Tuning systems for qanun: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>

: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2011-09-30 10:25:10 UTC</tt>.<br>

: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2011-09-30 16:33:13 UTC</tt>.<br>

: The original revision id was <tt>~~259988180~~</tt>.<br>

: The original revision id was <tt>260195654</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>

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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, 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.

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.

=Notation=

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The idea behind this system is as folows:

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 the approximation for 14/13.

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 the approximation for [[14_13|14/13]].

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

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

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||~ F || 384 || 406 || 422 || 435 || 447 || 460 || 476 ||= 498 || 520 || 536 || 549 || 561 || 574 || 590 || 612 ||

Mandal positions in ratios:

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

Ratios XXX</pre></div>

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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),<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 />

An notable property (of all systems) is that the second-highest mandal position of, say, the C string is 114-22=92 cents (the <a class="wiki_link" href="/135_128">major limma</a>), while the lowest mandal position on the following string (D in the example) is 214 (one wholetone above C) - 114 = 90 cents (the <a class="wiki_link" href="/256_243">pythagorean limma</a>, the same interval as between E and F) - 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 />

<h1 id="toc0"><a name="Notation"></a>Notation</h1>

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The idea behind this system is as folows:<br />

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 the approximation for 14/13.<br />

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 the approximation for <a class="wiki_link" href="/14_13">14/13</a>.<br />

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

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):<br />

<br />

22, 16, 13, 12, 13, 16, 22<br />

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<br />

Mandal positions in ratios:<br />

81/80, 1701/1664, 416/413, 3456/3481, 416/413, 1701/1664, 81/80<br />

<br />

Ratios XXX</body></html></pre></div>

@@ 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 10:25:10 UTC</tt>.<br>
+: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2011-09-30 16:33:13 UTC</tt>.<br>
-: The original revision id was <tt>259988180</tt>.<br>
+: The original revision id was <tt>260195654</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 23: / Line 23: @@
 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, 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.
+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.
 =Notation=
@@ Line 94: / Line 94: @@
 © J.J. Weiss
 The idea behind this system is as folows:
-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 the approximation for 14/13.
+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 the approximation for [[14_13|14/13]].
-The division of the apotome derived from this combines the known basic division into apotome, Zarlinian semitone and apotoms with an equal division into 3 parts, which yields the following mandal positions (cents):
+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):
 , 16, 13, 12, 13, 16, 22
@@ Line 110: / Line 110: @@
 ||~ F || 384 || 406 || 422 || 435 || 447 || 460 || 476 ||= 498 || 520 || 536 || 549 || 561 || 574 || 590 || 612 ||
+Mandal positions in ratios:
+/80, 1701/1664, 416/413, 3456/3481, 416/413, 1701/1664, 81/80
 Ratios XXX</pre></div>
@@ Line 135: / Line 137: @@
 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),&lt;br /&gt;
 &lt;br /&gt;
-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 &lt;a class="wiki_link" href="/32805_32768"&gt;schisma (2 cents)&lt;/a&gt;. So the interval of the schisma is present and can be played on a qanun in any of the tuning systems described here.&lt;br /&gt;
+An notable property (of all systems) is that the second-highest mandal position of, say, the C string is 114-22=92 cents (the &lt;a class="wiki_link" href="/135_128"&gt;major limma&lt;/a&gt;), while the lowest mandal position on the following string (D in the example) is 214 (one wholetone above C) - 114 = 90 cents (the &lt;a class="wiki_link" href="/256_243"&gt;pythagorean limma&lt;/a&gt;, the same interval as between E and F) - we have two notes differing by one &lt;a class="wiki_link" href="/32805_32768"&gt;schisma (2 cents)&lt;/a&gt;. So the interval of the schisma is present and can be played on a qanun in any of the tuning systems described here.&lt;br /&gt;
 &lt;br /&gt;
 &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;
@@ Line 200: / Line 202: @@
   © J.J. Weiss&lt;br /&gt;
 The idea behind this system is as folows:&lt;br /&gt;
-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 the approximation for 14/13.&lt;br /&gt;
+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 the approximation for &lt;a class="wiki_link" href="/14_13"&gt;14/13&lt;/a&gt;.&lt;br /&gt;
-The division of the apotome derived from this combines the known basic division into apotome, Zarlinian semitone and apotoms with an equal division into 3 parts, which yields the following mandal positions (cents):&lt;br /&gt;
+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):&lt;br /&gt;
 &lt;br /&gt;
 , 16, 13, 12, 13, 16, 22&lt;br /&gt;
@@ Line 386: / Line 388: @@
 &lt;br /&gt;
+Mandal positions in ratios:&lt;br /&gt;
+/80, 1701/1664, 416/413, 3456/3481, 416/413, 1701/1664, 81/80&lt;br /&gt;
 &lt;br /&gt;
 Ratios XXX&lt;/body&gt;&lt;/html&gt;</pre></div>