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Kite's thoughts on pergens: Difference between revisions

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**Imported revision 628010623 - 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>2018-03-26 04:03:26 UTC</tt>.<br>
: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-03-27 16:18:40 UTC</tt>.<br>
: The original revision id was <tt>628010623</tt>.<br>
: The original revision id was <tt>628074207</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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||= (P8/3, P12/4) ||= third-8ve, quarter-12th ||= 15, 18b, 30* ||
||= (P8/3, P12/4) ||= third-8ve, quarter-12th ||= 15, 18b, 30* ||
||= (P8/4, P4/4) ||= quarter-everything ||= 20, 28 ||
||= (P8/4, P4/4) ||= quarter-everything ||= 20, 28 ||
The edos that support the fewest pergens are prime-number edos like 11edo or 13edo. The most "pergen-friendly" edos tend to be ones in which the circle of 5ths doesn't reach every edostep. For example, 24edo supports all half-split pergens, since both P8 and P5 map to an even number of edosteps. 72edo supports all half-splits and all third-splits. 15, 21 and 36 edo support many but not all third-splits (not those with m = 2 or n = 2).


See the screenshots in the next section for examples of which pergens are supported by a specific edo.
See the screenshots in the next section for which pergens are supported by 12, 15 and 19 edo.


__**EDO-pair names**__
__**EDO-pair names**__
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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. Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red.
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. Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red.


Screenshots of the first 60 pergens:
Screenshots of the first 69 pergens:
[[image:alt-pergenLister 1.png width="800" height="427"]]
[[image:alt-pergenLister 1.png width="800" height="427"]]
[[image:xenharmonic/alt-pergenLister 2.png width="800" height="455"]]
[[image:xenharmonic/alt-pergenLister 2.png width="800" height="455"]]
Pergens supported by 12edo:
The first 29 pergens supported by 12edo:
[[image:alt-pergenLister 12edo.png width="800" height="449"]]
[[image:alt-pergenLister 12edo.png width="800" height="449"]]
Pergens supported by 15edo. A red asterisk means partial support.
Some of the pergens supported by 15edo. A red asterisk means partial support.
[[image:alt-pergenLister 15edo.png width="800" height="493"]]
[[image:alt-pergenLister 15edo.png width="800" height="493"]]
Pergens supported by 19edo:
Pergens supported by 19edo. Edos that are a prime number support only 1 half-split pergen, 1 third-split, 1 quarter-split, etc.
[[image:alt-pergenLister 19edo.png width="800" height="459"]]
[[image:alt-pergenLister 19edo.png width="800" height="459"]]


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&lt;/table&gt;
&lt;/table&gt;


The edos that support the fewest pergens are prime-number edos like 11edo or 13edo. The most &amp;quot;pergen-friendly&amp;quot; edos tend to be ones in which the circle of 5ths doesn't reach every edostep. For example, 24edo supports all half-split pergens, since both P8 and P5 map to an even number of edosteps. 72edo supports all half-splits and all third-splits. 15, 21 and 36 edo support many but not all third-splits (not those with m = 2 or n = 2).&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
See the screenshots in the next section for examples of which pergens are supported by a specific edo.&lt;br /&gt;
See the screenshots in the next section for which pergens are supported by 12, 15 and 19 edo.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;u&gt;&lt;strong&gt;EDO-pair names&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
&lt;u&gt;&lt;strong&gt;EDO-pair names&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
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This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block. It includes alternate enharmonics for many pergens.&lt;br /&gt;
This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block. It includes alternate enharmonics for many pergens.&lt;br /&gt;
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Screenshots of the first 2 pages:&lt;br /&gt;
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;
&lt;!-- ws:start:WikiTextUrlRule:8164:http://www.tallkite.com/misc_files/alt-pergenLister.zip --&gt;&lt;a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergenLister.zip" rel="nofollow"&gt;http://www.tallkite.com/misc_files/alt-pergenLister.zip&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:8164 --&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextUrlRule:8165:http://www.tallkite.com/misc_files/alt-pergenLister.zip --&gt;&lt;a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergenLister.zip" rel="nofollow"&gt;http://www.tallkite.com/misc_files/alt-pergenLister.zip&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:8165 --&gt;&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. 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. 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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&lt;br /&gt;
Screenshots of the first 60 pergens:&lt;br /&gt;
Screenshots of the first 69 pergens:&lt;br /&gt;
&lt;!-- ws:start:WikiTextLocalImageRule:4929:&amp;lt;img src=&amp;quot;/file/view/alt-pergenLister%201.png/627986177/800x427/alt-pergenLister%201.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 427px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/alt-pergenLister%201.png/627986177/800x427/alt-pergenLister%201.png" alt="alt-pergenLister 1.png" title="alt-pergenLister 1.png" style="height: 427px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:4929 --&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextLocalImageRule:4929:&amp;lt;img src=&amp;quot;/file/view/alt-pergenLister%201.png/627986177/800x427/alt-pergenLister%201.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 427px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/alt-pergenLister%201.png/627986177/800x427/alt-pergenLister%201.png" alt="alt-pergenLister 1.png" title="alt-pergenLister 1.png" style="height: 427px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:4929 --&gt;&lt;br /&gt;
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Pergens supported by 12edo:&lt;br /&gt;
The first 29 pergens supported by 12edo:&lt;br /&gt;
&lt;!-- ws:start:WikiTextLocalImageRule:4931:&amp;lt;img src=&amp;quot;/file/view/alt-pergenLister%2012edo.png/627986189/800x449/alt-pergenLister%2012edo.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 449px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/alt-pergenLister%2012edo.png/627986189/800x449/alt-pergenLister%2012edo.png" alt="alt-pergenLister 12edo.png" title="alt-pergenLister 12edo.png" style="height: 449px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:4931 --&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextLocalImageRule:4931:&amp;lt;img src=&amp;quot;/file/view/alt-pergenLister%2012edo.png/627986189/800x449/alt-pergenLister%2012edo.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 449px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/alt-pergenLister%2012edo.png/627986189/800x449/alt-pergenLister%2012edo.png" alt="alt-pergenLister 12edo.png" title="alt-pergenLister 12edo.png" style="height: 449px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:4931 --&gt;&lt;br /&gt;
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;
&lt;!-- ws:start:WikiTextLocalImageRule:4932:&amp;lt;img src=&amp;quot;/file/view/alt-pergenLister%2015edo.png/627986197/800x493/alt-pergenLister%2015edo.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 493px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/alt-pergenLister%2015edo.png/627986197/800x493/alt-pergenLister%2015edo.png" alt="alt-pergenLister 15edo.png" title="alt-pergenLister 15edo.png" style="height: 493px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:4932 --&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextLocalImageRule:4932:&amp;lt;img src=&amp;quot;/file/view/alt-pergenLister%2015edo.png/627986197/800x493/alt-pergenLister%2015edo.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 493px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/alt-pergenLister%2015edo.png/627986197/800x493/alt-pergenLister%2015edo.png" alt="alt-pergenLister 15edo.png" title="alt-pergenLister 15edo.png" style="height: 493px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:4932 --&gt;&lt;br /&gt;
Pergens supported by 19edo:&lt;br /&gt;
Pergens supported by 19edo. Edos that are a prime number support only 1 half-split pergen, 1 third-split, 1 quarter-split, etc.&lt;br /&gt;
&lt;!-- ws:start:WikiTextLocalImageRule:4933:&amp;lt;img src=&amp;quot;/file/view/alt-pergenLister%2019edo.png/627986203/800x459/alt-pergenLister%2019edo.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 459px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/alt-pergenLister%2019edo.png/627986203/800x459/alt-pergenLister%2019edo.png" alt="alt-pergenLister 19edo.png" title="alt-pergenLister 19edo.png" style="height: 459px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:4933 --&gt;&lt;br /&gt;
&lt;!-- ws:start:WikiTextLocalImageRule:4933:&amp;lt;img src=&amp;quot;/file/view/alt-pergenLister%2019edo.png/627986203/800x459/alt-pergenLister%2019edo.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; style=&amp;quot;height: 459px; width: 800px;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/alt-pergenLister%2019edo.png/627986203/800x459/alt-pergenLister%2019edo.png" alt="alt-pergenLister 19edo.png" title="alt-pergenLister 19edo.png" style="height: 459px; width: 800px;" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:4933 --&gt;&lt;br /&gt;
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