulation correct?)&lt;br /&gt;
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XXX&lt;br /&gt;
XXX&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc11"&gt;&lt;a name="Newer systems-Symmetrical and super-symmetrical models-Unequal division of 65/54"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;Unequal division of 65/54&lt;/h3&gt;
&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc12"&gt;&lt;a name="Newer systems-Symmetrical and super-symmetrical models-Unequal division of 65/54"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;Unequal division of 65/54&lt;/h3&gt;
  J.J. Weiss&lt;br /&gt;
  J.J. Weiss&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
XXX&lt;br /&gt;
XXX&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc12"&gt;&lt;a name="Newer systems-Symmetrical and 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;
&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc13"&gt;&lt;a name="Newer systems-Symmetrical and 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;
  J.J. Weiss&lt;br /&gt;
  J.J. Weiss&lt;br /&gt;
This is the simplest variant for luthiers...&lt;br /&gt;
This is the simplest variant for luthiers...&lt;br /&gt;
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&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-Symmetrical and 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;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc14"&gt;&lt;a name="Newer systems-Symmetrical and 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;
  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;
&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;
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Interval table of just intervals XXX&lt;br /&gt;
Interval table of just intervals XXX&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc14"&gt;&lt;a name="Newer systems-Symmetrical and 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;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc15"&gt;&lt;a name="Newer systems-Symmetrical and 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;
  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;
&lt;span style="color: #00000a; font-family: Tahoma;"&gt;14/13: 128.3 cents&lt;/span&gt;&lt;br /&gt;
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Interval table XXX&lt;br /&gt;
Interval table XXX&lt;br /&gt;
&lt;br /&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-Symmetrical and 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;
&lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc16"&gt;&lt;a name="Newer systems-Symmetrical and 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;
  J.J. Weiss&lt;br /&gt;
  J.J. Weiss&lt;br /&gt;
11/10: 165 cents&lt;br /&gt;
11/10: 165 cents&lt;br /&gt;
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Interval table XXX&lt;br /&gt;
Interval table XXX&lt;br /&gt;
&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-Symmetrical and 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;
&lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc17"&gt;&lt;a name="Newer systems-Symmetrical and 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;
  J.J. Weiss&lt;br /&gt;
  J.J. Weiss&lt;br /&gt;
35/32: 155.14 cents&lt;br /&gt;
35/32: 155.14 cents&lt;br /&gt;
XXX&lt;br /&gt;
XXX&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc17"&gt;&lt;a name="Newer systems-System Jacques Dudon (2006)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&gt;System Jacques Dudon (2006)&lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc18"&gt;&lt;a name="Newer systems-System Jacques Dudon (2006)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;System Jacques Dudon (2006)&lt;/h2&gt;
  &lt;br /&gt;
  &lt;br /&gt;
Mandal positions (cents): &lt;span style="color: #00000a; font-family: Tahoma;"&gt;21,5 | 14,5 | 14,5 | 14,5 | 15 | 12 | 21,5&lt;/span&gt;&lt;br /&gt;
Mandal positions (cents): &lt;span style="color: #00000a; font-family: Tahoma;"&gt;21,5 | 14,5 | 14,5 | 14,5 | 15 | 12 | 21,5&lt;/span&gt;&lt;br /&gt;
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Interval table of just intervals XXX&lt;br /&gt;
Interval table of just intervals XXX&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc18"&gt;&lt;a name="Newer systems-System Jacques Dudon (2006)-Arithmetic system"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;Arithmetic system&lt;/h3&gt;
&lt;!-- ws:start:WikiTextHeadingRule:38:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc19"&gt;&lt;a name="Newer systems-System Jacques Dudon (2006)-Arithmetic system"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:38 --&gt;Arithmetic system&lt;/h3&gt;
  &lt;br /&gt;
  &lt;br /&gt;
Mandal positions (cents): &lt;span style="color: #00000a; font-family: Tahoma;"&gt;21,5 | 12 | 15 | 14,5 | 14,5 | 14,5 | 21,5 &lt;/span&gt;&lt;br /&gt;
Mandal positions (cents): &lt;span style="color: #00000a; font-family: Tahoma;"&gt;21,5 | 12 | 15 | 14,5 | 14,5 | 14,5 | 21,5 &lt;/span&gt;&lt;br /&gt;