Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 625080991 - Original comment: ** |
Wikispaces>TallKite **Imported revision 625110387 - 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>2018-01-19 | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-19 15:01:37 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>625110387</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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Scales can be named similar to Meantone[7], as (P8, P5) [7] = unsplit heptatonic, or (P8, P5/2) [7] = half-fifth pentatonic, etc. The number of notes in the scale tend to be a multiple of m, so that half-octave pergens have scales with an even number of notes. | Scales can be named similar to Meantone[7], as (P8, P5) [7] = unsplit heptatonic, or (P8, P5/2) [7] = half-fifth pentatonic, etc. The number of notes in the scale tend to be a multiple of m, so that half-octave pergens have scales with an even number of notes. | ||
== == | == == | ||
==MOS scales== | ==Pergens and MOS scales== | ||
MOS scales tend to correspond to just one or two pergens. The table below shows the pergen that best corresponds to each MOS scale, as well as | MOS scales tend to correspond to just one or two pergens. The table below shows the pergen that best corresponds to each MOS scale, as well as some others that could also generate the scale. The best pergen is the one that makes a reasonable L/s ratio. A ratio of 3 or more makes a scale that's too lopsided. | ||
||||||~ Tetratonic MOS scales ||~ secondary examples || | ||||||~ Tetratonic MOS scales ||~ secondary examples || | ||
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||= (P8/4, P5/3) ||= quarter-8ve third-5th ||= 8 = 4L 4s ||= 12 = 4L 8s ||= ||= || || || | ||= (P8/4, P5/3) ||= quarter-8ve third-5th ||= 8 = 4L 4s ||= 12 = 4L 8s ||= ||= || || || | ||
||= (P8/4, P11/3) ||= quarter-8ve third-11th ||= 8 = 4L 4s ||= 12 = 4L 8s ||= ||= || || || | ||= (P8/4, P11/3) ||= quarter-8ve third-11th ||= 8 = 4L 4s ||= 12 = 4L 8s ||= ||= || || || | ||
||= (P8/3, P4/4) ||= third-8ve quarter-4th ||= | ||= (P8/3, P4/4) ||= third-8ve quarter-4th ||= 6 = 3L 3s ||= 9 = 3L 6s ||= 12 = 9L 3s ||= || || || | ||
||= (P8/3, P5/4) ||= third-8ve quarter-5th ||= | ||= (P8/3, P5/4) ||= third-8ve quarter-5th ||= 6 = 3L 3s ||= 9 = 6L 3s ||= ||= || || || | ||
||= (P8/3, P11/4) ||= third-8ve quarter-11th ||= | ||= (P8/3, P11/4) ||= third-8ve quarter-11th ||= 6 = 3L 3s ||= 9 = 3L 6s ||= 12 = 3L 9s ||= || || || | ||
||= (P8/3, P12/4) ||= third-8ve quarter-12th ||= | ||= (P8/3, P12/4) ||= third-8ve quarter-12th ||= 6 = 3L 3s ||= 9 = 3L 6s ||= 12 = 3L 9s ||= || || || | ||
||= (P8/4, P4/4) ||= quarter-everything ||= | ||= (P8/4, P4/4) ||= quarter-everything ||= 8 = 4L 4s ||= 12 = 8L 4s ||= ||= || || || | ||
==Pergens and EDOs== | ==Pergens and EDOs== | ||
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Just as edos are said to support temperaments, they can support pergens. If a temperament is supported, its pergen is too, and vice versa. An edo can't suppoprt a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. A pergen is somewhat unsupported by an edo if the period and generator can only generate a subset of the edo. For example, (P8, P5) is somewhat unsupported by 15edo, because any chain-of-5ths scale could only make a 5-edo subset. | Just as edos are said to support temperaments, they can support pergens. If a temperament is supported, its pergen is too, and vice versa. An edo can't suppoprt a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. A pergen is somewhat unsupported by an edo if the period and generator can only generate a subset of the edo. For example, (P8, P5) is somewhat unsupported by 15edo, because any chain-of-5ths scale could only make a 5-edo subset. | ||
How many pergens are fully supported by a given edo? Surprisingly, an infinite number! There are | How many pergens are fully supported by a given edo? Surprisingly, an infinite number! There are infinitely many possible multigens, each one divisible by its keyspan. For example, 12edo supports, among other pergens, this series: (P8, P5/7), (P8, P12/19), (P8, WWP5/31),... (P8, (i-1,1)/n), where n = 12i+7. | ||
How many edos support a given pergen? | How many edos support a given pergen? In general, an infinite number. For (P8/m, M/n) to be supported by N-edo, N must be divisible by m, and k by n, where k is the multigen's N-edo keyspan. To be fully supported, N/m and k/n must be coprime. A pergen of the form (P8/m, P5) is only fully supported by m-edo. | ||
Given an edo, a period, and a generator, what is the pergen? For 12edo, if P = 12\12 and G = 1\12, it could be either (P8, P4/5) or (P8, P5/7). It hasn't yet been rigorously proven that every period/generator pair results from a valid pergen. It isn't yet known if there are period/generator pairs that require a true double pergen, or if all such pairs can result from either a false double or single-split pergen. | Given an edo, a period, and a generator, what is the pergen? For 12edo, if P = 12\12 and G = 1\12 (one semitone), it could be either (P8, P4/5) or (P8, P5/7). It hasn't yet been rigorously proven that every period/generator pair results from a valid pergen. It isn't yet known if there are period/generator pairs that require a true double pergen, or if all such pairs can result from either a false double or single-split pergen. | ||
This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. Since both of these edos are incompatible with heptatonic notation, 13edo's half-5th pergen is actually notated as a half-upfifth. 13b-edo and 18b-edo are listed as well. | This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. Since both of these edos are incompatible with heptatonic notation, 13edo's half-5th pergen is actually notated as a half-upfifth. 13b-edo and 18b-edo are listed as well. | ||
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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 || | ||
==Combining pergens== | |||
Tempering out 250/243 creates third-4th, and 49/48 creates half-4th, and tempering out both commas creates sixth-4th. Therefore (P8, P4/3) + (P8, P4/2) = (P8, P4/6). | |||
General rules for combining pergens: | |||
* (P8/m, M/n) + (P8, P5) = (P8/m, M/n) | |||
* (P8/m, P5) + (P8, M/n) = (P8/m, M/n) | |||
* (P8/m, P5) + (P8/m', P5) = (P8/m", P5), where m" = LCM (m,m') | |||
* (P8, M/n) + (P8, M/n') = (P8, M/n"), where n" = LCM (n,n') | |||
However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious. Further study is needed. | |||
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<!-- ws:end:WikiTextTocRule:105 --><!-- ws:start:WikiTextTocRule:106: --><div style="margin-left: 2em;"><a href="#Further Discussion-Chord names and scale names">Chord names and scale names</a></div> | <!-- ws:end:WikiTextTocRule:105 --><!-- ws:start:WikiTextTocRule:106: --><div style="margin-left: 2em;"><a href="#Further Discussion-Chord names and scale names">Chord names and scale names</a></div> | ||
<!-- ws:end:WikiTextTocRule:106 --><!-- ws:start:WikiTextTocRule:107: --><div style="margin-left: 2em;"><a href="#toc14"> </a></div> | <!-- ws:end:WikiTextTocRule:106 --><!-- ws:start:WikiTextTocRule:107: --><div style="margin-left: 2em;"><a href="#toc14"> </a></div> | ||
<!-- ws:end:WikiTextTocRule:107 --><!-- ws:start:WikiTextTocRule:108: --><div style="margin-left: 2em;"><a href="#Further Discussion-MOS scales">MOS scales</a></div> | <!-- ws:end:WikiTextTocRule:107 --><!-- ws:start:WikiTextTocRule:108: --><div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and MOS scales">Pergens and MOS scales</a></div> | ||
<!-- ws:end:WikiTextTocRule:108 --><!-- ws:start:WikiTextTocRule:109: --><div style="margin-left: 2em;"><a href="#Further Discussion- | <!-- ws:end:WikiTextTocRule:108 --><!-- ws:start:WikiTextTocRule:109: --><div style="margin-left: 2em;"><a href="#Further Discussion-Pergens and EDOs">Pergens and EDOs</a></div> | ||
<!-- ws:end:WikiTextTocRule:109 --><!-- ws:start:WikiTextTocRule:110: --><div style="margin-left: 2em;"><a href="#Further Discussion- | <!-- ws:end:WikiTextTocRule:109 --><!-- ws:start:WikiTextTocRule:110: --><div style="margin-left: 2em;"><a href="#Further Discussion-Combining pergens">Combining pergens</a></div> | ||
<!-- ws:end:WikiTextTocRule:110 --><!-- ws:start:WikiTextTocRule:111: --><div style="margin-left: 2em;"><a href="#Further Discussion-Supplemental materials">Supplemental materials</a></div> | <!-- ws:end:WikiTextTocRule:110 --><!-- ws:start:WikiTextTocRule:111: --><div style="margin-left: 2em;"><a href="#Further Discussion-Supplemental materials">Supplemental materials</a></div> | ||
<!-- ws:end:WikiTextTocRule:111 --><!-- ws:start:WikiTextTocRule:112: --><div style="margin-left: 2em;"><a href="#Further Discussion-Misc notes">Misc notes</a></div> | <!-- ws:end:WikiTextTocRule:111 --><!-- ws:start:WikiTextTocRule:112: --><div style="margin-left: 2em;"><a href="#Further Discussion-Misc notes">Misc notes</a></div> | ||
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Scales can be named similar to Meantone[7], as (P8, P5) [7] = unsplit heptatonic, or (P8, P5/2) [7] = half-fifth pentatonic, etc. The number of notes in the scale tend to be a multiple of m, so that half-octave pergens have scales with an even number of notes.<br /> | Scales can be named similar to Meantone[7], as (P8, P5) [7] = unsplit heptatonic, or (P8, P5/2) [7] = half-fifth pentatonic, etc. The number of notes in the scale tend to be a multiple of m, so that half-octave pergens have scales with an even number of notes.<br /> | ||
<!-- ws:start:WikiTextHeadingRule:80:&lt;h2&gt; --><h2 id="toc14"><!-- ws:end:WikiTextHeadingRule:80 --> </h2> | <!-- ws:start:WikiTextHeadingRule:80:&lt;h2&gt; --><h2 id="toc14"><!-- ws:end:WikiTextHeadingRule:80 --> </h2> | ||
<!-- ws:start:WikiTextHeadingRule:82:&lt;h2&gt; --><h2 id="toc15"><a name="Further Discussion-MOS scales"></a><!-- ws:end:WikiTextHeadingRule:82 -->MOS scales</h2> | <!-- ws:start:WikiTextHeadingRule:82:&lt;h2&gt; --><h2 id="toc15"><a name="Further Discussion-Pergens and MOS scales"></a><!-- ws:end:WikiTextHeadingRule:82 -->Pergens and MOS scales</h2> | ||
<br /> | <br /> | ||
MOS scales tend to correspond to just one or two pergens. The table below shows the pergen that best corresponds to each MOS scale, as well as | MOS scales tend to correspond to just one or two pergens. The table below shows the pergen that best corresponds to each MOS scale, as well as some others that could also generate the scale. The best pergen is the one that makes a reasonable L/s ratio. A ratio of 3 or more makes a scale that's too lopsided.<br /> | ||
<br /> | <br /> | ||
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<td style="text-align: center;">third-8ve quarter-4th<br /> | <td style="text-align: center;">third-8ve quarter-4th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">6 = 3L 3s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">9 = 3L 6s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">12 = 9L 3s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
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<td style="text-align: center;">third-8ve quarter-5th<br /> | <td style="text-align: center;">third-8ve quarter-5th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">6 = 3L 3s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">9 = 6L 3s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
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<td style="text-align: center;">third-8ve quarter-11th<br /> | <td style="text-align: center;">third-8ve quarter-11th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">6 = 3L 3s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">9 = 3L 6s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">12 = 3L 9s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
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<td style="text-align: center;">third-8ve quarter-12th<br /> | <td style="text-align: center;">third-8ve quarter-12th<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">6 = 3L 3s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">9 = 3L 6s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">12 = 3L 9s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
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<td style="text-align: center;">quarter-everything<br /> | <td style="text-align: center;">quarter-everything<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">8 = 4L 4s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">12 = 8L 4s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
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<!-- ws:start:WikiTextHeadingRule:84:&lt;h2&gt; --><h2 id="toc16"><a name="Further Discussion-Pergens and EDOs"></a><!-- ws:end:WikiTextHeadingRule:84 -->Pergens and EDOs</h2> | |||
<!-- ws:start:WikiTextHeadingRule:84:&lt;h2&gt; --><h2 id="toc16 | |||
<br /> | <br /> | ||
Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos, but fewer than a hundred have been explored. There are an infinite number of pergens, but fewer than a hundred will suffice most composers.<br /> | Pergens have much in common with edos. Pergens of rank-2 assume only primes 2 and 3, edos assume only prime 2. There are an infinite number of edos, but fewer than a hundred have been explored. There are an infinite number of pergens, but fewer than a hundred will suffice most composers.<br /> | ||
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Just as edos are said to support temperaments, they can support pergens. If a temperament is supported, its pergen is too, and vice versa. An edo can't suppoprt a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. A pergen is somewhat unsupported by an edo if the period and generator can only generate a subset of the edo. For example, (P8, P5) is somewhat unsupported by 15edo, because any chain-of-5ths scale could only make a 5-edo subset.<br /> | Just as edos are said to support temperaments, they can support pergens. If a temperament is supported, its pergen is too, and vice versa. An edo can't suppoprt a pergen if the split is impossible. For example, all odd-numbered edos are incompatible with half-octave pergens. A pergen is somewhat unsupported by an edo if the period and generator can only generate a subset of the edo. For example, (P8, P5) is somewhat unsupported by 15edo, because any chain-of-5ths scale could only make a 5-edo subset.<br /> | ||
<br /> | <br /> | ||
How many pergens are fully supported by a given edo? Surprisingly, an infinite number! There are | How many pergens are fully supported by a given edo? Surprisingly, an infinite number! There are infinitely many possible multigens, each one divisible by its keyspan. For example, 12edo supports, among other pergens, this series: (P8, P5/7), (P8, P12/19), (P8, WWP5/31),... (P8, (i-1,1)/n), where n = 12i+7.<br /> | ||
<br /> | <br /> | ||
How many edos support a given pergen? | How many edos support a given pergen? In general, an infinite number. For (P8/m, M/n) to be supported by N-edo, N must be divisible by m, and k by n, where k is the multigen's N-edo keyspan. To be fully supported, N/m and k/n must be coprime. A pergen of the form (P8/m, P5) is only fully supported by m-edo.<br /> | ||
<br /> | <br /> | ||
Given an edo, a period, and a generator, what is the pergen? For 12edo, if P = 12\12 and G = 1\12, it could be either (P8, P4/5) or (P8, P5/7). It hasn't yet been rigorously proven that every period/generator pair results from a valid pergen. It isn't yet known if there are period/generator pairs that require a true double pergen, or if all such pairs can result from either a false double or single-split pergen.<br /> | Given an edo, a period, and a generator, what is the pergen? For 12edo, if P = 12\12 and G = 1\12 (one semitone), it could be either (P8, P4/5) or (P8, P5/7). It hasn't yet been rigorously proven that every period/generator pair results from a valid pergen. It isn't yet known if there are period/generator pairs that require a true double pergen, or if all such pairs can result from either a false double or single-split pergen.<br /> | ||
<br /> | <br /> | ||
This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. Since both of these edos are incompatible with heptatonic notation, 13edo's half-5th pergen is actually notated as a half-upfifth. 13b-edo and 18b-edo are listed as well.<br /> | This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. Since both of these edos are incompatible with heptatonic notation, 13edo's half-5th pergen is actually notated as a half-upfifth. 13b-edo and 18b-edo are listed as well.<br /> | ||
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</table> | </table> | ||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:86:&lt;h2&gt; --><h2 id="toc17"><a name="Further Discussion-Combining pergens"></a><!-- ws:end:WikiTextHeadingRule:86 -->Combining pergens</h2> | |||
<br /> | |||
Tempering out 250/243 creates third-4th, and 49/48 creates half-4th, and tempering out both commas creates sixth-4th. Therefore (P8, P4/3) + (P8, P4/2) = (P8, P4/6).<br /> | |||
<br /> | |||
General rules for combining pergens:<br /> | |||
<ul><li>(P8/m, M/n) + (P8, P5) = (P8/m, M/n)</li><li>(P8/m, P5) + (P8, M/n) = (P8/m, M/n)</li><li>(P8/m, P5) + (P8/m', P5) = (P8/m&quot;, P5), where m&quot; = LCM (m,m')</li><li>(P8, M/n) + (P8, M/n') = (P8, M/n&quot;), where n&quot; = LCM (n,n')</li></ul><br /> | |||
However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious. Further study is needed.<br /> | |||
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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.<br /> | This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block.<br /> | ||
<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:4899:http://www.tallkite.com/misc_files/pergens.pdf --><a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/pergens.pdf" rel="nofollow">http://www.tallkite.com/misc_files/pergens.pdf</a><!-- ws:end:WikiTextUrlRule:4899 --><br /> | ||
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Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.<br /> | Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.<br /> | ||
<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:4900:http://www.tallkite.com/misc_files/alt-pergensLister.zip --><a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergensLister.zip" rel="nofollow">http://www.tallkite.com/misc_files/alt-pergensLister.zip</a><!-- ws:end:WikiTextUrlRule:4900 --><br /> | ||
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Screenshot of the first 38 pergens:<br /> | Screenshot of the first 38 pergens:<br /> | ||