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Kite's thoughts on pergens: Difference between revisions

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**Imported revision 630796453 - Original comment: **
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**Imported revision 630796511 - Original comment: **
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-07-03 17:22:22 UTC</tt>.<br>
: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-07-03 17:38:56 UTC</tt>.<br>
: The original revision id was <tt>630796453</tt>.<br>
: The original revision id was <tt>630796511</tt>.<br>
: The revision comment was: <tt></tt><br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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__**Pergens Within An EDO**__
__**Pergens Within An EDO**__


See the screenshots in the Supplemental Materials section for a complete list of which pergens are supported by 12, 15 and 19 edo. The list includes multiple pergens per period/generator combination. The next table shows only the simplest pergen for each combination. Combinations are excluded if the period and generator are not coprime, because the scale is contained in a smaller edo. Periods of 1 or 2 edosteps are excluded as too trivial, because every step of the edo appears in the genchains, even if the chains are only one step long. Generators of 1 edostep are excluded when the octave is unsplit, because a single run of edosteps isn't a very interesting scale.
See the screenshots in the Supplemental Materials section for a complete list of which pergens are supported by 12, 15 and 19 edo. The list includes multiple pergens per period/generator combination. The next table shows only the simplest pergen for each combination. Combinations are excluded if the period and generator are not coprime, because the scale is contained in a smaller edo. For example, 15edo has no unsplit pergen, because that is a 5edo scale. Periods of 1 or 2 edosteps are excluded as too trivial, because every step of the edo appears in the genchains, even if the chains are only one step long. Generators of 1 edostep are excluded for edos &gt;= 12 when the octave is unsplit, because a solid run of edosteps isn't a very interesting scale.


To find the pergen, look up the edo, then find the row that corresponds to the period, then the column that corresponds to the generator.
To find the pergen, find the edo, then find the row that corresponds to the period, then find the column that corresponds to the generator.


||~ EDO ||~ 5th ||~ P ||||||||||~ Generator ||~  ||~  ||~   ||~  ||~  ||~  ||~  ||~  ||~  ||~  ||
||~ EDO ||~ Period ||||||||||||||||||~ Generator in edosteps ||~  ||~  ||~  ||~  ||~  ||~  ||
||~  ||~  ||~  ||~ 1 ||~ 2 ||~ 3 ||~ 4 ||~ 5 ||~ 6 ||~ 7 ||~ 8 ||~ 9 ||~ 10 ||~ 11 ||~ 12 ||~ 13 ||~ 14 ||~ 15 ||
||~  ||~  ||~ 1 ||~ 2 ||~ 3 ||~ 4 ||~ 5 ||~ 6 ||~ 7 ||~ 8 ||~ 9 ||~ 10 ||~ 11 ||~ 12 ||~ 13 ||~ 14 ||~ 15 ||
||= 5 ||= 3 ||= P8 ||= P4/2 ||= P5 ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||~ 5 ||~ P8 ||= P4/2 ||= P5 ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||= 6 ||= 4 ||= P8 ||= P4/2 ||= - ||= - ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||~ 6 ||~ P8 ||= P4/2 ||= - ||= - ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||=  ||=  ||= P8/2 ||= P5 ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||~ " ||~ P8/2 ||= P5 ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||= 7 ||= 4 ||= P8 ||= P4/2 ||= P5/2 ||= P5 ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||~ 7 ||~ P8 ||= P4/3 ||= P5/2 ||= P5 ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||= 8 ||= 5 ||= P8 ||=   ||=   ||= P5 ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||~ 8 ||~ P8 ||= P4/3 ||= - ||= P5 ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||=  ||=  ||= P8/2 ||=   ||=   ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||~ " ||~ P8/2 ||= P5 ||= - ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||= 9 ||= 5 ||= P8 ||=   ||=   ||=   ||=   ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||~ 9 ||~ P8 ||= P4/4 ||= P4/2 ||= - ||= P5 ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||=  ||=  ||= P8/3 ||=   ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||~ " ||~ P8/3 ||= P5 ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||= 10 ||= 6 ||= P8 ||=   ||=   ||=   ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||~ 10 ||~ P8 ||= P4/4 ||= - ||= P5/2 ||= P5 ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||=  ||=  ||= P8/2 ||=   ||=   ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||~ " ||~ P8/2 ||= P5 ||= P4/2 ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||= 11 ||= 6 ||= P8 ||=   ||=   ||=   ||=   ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||~ 11 ||~ P8 ||= P4/5 ||= P5/3 ||= P5/2 ||= P11/4 ||= P5 ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||= 12 ||= 7 ||= P8 ||= P4/5 ||= - ||= - ||= - ||= P5 ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||~ 12 ||~ P8 ||= P4/5 ||= - ||= - ||= - ||= P5 ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||=  ||=  ||= P8/2 ||= P5 ||= -- ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||~ " ||~ P8/2 ||= P5 ||= - ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||=  ||=  ||= P8/3 ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||~ " ||~ P8/3 ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||=  ||=  ||= P8/4 ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||~ " ||~ P8/4 ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||=  ||=  ||=   ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||~ 13 ||~   ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||=  ||=  ||=   ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||~ 13b ||~   ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||=   ||=  ||=   ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||~   ||~   ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||=   ||=  ||=   ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||~   ||~   ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||=   ||=  ||=   ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||~   ||~   ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||=   ||=  ||=   ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||~   ||~   ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||=   ||=  ||=   ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||~   ||~   ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||=   ||=  ||=   ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||~   ||~   ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||=   ||=  ||=   ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||~   ||~   ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||=   ||=  ||=   ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||~   ||~   ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||=   ||=  ||=   ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||~   ||~   ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||=   ||=  ||=   ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||~   ||~   ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||=   ||=  ||=   ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||~   ||~   ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||=   ||=  ||=   ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
||~   ||~   ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||




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&lt;u&gt;&lt;span style="font-size: 110%;"&gt;Mizarian Porcupine Overture by Herman Miller (P8, P4/3)&lt;/span&gt;&lt;/u&gt;&lt;br /&gt;
&lt;u&gt;&lt;span style="font-size: 110%;"&gt;Mizarian Porcupine Overture by Herman Miller (P8, P4/3)&lt;/span&gt;&lt;/u&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextLocalImageRule:6303:&amp;lt;img src=&amp;quot;/file/view/Mizarian%20Porcupine%20Overture.png/628798699/800x692/Mizarian%20Porcupine%20Overture.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 692px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Mizarian%20Porcupine%20Overture.png/628798699/800x692/Mizarian%20Porcupine%20Overture.png" alt="Mizarian Porcupine Overture.png" title="Mizarian Porcupine Overture.png" style="height: 692px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:6303 --&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextLocalImageRule:6233:&amp;lt;img src=&amp;quot;/file/view/Mizarian%20Porcupine%20Overture.png/628798699/800x692/Mizarian%20Porcupine%20Overture.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 692px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/Mizarian%20Porcupine%20Overture.png/628798699/800x692/Mizarian%20Porcupine%20Overture.png" alt="Mizarian Porcupine Overture.png" title="Mizarian Porcupine Overture.png" style="height: 692px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:6233 --&gt;&lt;br /&gt;
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&lt;br /&gt;
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Pergen squares can be generalized to any prime subgroup by representing the notes as dots. Below are the first 32 rank-2 pergens in a completely JI-agnostic format. A is the interval of equivalence, the period of the unsplit pergen. B is the generator of the unsplit pergen. For 2.3 pergens, A = 8ve and B = 5th. The (A, (A-B)/2) square corresponds to (P8, P4/2). In the 2.5 subgroup, B = 5/4. In Bohlen-Peirce, A = 3/1 and B = 5/3. True doubles are in red. The true/false property of a pergen is independent of the prime subgroup. Imperfect multigens are in green. Imperfect is generalized to other subgroups as requiring multiples of B in the pergen.&lt;br /&gt;
Pergen squares can be generalized to any prime subgroup by representing the notes as dots. Below are the first 32 rank-2 pergens in a completely JI-agnostic format. A is the interval of equivalence, the period of the unsplit pergen. B is the generator of the unsplit pergen. For 2.3 pergens, A = 8ve and B = 5th. The (A, (A-B)/2) square corresponds to (P8, P4/2). In the 2.5 subgroup, B = 5/4. In Bohlen-Peirce, A = 3/1 and B = 5/3. True doubles are in red. The true/false property of a pergen is independent of the prime subgroup. Imperfect multigens are in green. Imperfect is generalized to other subgroups as requiring multiples of B in the pergen.&lt;br /&gt;
&lt;!-- ws:start:WikiTextLocalImageRule:6304:&amp;lt;img src=&amp;quot;/file/view/pergen%20squares.png/627986281/pergen%20squares.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/pergen%20squares.png/627986281/pergen%20squares.png" alt="pergen squares.png" title="pergen squares.png" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:6304 --&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextLocalImageRule:6234:&amp;lt;img src=&amp;quot;/file/view/pergen%20squares.png/627986281/pergen%20squares.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/pergen%20squares.png/627986281/pergen%20squares.png" alt="pergen squares.png" title="pergen squares.png" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:6234 --&gt;&lt;br /&gt;
A similar chart could be made for all rank-3 pergens, using pergen cubes.&lt;br /&gt;
A similar chart could be made for all rank-3 pergens, using pergen cubes.&lt;br /&gt;
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&lt;u&gt;&lt;strong&gt;Pergens Within An EDO&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
&lt;u&gt;&lt;strong&gt;Pergens Within An EDO&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
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&lt;br /&gt;
See the screenshots in the Supplemental Materials section for a complete list of which pergens are supported by 12, 15 and 19 edo. The list includes multiple pergens per period/generator combination. The next table shows only the simplest pergen for each combination. Combinations are excluded if the period and generator are not coprime, because the scale is contained in a smaller edo. Periods of 1 or 2 edosteps are excluded as too trivial, because every step of the edo appears in the genchains, even if the chains are only one step long. Generators of 1 edostep are excluded when the octave is unsplit, because a single run of edosteps isn't a very interesting scale.&lt;br /&gt;
See the screenshots in the Supplemental Materials section for a complete list of which pergens are supported by 12, 15 and 19 edo. The list includes multiple pergens per period/generator combination. The next table shows only the simplest pergen for each combination. Combinations are excluded if the period and generator are not coprime, because the scale is contained in a smaller edo. For example, 15edo has no unsplit pergen, because that is a 5edo scale. Periods of 1 or 2 edosteps are excluded as too trivial, because every step of the edo appears in the genchains, even if the chains are only one step long. Generators of 1 edostep are excluded for edos &amp;gt;= 12 when the octave is unsplit, because a solid run of edosteps isn't a very interesting scale.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
To find the pergen, look up the edo, then find the row that corresponds to the period, then the column that corresponds to the generator.&lt;br /&gt;
To find the pergen, find the edo, then find the row that corresponds to the period, then find the column that corresponds to the generator.&lt;br /&gt;
&lt;br /&gt;
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         &lt;th&gt;EDO&lt;br /&gt;
         &lt;th&gt;EDO&lt;br /&gt;
&lt;/th&gt;
&lt;/th&gt;
         &lt;th&gt;5th&lt;br /&gt;
         &lt;th&gt;Period&lt;br /&gt;
&lt;/th&gt;
&lt;/th&gt;
        &lt;th&gt;P&lt;br /&gt;
         &lt;th colspan="9"&gt;Generator in edosteps&lt;br /&gt;
&lt;/th&gt;
         &lt;th colspan="5"&gt;Generator&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;&lt;br /&gt;
&lt;/th&gt;
&lt;/th&gt;
         &lt;th&gt;&lt;br /&gt;
         &lt;th&gt;&lt;br /&gt;
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     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
        &lt;th&gt;&lt;br /&gt;
&lt;/th&gt;
         &lt;th&gt;&lt;br /&gt;
         &lt;th&gt;&lt;br /&gt;
&lt;/th&gt;
&lt;/th&gt;
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     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;5&lt;br /&gt;
         &lt;th&gt;5&lt;br /&gt;
&lt;/td&gt;
&lt;/th&gt;
         &lt;td style="text-align: center;"&gt;3&lt;br /&gt;
         &lt;th&gt;P8&lt;br /&gt;
&lt;/td&gt;
&lt;/th&gt;
        &lt;td style="text-align: center;"&gt;P8&lt;br /&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;P4/2&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;P4/2&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
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     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;6&lt;br /&gt;
         &lt;th&gt;6&lt;br /&gt;
&lt;/td&gt;
&lt;/th&gt;
         &lt;td style="text-align: center;"&gt;4&lt;br /&gt;
         &lt;th&gt;P8&lt;br /&gt;
&lt;/td&gt;
&lt;/th&gt;
        &lt;td style="text-align: center;"&gt;P8&lt;br /&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;P4/2&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;P4/2&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
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     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;th&gt;&amp;quot;&lt;br /&gt;
&lt;/td&gt;
&lt;/th&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;th&gt;P8/2&lt;br /&gt;
&lt;/td&gt;
&lt;/th&gt;
         &lt;td style="text-align: center;"&gt;P8/2&lt;br /&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;P5&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;P5&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
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     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;7&lt;br /&gt;
         &lt;th&gt;7&lt;br /&gt;
&lt;/td&gt;
&lt;/th&gt;
        &lt;td style="text-align: center;"&gt;4&lt;br /&gt;
         &lt;th&gt;P8&lt;br /&gt;
&lt;/td&gt;
&lt;/th&gt;
         &lt;td style="text-align: center;"&gt;P8&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;P4/3&lt;br /&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;P4/2&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;P5/2&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;P5/2&lt;br /&gt;
Line 6,608: Line 6,588:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;8&lt;br /&gt;
        &lt;th&gt;8&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;P8&lt;br /&gt;
&lt;/th&gt;
         &lt;td style="text-align: center;"&gt;P4/3&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;5&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;-&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;P8&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;P5&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
Line 6,618: Line 6,602:
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;P5&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
Line 6,646: Line 6,624:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
        &lt;th&gt;&amp;quot;&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;P8/2&lt;br /&gt;
&lt;/th&gt;
         &lt;td style="text-align: center;"&gt;P5&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;-&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;P8/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
Line 6,684: Line 6,660:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;9&lt;br /&gt;
        &lt;th&gt;9&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;P8&lt;br /&gt;
&lt;/th&gt;
         &lt;td style="text-align: center;"&gt;P4/4&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;5&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;P4/2&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;P8&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;-&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;P5&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
Line 6,722: Line 6,696:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;th&gt;&amp;quot;&lt;br /&gt;
&lt;/td&gt;
&lt;/th&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;th&gt;P8/3&lt;br /&gt;
&lt;/td&gt;
&lt;/th&gt;
         &lt;td style="text-align: center;"&gt;P8/3&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;P5&lt;br /&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
Line 6,760: Line 6,732:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;10&lt;br /&gt;
        &lt;th&gt;10&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;P8&lt;br /&gt;
&lt;/th&gt;
         &lt;td style="text-align: center;"&gt;P4/4&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;6&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;-&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;P8&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;P5/2&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;P5&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
Line 6,790: Line 6,766:
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
     &lt;/tr&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
        &lt;th&gt;&amp;quot;&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;P8/2&lt;br /&gt;
&lt;/th&gt;
         &lt;td style="text-align: center;"&gt;P5&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;P4/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;P8/2&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
Line 6,836: Line 6,804:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;11&lt;br /&gt;
        &lt;th&gt;11&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;P8&lt;br /&gt;
&lt;/th&gt;
         &lt;td style="text-align: center;"&gt;P4/5&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;6&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;P5/3&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;P8&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;P5/2&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;P11/4&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;P5&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
Line 6,874: Line 6,840:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;12&lt;br /&gt;
         &lt;th&gt;12&lt;br /&gt;
&lt;/td&gt;
&lt;/th&gt;
         &lt;td style="text-align: center;"&gt;7&lt;br /&gt;
         &lt;th&gt;P8&lt;br /&gt;
&lt;/td&gt;
&lt;/th&gt;
        &lt;td style="text-align: center;"&gt;P8&lt;br /&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;P4/5&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;P4/5&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
Line 6,912: Line 6,876:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;th&gt;&amp;quot;&lt;br /&gt;
&lt;/td&gt;
&lt;/th&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;th&gt;P8/2&lt;br /&gt;
&lt;/td&gt;
&lt;/th&gt;
         &lt;td style="text-align: center;"&gt;P8/2&lt;br /&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;P5&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;P5&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;--&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;-&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
Line 6,950: Line 6,912:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
        &lt;th&gt;&amp;quot;&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;P8/3&lt;br /&gt;
&lt;/th&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;P8/3&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
Line 6,982: Line 6,946:
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
     &lt;/tr&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;th&gt;&amp;quot;&lt;br /&gt;
&lt;/td&gt;
&lt;/th&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;th&gt;P8/4&lt;br /&gt;
&lt;/td&gt;
&lt;/th&gt;
         &lt;td style="text-align: center;"&gt;P8/4&lt;br /&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
Line 7,026: Line 6,984:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
        &lt;th&gt;13&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;&lt;br /&gt;
&lt;/th&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
Line 7,056: Line 7,018:
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;th&gt;13b&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;&lt;br /&gt;
&lt;/th&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
Line 7,062: Line 7,030:
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
Line 7,088: Line 7,054:
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;th&gt;&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;&lt;br /&gt;
&lt;/th&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
Line 7,100: Line 7,072:
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
Line 7,120: Line 7,090:
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;th&gt;&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;&lt;br /&gt;
&lt;/th&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
Line 7,138: Line 7,114:
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
Line 7,152: Line 7,126:
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;th&gt;&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;&lt;br /&gt;
&lt;/th&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
Line 7,176: Line 7,156:
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
Line 7,184: Line 7,162:
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;th&gt;&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;&lt;br /&gt;
&lt;/th&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
Line 7,216: Line 7,200:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;th&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/th&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;th&gt;&lt;br /&gt;
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Screenshots of the first 2 pages:&lt;br /&gt;
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Alt-pergenLister lists out thousands of rank-2 pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo or edo pair. Written in Jesusonic, runs inside Reaper.&lt;br /&gt;
Alt-pergenLister lists out thousands of rank-2 pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo or edo pair. Written in Jesusonic, runs inside Reaper.&lt;br /&gt;
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The first section (PERGEN and Per/Gen cents) describes each pergen without regard to notational issues. The period and generator's cents are given, assuming a 5th of 700¢ + c. The generator is reduced, e.g. (P8/2, P5) has a generator of 100¢ + c, not 700¢ + c. The next two sections show a possible notation for P and G. The last section shows the unreduced pergen, and for false doubles, a possible single-pair notation. Horizontal lines group the pergens into blocks (half-splits, third-splits, etc). Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red.&lt;br /&gt;
The first section (PERGEN and Per/Gen cents) describes each pergen without regard to notational issues. The period and generator's cents are given, assuming a 5th of 700¢ + c. The generator is reduced, e.g. (P8/2, P5) has a generator of 100¢ + c, not 700¢ + c. The next two sections show a possible notation for P and G. The last section shows the unreduced pergen, and for false doubles, a possible single-pair notation. Horizontal lines group the pergens into blocks (half-splits, third-splits, etc). Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red.&lt;br /&gt;
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Screenshots of the first 69 pergens:&lt;br /&gt;
Screenshots of the first 69 pergens:&lt;br /&gt;
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The first 29 pergens supported by 12edo:&lt;br /&gt;
The first 29 pergens supported by 12edo:&lt;br /&gt;
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Some of the pergens supported by 15edo. A red asterisk means partial support.&lt;br /&gt;
Some of the pergens supported by 15edo. A red asterisk means partial support.&lt;br /&gt;
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Pergens supported by 19edo. Edos that are a prime number support only 1 pergen per block.&lt;br /&gt;
Pergens supported by 19edo. Edos that are a prime number support only 1 pergen per block.&lt;br /&gt;
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Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:&lt;br /&gt;
Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:&lt;br /&gt;
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