13edo: 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:~~xenwolf~~|~~xenwolf~~]] and made on <tt>2016-02-~~22 16~~:18:53 UTC</tt>.

: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-10-30 14:42:22 UTC</tt>.

: The original revision id was <tt>~~575535679~~</tt>.

: The original revision id was <tt>597493228</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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As a temperament of 21-odd-limit Just Intonation, 13edo has excellent approximations to the 11th and 21st harmonics, and reasonable approximations to the 5th, 9th, 13th, 17th, and 19th harmonics. For most purposes it does not offer acceptable approximations to the 3rd, 7th, or 15th. The lack of reasonable approximation to the 3rd harmonic makes 13edo unsuitable for common-practice music, but its good approximations to ratios of 11, 13, and 21 make it a very xenharmonic tuning, as these identities are not remotely represented in 12edo. Despite its reputation for dissonance, it is an excellent rank-1 temperament on the 2.5.9.11.13.17.19.21 subgroup, and has a substantial repertoire of complex consonances for its small size.

|| **Degree** || Cents

|| **Degree** || Cents (coarse/fine)

DMS ||= Approximated 21-limit Ratios* || Erv Wilson || Archaeotonic || Oneirotonic || [[26edo]] names || Kentaku ||

|| 0 || 0 ||= 1/1 || H || C || C || C || J ||

|| 1 || 92.~~3077~~

|| 1 || 92.31, 110.77

27º41'32" ||= 17/16, 18/17, 19/18, 20/19, 21/20, 22/21 || β || C#/Db || C#/Db || Cx/Dbb || J#/Kb ||

|| 2 || 184.~~6154~~

|| 2 || 184.615, 221.54

55°22'5" ||= 9/8, 10/9, 11/10, 19/17, 21/19 || A || D || D || D || K ||

|| 3 || 276.~~9231~~

|| 3 || 276.92, 332.31

83°4'37" ||= 7/6, 13/11, 20/17, 19/16, 22/19 || δ || D#/Eb || D#/Eb || Dx/Ebb || K#/Lb ||

|| 4 || 369.~~2308~~

|| 4 || 369.23, 443.08

110°46'9" ||= 5/4, 11/9, 16/13, 26/21 || C || E || E || E || L ||

|| 5 || 461.~~5385~~

|| 5 || 461.54, 553.85

138°27'42" ||= 13/10, 17/13, 21/16, 22/17 || B || E#/Fb || F || Ex/Fb || M ||

|| 6 || 553.84

|| 6 || 553.85, 664.615

166°9'14" ||= 11/8, 18/13, 26/19 || ε || F || F#/Gb || F# || M#/Nb ||

|| 7 || 646.15

|| 7 || 646.15, 775.385

193°50'46" ||= 16/11, 13/9, 19/13 || D || F#/Gb || G || Gb || N ||

|| 8 || 738.46

|| 8 || 738.46, 886.15

221°32'18" ||= 17/11, 20/13, 26/17, 32/21 || γ || G || G#/Hb || G# || N#/Ob ||

|| 9 || 830.77

|| 9 || 830.77, 996.92

249°13'51" ||= 8/5, 13/8, 18/11, 21/13 || F || G#/Ab || H || Ab || O ||

|| 10 || 923.08

|| 10 || 923.08, 1107.69

276°55'37" ||= [[17_10|17/10]], [[12_7|12/7]], [[22_13|22/13]], [[19_11|19/11]] || E || A || A || A# || P ||

|| 11 || 1015.38

|| 11 || 1015.385, 1218.46

304°37'55" ||= 9/5, 16/9, 20/11, 34/19, 38/21 || α || A#/Bb || A#/Bb || Bb || P#/Qb ||

|| 12 || 1107.69

|| 12 || 1107.69, 1329.23

332°18'28" ||= 17/9, 19/10, 21/11, 32/17, 36/19, 40/21 || G || B/Cb || B || B#/Cbb || Q ||

|| 13 || 1200

|| 13 || 1200, 1440

360° ||= 2/1 || H || C/B# || C || C || J ||

*based on treating 13-EDO as a 2.5.9.11.13.21 subgroup temperament; other approaches are possible.

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=Mapping to Standard Keyboards=

The 5L+3s scale (Oneirotonic) can be mapped to the standard keyboard effectively, although somewhat awkwardly. Consider the sequence of 730-cent intervals that it derives from: 1 6 11 3 8 (13) 5 10 2 7 12 4 9 1. One of these must be absent, so it might as well be the last. So, there are at most five of the full octatonic scales on different keys. Of the four mappings that keep the major pentatonic on the white keys, which ironically look like ordinary minor-pentatonics, the latter which begins on B might be the most straightforward to learn and use.

The 5L+3s scale (Oneirotonic) can be mapped to the standard keyboard effectively, although somewhat awkwardly. Consider the sequence of 730-cent intervals that it derives from: 1 6 11 3 [[tel:8 (13) 5 10 2 7 12 4 9 1|8 (13) 5 10 2 7 12 4 9 1]]. One of these must be absent, so it might as well be the last. So, there are at most five of the full octatonic scales on different keys. Of the four mappings that keep the major pentatonic on the white keys, which ironically look like ordinary minor-pentatonics, the latter which begins on B might be the most straightforward to learn and use.

|| 1 || 6 || 11 || 3 || 8 || (13) || 5 || 10 || 2 || 7 || 12 || 4 || 9 || 1 || Place in Chain of 730 cent intervals ||

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=Commas=

13 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes the val < 13 21 30 36 45 48 |.)

13 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes the val < [[tel:13 21 30 36 45 48|13 21 30 36 45 48]] |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||

||= 2109375/2097152 ||< | -21 3 7 > ||> 10.06 ||= Semicomma ||= Fokker Comma ||= ||

||= [[tel:2109375/2097152|2109375/2097152]] ||< | -21 3 7 > ||> 10.06 ||= Semicomma ||= Fokker Comma ||= ||

||= 1029/1000 ||< | -3 1 -3 3 > ||> 49.49 ||= Keega ||= ||= ||

||= 525/512 ||< | -9 1 2 1 > ||> 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||= ||

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<td>Degree

</td>

<td>Cents

<td>Cents (coarse/fine)

DMS

</td>

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<td>1

</td>

<td>92.~~3077~~

<td>92.31, 110.77

27º41'32&quot;

</td>

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<td>2

</td>

<td>184.~~6154~~

<td>184.615, 221.54

55°22'5&quot;

</td>

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<td>3

</td>

<td>276.~~9231~~

<td>276.92, 332.31

83°4'37&quot;

</td>

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<td>4

</td>

<td>369.~~2308~~

<td>369.23, 443.08

110°46'9&quot;

</td>

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<td>5

</td>

<td>461.~~5385~~

<td>461.54, 553.85

138°27'42&quot;

</td>

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<td>6

</td>

<td>553.84

<td>553.85, 664.615

166°9'14&quot;

</td>

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<td>7

</td>

<td>646.15

<td>646.15, 775.385

193°50'46&quot;

</td>

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<td>8

</td>

<td>738.46

<td>738.46, 886.15

221°32'18&quot;

</td>

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<td>9

</td>

<td>830.77

<td>830.77, 996.92

249°13'51&quot;

</td>

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<td>10

</td>

<td>923.08

<td>923.08, 1107.69

276°55'37&quot;

</td>

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<td>11

</td>

<td>1015.38

<td>1015.385, 1218.46

304°37'55&quot;

</td>

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<td>12

</td>

<td>1107.69

<td>1107.69, 1329.23

332°18'28&quot;

</td>

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<td>13

</td>

<td>1200

<td>1200, 1440

360°

</td>

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<h1 id="toc10"><a name="Mapping to Standard Keyboards"></a>Mapping to Standard Keyboards</h1>

The 5L+3s scale (Oneirotonic) can be mapped to the standard keyboard effectively, although somewhat awkwardly. Consider the sequence of 730-cent intervals that it derives from: 1 6 11 3 8 (13) 5 10 2 7 12 4 9 1. One of these must be absent, so it might as well be the last. So, there are at most five of the full octatonic scales on different keys. Of the four mappings that keep the major pentatonic on the white keys, which ironically look like ordinary minor-pentatonics, the latter which begins on B might be the most straightforward to learn and use.

The 5L+3s scale (Oneirotonic) can be mapped to the standard keyboard effectively, although somewhat awkwardly. Consider the sequence of 730-cent intervals that it derives from: 1 6 11 3 <a class="wiki_link" href="http://tel.wikispaces.com/8%20%2813%29%205%2010%202%207%2012%204%209%201">8 (13) 5 10 2 7 12 4 9 1</a>. One of these must be absent, so it might as well be the last. So, there are at most five of the full octatonic scales on different keys. Of the four mappings that keep the major pentatonic on the white keys, which ironically look like ordinary minor-pentatonics, the latter which begins on B might be the most straightforward to learn and use.

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<h1 id="toc11"><a name="Commas"></a>Commas</h1>

13 EDO <a class="wiki_link" href="/tempering%20out">tempers out</a> the following <a class="wiki_link" href="/comma">comma</a>s. (Note: This assumes the val &lt; 13 21 30 36 45 48 |.)

13 EDO <a class="wiki_link" href="/tempering%20out">tempers out</a> the following <a class="wiki_link" href="/comma">comma</a>s. (Note: This assumes the val &lt; <a class="wiki_link" href="http://tel.wikispaces.com/13%2021%2030%2036%2045%2048">13 21 30 36 45 48</a> |.)

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</tr>

<tr>

<td style="text-align: center;">2109375/2097152

<td style="text-align: center;">[[tel:2109375/2097152|2109375/2097152]]

</td>

<td style="text-align: left;">| -21 3 7 &gt;

@@ 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:xenwolf|xenwolf]] and made on <tt>2016-02-22 16:18:53 UTC</tt>.<br>
+: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-10-30 14:42:22 UTC</tt>.<br>
-: The original revision id was <tt>575535679</tt>.<br>
+: The original revision id was <tt>597493228</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 18: / Line 18: @@
 As a temperament of 21-odd-limit Just Intonation, 13edo has excellent approximations to the 11th and 21st harmonics, and reasonable approximations to the 5th, 9th, 13th, 17th, and 19th harmonics. For most purposes it does not offer acceptable approximations to the 3rd, 7th, or 15th. The lack of reasonable approximation to the 3rd harmonic makes 13edo unsuitable for common-practice music, but its good approximations to ratios of 11, 13, and 21 make it a very xenharmonic tuning, as these identities are not remotely represented in 12edo. Despite its reputation for dissonance, it is an excellent rank-1 temperament on the 2.5.9.11.13.17.19.21 subgroup, and has a substantial repertoire of complex consonances for its small size.
-|| **Degree** || Cents
+|| **Degree** || Cents (coarse/fine)
 &lt;span style="background-color: #ffffff;"&gt;DMS&lt;/span&gt; ||= Approximated 21-limit Ratios* || Erv Wilson || Archaeotonic || Oneirotonic || [[26edo]] names || Kentaku ||
 || 0 || 0 ||= 1/1 || H || C || C || C || J ||
-|| 1 || 92.3077
+|| 1 || 92.31, 110.77
 º41'32" ||= 17/16, 18/17, 19/18, 20/19, 21/20, 22/21 || &lt;span style="background-color: #ffffff; color: #333333; font-family: Verdana; font-size: 12px;"&gt;β&lt;/span&gt; || C#/Db || C#/Db || Cx/Dbb || J#/Kb ||
-|| 2 || 184.6154
+|| 2 || 184.615, 221.54
 °22'5" ||= 9/8, 10/9, 11/10, 19/17, 21/19 || A || D || D || D || K ||
-|| 3 || 276.9231
+|| 3 || 276.92, 332.31
 °4'37" ||= 7/6, 13/11, 20/17, 19/16, 22/19 || &lt;span style="background-color: #ffffff; color: #333333; font-family: Verdana; font-size: 12px;"&gt;δ&lt;/span&gt; || D#/Eb || D#/Eb || Dx/Ebb || K#/Lb ||
-|| 4 || 369.2308
+|| 4 || 369.23, 443.08
 °46'9" ||= 5/4, 11/9, 16/13, 26/21 || C || E || E || E || L ||
-|| 5 || 461.5385
+|| 5 || 461.54, 553.85
 °27'42" ||= 13/10, 17/13, 21/16, 22/17 || B || E#/Fb || F || Ex/Fb || M ||
-|| 6 || 553.84
+|| 6 || 553.85, 664.615
 °9'14" ||= 11/8, 18/13, 26/19 || &lt;span style="background-color: #ffffff; color: #333333; font-family: Verdana; font-size: 12px;"&gt;ε&lt;/span&gt; || F || F#/Gb || F# || M#/Nb ||
-|| 7 || 646.15
+|| 7 || 646.15, 775.385
 °50'46" ||= 16/11, 13/9, 19/13 || D || F#/Gb || G || Gb || N ||
-|| 8 || 738.46
+|| 8 || 738.46, 886.15
 °32'18" ||= 17/11, 20/13, 26/17, 32/21 || &lt;span style="background-color: #ffffff; color: #333333; font-family: Verdana; font-size: 12px;"&gt;γ&lt;/span&gt; || G || G#/Hb || G# || N#/Ob ||
-|| 9 || 830.77
+|| 9 || 830.77, 996.92
 °13'51" ||= 8/5, 13/8, 18/11, 21/13 || F || G#/Ab || H || Ab || O ||
-|| 10 || 923.08
+|| 10 || 923.08, 1107.69
 °55'37" ||= [[17_10|17/10]], [[12_7|12/7]], [[22_13|22/13]], [[19_11|19/11]] || E || A || A || A# || P ||
-|| 11 || 1015.38
+|| 11 || 1015.385, 1218.46
 °37'55" ||= 9/5, 16/9, 20/11, 34/19, 38/21 || &lt;span style="background-color: #ffffff; color: #333333; font-family: Verdana; font-size: 12px;"&gt;α&lt;/span&gt; || A#/Bb || A#/Bb || Bb || P#/Qb ||
-|| 12 || 1107.69
+|| 12 || 1107.69, 1329.23
 °18'28" ||= 17/9, 19/10, 21/11, 32/17, 36/19, 40/21 || G || B/Cb || B || B#/Cbb || Q ||
-|| 13 || 1200
+|| 13 || 1200, 1440
 ° ||= 2/1 || H || C/B# || C || C || J ||
 *based on treating 13-EDO as a 2.5.9.11.13.21 subgroup temperament; other approaches are possible.
@@ Line 120: / Line 120: @@
 =Mapping to Standard Keyboards=
-The 5L+3s scale (Oneirotonic) can be mapped to the standard keyboard effectively, although somewhat awkwardly. Consider the sequence of 730-cent intervals that it derives from: 1 6 11 3 8 (13) 5 10 2 7 12 4 9 1. One of these must be absent, so it might as well be the last. So, there are at most five of the full octatonic scales on different keys. Of the four mappings that keep the major pentatonic on the white keys, which ironically look like ordinary minor-pentatonics, the latter which begins on B might be the most straightforward to learn and use.
+The 5L+3s scale (Oneirotonic) can be mapped to the standard keyboard effectively, although somewhat awkwardly. Consider the sequence of 730-cent intervals that it derives from: 1 6 11 3 [[tel:8 (13) 5 10 2 7 12 4 9 1|8 (13) 5 10 2 7 12 4 9 1]]. One of these must be absent, so it might as well be the last. So, there are at most five of the full octatonic scales on different keys. Of the four mappings that keep the major pentatonic on the white keys, which ironically look like ordinary minor-pentatonics, the latter which begins on B might be the most straightforward to learn and use.
 || 1 || 6 || 11 || 3 || 8 || (13) || 5 || 10 || 2 || 7 || 12 || 4 || 9 || 1 || Place in Chain of 730 cent intervals ||
@@ Line 138: / Line 138: @@
 =Commas=
-EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes the val &lt; 13 21 30 36 45 48 |.)
+EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes the val &lt; [[tel:13 21 30 36 45 48|13 21 30 36 45 48]] |.)
 ||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
-||= 2109375/2097152 ||&lt; | -21 3 7 &gt; ||&gt; 10.06 ||= Semicomma ||= Fokker Comma ||=   ||
+||= [[tel:2109375/2097152|2109375/2097152]] ||&lt; | -21 3 7 &gt; ||&gt; 10.06 ||= Semicomma ||= Fokker Comma ||=   ||
 ||= 1029/1000 ||&lt; | -3 1 -3 3 &gt; ||&gt; 49.49 ||= Keega ||=   ||=   ||
 ||= 525/512 ||&lt; | -9 1 2 1 &gt; ||&gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||=   ||
@@ Line 203: / Line 203: @@
          &lt;td&gt;&lt;strong&gt;Degree&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Cents&lt;br /&gt;
+         &lt;td&gt;Cents (coarse/fine)&lt;br /&gt;
 &lt;span style="background-color: #ffffff;"&gt;DMS&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 240: / Line 240: @@
          &lt;td&gt;1&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;92.3077&lt;br /&gt;
+         &lt;td&gt;92.31, 110.77&lt;br /&gt;
 º41'32&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 259: / Line 259: @@
          &lt;td&gt;2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;184.6154&lt;br /&gt;
+         &lt;td&gt;184.615, 221.54&lt;br /&gt;
 °22'5&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 278: / Line 278: @@
          &lt;td&gt;3&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;276.9231&lt;br /&gt;
+         &lt;td&gt;276.92, 332.31&lt;br /&gt;
 °4'37&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 297: / Line 297: @@
          &lt;td&gt;4&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;369.2308&lt;br /&gt;
+         &lt;td&gt;369.23, 443.08&lt;br /&gt;
 °46'9&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 316: / Line 316: @@
          &lt;td&gt;5&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;461.5385&lt;br /&gt;
+         &lt;td&gt;461.54, 553.85&lt;br /&gt;
 °27'42&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 335: / Line 335: @@
          &lt;td&gt;6&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;553.84&lt;br /&gt;
+         &lt;td&gt;553.85, 664.615&lt;br /&gt;
 °9'14&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 354: / Line 354: @@
          &lt;td&gt;7&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;646.15&lt;br /&gt;
+         &lt;td&gt;646.15, 775.385&lt;br /&gt;
 °50'46&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 373: / Line 373: @@
          &lt;td&gt;8&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;738.46&lt;br /&gt;
+         &lt;td&gt;738.46, 886.15&lt;br /&gt;
 °32'18&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 392: / Line 392: @@
          &lt;td&gt;9&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;830.77&lt;br /&gt;
+         &lt;td&gt;830.77, 996.92&lt;br /&gt;
 °13'51&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 411: / Line 411: @@
          &lt;td&gt;10&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;923.08&lt;br /&gt;
+         &lt;td&gt;923.08, 1107.69&lt;br /&gt;
 °55'37&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 430: / Line 430: @@
          &lt;td&gt;11&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;1015.38&lt;br /&gt;
+         &lt;td&gt;1015.385, 1218.46&lt;br /&gt;
 °37'55&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 449: / Line 449: @@
          &lt;td&gt;12&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;1107.69&lt;br /&gt;
+         &lt;td&gt;1107.69, 1329.23&lt;br /&gt;
 °18'28&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 468: / Line 468: @@
          &lt;td&gt;13&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;1200&lt;br /&gt;
+         &lt;td&gt;1200, 1440&lt;br /&gt;
 °&lt;br /&gt;
 &lt;/td&gt;
@@ Line 559: / Line 559: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:29:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc10"&gt;&lt;a name="Mapping to Standard Keyboards"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:29 --&gt;Mapping to Standard Keyboards&lt;/h1&gt;
-  The 5L+3s scale (Oneirotonic) can be mapped to the standard keyboard effectively, although somewhat awkwardly. Consider the sequence of 730-cent intervals that it derives from: 1 6 11 3 8 (13) 5 10 2 7 12 4 9 1. One of these must be absent, so it might as well be the last. So, there are at most five of the full octatonic scales on different keys. Of the four mappings that keep the major pentatonic on the white keys, which ironically look like ordinary minor-pentatonics, the latter which begins on B might be the most straightforward to learn and use.&lt;br /&gt;
+  The 5L+3s scale (Oneirotonic) can be mapped to the standard keyboard effectively, although somewhat awkwardly. Consider the sequence of 730-cent intervals that it derives from: 1 6 11 3 &lt;a class="wiki_link" href="http://tel.wikispaces.com/8%20%2813%29%205%2010%202%207%2012%204%209%201"&gt;8 (13) 5 10 2 7 12 4 9 1&lt;/a&gt;. One of these must be absent, so it might as well be the last. So, there are at most five of the full octatonic scales on different keys. Of the four mappings that keep the major pentatonic on the white keys, which ironically look like ordinary minor-pentatonics, the latter which begins on B might be the most straightforward to learn and use.&lt;br /&gt;
 &lt;br /&gt;
@@ Line 954: / Line 954: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:31:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc11"&gt;&lt;a name="Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:31 --&gt;Commas&lt;/h1&gt;
-EDO &lt;a class="wiki_link" href="/tempering%20out"&gt;tempers out&lt;/a&gt; the following &lt;a class="wiki_link" href="/comma"&gt;comma&lt;/a&gt;s. (Note: This assumes the val &amp;lt; 13 21 30 36 45 48 |.)&lt;br /&gt;
+EDO &lt;a class="wiki_link" href="/tempering%20out"&gt;tempers out&lt;/a&gt; the following &lt;a class="wiki_link" href="/comma"&gt;comma&lt;/a&gt;s. (Note: This assumes the val &amp;lt; &lt;a class="wiki_link" href="http://tel.wikispaces.com/13%2021%2030%2036%2045%2048"&gt;13 21 30 36 45 48&lt;/a&gt; |.)&lt;br /&gt;
@@ Line 973: / Line 973: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;2109375/2097152&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;[[tel:2109375/2097152|2109375/2097152]]&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: left;"&gt;| -21 3 7 &amp;gt;&lt;br /&gt;