Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 625289603 - Original comment: ** |
Wikispaces>TallKite **Imported revision 625347349 - 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- | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-25 03:53:54 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>625347349</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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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 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? Presumably, an infinite number. For (P8/m, M/n) to be supported by N-edo, N must be a multiple of m, and k must be divisible by n, where k is the multigen's N-edo keyspan. To be fully supported, N/m and k/n must be coprime. | ||
Given an edo, a period, and a generator, what is the pergen? There is more than one right answer. For | Given an edo, a period, and a generator, what is the pergen? There is more than one right answer. For 10edo with P = 5\10 and G = 2\10, it could be either (P8/2, P4/2) or (P8/2, P5/3). 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. | ||
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, M/n) + (P8, M/n') = (P8, M/n"), where n" = LCM (n,n') | * (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 | However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious. | ||
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Next we loop through all combinations of x and z in such a way that larger values of x and z come last: | Next we loop through all combinations of x and z in such a way that larger values of x and z come last: | ||
i = 1; loop (maxFraction, | i = 1; loop (maxFraction, | ||
> j = 1; loop (i - 1, | > j = 1; loop (i - 1, | ||
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if the pergen is false but isn't explicitly false, is the unreduced pergen explicitly false? | if the pergen is false but isn't explicitly false, is the unreduced pergen explicitly false? | ||
If m > |b| but GCD (m,n) = b, is the unreduced pergen explicitly false? | If m > |b| but GCD (m,n) = b, is the unreduced pergen explicitly false? | ||
(P8/m, (a,b)/n) unreduced is (P8/m, (n-am, -bm) / mn) = (P8/m, ( | (P8/m, (a,b)/n) unreduced is (P8/m, (n-am, -bm) / mn) | ||
Let p = m/b and q = n/b, p and q are coprime, |p| > 1 and |q| > 1 | |||
[// | Simplify by dividing by b to get (P8/m, (q - ap, -m) / qm) = (P8/m, (a',b')/n') | ||
[//unfinished proof//] | |||
Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. | Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. | ||
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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 /> | 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? Presumably, an infinite number. For (P8/m, M/n) to be supported by N-edo, N must be a multiple of m, and k must be divisible by n, where k is the multigen's N-edo keyspan. To be fully supported, N/m and k/n must be coprime.<br /> | ||
<br /> | <br /> | ||
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? There is more than one right answer. For 10edo with P = 5\10 and G = 2\10, it could be either (P8/2, P4/2) or (P8/2, P5/3). 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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General rules for combining pergens:<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 /> | <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 | However, (P8/2, M2/4) + (P8, P4/2) = (P8/4, P4/2), so the sum isn't always obvious.<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:4884: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:4884 --><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:4885: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:4885 --><br /> | ||
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Screenshot of the first 38 pergens:<br /> | Screenshot of the first 38 pergens:<br /> | ||
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Next we loop through all combinations of x and z in such a way that larger values of x and z come last:<br /> | Next we loop through all combinations of x and z in such a way that larger values of x and z come last:<br /> | ||
<br /> | |||
i = 1; loop (maxFraction,<br /> | i = 1; loop (maxFraction,<br /> | ||
<ul class="quotelist"><li>j = 1; loop (i - 1,<ul class="quotelist"><li>makeMapping (i, j); makeMapping (i, -j);</li><li>makeMapping (j, i); makeMapping (j, -i);</li><li>j += 1;</li></ul></li><li>);</li><li>makeMapping (i, i); makeMapping (i, -i);</li><li>i += 1;</li></ul>);<br /> | <ul class="quotelist"><li>j = 1; loop (i - 1,<ul class="quotelist"><li>makeMapping (i, j); makeMapping (i, -j);</li><li>makeMapping (j, i); makeMapping (j, -i);</li><li>j += 1;</li></ul></li><li>);</li><li>makeMapping (i, i); makeMapping (i, -i);</li><li>i += 1;</li></ul>);<br /> | ||
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if the pergen is false but isn't explicitly false, is the unreduced pergen explicitly false?<br /> | if the pergen is false but isn't explicitly false, is the unreduced pergen explicitly false?<br /> | ||
If m &gt; |b| but GCD (m,n) = b, is the unreduced pergen explicitly false?<br /> | If m &gt; |b| but GCD (m,n) = b, is the unreduced pergen explicitly false?<br /> | ||
(P8/m, (a,b)/n) unreduced is (P8/m, (n-am, -bm) / mn) = | (P8/m, (a,b)/n) unreduced is (P8/m, (n-am, -bm) / mn)<br /> | ||
Simplify | Let p = m/b and q = n/b, p and q are coprime, |p| &gt; 1 and |q| &gt; 1<br /> | ||
[<em> | Simplify by dividing by b to get (P8/m, (q - ap, -m) / qm) = (P8/m, (a',b')/n')<br /> | ||
[<em>unfinished proof</em>]<br /> | |||
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Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana.<br /> | Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana.<br /> | ||