Pergen names: 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:TallKite|TallKite]] and made on <tt>2017-11-18 08:15:02 UTC</tt>.

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-11-18 09:12:20 UTC</tt>.

: The original revision id was <tt>~~621914201~~</tt>.

: The original revision id was <tt>621915379</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 41:

For non-standard prime groups, the period uses the first prime only, and the multi-gen uses the first two primes only. [[Kite's color notation|Color notation ]]is used to indicate primes higher than 3. For example, 2.5.7 with 50/49 tempered out is {P8/2, y3} = half-octave, yellow-third.

Rank-3 pergen ~~names~~ have three intervals, any of which may be split. The unsplit 2.3.5... subgroup is {P8, P5, y3} = fifth-and-third-based. 2.3.5.7 with 50/49 tempered out is {P8/2, P5, y3}.

Rank-3 pergen sets have three intervals, any of which may be split. The unsplit 2.3.5... subgroup is {P8, P5, y3} = fifth-and-third-based. 2.3.5.7 with 50/49 tempered out is {P8/2, P5, y3}.

Rank-4 temperaments ~~can be named similarly~~. Rank-1 temperaments could have pergen ~~names~~, such as {P8/12} for 12-edo or {P12/13} for 13-ed3, but there's no particular reason to do so.</pre></div>

Rank-4 temperaments have pergen sets of four intervals. Rank-1 temperaments could have pergen sets of one, such as {P8/12} for 12-edo or {P12/13} for 13-ed3, but there's no particular reason to do so.

To find a temperament's pergen set, first find the PGM, the period generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each generator, counting the period as a special type of generator. Choose your generators so that all entries below the diagonal are zero. (This is not a problem with rank-2 temperaments.) You can use x31.com to find this matrix. Next make a square matrix by discarding columns for the higher primes. Then invert the matrix to get the monzos for each generator. Add/subtract periods from the generator to get alternate generators. Add/subtract both periods and generators from the 2nd generator to get more alternates. Choose the alternates to minimize the splitting and the cents.

Example: 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. x31.com gives us this matrix:

||~ ||~ 2/1 ||~ 3/1 ||~ 5/1 ||~ 7/1 ||

||~ period ||= 1 ||= 1 ||= 1 ||= 2 ||

||~ gen1 ||= 0 ||= 2 ||= 1 ||= 1 ||

||~ gen2 ||= 0 ||= 0 ||= 2 ||= 1 ||

Thus 2/1 = P, 3/1 = P + 2 G1, 5/1 = P + G1 + 2 G2, and 7/1 = 2 P + G1 + G2.

Discard the last column to make a square matrix:

||~ ||~ 2/1 ||~ 3/1 ||~ 5/1 ||

||~ period ||= 1 ||= 1 ||= 1 ||

||~ gen1 ||= 0 ||= 2 ||= 1 ||

||~ gen2 ||= 0 ||= 0 ||= 2 ||

Invert it. "/4" means that each entry is to be divided by the determinant of the last matrix, which is 4.

||~ ||~ period ||~ gen1 ||~ gen2 ||~ ||

||~ 2/1 ||= 4 ||= -2 ||= -1 || ||

||~ 3/1 ||= 0 ||= 2 ||= -1 || ||

||~ 5/1 ||= 0 ||= 0 ||= 2 || /4 ||

Thus the period = (4, 0, 0)/4 = (1, 0, 0) = 2/1, gen1 = (-2, 2, 0)/4 = (-1, 1, 0)/2 = P5/2, and gen2 = (-1, -1, 2)/4 = [25/6]/4.

Next, search for alternate generators... (to be continued)

The primes can be octave-reduced, if preferred:

||~ ||~ 2/1 ||~ 3/2 ||~ 5/4 ||~ 7/4 ||

||~ period = 2/1 ||= 1 ||= 0 ||= 0 ||= -1 ||

||~ gen1 = 3/2 ||= 0 ||= 1 ||= 0 ||= 2 ||

||~ gen2 = 5/4 ||= 0 ||= 0 ||= 1 ||= 2 ||</pre></div>

<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>pergen names</title></head><body>Pergen (pronounced &quot;peer-gen&quot;) sets are a way of identifying rank-2 and rank-3 regular temperaments by their periods and generators. They are somewhat JI-agnostic in that they don't use higher primes. Rank-2 names only use the first two primes in the prime subgroup, and rank-3 names only use the first three primes.

Line 270:

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For non-standard prime groups, the period uses the first prime only, and the multi-gen uses the first two primes only. <a class="wiki_link" href="/Kite%27s%20color%20notation">Color notation </a>is used to indicate primes higher than 3. For example, 2.5.7 with 50/49 tempered out is {P8/2, y3} = half-octave, yellow-third.

Rank-3 pergen ~~names~~ have three intervals, any of which may be split. The unsplit 2.3.5... subgroup is {P8, P5, y3} = fifth-and-third-based. 2.3.5.7 with 50/49 tempered out is {P8/2, P5, y3}.

Rank-3 pergen sets have three intervals, any of which may be split. The unsplit 2.3.5... subgroup is {P8, P5, y3} = fifth-and-third-based. 2.3.5.7 with 50/49 tempered out is {P8/2, P5, y3}.

Rank-4 temperaments have pergen sets of four intervals. Rank-1 temperaments could have pergen sets of one, such as {P8/12} for 12-edo or {P12/13} for 13-ed3, but there's no particular reason to do so.

To find a temperament's pergen set, first find the PGM, the period generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each generator, counting the period as a special type of generator. Choose your generators so that all entries below the diagonal are zero. (This is not a problem with rank-2 temperaments.) You can use x31.com to find this matrix. Next make a square matrix by discarding columns for the higher primes. Then invert the matrix to get the monzos for each generator. Add/subtract periods from the generator to get alternate generators. Add/subtract both periods and generators from the 2nd generator to get more alternates. Choose the alternates to minimize the splitting and the cents.

Example: 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. x31.com gives us this matrix:

<tr>

<th>

</th>

<th>2/1

</th>

<th>3/1

</th>

<th>5/1

</th>

<th>7/1

</th>

</tr>

<tr>

<th>period

</th>

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

</td>

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

</td>

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

</td>

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

</td>

</tr>

<tr>

<th>gen1

</th>

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

</td>

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

</td>

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

</td>

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

</td>

</tr>

<tr>

<th>gen2

</th>

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

</td>

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

</td>

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

</td>

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

</td>

</tr>

</table>

Thus 2/1 = P, 3/1 = P + 2 G1, 5/1 = P + G1 + 2 G2, and 7/1 = 2 P + G1 + G2.

Discard the last column to make a square matrix:

<tr>

<th>

</th>

<th>2/1

</th>

<th>3/1

</th>

<th>5/1

</th>

</tr>

<tr>

<th>period

</th>

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

</td>

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

</td>

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

</td>

</tr>

<tr>

<th>gen1

</th>

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

</td>

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

</td>

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

</td>

</tr>

<tr>

<th>gen2

</th>

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

</td>

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

</td>

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

</td>

</tr>

</table>

Invert it. &quot;/4&quot; means that each entry is to be divided by the determinant of the last matrix, which is 4.

<tr>

<th>

</th>

<th>period

</th>

<th>gen1

</th>

<th>gen2

</th>

<th>

</th>

</tr>

<tr>

<th>2/1

</th>

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

</td>

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

</td>

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

</td>

<td>

</td>

</tr>

<tr>

<th>3/1

</th>

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

</td>

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

</td>

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

</td>

<td>

</td>

</tr>

<tr>

<th>5/1

</th>

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

</td>

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

</td>

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

</td>

<td>/4

</td>

</tr>

</table>

Thus the period = (4, 0, 0)/4 = (1, 0, 0) = 2/1, gen1 = (-2, 2, 0)/4 = (-1, 1, 0)/2 = P5/2, and gen2 = (-1, -1, 2)/4 = [25/6]/4.

Next, search for alternate generators... (to be continued)

~~Rank~~-4 ~~temperaments can be named similarly. Rank~~-1 ~~temperaments could have pergen names, such as {P8~~/~~12} for 12~~-~~edo or {P12~~/~~13} for 13~~-~~ed3, but there's no particular reason to do so.~~</body></html></pre></div>

The primes can be octave-reduced, if preferred:

<tr>

<th>

</th>

<th>2/1

</th>

<th>3/2

</th>

<th>5/4

</th>

<th>7/4

</th>

</tr>

<tr>

<th>period = 2/1

</th>

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

</td>

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

</td>

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

</td>

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

</td>

</tr>

<tr>

<th>gen1 = 3/2

</th>

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

</td>

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

</td>

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

</td>

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

</td>

</tr>

<tr>

<th>gen2 = 5/4

</th>

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

</td>

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

</td>

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

</td>

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

</td>

</tr>

</table>

</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:TallKite|TallKite]] and made on <tt>2017-11-18 08:15:02 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-11-18 09:12:20 UTC</tt>.<br>
-: The original revision id was <tt>621914201</tt>.<br>
+: The original revision id was <tt>621915379</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 41: / Line 41: @@
 For non-standard prime groups, the period uses the first prime only, and the multi-gen uses the first two primes only. [[Kite's color notation|Color notation ]]is used to indicate primes higher than 3. For example, 2.5.7 with 50/49 tempered out is {P8/2, y3} = half-octave, yellow-third.
-Rank-3 pergen names have three intervals, any of which may be split. The unsplit 2.3.5... subgroup is {P8, P5, y3} = fifth-and-third-based. 2.3.5.7 with 50/49 tempered out is {P8/2, P5, y3}.
+Rank-3 pergen sets have three intervals, any of which may be split. The unsplit 2.3.5... subgroup is {P8, P5, y3} = fifth-and-third-based. 2.3.5.7 with 50/49 tempered out is {P8/2, P5, y3}.
-Rank-4 temperaments can be named similarly. Rank-1 temperaments could have pergen names, such as {P8/12} for 12-edo or {P12/13} for 13-ed3, but there's no particular reason to do so.</pre></div>
+Rank-4 temperaments have pergen sets of four intervals. Rank-1 temperaments could have pergen sets of one, such as {P8/12} for 12-edo or {P12/13} for 13-ed3, but there's no particular reason to do so.
+To find a temperament's pergen set, first find the PGM, the period generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each generator, counting the period as a special type of generator. Choose your generators so that all entries below the diagonal are zero. (This is not a problem with rank-2 temperaments.) You can use x31.com to find this matrix. Next make a square matrix by discarding columns for the higher primes. Then invert the matrix to get the monzos for each generator. Add/subtract periods from the generator to get alternate generators. Add/subtract both periods and generators from the 2nd generator to get more alternates. Choose the alternates to minimize the splitting and the cents.
+Example: 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. x31.com gives us this matrix:
+||~   ||~ 2/1 ||~ 3/1 ||~ 5/1 ||~ 7/1 ||
+||~ period ||= 1 ||= 1 ||= 1 ||= 2 ||
+||~ gen1 ||= 0 ||= 2 ||= 1 ||= 1 ||
+||~ gen2 ||= 0 ||= 0 ||= 2 ||= 1 ||
+Thus 2/1 = P, 3/1 = P + 2 G1, 5/1 = P + G1 + 2 G2, and 7/1 = 2 P + G1 + G2.
+Discard the last column to make a square matrix:
+||~   ||~ 2/1 ||~ 3/1 ||~ 5/1 ||
+||~ period ||= 1 ||= 1 ||= 1 ||
+||~ gen1 ||= 0 ||= 2 ||= 1 ||
+||~ gen2 ||= 0 ||= 0 ||= 2 ||
+Invert it. "/4" means that each entry is to be divided by the determinant of the last matrix, which is 4.
+||~   ||~ period ||~ gen1 ||~ gen2 ||~   ||
+||~ 2/1 ||= 4 ||= -2 ||= -1 ||   ||
+||~ 3/1 ||= 0 ||= 2 ||= -1 ||   ||
+||~ 5/1 ||= 0 ||= 0 ||= 2 || /4 ||
+Thus the period = (4, 0, 0)/4 = (1, 0, 0) = 2/1, gen1 = (-2, 2, 0)/4 = (-1, 1, 0)/2 = P5/2, and gen2 = (-1, -1, 2)/4 = [25/6]/4.
+Next, search for alternate generators... (to be continued)
+The primes can be octave-reduced, if preferred:
+||~   ||~ 2/1 ||~ 3/2 ||~ 5/4 ||~ 7/4 ||
+||~ period = 2/1 ||= 1 ||= 0 ||= 0 ||= -1 ||
+||~ gen1 = 3/2 ||= 0 ||= 1 ||= 0 ||= 2 ||
+||~ gen2 = 5/4 ||= 0 ||= 0 ||= 1 ||= 2 ||</pre></div>
 <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">&lt;html&gt;&lt;head&gt;&lt;title&gt;pergen names&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;strong&gt;Pergen&lt;/strong&gt; (pronounced &amp;quot;peer-gen&amp;quot;) sets are a way of identifying rank-2 and rank-3 regular temperaments by their periods and generators. They are somewhat JI-agnostic in that they don't use higher primes. Rank-2 names only use the first two primes in the prime subgroup, and rank-3 names only use the first three primes.&lt;br /&gt;
@@ Line 270: / Line 301: @@
 For non-standard prime groups, the period uses the first prime only, and the multi-gen uses the first two primes only. &lt;a class="wiki_link" href="/Kite%27s%20color%20notation"&gt;Color notation &lt;/a&gt;is used to indicate primes higher than 3. For example, 2.5.7 with 50/49 tempered out is {P8/2, y3} = half-octave, yellow-third.&lt;br /&gt;
 &lt;br /&gt;
-Rank-3 pergen names have three intervals, any of which may be split. The unsplit 2.3.5... subgroup is {P8, P5, y3} = fifth-and-third-based. 2.3.5.7 with 50/49 tempered out is {P8/2, P5, y3}.&lt;br /&gt;
+Rank-3 pergen sets have three intervals, any of which may be split. The unsplit 2.3.5... subgroup is {P8, P5, y3} = fifth-and-third-based. 2.3.5.7 with 50/49 tempered out is {P8/2, P5, y3}.&lt;br /&gt;
+&lt;br /&gt;
+Rank-4 temperaments have pergen sets of four intervals. Rank-1 temperaments could have pergen sets of one, such as {P8/12} for 12-edo or {P12/13} for 13-ed3, but there's no particular reason to do so.&lt;br /&gt;
+&lt;br /&gt;
+To find a temperament's pergen set, first find the PGM, the period generator mapping. This is a matrix with a column for each prime in the subgroup, and a row for each generator, counting the period as a special type of generator. Choose your generators so that all entries below the diagonal are zero. (This is not a problem with rank-2 temperaments.) You can use x31.com to find this matrix. Next make a square matrix by discarding columns for the higher primes. Then invert the matrix to get the monzos for each generator. Add/subtract periods from the generator to get alternate generators. Add/subtract both periods and generators from the 2nd generator to get more alternates. Choose the alternates to minimize the splitting and the cents.&lt;br /&gt;
+&lt;br /&gt;
+Example: 2.3.5.7 with 2401/2400 = (-5,-1,-2,4) tempered out. x31.com gives us this matrix:&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;th&gt;&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;2/1&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;3/1&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;5/1&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;7/1&lt;br /&gt;
+&lt;/th&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;th&gt;period&lt;br /&gt;
+&lt;/th&gt;
+        &lt;td style="text-align: center;"&gt;1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;2&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;th&gt;gen1&lt;br /&gt;
+&lt;/th&gt;
+        &lt;td style="text-align: center;"&gt;0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;1&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;th&gt;gen2&lt;br /&gt;
+&lt;/th&gt;
+        &lt;td style="text-align: center;"&gt;0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;1&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+Thus 2/1 = P, 3/1 = P + 2 G1, 5/1 = P + G1 + 2 G2, and 7/1 = 2 P + G1 + G2.&lt;br /&gt;
+&lt;br /&gt;
+Discard the last column to make a square matrix:&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;th&gt;&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;2/1&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;3/1&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;5/1&lt;br /&gt;
+&lt;/th&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;th&gt;period&lt;br /&gt;
+&lt;/th&gt;
+        &lt;td style="text-align: center;"&gt;1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;1&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;th&gt;gen1&lt;br /&gt;
+&lt;/th&gt;
+        &lt;td style="text-align: center;"&gt;0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;1&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;th&gt;gen2&lt;br /&gt;
+&lt;/th&gt;
+        &lt;td style="text-align: center;"&gt;0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;2&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+Invert it. &amp;quot;/4&amp;quot; means that each entry is to be divided by the determinant of the last matrix, which is 4.&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;th&gt;&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;period&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;gen1&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;gen2&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;&lt;br /&gt;
+&lt;/th&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;th&gt;2/1&lt;br /&gt;
+&lt;/th&gt;
+        &lt;td style="text-align: center;"&gt;4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;-2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;-1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;th&gt;3/1&lt;br /&gt;
+&lt;/th&gt;
+        &lt;td style="text-align: center;"&gt;0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;-1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;th&gt;5/1&lt;br /&gt;
+&lt;/th&gt;
+        &lt;td style="text-align: center;"&gt;0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;/4&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+Thus the period = (4, 0, 0)/4 = (1, 0, 0) = 2/1, gen1 = (-2, 2, 0)/4 = (-1, 1, 0)/2 = P5/2, and gen2 = (-1, -1, 2)/4 = [25/6]/4.&lt;br /&gt;
+&lt;br /&gt;
+Next, search for alternate generators... (to be continued)&lt;br /&gt;
 &lt;br /&gt;
-Rank-4 temperaments can be named similarly. Rank-1 temperaments could have pergen names, such as {P8/12} for 12-edo or {P12/13} for 13-ed3, but there's no particular reason to do so.&lt;/body&gt;&lt;/html&gt;</pre></div>
+&lt;br /&gt;
+&lt;br /&gt;
+The primes can be octave-reduced, if preferred:&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;th&gt;&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;2/1&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;3/2&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;5/4&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;7/4&lt;br /&gt;
+&lt;/th&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;th&gt;period = 2/1&lt;br /&gt;
+&lt;/th&gt;
+        &lt;td style="text-align: center;"&gt;1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;-1&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;th&gt;gen1 = 3/2&lt;br /&gt;
+&lt;/th&gt;
+        &lt;td style="text-align: center;"&gt;0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;2&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;th&gt;gen2 = 5/4&lt;br /&gt;
+&lt;/th&gt;
+        &lt;td style="text-align: center;"&gt;0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;2&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+&lt;/body&gt;&lt;/html&gt;</pre></div>