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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<span style="font-size: 150%;">'''Tuning systems for the qanun'''</span>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
__FORCETOC__
: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2011-08-22 05:47:48 UTC</tt>.<br>
: The original revision id was <tt>247611167</tt>.<br>
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">&lt;span style="font-size: 150%;"&gt;**Tuning systems for the qanun**&lt;/span&gt;
[[toc]]
Julien Jalaleddine Weiss, used with permission.
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.


=Explanation=
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 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 lowering the mandals/orabs.
Online version of Stefan Pohlit's dissertation: see [http://stefanpohlit.com/dissertation.engl..htm http://stefanpohlit.com/dissertation.engl..htm]


The first table contains the cent values and the second the just intervals, sometimes differing between ascending and descending ratios.
The tuning tables on this page are specifically designed for the tuning system of the [[qanun|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:


Any given configuration of mandal/orab positions, resulting in a certain set of pitches that can be played at a given time
The empty strings of the qanun are tuned to a pythagorean diatonic scale, with a major third of [[81/64|81/64]], a major sixth of [[27/16|27/16]] and a major seventh of [[243/128|243/128]].
(base for a maqam tetrachord) is represented by a choice of one cell in each row.


=Older systems=
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.
==First System J.J.Weiss==
Inequal subdivision of the [[2187_2048|apotome (2187/2048, 113.7 cents)]] into 7 parts (8 mandals/orabs), in the following manner:
* Subdivision 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.
* Further 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.
Each apotome is divided into 7 unequal 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.
Luthier: Ejder Gulec


Interval Table (cents):
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, altogether 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 [[tetrachord|tetrachords]] - reside),
|| 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):
An notable property (of all systems) is that the second-highest mandal position of, say, the C string is 114-22=92 cents (the [[135/128|major limma]]), while the lowest mandal position on the following string (D in the example) is 214 (one wholetone above C) - 114 = 90 cents (the [[256/243|pythagorean limma]], the same interval as between E and F) - 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.
|| 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.)
XXX || 27/22 (asc.)
XXX || 99/80, 26/21 (asc.)
XXX || 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==  
=Notation=
XXX
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|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.


==Variant with 128/119 ascending/descending==
For the steps in between, additional symbols are used - altogether 7 symbols for raising pitches and 7 for lowering pitches.
XXX


==Variant with 43/40 ascending/descending==
This gives 15 potential different pitches per base note, corresponding to the mandals. Seven base notes (C, D, E, F, G, A, B or Do, Re, Mi, Fa, Sol, La, Si), corresponding to the strings, lead to a notation system of 7*15=105 pitches, in accordance with the real playing capabilities of the qanun. 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.
XXX


=Newer systems=
[[:File:Tableaux_JJW_VIII-2011.pdf|Tableaux JJW VIII-2011.pdf]]
==System 2, better suited for ottoman maqams==
J.J. Weiss, Qanun no. 9. Luthier: Kenan Ozten.


Mandal positions (cents): &lt;span style="color: #00000a; font-family: Tahoma;"&gt;22|16|15|12|13|14|22&lt;/span&gt;
(used with permission J. J. Weiss/S. Pohlit)


Interval table XXX
=System 1=
© J.J.Weiss. Luthier: Ejder Gulec.


==Symmetrical and super-symmetrical models==
Subdivision of 25/24 into 65/64 (26 cents), 144/143 (12 cents) and 55/54 (32 cents).
Characteristics: XXX


===Symmetrical model===
65/64 and 55/54 are each split into two roughly equal parts.
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): &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;
This gives the following rational intervals between the mandals:


Interval table XXX
81/80, 245/243, 3159/3136, 144/143, 121/120, 100/99, 81/80


===Super-symmetrical model===
In cents (approximations):
J.J. Weiss
XXX


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;
22, 13, 13, 12, 16, 16, 22


Interval table XXX
Rational intervals each string can be detuned (approximations in cents in parentheses):


===Unequal division of 65/54===
81/80 (22), 49/48 (35), 1053/1024 (48), 729/704 (60), 2673/2560 (76), 135/128 (92), 2187/2048 (114)
J.J. Weiss
Unequal division of 65/54 (320.98 cents)


XXX
Intervals ratios, ascending from C:


===Equal division of the Zarlinian semitone===
<ul><li>On the D string (from Db to D):
J.J. Weiss
Equal division of [[25_24|25/24 (70.67 cents)]] into 5 parts (to make it easier for luthiers)


Mandal positions (cents): &lt;span style="color: #00000a; font-family: Tahoma;"&gt;22|14|14|14|14|14|22&lt;/span&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)</li><li>On the E string (from Eb to E):


Interval table XXX
32/27 (294), 6/5 (316), '''98/81 (330)''', '''39/32 (342)''', '''27/22 (354)''', '''99/80 (370)''', 5/4 (386), 81/64 (408)</li></ul>


===Ascending/descending with 54/49===
Interval ratios, descending from F:
J.J. Weiss
XXX


Mandal positions (cents): &lt;span style="color: #00000a; font-family: Tahoma;"&gt;22|14|17|13|13|13|22&lt;/span&gt;
<ul><li>On the E string (from Eb to E):


Interval table XXX
9/8 (204), 10/9 (182), '''54/49 (169)''', '''128/117 (156)''', '''88/81 (144)''', '''320/297 (129)''', 16/15 (112), 256/243 (90)</li><li>On the D string (from Db to D):


===Ascending/descending with 14/13===
81/64 (408), 5/4 (386), '''243/196 (372)''', '''16/13 (360)''', '''11/9 (348)''', '''40/33 (333)''', 6/5 (316), 32/27 (294)</li></ul>
J.J. Weiss
&lt;span style="color: #00000a; font-family: Tahoma;"&gt;XXX&lt;/span&gt;


Mandal positions (cents): &lt;span style="color: #00000a; font-family: Tahoma;"&gt;22, 16, 13, 12, 13, 16, 22&lt;/span&gt;
A complete list of all intervals available within one octave can be found in the above-mentioned [[:File:Tableaux_JJW_VIII-2011.pdf|document]] (on the first page).


Interval table XXX
=System 2, better suited for ottoman maqams=
© J.J. Weiss. Qanun no. 9, luthier: Kenan Ozten.


===Ascending/descending with 11/10===
Mandal positions in ratios:
J.J. Weiss
XXX


Mandal positions (cents): &lt;span style="color: #00000a; font-family: Tahoma;"&gt;22|16|15|13|13|13|22&lt;/span&gt;
81/80, 105/104, 572/567, 144/143, 1547/1536, 120/119, 81/80


Interval table XXX
In cents (approximations):


===Ascending/descending with 35/32===
<span style="color: #00000a; font-family: Tahoma;">22|16|15|12|13|14|22</span>
J.J. Weiss
XXX


==System Jacques Dudon (2006)==
Rational intervals each string can be detuned (approximations in cents in parentheses):


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;
81/80 (22), 1701/1664 (38), 33/32 (54), 27/26 (66), 243/232 (78), 135/128 (92), 2187/2048 (114)


Interval table XXX
Intervals ratios, ascending from C:


===Arithmetic system===
<ul><li>On the D string (from Db to D):


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;
256/243 (90), 16/15 (112), '''14/13 (128), 88/81 (144), 128/117 or 35/32 (156), 119/108 (168)''', 10/9 (182), 9/8 (204)</li><li>On the E string (from Eb to E):


32/27 (294), 6/5 (316), '''63/52 (332), 11/9 (348), 16/13 or 315/256 (360), 119/96 (372)''', 5/4 (386), 81/64 (408)</li></ul>


Interval table XXX</pre></div>
Interval ratios descending from F:
<h4>Original HTML content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;tuning systems for qanun&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;span style="font-size: 150%;"&gt;&lt;strong&gt;Tuning systems for the qanun&lt;/strong&gt;&lt;/span&gt;&lt;br /&gt;
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&lt;!-- ws:end:WikiTextTocRule:39 --&gt;&lt;!-- ws:start:WikiTextTocRule:40: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Older systems"&gt;Older systems&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:40 --&gt;&lt;!-- ws:start:WikiTextTocRule:41: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Older systems-First System J.J.Weiss"&gt;First System J.J.Weiss&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:41 --&gt;&lt;!-- ws:start:WikiTextTocRule:42: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Older systems-Variant with 128/119"&gt;Variant with 128/119&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:42 --&gt;&lt;!-- ws:start:WikiTextTocRule:43: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Older systems-Variant with 128/119 ascending/descending"&gt;Variant with 128/119 ascending/descending&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:43 --&gt;&lt;!-- ws:start:WikiTextTocRule:44: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Older systems-Variant with 43/40 ascending/descending"&gt;Variant with 43/40 ascending/descending&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:44 --&gt;&lt;!-- ws:start:WikiTextTocRule:45: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Newer systems"&gt;Newer systems&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:45 --&gt;&lt;!-- ws:start:WikiTextTocRule:46: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Newer systems-System 2, better suited for ottoman maqams"&gt;System 2, better suited for ottoman maqams&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:46 --&gt;&lt;!-- ws:start:WikiTextTocRule:47: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Newer systems-Symmetrical and super-symmetrical models"&gt;Symmetrical and super-symmetrical models&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:47 --&gt;&lt;!-- ws:start:WikiTextTocRule:48: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Newer systems-Symmetrical and super-symmetrical models-Symmetrical model"&gt;Symmetrical model&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:48 --&gt;&lt;!-- ws:start:WikiTextTocRule:49: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Newer systems-Symmetrical and super-symmetrical models-Super-symmetrical model"&gt;Super-symmetrical model&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:49 --&gt;&lt;!-- ws:start:WikiTextTocRule:50: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Newer systems-Symmetrical and super-symmetrical models-Unequal division of 65/54"&gt;Unequal division of 65/54&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:50 --&gt;&lt;!-- ws:start:WikiTextTocRule:51: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Newer systems-Symmetrical and super-symmetrical models-Equal division of the Zarlinian semitone"&gt;Equal division of the Zarlinian semitone&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:51 --&gt;&lt;!-- ws:start:WikiTextTocRule:52: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Newer systems-Symmetrical and super-symmetrical models-Ascending/descending with 54/49"&gt;Ascending/descending with 54/49&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:52 --&gt;&lt;!-- ws:start:WikiTextTocRule:53: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Newer systems-Symmetrical and super-symmetrical models-Ascending/descending with 14/13"&gt;Ascending/descending with 14/13&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:53 --&gt;&lt;!-- ws:start:WikiTextTocRule:54: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Newer systems-Symmetrical and super-symmetrical models-Ascending/descending with 11/10"&gt;Ascending/descending with 11/10&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:54 --&gt;&lt;!-- ws:start:WikiTextTocRule:55: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Newer systems-Symmetrical and super-symmetrical models-Ascending/descending with 35/32"&gt;Ascending/descending with 35/32&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:55 --&gt;&lt;!-- ws:start:WikiTextTocRule:56: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Newer systems-System Jacques Dudon (2006)"&gt;System Jacques Dudon (2006)&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:56 --&gt;&lt;!-- ws:start:WikiTextTocRule:57: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Newer systems-System Jacques Dudon (2006)-Arithmetic system"&gt;Arithmetic system&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:57 --&gt;&lt;!-- ws:start:WikiTextTocRule:58: --&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:58 --&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;
Practice of the Ensemble “Al-Kindi” and in Contemporary Western Music. PhD Thesis, MIAM/Istanbul Technical&lt;br /&gt;
University.&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"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Explanation&lt;/h1&gt;
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 &amp;quot;mandals/orabs&amp;quot;).&lt;br /&gt;
&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 lowering the mandals/orabs.&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.&lt;br /&gt;
&lt;br /&gt;
Any given configuration of mandal/orab 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: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;!-- 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;
Inequal subdivision of the &lt;a class="wiki_link" href="/2187_2048"&gt;apotome (2187/2048, 113.7 cents)&lt;/a&gt; into 7 parts (8 mandals/orabs), in the following manner:&lt;br /&gt;
&lt;ul&gt;&lt;li&gt;Subdivision 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.&lt;/li&gt;&lt;li&gt;Further subdivision of 25/24 into 65/64 (26 cents), 144/143 (12 cents) and 55/54 (32 cents).&lt;/li&gt;&lt;li&gt;65/64 and 55/54 are each split into two.&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
This gives the following interval positions of the mandals: 22, 13, 13, 12, 16, 16, 22 cents.&lt;br /&gt;
Luthier: Ejder Gulec&lt;br /&gt;
&lt;br /&gt;
Interval Table (cents):&lt;br /&gt;


<ul><li>On the E string (from Eb to E):


&lt;table class="wiki_table"&gt;
9/8 (204), 10/9 (182), '''208/189 (166), 12/11 (150), 13/12 (138), 128/119 (126)''', 16/15 (112), 256/243 (90)</li><li>On the D string (from Db to D):
    &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;35&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;48&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;60&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;76&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;125&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;138&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;150&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;166&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;330&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;342&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;354&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;370&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;420&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;432&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;444&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;460&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;533&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;546&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;558&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;574&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;
81/64 (408), 5/4 (386), '''26/21 (370), 27/22 (354), 39/32 (342), 144/119 (330)''', 6/5 (316), 32/27 (294)</li></ul>
Interval table of just intervals (ascending, descending):&lt;br /&gt;


A complete list of all intervals available within one octave can be found in the above-mentioned [[:File:Tableaux_JJW_VIII-2011.pdf|document]] (on the second page).


&lt;table class="wiki_table"&gt;
=Other models=
    &lt;tr&gt;
Julien Weiss has developed a number of other systems besides the two described above. A notable class of these are so-called super-symmetrical systems, which have the property that the intervals ascending from C and the intervals descending from F (which show slight differences in the previous two systems, marked in '''bold''' above) are the same.
        &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;49/48&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1053/1024&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;729/704&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2673/2560&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;784/729, 128/119, 43/40 (asc.)&lt;br /&gt;
320/297 (desc.)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;13/12 (asc.)&lt;br /&gt;
88/81 (desc.)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;12/11 (asc.)&lt;br /&gt;
128/117 (desc.)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;11/10, 208/189 (asc.)&lt;br /&gt;
54/49 (desc.)&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;98/81 (asc.)&lt;br /&gt;
40/33 (desc.)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;39/32 (asc.)&lt;br /&gt;
XXX&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;27/22 (asc.)&lt;br /&gt;
XXX&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;99/80, 26/21 (asc.)&lt;br /&gt;
XXX&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;25088/19683&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;104/81&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;XXX&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;176/135&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;351/258&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;XXX&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;243/176&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;891/640&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;
3 examples are described below. For more and detailed descriptions see chapter 3.4 and appendix I in [http://stefanpohlit.com/dissertation.engl..htm Stefan Pohlit's dissertation] .
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Older systems-Variant with 128/119"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Variant with 128/119&lt;/h2&gt;
 
XXX&lt;br /&gt;
==Super-symmetric model with non-aliquot division of 65/64==
&lt;br /&gt;
© J.J. Weiss
&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;
 
XXX&lt;br /&gt;
Similar to [[tuning_systems_for_qanun#System 1|system 1]], but with 65/64 (26.84 cents) divided into two non-equal parts (14 and 12 cents instead of 13 and 13).
&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;
Mandal positions in ratios:
XXX&lt;br /&gt;
 
&lt;br /&gt;
81/80 (22), 120/119 (14), 1547/1536 (12), <span style="color: #00000a; font-family: 'Tahoma','sans-serif';">512/507 (17), </span>1547/1536 (12), 120/119 (14)<span style="color: #00000a; font-family: 'Tahoma','sans-serif';">, 81/80 (22)</span>
&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;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;
Table of pitches from C to F (approximations in cents):
J.J. Weiss, Qanun no. 9. Luthier: Kenan Ozten.&lt;br /&gt;
 
&lt;br /&gt;
{| class="wikitable"
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;
! | String
Interval table XXX&lt;br /&gt;
! | b
&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;
! |
Characteristics: XXX&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;
! |
J.J. Weiss&lt;br /&gt;
! |
Advantage: marked contrast between Segah of Ushaq and Segah of arabic Rast.&lt;br /&gt;
! | Base note
Equal division of 65/54 (320.98 cents)&lt;br /&gt;
! |
&lt;br /&gt;
! |
Mandal positions (cents): &lt;span style="font-family: Tahoma;"&gt;22|13|13|1&lt;/span&gt;&lt;span style="color: #00000a; font-family: Tahoma;"&gt;8&lt;/span&gt;&lt;span style="font-family: Tahoma;"&gt;|13|13|22&lt;/span&gt;&lt;br /&gt;
! |
&lt;br /&gt;
! |
Interval table XXX&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;
! | #
J.J. Weiss&lt;br /&gt;
|-
XXX&lt;br /&gt;
! | C
&lt;br /&gt;
| |
Mandal positions (cents): &lt;span style="font-family: Tahoma;"&gt;22|13|13|1&lt;/span&gt;&lt;span style="color: #00000a; font-family: Tahoma;"&gt;8&lt;/span&gt;&lt;span style="font-family: Tahoma;"&gt;|13|13&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 XXX&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;
| |
J.J. Weiss&lt;br /&gt;
| |
Unequal division of 65/54 (320.98 cents)&lt;br /&gt;
| style="text-align:center;" | 0
&lt;br /&gt;
| | 22
XXX&lt;br /&gt;
| | 36
&lt;br /&gt;
| | 48
&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;
| | 66
J.J. Weiss&lt;br /&gt;
| | 78
Equal division of &lt;a class="wiki_link" href="/25_24"&gt;25/24 (70.67 cents)&lt;/a&gt; into 5 parts (to make it easier for luthiers)&lt;br /&gt;
| | 92
&lt;br /&gt;
| | 114
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;
! | D
Interval table XXX&lt;br /&gt;
| | 90
&lt;br /&gt;
| | 112
&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;
| | 126
J.J. Weiss&lt;br /&gt;
| | 138
XXX&lt;br /&gt;
| | 156
&lt;br /&gt;
| | 168
Mandal positions (cents): &lt;span style="color: #00000a; font-family: Tahoma;"&gt;22|14|17|13|13|13|22&lt;/span&gt;&lt;br /&gt;
| | 182
&lt;br /&gt;
| style="text-align:center;" | 204
Interval table XXX&lt;br /&gt;
| | 226
&lt;br /&gt;
| | 240
&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;
| | 252
J.J. Weiss&lt;br /&gt;
| | 270
&lt;span style="color: #00000a; font-family: Tahoma;"&gt;XXX&lt;/span&gt;&lt;br /&gt;
| | 282
&lt;br /&gt;
| | 296
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;
| | 318
&lt;br /&gt;
|-
Interval table XXX&lt;br /&gt;
! | E
&lt;br /&gt;
| | 294
&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;
| | 316
J.J. Weiss&lt;br /&gt;
| | 330
XXX&lt;br /&gt;
| | 342
&lt;br /&gt;
| | 360
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;
| | 372
&lt;br /&gt;
| | 386
Interval table XXX&lt;br /&gt;
| style="text-align:center;" | 408
&lt;br /&gt;
| | 430
&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;
| | 444
J.J. Weiss&lt;br /&gt;
| | 456
XXX&lt;br /&gt;
| | 474
&lt;br /&gt;
| | 486
&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;
| | 500
&lt;br /&gt;
| | 522
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;
! | F
Interval table XXX&lt;br /&gt;
| | 384
&lt;br /&gt;
| | 406
&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;
| | 420
&lt;br /&gt;
| | 433
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;
| | 450
&lt;br /&gt;
| | 462
&lt;br /&gt;
| | 476
Interval table XXX&lt;/body&gt;&lt;/html&gt;</pre></div>
| style="text-align:center;" | 498
| | 520
| | 534
| | 546
| | 464
| | 476
| | 590
| | 612
|}
 
Interval ratios, ascending from C:
 
<ul><li>On the D string (from Db to D):
 
245/243 (90), 16/15 (112), '''128/119 (126), 13/12 (138), 128/117 (156), 119/108 (168),''' 10/9 (182), 9/8 (204)
 
On the E string (from Eb to E):
 
32/27 (294), 6/5 (316), '''144/119 (330), 39/32 (342), 16/13 (360), 119/96 (372),''' 5/4 (386), 81/64 (408)</li></ul>
 
Interval ratios, descending from F:
 
<ul><li>On the E string (from Eb to E):
 
9(8 /204), 10/9 (182), '''119/108 (168), 128/117 (156), 13/12 (138), 128/119 (126)''', 16/15 (112), 256/243 (90)</li><li>On the D string (from Db to D)
 
X81/64 (408), 5/4 (386), '''119/96 (372), 16/13 (360), 39/32 (342), 144/119 (330)''', 6/5 (316), 32/27 (294)</li></ul>
 
Ascending and descending intervals are indeed the same, which is what "super-symmetrical" means in this context.
 
==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>
 
Mandal positions in ratios:
 
<span style="color: #00000a; font-family: Tahoma;">81/80, 125/124, 124/123, 123/122, 122/121, 121/120, 81/80</span>
 
Table of pitches from C to F (approximations in cents):
 
{| class="wikitable"
|-
! | String
! | b
! |
! |
! |
! |  
! |  
! |
! | Base note
! |  
! |  
! |  
! |
! |
! |
! | #
|-
! | C
| |
| |
| |
| |
| |
| |
| |
| style="text-align:center;" | 0
| | 22
| | 36
| | 50
| | 64
| | 78
| | 92
| | 114
|-
! | D
| | 90
| | 112
| | 126
| | 140
| | 154
| | 168
| | 182
| style="text-align:center;" | 204
| | 226
| | 240
| | 254
| | 268
| | 282
| | 296
| | 318
|-
! | E
| | 294
| | 316
| | 330
| | 344
| | 358
| | 372
| | 386
| style="text-align:center;" | 408
| | 430
| | 444
| | 458
| | 472
| | 486
| | 500
| | 522
|-
! | F
| | 384
| | 406
| | 420
| | 434
| | 448
| | 462
| | 476
| style="text-align:center;" | 498
| | 520
| | 534
| | 548
| | 562
| | 576
| | 590
| | 612
|}
 
Interval ratios, ascending from C:
 
<ul><li>On the D string (from Db to D):
 
256/243 (90), 16/15 (112), '''100/93 (126), 400/369 (140), 200/183 (153.78), 400/363 (168).''' 10/9 (182), 9/8 (204)</li><li>On the E string (from Eb to E):
 
32/27 (294), 6/5 (316), '''75/62 (329.54), 50/41 (343.56), 75/61 (357.69), 150/121 (371.94),''' 5/4 (386), 81/64 (408)</li></ul>
 
Interval ratios descending from F:
 
<ul><li>On the E string (from Eb to E):
 
9/8, 10/9, '''248/225 (168.49), 82/75 (154.47), 244/225 (140.34), 242/225 (126.09'''), 16/15, 256/243or approximating ratios: XXX</li><li>On the D string (from Db to D):
 
81/64, 5/4, '''31/25 (372.40), 49/40 (351.33), 61/50 (344.25), 121/100 (330)''', 6/5, 32/27Or approximatiing ratios: XXX</li></ul>
 
==Super-symmetrical model with 14/13==
© J.J. Weiss
 
The idea behind this system is as follows:
 
Dividing the apotome (114 cents) into 3 equal parts gives 38 cents, and adding this to the pythagorean limma (90 cents) gives 128 cents, which is an approximation for [[14/13|14/13]] (two-third tone, a favorite interval of [http://en.wikipedia.org/wiki/Avicenna Avicenna/Ibn Sina]).
 
The division of the apotome derived from this combines the known basic division into apotome, Zarlinian semitone and apotome with an equal division into 3 parts, which yields the following mandal positions (cents):
 
22, 16, 13, 12, 13, 16, 22
 
(Observe that 22+16 = 38, as well as 13+12+13.)
 
Mandal positions in ratios:
 
81/80, 1701/1664, 416/413, 3456/3481, 416/413, 1701/1664, 81/80
 
Since the pythagorean limma appears prominently in the basic framework anyway (as semitone from E to F and from B to C as well as one apotome minus a syntonic comma several times on each string), 14/13 also appears at various positions.
 
Table of pitches from C to F (approximations in cents):
 
{| class="wikitable"
|-
! | String
! | b
! |
! |
! |
! |
! |
! |
! | Base note
! |
! |
! |
! |
! |
! |
! | #
|-
! | C
| |
| |
| |
| |
| |
| |
| |
| style="text-align:center;" | 0
| | 22
| | 38
| | 51
| | 63
| | 76
| | 92
| | 114
|-
! | D
| | 90
| | 112
| | 128
| | 141
| | 153
| | 166
| | 182
| style="text-align:center;" | 204
| | 226
| | 242
| | 255
| | 267
| | 280
| | 296
| | 318
|-
! | E
| | 294
| | 316
| | 329
| | 341
| | 354
| | 370
| | 386
| style="text-align:center;" | 408
| | 430
| | 446
| | 459
| | 471
| | 484
| | 500
| | 522
|-
! | F
| | 384
| | 406
| | 422
| | 435
| | 447
| | 460
| | 476
| style="text-align:center;" | 498
| | 520
| | 536
| | 549
| | 561
| | 574
| | 590
| | 612
|}
 
Interval ratios, ascending from C:
 
<ul><li>On the D string (from Db to D):
 
256/243 (90), 16/15/112), 14/13 (128), 64/59 (141), 59/54 /153), 209/189 (166), 10(9 (182), 9/8 (204)</li><li>On the E string (from Eb to E):
 
32/27 (294), 6/5 (316), 63/52 (332), 72/59 (345), 59/48 (357), 26/21 (370), 5/4 (386), 81/64 (408)</li></ul>
 
{{Navbox notation}}
 
{{Todo| cleanup }}
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[[Category:Qanun]]

Latest revision as of 18:26, 11 February 2025

Tuning systems for the qanun

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 81/64, a major sixth of 27/16 and a major seventh of 243/128.

The possible pitches of a string obtained via raising/lowering the mandals lie within two 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 unequal parts, which requires 14 mandals per string. The first rough subdivision of the apotome is always into one syntonic comma (81/80, 21.5 cents), one 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, altogether 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),

An notable property (of all systems) is that the second-highest mandal position of, say, the C string is 114-22=92 cents (the major limma), while the lowest mandal position on the following string (D in the example) is 214 (one wholetone above C) - 114 = 90 cents (the pythagorean limma, the same interval as between E and F) - we have two notes differing by one 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, corresponding to the mandals. Seven base notes (C, D, E, F, G, A, B or Do, Re, Mi, Fa, Sol, La, Si), corresponding to the strings, lead to a notation system of 7*15=105 pitches, in accordance with the real playing capabilities of the qanun. 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.

Tableaux JJW VIII-2011.pdf

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

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 (approximations in cents in parentheses):

81/80 (22), 49/48 (35), 1053/1024 (48), 729/704 (60), 2673/2560 (76), 135/128 (92), 2187/2048 (114)

Intervals ratios, ascending from C:

  • 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), 99/80 (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)

A complete list of all intervals available within one octave can be found in the above-mentioned document (on the first page).

System 2, better suited for ottoman maqams

© J.J. Weiss. Qanun no. 9, luthier: Kenan Ozten.

Mandal positions in ratios:

81/80, 105/104, 572/567, 144/143, 1547/1536, 120/119, 81/80

In cents (approximations):

22|16|15|12|13|14|22

Rational intervals each string can be detuned (approximations in cents in parentheses):

81/80 (22), 1701/1664 (38), 33/32 (54), 27/26 (66), 243/232 (78), 135/128 (92), 2187/2048 (114)

Intervals ratios, ascending from C:

  • On the D string (from Db to D): 256/243 (90), 16/15 (112), 14/13 (128), 88/81 (144), 128/117 or 35/32 (156), 119/108 (168), 10/9 (182), 9/8 (204)
  • On the E string (from Eb to E): 32/27 (294), 6/5 (316), 63/52 (332), 11/9 (348), 16/13 or 315/256 (360), 119/96 (372), 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), 208/189 (166), 12/11 (150), 13/12 (138), 128/119 (126), 16/15 (112), 256/243 (90)
  • On the D string (from Db to D): 81/64 (408), 5/4 (386), 26/21 (370), 27/22 (354), 39/32 (342), 144/119 (330), 6/5 (316), 32/27 (294)

A complete list of all intervals available within one octave can be found in the above-mentioned document (on the second page).

Other models

Julien Weiss has developed a number of other systems besides the two described above. A notable class of these are so-called super-symmetrical systems, which have the property that the intervals ascending from C and the intervals descending from F (which show slight differences in the previous two systems, marked in bold above) are the same.

3 examples are described below. For more and detailed descriptions see chapter 3.4 and appendix I in Stefan Pohlit's dissertation .

Super-symmetric model with non-aliquot division of 65/64

© J.J. Weiss

Similar to system 1, but with 65/64 (26.84 cents) divided into two non-equal parts (14 and 12 cents instead of 13 and 13).

Mandal positions in ratios:

81/80 (22), 120/119 (14), 1547/1536 (12), 512/507 (17), 1547/1536 (12), 120/119 (14), 81/80 (22)

Table of pitches from C to F (approximations in cents):

String b Base note #
C 0 22 36 48 66 78 92 114
D 90 112 126 138 156 168 182 204 226 240 252 270 282 296 318
E 294 316 330 342 360 372 386 408 430 444 456 474 486 500 522
F 384 406 420 433 450 462 476 498 520 534 546 464 476 590 612

Interval ratios, ascending from C:

  • On the D string (from Db to D): 245/243 (90), 16/15 (112), 128/119 (126), 13/12 (138), 128/117 (156), 119/108 (168), 10/9 (182), 9/8 (204) On the E string (from Eb to E): 32/27 (294), 6/5 (316), 144/119 (330), 39/32 (342), 16/13 (360), 119/96 (372), 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), 119/108 (168), 128/117 (156), 13/12 (138), 128/119 (126), 16/15 (112), 256/243 (90)
  • On the D string (from Db to D) X81/64 (408), 5/4 (386), 119/96 (372), 16/13 (360), 39/32 (342), 144/119 (330), 6/5 (316), 32/27 (294)

Ascending and descending intervals are indeed the same, which is what "super-symmetrical" means in this context.

Equal division of the Zarlinian semitone

© J.J. Weiss

This is the simplest variant for luthiers...

Mandal positions (cents): 22|14|14|14|14|14|22

Mandal positions in ratios:

81/80, 125/124, 124/123, 123/122, 122/121, 121/120, 81/80

Table of pitches from C to F (approximations in cents):

String b Base note #
C 0 22 36 50 64 78 92 114
D 90 112 126 140 154 168 182 204 226 240 254 268 282 296 318
E 294 316 330 344 358 372 386 408 430 444 458 472 486 500 522
F 384 406 420 434 448 462 476 498 520 534 548 562 576 590 612

Interval ratios, ascending from C:

  • On the D string (from Db to D): 256/243 (90), 16/15 (112), 100/93 (126), 400/369 (140), 200/183 (153.78), 400/363 (168). 10/9 (182), 9/8 (204)
  • On the E string (from Eb to E): 32/27 (294), 6/5 (316), 75/62 (329.54), 50/41 (343.56), 75/61 (357.69), 150/121 (371.94), 5/4 (386), 81/64 (408)

Interval ratios descending from F:

  • On the E string (from Eb to E): 9/8, 10/9, 248/225 (168.49), 82/75 (154.47), 244/225 (140.34), 242/225 (126.09), 16/15, 256/243or approximating ratios: XXX
  • On the D string (from Db to D): 81/64, 5/4, 31/25 (372.40), 49/40 (351.33), 61/50 (344.25), 121/100 (330), 6/5, 32/27Or approximatiing ratios: XXX

Super-symmetrical model with 14/13

© J.J. Weiss

The idea behind this system is as follows:

Dividing the apotome (114 cents) into 3 equal parts gives 38 cents, and adding this to the pythagorean limma (90 cents) gives 128 cents, which is an approximation for 14/13 (two-third tone, a favorite interval of Avicenna/Ibn Sina).

The division of the apotome derived from this combines the known basic division into apotome, Zarlinian semitone and apotome with an equal division into 3 parts, which yields the following mandal positions (cents):

22, 16, 13, 12, 13, 16, 22

(Observe that 22+16 = 38, as well as 13+12+13.)

Mandal positions in ratios:

81/80, 1701/1664, 416/413, 3456/3481, 416/413, 1701/1664, 81/80

Since the pythagorean limma appears prominently in the basic framework anyway (as semitone from E to F and from B to C as well as one apotome minus a syntonic comma several times on each string), 14/13 also appears at various positions.

Table of pitches from C to F (approximations in cents):

String b Base note #
C 0 22 38 51 63 76 92 114
D 90 112 128 141 153 166 182 204 226 242 255 267 280 296 318
E 294 316 329 341 354 370 386 408 430 446 459 471 484 500 522
F 384 406 422 435 447 460 476 498 520 536 549 561 574 590 612

Interval ratios, ascending from C:

  • On the D string (from Db to D): 256/243 (90), 16/15/112), 14/13 (128), 64/59 (141), 59/54 /153), 209/189 (166), 10(9 (182), 9/8 (204)
  • On the E string (from Eb to E): 32/27 (294), 6/5 (316), 63/52 (332), 72/59 (345), 59/48 (357), 26/21 (370), 5/4 (386), 81/64 (408)


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