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 logic behind the systems is as follows:

The empty strings of the qanun are tuned to 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]].

The possible pitches of a string obtained via raising/lowering the mandals lie within two [[2187_2048|apotomes (2187/2048, 113.7 cents)]]. The base note is assumed in the middle. The mandals allow raising and lowering this note by maximally one apotome.

Each apotome is divided into 7 parts, which requires 14 mandals per string. 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 tuning systems are all described by a series of cent values, which describe the subdivision of one apotome. According to the system sketched above, the first and the last value are always 22 cents (or 21.5 cents). This subdivision pattern occurs twice on each string (one apotome up and one down), alltogether 14 times per octave. This is followed by listings of some important rational intervals that are possible in this tuning, mainly in the range of a fourth (the range where the ajnas - maqam tetrachords - reside), (XXX STILL TO DO).

An notable property (of all systems) is that the second-highest mandal position of, say, the C string is 114-22=92 cents, while the lowest mandal position on the following string (D in the example) is 214 (one wholetone above C) - 114 = 90 cents - we have two notes differing by one [[32805_32768|schisma (2 cents)]]. So the interval of the schisma is present and can be played on a qanun in any of the tuning systems described here.

=Notation= 
The notes without accidentals stand for the pythagorean intervals of the base tuning of the qanun). Raising a pitch by an apotome is notated with "#", lowering a pitch by the same amount is notated with "b". Sharps are higher than flats (unlike in [[meantone]] systems): C# is one apotome (114 cents) above C, while Db is 9/8 (214 cents) minus one apotome = 90 cents. Both properties indicate that the framework is essentially pythagorean. The tuning system as a whole, however, is not.

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, in accordance with the real playing capabilities of the qanun. 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, which also gives all the pitches in one octave (in ratios and cents) that can be played by system 1 and 2.
[[file:Tableaux JJW VIII-2011.pdf]]

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

=Older systems= 
==System 1 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 roughly equal parts.

This gives the following rational intervals between the mandals:
81/80, 245/243, 3159/3136, 144/143, 121/120, 100/99, 81/80

In cents (approximations):
22, 13, 13, 12, 16, 16, 22

Rational intervals each string can be detuned:
81/80, 49/48, 1053/1024, 729/704, 2673/2560, 135/128, 2187/2048

Intervals ratios, ascending from C (cent values in parentheses):
* On the D string (from Db to D):
> 256/243 (90), 16/15 (112), **784/729 (126)**, **13/12 (138)**, **12/11 (150)**, **11/10 (166)**, 10/9 (182), 9/8 (204)
* On the E string (from Eb to E):
> 32/27 (294), 6/5 (316), **98/81 (330)**, **39/32 (342)**, **27/22 (354)**, **89/80 (370)**, 5/4 (386), 81/64 (408)

Interval ratios, descending from F:
* On the E string (from Eb to E):
> 9/8 (204), 10/9 (182), **54/49 (169)**, **128/117 (156)**, **88/81 (144)**, **320/297 (129)**, 16/15 (112), 256/243 (90)
* On the D string (from Db to D):
> 81/64 (408), 5/4 (386), **243/196 (372)**, **16/13 (360)**, **11/9 (348)**, **40/33 (333)**, 6/5 (316), 32/27 (294)

===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>

Intervals ratios 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>

Intervals ratios 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>

Intervals ratios XXX

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

Intervals ratios 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...

Intervals ratios 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>

Intervals ratios 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

Intervals ratios 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>

Intervals ratios 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>

Intervals ratios 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-System 1 J.J.Weiss">System 1 J.J.Weiss</a></div>
<!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --><div style="margin-left: 3em;"><a href="#Older systems-System 1 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-System 1 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-System 1 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). The logic behind the systems is as follows:<br />
<br />
The empty strings of the qanun are tuned to 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 />
The possible pitches of a string obtained via raising/lowering the mandals lie within two <a class="wiki_link" href="/2187_2048">apotomes (2187/2048, 113.7 cents)</a>. The base note is assumed in the middle. The mandals allow raising and lowering this note by maximally one apotome.<br />
<br />
Each apotome is divided into 7 parts, which requires 14 mandals per string. 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 />
The tuning systems are all described by a series of cent values, which describe the subdivision of one apotome. According to the system sketched above, the first and the last value are always 22 cents (or 21.5 cents). This subdivision pattern occurs twice on each string (one apotome up and one down), alltogether 14 times per octave. This is followed by listings of some important rational intervals that are possible in this tuning, mainly in the range of a fourth (the range where the ajnas - maqam tetrachords - reside), (XXX STILL TO DO).<br />
<br />
An notable property (of all systems) is that the second-highest mandal position of, say, the C string is 114-22=92 cents, while the lowest mandal position on the following string (D in the example) is 214 (one wholetone above C) - 114 = 90 cents - we have two notes differing by one <a class="wiki_link" href="/32805_32768">schisma (2 cents)</a>. So 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:0:&lt;h1&gt; --><h1 id="toc0"><a name="Notation"></a><!-- ws:end:WikiTextHeadingRule:0 -->Notation</h1>
 The notes without accidentals stand for the pythagorean intervals of the base tuning of the qanun). Raising a pitch by an apotome is notated with &quot;#&quot;, lowering a pitch by the same amount is notated with &quot;b&quot;. Sharps are higher than flats (unlike in <a class="wiki_link" href="/meantone">meantone</a> systems): C# is one apotome (114 cents) above C, while Db is 9/8 (214 cents) minus one apotome = 90 cents. Both properties indicate that the framework is essentially pythagorean. The tuning system as a whole, however, is not.<br />
<br />
For the steps in between, additional symbols are used - altogether 7 symbols for raising pitches and 7 for lowering pitches.<br />
<br />
This gives 15 potential different pitches per base note, in accordance with the real playing capabilities of the qanun. 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, which also gives all the pitches in one octave (in ratios and cents) that can be played by system 1 and 2.<br />
<!-- ws:start:WikiTextFileRule:78:&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:78 --><br />
<br />
(used with permission J. J. Weiss/S. Pohlit)<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-System 1 J.J.Weiss"></a><!-- ws:end:WikiTextHeadingRule:4 -->System 1 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 roughly equal parts.<br />
<br />
This gives the following rational intervals between the mandals:<br />
81/80, 245/243, 3159/3136, 144/143, 121/120, 100/99, 81/80<br />
<br />
In cents (approximations):<br />
22, 13, 13, 12, 16, 16, 22<br />
<br />
Rational intervals each string can be detuned:<br />
81/80, 49/48, 1053/1024, 729/704, 2673/2560, 135/128, 2187/2048<br />
<br />
Intervals ratios, ascending from C (cent values in parentheses):<br />
<ul><li>On the D string (from Db to D):<br />
256/243 (90), 16/15 (112), <strong>784/729 (126)</strong>, <strong>13/12 (138)</strong>, <strong>12/11 (150)</strong>, <strong>11/10 (166)</strong>, 10/9 (182), 9/8 (204)</li><li>On the E string (from Eb to E):<br />
32/27 (294), 6/5 (316), <strong>98/81 (330)</strong>, <strong>39/32 (342)</strong>, <strong>27/22 (354)</strong>, <strong>89/80 (370)</strong>, 5/4 (386), 81/64 (408)</li></ul><br />
Interval ratios, descending from F:<br />
<ul><li>On the E string (from Eb to E):<br />
9/8 (204), 10/9 (182), <strong>54/49 (169)</strong>, <strong>128/117 (156)</strong>, <strong>88/81 (144)</strong>, <strong>320/297 (129)</strong>, 16/15 (112), 256/243 (90)</li><li>On the D string (from Db to D):<br />
81/64 (408), 5/4 (386), <strong>243/196 (372)</strong>, <strong>16/13 (360)</strong>, <strong>11/9 (348)</strong>, <strong>40/33 (333)</strong>, 6/5 (316), 32/27 (294)</li></ul><br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="Older systems-System 1 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-System 1 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-System 1 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 />
Intervals ratios 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 />
Intervals ratios 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 />
Intervals ratios 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 />
Intervals ratios 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 />
Intervals ratios 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 />
Intervals ratios 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 />
Intervals ratios 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 />
Intervals ratios 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 />
Intervals ratios XXX</body></html>

@@ Line 1: / Line 1: @@
 <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>2011-09-30 05:53:56 UTC</tt>.<br>
+: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2011-09-30 06:04:18 UTC</tt>.<br>
-: The original revision id was <tt>259921722</tt>.<br>
+: The original revision id was <tt>259923008</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 44: / Line 44: @@
 /80, 245/243, 3159/3136, 144/143, 121/120, 100/99, 81/80
 In cents (approximations):
 , 13, 13, 12, 16, 16, 22
@@ Line 52: / Line 52: @@
 Intervals ratios, ascending from C (cent values in parentheses):
 * On the D string (from Db to D):
-&gt; 256/243 (90), 16/15 (112), 784/729 (126), 13/12 (138), 12/11 (150), 11/10 (166), 10/9 (182), 9/8 (204)
+&gt; 256/243 (90), 16/15 (112), **784/729 (126)**, **13/12 (138)**, **12/11 (150)**, **11/10 (166)**, 10/9 (182), 9/8 (204)
 * On the E string (from Eb to E):
-&gt; 32/27 (294), 6/5 (316), 98/81 (330), 39/32 (342), 27/22 (354),8 9/80 (370), 5/4 (386), 81/64 (408)
+&gt; 32/27 (294), 6/5 (316), **98/81 (330)**, **39/32 (342)**, **27/22 (354)**, **89/80 (370)**, 5/4 (386), 81/64 (408)
 Interval ratios, descending from F:
-XXX
+* On the E string (from Eb to E):
+&gt; 9/8 (204), 10/9 (182), **54/49 (169)**, **128/117 (156)**, **88/81 (144)**, **320/297 (129)**, 16/15 (112), 256/243 (90)
+* On the D string (from Db to D):
+&gt; 81/64 (408), 5/4 (386), **243/196 (372)**, **16/13 (360)**, **11/9 (348)**, **40/33 (333)**, 6/5 (316), 32/27 (294)
 ===Variant with 128/119===
@@ Line 200: / Line 203: @@
 &lt;br /&gt;
 This gives 15 potential different pitches per base note, in accordance with the real playing capabilities of the qanun. 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, which also gives all the pitches in one octave (in ratios and cents) that can be played by system 1 and 2.&lt;br /&gt;
-&lt;!-- ws:start:WikiTextFileRule:70:&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:70 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextFileRule:78:&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:78 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 (used with permission J. J. Weiss/S. Pohlit)&lt;br /&gt;
@@ Line 213: / Line 216: @@
 /80, 245/243, 3159/3136, 144/143, 121/120, 100/99, 81/80&lt;br /&gt;
 &lt;br /&gt;
-In cents (approximations): &lt;br /&gt;
+In cents (approximations):&lt;br /&gt;
 , 13, 13, 12, 16, 16, 22&lt;br /&gt;
 &lt;br /&gt;
@@ Line 221: / Line 224: @@
 Intervals ratios, ascending from C (cent values in parentheses):&lt;br /&gt;
 &lt;ul&gt;&lt;li&gt;On the D string (from Db to D):&lt;br /&gt;
-/243 (90), 16/15 (112), 784/729 (126), 13/12 (138), 12/11 (150), 11/10 (166), 10/9 (182), 9/8 (204)&lt;/li&gt;&lt;li&gt;On the E string (from Eb to E):&lt;br /&gt;
+/243 (90), 16/15 (112), &lt;strong&gt;784/729 (126)&lt;/strong&gt;, &lt;strong&gt;13/12 (138)&lt;/strong&gt;, &lt;strong&gt;12/11 (150)&lt;/strong&gt;, &lt;strong&gt;11/10 (166)&lt;/strong&gt;, 10/9 (182), 9/8 (204)&lt;/li&gt;&lt;li&gt;On the E string (from Eb to E):&lt;br /&gt;
-/27 (294), 6/5 (316), 98/81 (330), 39/32 (342), 27/22 (354),8 9/80 (370), 5/4 (386), 81/64 (408)&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
+/27 (294), 6/5 (316), &lt;strong&gt;98/81 (330)&lt;/strong&gt;, &lt;strong&gt;39/32 (342)&lt;/strong&gt;, &lt;strong&gt;27/22 (354)&lt;/strong&gt;, &lt;strong&gt;89/80 (370)&lt;/strong&gt;, 5/4 (386), 81/64 (408)&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
 Interval ratios, descending from F:&lt;br /&gt;
-XXX&lt;br /&gt;
+&lt;ul&gt;&lt;li&gt;On the E string (from Eb to E):&lt;br /&gt;
-&lt;br /&gt;
+/8 (204), 10/9 (182), &lt;strong&gt;54/49 (169)&lt;/strong&gt;, &lt;strong&gt;128/117 (156)&lt;/strong&gt;, &lt;strong&gt;88/81 (144)&lt;/strong&gt;, &lt;strong&gt;320/297 (129)&lt;/strong&gt;, 16/15 (112), 256/243 (90)&lt;/li&gt;&lt;li&gt;On the D string (from Db to D):&lt;br /&gt;
+/64 (408), 5/4 (386), &lt;strong&gt;243/196 (372)&lt;/strong&gt;, &lt;strong&gt;16/13 (360)&lt;/strong&gt;, &lt;strong&gt;11/9 (348)&lt;/strong&gt;, &lt;strong&gt;40/33 (333)&lt;/strong&gt;, 6/5 (316), 32/27 (294)&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="Older systems-System 1 J.J.Weiss-Variant with 128/119"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Variant with 128/119&lt;/h3&gt;
 /119: 126.2 cents&lt;br /&gt;

Tuning systems for qanun: Difference between revisions

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