21edo: Difference between revisions

Line 1:

<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:~~Gedankenwelt~~|~~Gedankenwelt~~]] and made on <tt>~~2015~~-12-~~12 05~~:20:20 UTC</tt>.

: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-10-30 18:56:53 UTC</tt>.

: The original revision id was <tt>~~569913991~~</tt>.

: The original revision id was <tt>597508726</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.

Line 19:

The patent val for 21edo tempers out 128/125 and 2187/2000 in the 5-limit, and supplies the optimal patent val for the 5-limit [[Laconic Family|laconic]] temperament tempering out 2187/2000, and also the optimal patent val for 7-limit, 11-limit and 13-limit laconic, spartan and gorgo temperaments. These temperaments lead to some "interesting" mappings, where 10/9 is larger than 9/8, 11/9 is larger than 16/13, and 8/7 maps to the same interval as 10/9, for instance.

|| **Degree** || **Cents, DMS**

|| **Degree** || **Cents coarse/fine, DMS**

**Value** || **7-EDO**

**Notation** || **5L3s**

Line 26:

**Ratios *1** ||= **Approximate****Ratios *2** ||= **Approximate****Ratios *3** ||

|| 0 || 0 || C || C ||= Unison ||= 1/1 ||= 1/1 ||= 1/1 ||

|| 1 || 57.143

|| 1 || 57.143, 68.571

17°8'34" || C^/Dvv || C# ||= Subminor 2nd ||= 28/27, 30/29 ||= 35/34, 36/35 ||= 64/63 ||

|| 2 || 114.286

|| 2 || 114.286, 137.143

34°17'9" || C^^/Dv || Db ||= Minor 2nd ||= 16/15, 15/14, 29/27 ||= 18/17 ||= 16/15, 25/24 ||

|| 3 || 171.429

|| 3 || 171.429, 205.714

51°25'43" || D || D ||= Submajor 2nd ||= 10/9, 32/29 ||= 10/9,11/10 ||= 9/8 ||

|| 4 || 228.571

|| 4 || 228.571, 274.286

68°34'17" || D^/Evv || D# ||= Supermajor 2nd ||= 8/7 ||= 8/7 ||= 8/7, 10/9, 11/10 ||

|| 5 || 285.714

|| 5 || 285.714, 342.857

85°42'51" || D^^/Ev || Eb ||= Subminor 3rd ||= 27/23, 32/27 ||= 13/11, 20/17 ||= 6/5, 7/6 ||

|| 6 || 342.857

|| 6 || 342.857, 411.429

102°51'26" || E || E ||= Neutral 3rd ||= 28/23 ||= 11/9 ||= 16/13 ||

|| 7 || 400

|| 7 || 400, 480

120° || E^/Fvv || E#/Fb ||= Major 3rd ||= 29/23 ||= 44/35 ||= 5/4, 9/7, 11/9, 14/11 ||

|| 8 || 457.143

|| 8 || 457.143, 548.571

137°8'34" || E^^/Fv || F ||= Third-Fourth ||= 30/23 ||= 13/10, 17/13, 22/17 ||= 13/10 ||

|| 9 || 514.286

|| 9 || 514.286, 617.143

154°17'9" || F || F# ||= Acute 4th ||= 161/120, 256/189 ||= 35/26 ||= 4/3, 18/13 ||

|| 10 || 571.429

|| 10 || 571.429, 685.714

171°25'43" || F^/Gvv || Gb ||= Narrow Tritone ||= 32/23 ||= 18/13 ||= 7/5, 11/8 ||

|| 11 || 628.571

|| 11 || 628.571, 754.286

188°34'17" || F^^/Gv || G ||= Wide Tritone ||= 23/16 ||= 13/9 ||= 10/7, 16/11 ||

|| 12 || 685.714

|| 12 || 685.714, 824.857

205°42'51" || G || G# ||= Grave 5th ||= 189/128, 240/161 ||= 52/35 ||= 3/2, 13/9 ||

|| 13 || 742.857

|| 13 || 742.857, 891.428

222°51'26" || G^/Avv || Hb ||= Fifth-Sixth ||= 23/15 ||= 17/11, 20/13, 26/17 ||= 20/13 ||

|| 14 || 800

|| 14 || 800, 960

240° || G^^/Av || H ||= Minor 6th ||= 46/29 ||= 35/22 ||= 8/5, 11/7, 14/9, 18/11 ||

|| 15 || 857.143

|| 15 || 857.143, 1028.571

257°8'34" || A || H#/Ab ||= Neutral 6th ||= 23/14 ||= 18/11 ||= 13/8 ||

|| 16 || 914.286

|| 16 || 914.286, 1097.143

274°17'9" || A^/Bvv || A ||= Supermajor 6th ||= 27/16, 46/27 ||= 17/10, 22/13 ||= 5/3, 12/7 ||

|| 17 || 971.429

|| 17 || 971.429, 1165.714

391°25'43" || A^^/Bv || A# ||= Subminor 7th ||= 7/4 ||= 7/4 ||= 7/4, 9/5, 20/11 ||

|| 18 || 1028.571

|| 18 || 1028.571, 1234.286

308°34'17" || B || Bb ||= Supraminor 7th ||= 29/16, 9/5 ||= 9/5, 20/11 ||= 16/9 ||

|| 19 || 1085.714

|| 19 || 1085.714, 1302.857

325°42'51" || B^/Cvv || B ||= Major 7th ||= 15/8 ||= 17/9 ||= 15/8, 48/25 ||

|| 20 || 1142.857

|| 20 || 1142.857, 1371.429

342°8'34" || B^^/Cv || B#/Cb ||= Supermajor 7th ||= 27/14, 29/15 ||= 35/18, 68/35 ||= 63/32 ||

|| 21 || 1200

|| 21 || 1200, 1440

360° || C || C ||= Octave ||= 2/1 ||= 2/1 ||= 2/1 ||

Line 101:

||= Step Pattern ||= Cents ||= Name* ||

||= 3, 3, 3 ||= 0-171-343-514 ||= Equable diatonic ||

||= 3, 3, 3 ||= [[tel:0-171-343-514|0-171-343-514]] ||= Equable diatonic ||

||= 4, 3, 2 ||= 0-229-400-514 ||= Soft diatonic ||

||= 4, 3, 2 ||= [[tel:0-229-400-514|0-229-400-514]] ||= Soft diatonic ||

||= 4, 4, 1 ||= 0-229-457-514 ||= Intense diatonic ||

||= 4, 4, 1 ||= [[tel:0-229-457-514|0-229-457-514]] ||= Intense diatonic ||

||= 5, 3, 1 ||= 0-286-457-514 ||= Archytas chromatic ||

||= 5, 3, 1 ||= [[tel:0-286-457-514|0-286-457-514]] ||= Archytas chromatic ||

||= 5, 2, 2 ||= 0-286-400-514 ||= Weak chromatic ||

||= 5, 2, 2 ||= [[tel:0-286-400-514|0-286-400-514]] ||= Weak chromatic ||

||= 6, 2, 1 ||= 0-343-457-514 ||= Strong enharmonic ||

||= 6, 2, 1 ||= [[tel:0-343-457-514|0-343-457-514]] ||= Strong enharmonic ||

||= 7, 1, 1 ||= 0-400-457-514 ||= Pythagorean enharmonic ||

||= 7, 1, 1 ||= [[tel:0-400-457-514|0-400-457-514]] ||= Pythagorean enharmonic ||

*these names may not be correct in relating to the ancient Greek tetrachordal genera; please change them if you know better!

Line 129:

==13-limit Commas==

21 EDO tempers out the following 13-limit commas. (Note: This assumes the val < 21 33 49 59 73 78 |.)

21 EDO tempers out the following 13-limit commas. (Note: This assumes the val < [[tel:21 33 49 59 73 78|21 33 49 59 73 78]] |.)

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

||= 2187/2048 ||< | -11 7 > ||> 113.69 ||= Apotome ||= ||

||= 128/125 ||< | 7 0 -3 > ||> 41.06 ||= Diesis ||= Augmented Comma ||

||= 9931568/9752117 ||< | -25 7 6 > ||> 31.57 ||= Ampersand's Comma || ||

||= [[tel:9931568/9752117|9931568/9752117]] ||< | -25 7 6 > ||> 31.57 ||= Ampersand's Comma || ||

||= 9193891/9143623 ||< | 32 -7 -9 > ||> 9.49 ||= Escapade Comma || ||

||= [[tel:9193891/9143623|9193891/9143623]] ||< | 32 -7 -9 > ||> 9.49 ||= Escapade Comma || ||

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

||= 36/35 ||< | 2 2 -1 -1 > ||> 48.77 ||= Septimal Quarter Tone || ||

||= 9859966/9733137 ||< | -10 7 8 -7 > ||> 22.41 ||= Blackjackisma ||= ||

||= [[tel:9859966/9733137|9859966/9733137]] ||< | -10 7 8 -7 > ||> 22.41 ||= Blackjackisma ||= ||

||= 1029/1024 ||< | -10 1 0 3 > ||> 8.43 ||= Gamelisma ||= ||

||= 225/224 ||< | -5 2 2 -1 > ||> 7.71 ||= Septimal Kleisma ||= Marvel Comma ||

||= 16875/16807 ||< | 0 3 4 -5 > ||> 6.99 ||= Mirkwai ||= ||

||= 2401/2400 ||< | -5 -1 -2 4 > ||> 0.72 ||= Breedsma ||= ||

||= 394839/394762 ||< | 47 -7 -7 -7 > ||> 0.34 ||= Akjaysma ||= 5\7 Octave Comma ||

||= [[tel:394839/394762|394839/394762]] ||< | 47 -7 -7 -7 > ||> 0.34 ||= Akjaysma ||= 5\7 Octave Comma ||

||= 99/98 ||< | -1 2 0 -2 1 > ||> 17.58 ||= Mothwellsma ||= ||

||= 176/175 ||< | 4 0 -2 -1 1 > ||> 9.86 ||= Valinorsma ||= ||

Line 183:

<td>Degree

</td>

<td>Cents, DMS

<td>Cents coarse/fine, DMS

Value

</td>

Line 224:

<td>1

</td>

<td>57.143

<td>57.143, 68.571

17°8'34&quot;

</td>

Line 243:

<td>2

</td>

<td>114.286

<td>114.286, 137.143

34°17'9&quot;

</td>

Line 262:

<td>3

</td>

<td>171.429

<td>171.429, 205.714

51°25'43&quot;

</td>

Line 281:

<td>4

</td>

<td>228.571

<td>228.571, 274.286

68°34'17&quot;

</td>

Line 300:

<td>5

</td>

<td>285.714

<td>285.714, 342.857

85°42'51&quot;

</td>

Line 319:

<td>6

</td>

<td>342.857

<td>342.857, 411.429

102°51'26&quot;

</td>

Line 338:

<td>7

</td>

<td>400

<td>400, 480

120°

</td>

Line 357:

<td>8

</td>

<td>457.143

<td>457.143, 548.571

137°8'34&quot;

</td>

Line 376:

<td>9

</td>

<td>514.286

<td>514.286, 617.143

154°17'9&quot;

</td>

Line 395:

<td>10

</td>

<td>571.429

<td>571.429, 685.714

171°25'43&quot;

</td>

Line 414:

<td>11

</td>

<td>628.571

<td>628.571, 754.286

188°34'17&quot;

</td>

Line 433:

<td>12

</td>

<td>685.714

<td>685.714, 824.857

205°42'51&quot;

</td>

Line 452:

<td>13

</td>

<td>742.857

<td>742.857, 891.428

222°51'26&quot;

</td>

Line 471:

<td>14

</td>

<td>800

<td>800, 960

240°

</td>

Line 490:

<td>15

</td>

<td>857.143

<td>857.143, 1028.571

257°8'34&quot;

</td>

Line 509:

<td>16

</td>

<td>914.286

<td>914.286, 1097.143

274°17'9&quot;

</td>

Line 528:

<td>17

</td>

<td>971.429

<td>971.429, 1165.714

391°25'43&quot;

</td>

Line 547:

<td>18

</td>

<td>1028.571

<td>1028.571, 1234.286

308°34'17&quot;

</td>

Line 566:

<td>19

</td>

<td>1085.714

<td>1085.714, 1302.857

325°42'51&quot;

</td>

Line 585:

<td>20

</td>

<td>1142.857

<td>1142.857, 1371.429

342°8'34&quot;

</td>

Line 604:

<td>21

</td>

<td>1200

<td>1200, 1440

360°

</td>

Line 722:

<td style="text-align: center;">3, 3, 3

</td>

<td style="text-align: center;">0-171-343-514

</td>

<td style="text-align: center;">Equable diatonic

Line 730:

<td style="text-align: center;">4, 3, 2

</td>

<td style="text-align: center;">0-229-400-514

</td>

<td style="text-align: center;">Soft diatonic

Line 738:

<td style="text-align: center;">4, 4, 1

</td>

<td style="text-align: center;">0-229-457-514

</td>

<td style="text-align: center;">Intense diatonic

Line 746:

<td style="text-align: center;">5, 3, 1

</td>

<td style="text-align: center;">0-286-457-514

</td>

<td style="text-align: center;">Archytas chromatic

Line 754:

<td style="text-align: center;">5, 2, 2

</td>

<td style="text-align: center;">0-286-400-514

</td>

<td style="text-align: center;">Weak chromatic

Line 762:

<td style="text-align: center;">6, 2, 1

</td>

<td style="text-align: center;">0-343-457-514

</td>

<td style="text-align: center;">Strong enharmonic

Line 770:

<td style="text-align: center;">7, 1, 1

</td>

<td style="text-align: center;">0-400-457-514

</td>

<td style="text-align: center;">Pythagorean enharmonic

Line 880:

<h2 id="toc6"><a name="x21 equal divisions of the octave-13-limit Commas"></a>13-limit Commas</h2>

21 EDO tempers out the following 13-limit commas. (Note: This assumes the val &lt; 21 33 49 59 73 78 |.)

21 EDO tempers out the following 13-limit commas. (Note: This assumes the val &lt; <a class="wiki_link" href="http://tel.wikispaces.com/21%2033%2049%2059%2073%2078">21 33 49 59 73 78</a> |.)

Line 921:

</tr>

<tr>

<td style="text-align: center;">9931568/9752117

<td style="text-align: center;">[[tel:9931568/9752117|9931568/9752117]]

</td>

<td style="text-align: left;">| -25 7 6 &gt;

Line 933:

</tr>

<tr>

<td style="text-align: center;">9193891/9143623

<td style="text-align: center;">[[tel:9193891/9143623|9193891/9143623]]

</td>

<td style="text-align: left;">| 32 -7 -9 &gt;

Line 969:

</tr>

<tr>

<td style="text-align: center;">9859966/9733137

<td style="text-align: center;">[[tel:9859966/9733137|9859966/9733137]]

</td>

<td style="text-align: left;">| -10 7 8 -7 &gt;

Line 1,029:

</tr>

<tr>

<td style="text-align: center;">394839/394762

<td style="text-align: center;">[[tel:394839/394762|394839/394762]]

</td>

<td style="text-align: left;">| 47 -7 -7 -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:Gedankenwelt|Gedankenwelt]] and made on <tt>2015-12-12 05:20:20 UTC</tt>.<br>
+: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-10-30 18:56:53 UTC</tt>.<br>
-: The original revision id was <tt>569913991</tt>.<br>
+: The original revision id was <tt>597508726</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 19: / Line 19: @@
 The patent val for 21edo tempers out 128/125 and 2187/2000 in the 5-limit, and supplies the optimal patent val for the 5-limit [[Laconic Family|laconic]] temperament tempering out 2187/2000, and also the optimal patent val for 7-limit, 11-limit and 13-limit laconic, spartan and gorgo temperaments. These temperaments lead to some "interesting" mappings, where 10/9 is larger than 9/8, 11/9 is larger than 16/13, and 8/7 maps to the same interval as 10/9, for instance.
-|| **Degree** || **Cents, DMS**
+|| **Degree** || **Cents coarse/fine, DMS**
 **Value** || **7-EDO**
 **Notation** || **5L3s**
@@ Line 26: / Line 26: @@
 **Ratios *1** ||= &lt;span style="display: block; text-align: center;"&gt;**Approximate****Ratios *2**&lt;/span&gt; ||= &lt;span style="display: block; text-align: center;"&gt;**Approximate**&lt;/span&gt;**Ratios *3** ||
 || 0 || 0 || C || C ||= Unison ||= 1/1 ||= 1/1 ||= 1/1 ||
-|| 1 || 57.143
+|| 1 || 57.143, 68.571
 °&lt;span style="background-color: #ffffff;"&gt;8'34"&lt;/span&gt; || C^/Dvv || C# ||= Subminor 2nd ||= 28/27, 30/29 ||= 35/34, 36/35 ||= 64/63 ||
-|| 2 || 114.286
+|| 2 || 114.286, 137.143
 °&lt;span style="background-color: #ffffff;"&gt;17'9"&lt;/span&gt; || C^^/Dv || Db ||= Minor 2nd ||= 16/15, 15/14, 29/27 ||= 18/17 ||= 16/15, 25/24 ||
-|| 3 || 171.429
+|| 3 || 171.429, 205.714
 &lt;span style="background-color: #ffffff;"&gt;51°25'43"&lt;/span&gt; || D || D ||= Submajor 2nd ||= 10/9, 32/29 ||= 10/9,11/10 ||= 9/8 ||
-|| 4 || 228.571
+|| 4 || 228.571, 274.286
 °&lt;span style="background-color: #ffffff;"&gt;34'17"&lt;/span&gt; || D^/Evv || D# ||= Supermajor 2nd ||= 8/7 ||= 8/7 ||= 8/7, 10/9, 11/10 ||
-|| 5 || 285.714
+|| 5 || 285.714, 342.857
 °&lt;span style="background-color: #ffffff;"&gt;42'51"&lt;/span&gt; || D^^/Ev || Eb ||= Subminor 3rd ||= 27/23, 32/27 ||= 13/11, 20/17 ||= 6/5, 7/6 ||
-|| 6 || 342.857
+|| 6 || 342.857, 411.429
 &lt;span style="background-color: #ffffff;"&gt;102°51'26"&lt;/span&gt; || E || E ||= Neutral 3rd ||= 28/23 ||= 11/9 ||= 16/13 ||
-|| 7 || 400
+|| 7 || 400, 480
 ° || E^/Fvv || E#/Fb ||= Major 3rd ||= 29/23 ||= 44/35 ||= 5/4, 9/7, 11/9, 14/11 ||
-|| 8 || 457.143
+|| 8 || 457.143, 548.571
 °&lt;span style="background-color: #ffffff; line-height: 1.5;"&gt;8'34"&lt;/span&gt; || E^^/Fv || F ||= Third-Fourth ||= 30/23 ||= 13/10, 17/13, 22/17 ||= 13/10 ||
-|| 9 || 514.286
+|| 9 || 514.286, 617.143
 &lt;span style="background-color: #ffffff;"&gt;154°17'9"&lt;/span&gt; || F || F# ||= Acute 4th ||= 161/120, 256/189 ||= 35/26 ||= 4/3, 18/13 ||
-|| 10 || 571.429
+|| 10 || 571.429, 685.714
 °&lt;span style="background-color: #ffffff;"&gt;25'43"&lt;/span&gt; || F^/Gvv || Gb ||= Narrow Tritone ||= 32/23 ||= 18/13 ||= 7/5, 11/8 ||
-|| 11 || 628.571
+|| 11 || 628.571, 754.286
 °&lt;span style="background-color: #ffffff;"&gt;34'17"&lt;/span&gt; || F^^/Gv || G ||= Wide Tritone ||= 23/16 ||= 13/9 ||= 10/7, 16/11 ||
-|| 12 || 685.714
+|| 12 || 685.714, 824.857
 &lt;span style="background-color: #ffffff;"&gt;205°42'51"&lt;/span&gt; || G || G# ||= Grave 5th ||= 189/128, 240/161 ||= 52/35 ||= 3/2, 13/9 ||
-|| 13 || 742.857
+|| 13 || 742.857, 891.428
 °&lt;span style="background-color: #ffffff;"&gt;51'26"&lt;/span&gt; || G^/Avv || Hb ||= Fifth-Sixth ||= 23/15 ||= 17/11, 20/13, 26/17 ||= 20/13 ||
-|| 14 || 800
+|| 14 || 800, 960
 ° || G^^/Av || H ||= Minor 6th ||= 46/29 ||= 35/22 ||= 8/5, 11/7, 14/9, 18/11 ||
-|| 15 || 857.143
+|| 15 || 857.143, 1028.571
 &lt;span style="background-color: #ffffff;"&gt;257°8'34"&lt;/span&gt; || A || H#/Ab ||= Neutral 6th ||= 23/14 ||= 18/11 ||= 13/8 ||
-|| 16 || 914.286
+|| 16 || 914.286, 1097.143
 °&lt;span style="background-color: #ffffff;"&gt;17'9"&lt;/span&gt; || A^/Bvv || A ||= Supermajor 6th ||= 27/16, 46/27 ||= 17/10, 22/13 ||= 5/3, 12/7 ||
-|| 17 || 971.429
+|| 17 || 971.429, 1165.714
 °&lt;span style="background-color: #ffffff;"&gt;25'43"&lt;/span&gt; || A^^/Bv || A# ||= Subminor 7th ||= 7/4 ||= 7/4 ||= 7/4, 9/5, 20/11 ||
-|| 18 || 1028.571
+|| 18 || 1028.571, 1234.286
 &lt;span style="background-color: #ffffff;"&gt;308°34'17"&lt;/span&gt; || B || Bb ||= Supraminor 7th ||= 29/16, 9/5 ||= 9/5, 20/11 ||= 16/9 ||
-|| 19 || 1085.714
+|| 19 || 1085.714, 1302.857
 °&lt;span style="background-color: #ffffff;"&gt;42'51"&lt;/span&gt; || B^/Cvv || B ||= Major 7th ||= 15/8 ||= 17/9 ||= 15/8, 48/25 ||
-|| 20 || 1142.857
+|| 20 || 1142.857, 1371.429
 °&lt;span style="background-color: #ffffff;"&gt;8'34"&lt;/span&gt; || B^^/Cv || B#/Cb ||= Supermajor 7th ||= 27/14, 29/15 ||= 35/18, 68/35 ||= 63/32 ||
-|| 21 || 1200
+|| 21 || 1200, 1440
 ° || C || C ||= Octave ||= 2/1 ||= 2/1 ||= 2/1 ||
@@ Line 101: / Line 101: @@
 ||= Step Pattern ||= Cents ||= Name* ||
-||= 3, 3, 3 ||= 0-171-343-514 ||= Equable diatonic ||
+||= 3, 3, 3 ||= [[tel:0-171-343-514|0-171-343-514]] ||= Equable diatonic ||
-||= 4, 3, 2 ||= 0-229-400-514 ||= Soft diatonic ||
+||= 4, 3, 2 ||= [[tel:0-229-400-514|0-229-400-514]] ||= Soft diatonic ||
-||= 4, 4, 1 ||= 0-229-457-514 ||= Intense diatonic ||
+||= 4, 4, 1 ||= [[tel:0-229-457-514|0-229-457-514]] ||= Intense diatonic ||
-||= 5, 3, 1 ||= 0-286-457-514 ||= Archytas chromatic ||
+||= 5, 3, 1 ||= [[tel:0-286-457-514|0-286-457-514]] ||= Archytas chromatic ||
-||= 5, 2, 2 ||= 0-286-400-514 ||= Weak chromatic ||
+||= 5, 2, 2 ||= [[tel:0-286-400-514|0-286-400-514]] ||= Weak chromatic ||
-||= 6, 2, 1 ||= 0-343-457-514 ||= Strong enharmonic ||
+||= 6, 2, 1 ||= [[tel:0-343-457-514|0-343-457-514]] ||= Strong enharmonic ||
-||= 7, 1, 1 ||= 0-400-457-514 ||= Pythagorean enharmonic ||
+||= 7, 1, 1 ||= [[tel:0-400-457-514|0-400-457-514]] ||= Pythagorean enharmonic ||
 *these names may not be correct in relating to the ancient Greek tetrachordal genera; please change them if you know better!
@@ Line 129: / Line 129: @@
 ==13-limit Commas==
-EDO tempers out the following 13-limit commas. (Note: This assumes the val &lt; 21 33 49 59 73 78 |.)
+EDO tempers out the following 13-limit commas. (Note: This assumes the val &lt; [[tel:21 33 49 59 73 78|21 33 49 59 73 78]] |.)
 ||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||
 ||= 2187/2048 ||&lt; | -11 7 &gt; ||&gt; 113.69 ||= Apotome ||=   ||
 ||= 128/125 ||&lt; | 7 0 -3 &gt; ||&gt; 41.06 ||= Diesis ||= Augmented Comma ||
-||= 9931568/9752117 ||&lt; | -25 7 6 &gt; ||&gt; 31.57 ||= Ampersand's Comma ||   ||
+||= [[tel:9931568/9752117|9931568/9752117]] ||&lt; | -25 7 6 &gt; ||&gt; 31.57 ||= Ampersand's Comma ||   ||
-||= 9193891/9143623 ||&lt; | 32 -7 -9 &gt; ||&gt; 9.49 ||= Escapade Comma ||   ||
+||= [[tel:9193891/9143623|9193891/9143623]] ||&lt; | 32 -7 -9 &gt; ||&gt; 9.49 ||= Escapade Comma ||   ||
 ||= 1029/1000 ||&lt; | -3 1 -3 3 &gt; ||&gt; 49.49 ||= Keega ||=   ||
 ||= 36/35 ||&lt; | 2 2 -1 -1 &gt; ||&gt; 48.77 ||= Septimal Quarter Tone ||   ||
-||= 9859966/9733137 ||&lt; | -10 7 8 -7 &gt; ||&gt; 22.41 ||= Blackjackisma ||=   ||
+||= [[tel:9859966/9733137|9859966/9733137]] ||&lt; | -10 7 8 -7 &gt; ||&gt; 22.41 ||= Blackjackisma ||=   ||
 ||= 1029/1024 ||&lt; | -10 1 0 3 &gt; ||&gt; 8.43 ||= Gamelisma ||=   ||
 ||= 225/224 ||&lt; | -5 2 2 -1 &gt; ||&gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||
 ||= 16875/16807 ||&lt; | 0 3 4 -5 &gt; ||&gt; 6.99 ||= Mirkwai ||=   ||
 ||= 2401/2400 ||&lt; | -5 -1 -2 4 &gt; ||&gt; 0.72 ||= Breedsma ||=   ||
-||= 394839/394762 ||&lt; | 47 -7 -7 -7 &gt; ||&gt; 0.34 ||= Akjaysma ||= 5\7 Octave Comma ||
+||= [[tel:394839/394762|394839/394762]] ||&lt; | 47 -7 -7 -7 &gt; ||&gt; 0.34 ||= Akjaysma ||= 5\7 Octave Comma ||
 ||= 99/98 ||&lt; | -1 2 0 -2 1 &gt; ||&gt; 17.58 ||= Mothwellsma ||=   ||
 ||= 176/175 ||&lt; | 4 0 -2 -1 1 &gt; ||&gt; 9.86 ||= Valinorsma ||=   ||
@@ Line 183: / Line 183: @@
          &lt;td&gt;&lt;strong&gt;Degree&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;Cents, DMS&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;Cents coarse/fine, DMS&lt;/strong&gt;&lt;br /&gt;
 &lt;strong&gt;Value&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 224: / Line 224: @@
          &lt;td&gt;1&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;57.143&lt;br /&gt;
+         &lt;td&gt;57.143, 68.571&lt;br /&gt;
 °&lt;span style="background-color: #ffffff;"&gt;8'34&amp;quot;&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 243: / Line 243: @@
          &lt;td&gt;2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;114.286&lt;br /&gt;
+         &lt;td&gt;114.286, 137.143&lt;br /&gt;
 °&lt;span style="background-color: #ffffff;"&gt;17'9&amp;quot;&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 262: / Line 262: @@
          &lt;td&gt;3&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;171.429&lt;br /&gt;
+         &lt;td&gt;171.429, 205.714&lt;br /&gt;
 &lt;span style="background-color: #ffffff;"&gt;51°25'43&amp;quot;&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 281: / Line 281: @@
          &lt;td&gt;4&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;228.571&lt;br /&gt;
+         &lt;td&gt;228.571, 274.286&lt;br /&gt;
 °&lt;span style="background-color: #ffffff;"&gt;34'17&amp;quot;&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 300: / Line 300: @@
          &lt;td&gt;5&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;285.714&lt;br /&gt;
+         &lt;td&gt;285.714, 342.857&lt;br /&gt;
 °&lt;span style="background-color: #ffffff;"&gt;42'51&amp;quot;&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 319: / Line 319: @@
          &lt;td&gt;6&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;342.857&lt;br /&gt;
+         &lt;td&gt;342.857, 411.429&lt;br /&gt;
 &lt;span style="background-color: #ffffff;"&gt;102°51'26&amp;quot;&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 338: / Line 338: @@
          &lt;td&gt;7&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;400&lt;br /&gt;
+         &lt;td&gt;400, 480&lt;br /&gt;
 °&lt;br /&gt;
 &lt;/td&gt;
@@ Line 357: / Line 357: @@
          &lt;td&gt;8&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;457.143&lt;br /&gt;
+         &lt;td&gt;457.143, 548.571&lt;br /&gt;
 °&lt;span style="background-color: #ffffff; line-height: 1.5;"&gt;8'34&amp;quot;&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 376: / Line 376: @@
          &lt;td&gt;9&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;514.286&lt;br /&gt;
+         &lt;td&gt;514.286, 617.143&lt;br /&gt;
 &lt;span style="background-color: #ffffff;"&gt;154°17'9&amp;quot;&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 395: / Line 395: @@
          &lt;td&gt;10&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;571.429&lt;br /&gt;
+         &lt;td&gt;571.429, 685.714&lt;br /&gt;
 °&lt;span style="background-color: #ffffff;"&gt;25'43&amp;quot;&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 414: / Line 414: @@
          &lt;td&gt;11&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;628.571&lt;br /&gt;
+         &lt;td&gt;628.571, 754.286&lt;br /&gt;
 °&lt;span style="background-color: #ffffff;"&gt;34'17&amp;quot;&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 433: / Line 433: @@
          &lt;td&gt;12&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;685.714&lt;br /&gt;
+         &lt;td&gt;685.714, 824.857&lt;br /&gt;
 &lt;span style="background-color: #ffffff;"&gt;205°42'51&amp;quot;&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 452: / Line 452: @@
          &lt;td&gt;13&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;742.857&lt;br /&gt;
+         &lt;td&gt;742.857, 891.428 &lt;br /&gt;
 °&lt;span style="background-color: #ffffff;"&gt;51'26&amp;quot;&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 471: / Line 471: @@
          &lt;td&gt;14&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;800&lt;br /&gt;
+         &lt;td&gt;800, 960&lt;br /&gt;
 °&lt;br /&gt;
 &lt;/td&gt;
@@ Line 490: / Line 490: @@
          &lt;td&gt;15&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;857.143&lt;br /&gt;
+         &lt;td&gt;857.143, 1028.571&lt;br /&gt;
 &lt;span style="background-color: #ffffff;"&gt;257°8'34&amp;quot;&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 509: / Line 509: @@
          &lt;td&gt;16&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;914.286&lt;br /&gt;
+         &lt;td&gt;914.286, 1097.143&lt;br /&gt;
 °&lt;span style="background-color: #ffffff;"&gt;17'9&amp;quot;&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 528: / Line 528: @@
          &lt;td&gt;17&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;971.429&lt;br /&gt;
+         &lt;td&gt;971.429, 1165.714&lt;br /&gt;
 °&lt;span style="background-color: #ffffff;"&gt;25'43&amp;quot;&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 547: / Line 547: @@
          &lt;td&gt;18&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;1028.571&lt;br /&gt;
+         &lt;td&gt;1028.571, 1234.286&lt;br /&gt;
 &lt;span style="background-color: #ffffff;"&gt;308°34'17&amp;quot;&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 566: / Line 566: @@
          &lt;td&gt;19&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;1085.714&lt;br /&gt;
+         &lt;td&gt;1085.714, 1302.857&lt;br /&gt;
 °&lt;span style="background-color: #ffffff;"&gt;42'51&amp;quot;&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 585: / Line 585: @@
          &lt;td&gt;20&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;1142.857&lt;br /&gt;
+         &lt;td&gt;1142.857, 1371.429&lt;br /&gt;
 °&lt;span style="background-color: #ffffff;"&gt;8'34&amp;quot;&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 604: / Line 604: @@
          &lt;td&gt;21&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 722: / Line 722: @@
          &lt;td style="text-align: center;"&gt;3, 3, 3&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;0-171-343-514&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="http://tel.wikispaces.com/0-171-343-514"&gt;0-171-343-514&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Equable diatonic&lt;br /&gt;
@@ Line 730: / Line 730: @@
          &lt;td style="text-align: center;"&gt;4, 3, 2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;0-229-400-514&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="http://tel.wikispaces.com/0-229-400-514"&gt;0-229-400-514&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Soft diatonic&lt;br /&gt;
@@ Line 738: / Line 738: @@
          &lt;td style="text-align: center;"&gt;4, 4, 1&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;0-229-457-514&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="http://tel.wikispaces.com/0-229-457-514"&gt;0-229-457-514&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Intense diatonic&lt;br /&gt;
@@ Line 746: / Line 746: @@
          &lt;td style="text-align: center;"&gt;5, 3, 1&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;0-286-457-514&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="http://tel.wikispaces.com/0-286-457-514"&gt;0-286-457-514&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Archytas chromatic&lt;br /&gt;
@@ Line 754: / Line 754: @@
          &lt;td style="text-align: center;"&gt;5, 2, 2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;0-286-400-514&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="http://tel.wikispaces.com/0-286-400-514"&gt;0-286-400-514&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Weak chromatic&lt;br /&gt;
@@ Line 762: / Line 762: @@
          &lt;td style="text-align: center;"&gt;6, 2, 1&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;0-343-457-514&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="http://tel.wikispaces.com/0-343-457-514"&gt;0-343-457-514&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Strong enharmonic&lt;br /&gt;
@@ Line 770: / Line 770: @@
          &lt;td style="text-align: center;"&gt;7, 1, 1&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;0-400-457-514&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;a class="wiki_link" href="http://tel.wikispaces.com/0-400-457-514"&gt;0-400-457-514&lt;/a&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Pythagorean enharmonic&lt;br /&gt;
@@ Line 880: / Line 880: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="x21 equal divisions of the octave-13-limit Commas"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;13-limit Commas&lt;/h2&gt;
-EDO tempers out the following 13-limit commas. (Note: This assumes the val &amp;lt; 21 33 49 59 73 78 |.)&lt;br /&gt;
+EDO tempers out the following 13-limit commas. (Note: This assumes the val &amp;lt; &lt;a class="wiki_link" href="http://tel.wikispaces.com/21%2033%2049%2059%2073%2078"&gt;21 33 49 59 73 78&lt;/a&gt; |.)&lt;br /&gt;
@@ Line 921: / Line 921: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;9931568/9752117&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;[[tel:9931568/9752117|9931568/9752117]]&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: left;"&gt;| -25 7 6 &amp;gt;&lt;br /&gt;
@@ Line 933: / Line 933: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;9193891/9143623&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;[[tel:9193891/9143623|9193891/9143623]]&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: left;"&gt;| 32 -7 -9 &amp;gt;&lt;br /&gt;
@@ Line 969: / Line 969: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;9859966/9733137&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;[[tel:9859966/9733137|9859966/9733137]]&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: left;"&gt;| -10 7 8 -7 &amp;gt;&lt;br /&gt;
@@ Line 1,029: / Line 1,029: @@
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td style="text-align: center;"&gt;394839/394762&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;[[tel:394839/394762|394839/394762]]&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: left;"&gt;| 47 -7 -7 -7 &amp;gt;&lt;br /&gt;