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:

: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2011-11-08 06:50:03 UTC</tt>.

: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2011-11-08 07:29:58 UTC</tt>.

: The original revision id was <tt>~~273056000~~</tt>.

: The original revision id was <tt>273064284</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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

||~ String ||~ b ||~ ||~ ||~ ||~ ||~ ||~ ||~ Base note ||~ ||~ ||~ ||~ ||~ ||~ ||~ # ||

||~ C || || || || || || || ||= 0 || 22 || || || || || 92 || 114 ||

||~ C || || || || || || || ||= 0 || 22 || 36 || 48 || 66 || 78 || 92 || 114 ||

||~ D || 90 || 112 || 126 || 138 || 156 || 168 || 182 ||= 204 || 226 || || || || || 196 || 318 ||

||~ D || 90 || 112 || 126 || 138 || 156 || 168 || 182 ||= 204 || 226 || 240 || 252 || 170 || 182 || 196 || 318 ||

||~ E || 294 || 316 || || || || || 386 ||= 408 || 430 || || || || || ~~500~~ || ~~522~~ ||

||~ E || 294 || 316 || 330 || 342 || 360 || 372 || 386 ||= 408 || 430 || 444 || 456 || 374 || 386 || 400 || 422 ||

||~ F || 384 || 406 || || || || || 476 ||= 498 || 520 || || || || || 590 || 612 ||

||~ F || 384 || 406 || 420 || 433 || 450 || 462 || 476 ||= 498 || 520 || 534 || 546 || 464 || 476 || 590 || 612 ||

Interval ratios, ascending from C:

Line 112:

> 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):

> ~~XXX~~

> 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:

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)

~~XXX~~

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

==Equal division of the Zarlinian semitone==

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Interval ratios descending from F:

* On the E string (from Eb to E):

XXX

> XX

* On the D string (from Db to D):

> XXX

==Super-symmetrical model with 14/13==

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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.

<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> <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>

<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> <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>

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

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<td>22

</td>

<td>

<td>36

</td>

<td>

<td>48

</td>

<td>

<td>66

</td>

<td>

<td>78

</td>

<td>92

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<td>226

</td>

<td>

<td>240

</td>

<td>

<td>252

</td>

<td>

<td>170

</td>

<td>

<td>182

</td>

<td>196

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<td>316

</td>

<td>

<td>330

</td>

<td>

<td>342

</td>

<td>

<td>360

</td>

<td>

<td>372

</td>

<td>386

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<td>430

</td>

<td>

<td>444

</td>

<td>

<td>456

</td>

<td>

<td>374

</td>

<td>

<td>386

</td>

<td>~~500~~

<td>400

</td>

<td>~~522~~

<td>422

</td>

</tr>

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<td>406

</td>

<td>

<td>420

</td>

<td>

<td>433

</td>

<td>

<td>450

</td>

<td>

<td>462

</td>

<td>476

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<td>520

</td>

<td>

<td>534

</td>

<td>

<td>546

</td>

<td>

<td>464

</td>

<td>

<td>476

</td>

<td>590

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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):

~~XXX~~</li></ul>

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:

Interval ratios, descending from F:

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

~~XXX~~

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 &quot;super-symmetrical&quot; means in this context.

<h2 id="toc5"><a name="Other models-Equal division of the Zarlinian semitone"></a>Equal division of the Zarlinian semitone</h2>

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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):

~~XXX~~

XX</li><li>On the D string (from Db to D):

XXX</li></ul>

<h2 id="toc6"><a name="Other models-Super-symmetrical model with 14/13"></a>Super-symmetrical model with 14/13</h2>

@@ 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>2011-11-08 06:50:03 UTC</tt>.<br>
+: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2011-11-08 07:29:58 UTC</tt>.<br>
-: The original revision id was <tt>273056000</tt>.<br>
+: The original revision id was <tt>273064284</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 103: / Line 103: @@
 Table of pitches from C to F (approximations in cents):
 ||~ String ||~ b ||~   ||~   ||~   ||~   ||~   ||~   ||~ Base note ||~   ||~   ||~   ||~   ||~   ||~   ||~ # ||
-||~ C ||   ||   ||   ||   ||   ||   ||   ||= 0 || 22 ||   ||   ||   ||   || 92 || 114 ||
+||~ C ||   ||   ||   ||   ||   ||   ||   ||= 0 || 22 || 36 || 48 || 66 || 78 || 92 || 114 ||
-||~ D || 90 || 112 || 126 || 138 || 156 || 168 || 182 ||= 204 || 226 ||   ||   ||   ||   || 196 || 318 ||
+||~ D || 90 || 112 || 126 || 138 || 156 || 168 || 182 ||= 204 || 226 || 240 || 252 || 170 || 182 || 196 || 318 ||
-||~ E || 294 || 316 ||   ||   ||   ||   || 386 ||= 408 || 430 ||   ||   ||   ||   || 500 || 522 ||
+||~ E || 294 || 316 || 330 || 342 || 360 || 372 || 386 ||= 408 || 430 || 444 || 456 || 374 || 386 || 400 || 422 ||
-||~ F || 384 || 406 ||   ||   ||   ||   || 476 ||= 498 || 520 ||   ||   ||   ||   || 590 || 612 ||
+||~ F || 384 || 406 || 420 || 433 || 450 || 462 || 476 ||= 498 || 520 || 534 || 546 || 464 || 476 || 590 || 612 ||
 Interval ratios, ascending from C:
@@ Line 112: / Line 112: @@
 &gt; 245/243 (90), 16/15 (112), **128/119 (126), 13/12 (138), 128/117 (156), 119/108 (168),** 10/9 (182), 9/8 (204)
 &gt; On the E string (from Eb to E):
-&gt; XXX
+&gt; 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:
+Interval ratios, descending from F:
+* On the E string (from Eb to E):
+&gt; 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)
+&gt; X81/64 (408), 5/4 (386), **119/96 (372), 16/13 (360), 39/32 (342), 144/119 (330)**, 6/5 (316), 32/27 (294)
-XXX
+Ascending and descending intervals are indeed the same, which is what "super-symmetrical" means in this context.
 ==Equal division of the Zarlinian semitone==
@@ Line 138: / Line 142: @@
 Interval ratios descending from F:
+* On the E string (from Eb to E):
-XXX
+&gt; XX
+* On the D string (from Db to D):
+&gt; XXX
 ==Super-symmetrical model with 14/13==
@@ Line 202: / Line 207: @@
 &lt;br /&gt;
 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.&lt;br /&gt;
-&lt;!-- ws:start:WikiTextFileRule:594:&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:594 --&gt;&lt;br /&gt;
+&lt;!-- ws:start:WikiTextFileRule:610:&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:610 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 (used with permission J. J. Weiss/S. Pohlit)&lt;br /&gt;
@@ Line 323: / Line 328: @@
          &lt;td&gt;22&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;36&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;48&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;66&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;78&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;92&lt;br /&gt;
@@ Line 357: / Line 362: @@
          &lt;td&gt;226&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;240&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;252&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;170&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;182&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;196&lt;br /&gt;
@@ Line 377: / Line 382: @@
          &lt;td&gt;316&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;330&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;342&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;360&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;372&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;386&lt;br /&gt;
@@ Line 391: / Line 396: @@
          &lt;td&gt;430&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;444&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;456&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;374&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;386&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;500&lt;br /&gt;
+         &lt;td&gt;400&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;522&lt;br /&gt;
+         &lt;td&gt;422&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 411: / Line 416: @@
          &lt;td&gt;406&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;420&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;433&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;450&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;462&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;476&lt;br /&gt;
@@ Line 425: / Line 430: @@
          &lt;td&gt;520&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;534&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;546&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;464&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;476&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;590&lt;br /&gt;
@@ Line 445: / Line 450: @@
 /243 (90), 16/15 (112), &lt;strong&gt;128/119 (126), 13/12 (138), 128/117 (156), 119/108 (168),&lt;/strong&gt; 10/9 (182), 9/8 (204)&lt;br /&gt;
 On the E string (from Eb to E):&lt;br /&gt;
-XXX&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
+/27 (294), 6/5 (316), &lt;strong&gt;144/119 (330), 39/32 (342), 16/13 (360), 119/96 (372),&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;
+Interval ratios, descending from F:&lt;br /&gt;
-&lt;br /&gt;
+&lt;ul&gt;&lt;li&gt;On the E string (from Eb to E):&lt;br /&gt;
-XXX&lt;br /&gt;
+(8 /204), 10/9 (182), &lt;strong&gt;119/108 (168), 128/117 (156), 13/12 (138), 128/119 (126)&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;
+X81/64 (408), 5/4 (386), &lt;strong&gt;119/96 (372), 16/13 (360), 39/32 (342), 144/119 (330)&lt;/strong&gt;, 6/5 (316), 32/27 (294)&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
+Ascending and descending intervals are indeed the same, which is what &amp;quot;super-symmetrical&amp;quot; means in this context.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Other models-Equal division of the Zarlinian semitone"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Equal division of the Zarlinian semitone&lt;/h2&gt;
@@ Line 638: / Line 645: @@
 /27 (294), 6/5 (316), &lt;strong&gt;75/62 (329.54), 50/41 (343.56), 75/61 (357.69), 150/121 (371.94),&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;
-&lt;br /&gt;
+&lt;ul&gt;&lt;li&gt;On the E string (from Eb to E):&lt;br /&gt;
-XXX&lt;br /&gt;
+XX&lt;/li&gt;&lt;li&gt;On the D string (from Db to D):&lt;br /&gt;
-&lt;br /&gt;
+XXX&lt;/li&gt;&lt;/ul&gt;&lt;br /&gt;
-&lt;br /&gt;
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   © J.J. Weiss&lt;br /&gt;