Tuning systems for qanun: Difference between revisions
Wikispaces>hstraub **Imported revision 262862548 - Original comment: ** |
Wikispaces>hstraub **Imported revision 262863912 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2011-10-08 16: | : This revision was by author [[User:hstraub|hstraub]] and made on <tt>2011-10-08 16:41:56 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>262863912</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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> 256/243 (90), 16/15 (112), **784/729 (126)**, **13/12 (138)**, **12/11 (150)**, **11/10 (166)**, 10/9 (182), 9/8 (204) | > 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): | * On the E string (from Eb to E): | ||
> 32/27 (294), 6/5 (316), **98/81 (330)**, **39/32 (342)**, **27/22 (354)**, ** | > 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: | Interval ratios, descending from F: | ||
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==Example: Super-symmetrical model with 14/13== | ==Example: Super-symmetrical model with 14/13== | ||
© J.J. Weiss | © J.J. Weiss | ||
The idea behind this system is as | 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 | 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]]. | ||
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): | 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 | 22, 16, 13, 12, 13, 16, 22 | ||
(Observe that 22+16 = 38, as | (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. | 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. | ||
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||~ F || 384 || 406 || 422 || 435 || 447 || 460 || 476 ||= 498 || 520 || 536 || 549 || 561 || 574 || 590 || 612 || | ||~ 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) | |||
XXX</pre></div> | |||
<h4>Original HTML content:</h4> | <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"><html><head><title>tuning systems for qanun</title></head><body><span style="font-size: 150%;"><strong>Tuning systems for the qanun</strong></span><br /> | <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"><html><head><title>tuning systems for qanun</title></head><body><span style="font-size: 150%;"><strong>Tuning systems for the qanun</strong></span><br /> | ||
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<br /> | <br /> | ||
This gives 15 potential different pitches per base note, correpsonding 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.<br /> | This gives 15 potential different pitches per base note, correpsonding 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.<br /> | ||
<!-- ws:start:WikiTextFileRule: | <!-- ws:start:WikiTextFileRule:229:&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:229 --><br /> | ||
<br /> | <br /> | ||
(used with permission J. J. Weiss/S. Pohlit)<br /> | (used with permission J. J. Weiss/S. Pohlit)<br /> | ||
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<ul><li>On the D string (from Db to D):<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 /> | 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> | 32/27 (294), 6/5 (316), <strong>98/81 (330)</strong>, <strong>39/32 (342)</strong>, <strong>27/22 (354)</strong>, <strong>99/80 (370)</strong>, 5/4 (386), 81/64 (408)</li></ul><br /> | ||
Interval ratios, descending from F:<br /> | Interval ratios, descending from F:<br /> | ||
<ul><li>On the E string (from Eb to E):<br /> | <ul><li>On the E string (from Eb to E):<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Other models-Example: Super-symmetrical model with 14/13"></a><!-- ws:end:WikiTextHeadingRule:8 -->Example: Super-symmetrical model with 14/13</h2> | <!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Other models-Example: Super-symmetrical model with 14/13"></a><!-- ws:end:WikiTextHeadingRule:8 -->Example: Super-symmetrical model with 14/13</h2> | ||
© J.J. Weiss<br /> | © J.J. Weiss<br /> | ||
The idea behind this system is as | The idea behind this system is as follows:<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 | 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 <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 apotome 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 /> | <br /> | ||
22, 16, 13, 12, 13, 16, 22<br /> | 22, 16, 13, 12, 13, 16, 22<br /> | ||
<br /> | <br /> | ||
(Observe that 22+16 = 38, as | (Observe that 22+16 = 38, as well as 13+12+13.)<br /> | ||
<br /> | |||
Mandal positions in ratios:<br /> | |||
81/80, 1701/1664, 416/413, 3456/3481, 416/413, 1701/1664, 81/80<br /> | |||
<br /> | <br /> | ||
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.<br /> | 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.<br /> | ||
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<br /> | <br /> | ||
Interval ratios, ascending from C:<br /> | |||
<ul><li>On the D string (from Db to D):<br /> | |||
<br /> | 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):<br /> | ||
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><br /> | |||
XXX</body></html></pre></div> | |||