Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 624832325 - Original comment: ** |
Wikispaces>TallKite **Imported revision 624838239 - 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>2018-01-14 08:25:57 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>624838239</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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Every 3-limit interval can be uniquely expressed as the combination of a keyspan and a stepspan. This combination is called a **gedra**, analogous to a monzo, but written in brackets not parentheses: 3/2 = (-1,1) is a 7-semitone 5th, thus (-1,1) = [7,4]. 9/8 = (-3,2) = [2,1] = a 2-semitone 1-step interval. The octave 2/1 = [12,7]. For any 3-limit interval with a monzo (a,b), there is a unique gedra [k,s], and vice versa: | Every 3-limit interval can be uniquely expressed as the combination of a keyspan and a stepspan. This combination is called a **gedra**, analogous to a monzo, but written in brackets not parentheses: 3/2 = (-1,1) is a 7-semitone 5th, thus (-1,1) = [7,4]. 9/8 = (-3,2) = [2,1] = a 2-semitone 1-step interval. The octave 2/1 = [12,7]. For any 3-limit interval with a monzo (a,b), there is a unique gedra [k,s], and vice versa: | ||
<span style="display: block; text-align: | <span style="display: block; text-align: left;">> k = 12a + 19b</span><span style="display: block; text-align: left;">> s = 7a + 11b</span> | ||
The matrix ((12,19) (7,11)) is unimodular, and can be inverted, and (a,b) can be derived from [k,s]: | The matrix ((12,19) (7,11)) is unimodular, and can be inverted, and (a,b) can be derived from [k,s]: | ||
<span style="display: block; text-align: center;">a = -11k + 19b</span><span style="display: block; text-align: center;">b = 7a - 12b</span> | <span style="display: block; text-align: center;">a = -11k + 19b</span><span style="display: block; text-align: center;">b = 7a - 12b</span> | ||
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http://www.tallkite.com/misc_files/pergens.pdf | http://www.tallkite.com/misc_files/pergens.pdf | ||
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. | |||
http://www.tallkite.com/misc_files/alt-pergensLister.zip | |||
Screenshot of the first 38 pergens: | |||
[[image:alt-pergenLister.png width="800" height="526"]] | |||
==Misc notes== | ==Misc notes== | ||
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Every 3-limit interval can be uniquely expressed as the combination of a keyspan and a stepspan. This combination is called a <strong>gedra</strong>, analogous to a monzo, but written in brackets not parentheses: 3/2 = (-1,1) is a 7-semitone 5th, thus (-1,1) = [7,4]. 9/8 = (-3,2) = [2,1] = a 2-semitone 1-step interval. The octave 2/1 = [12,7]. For any 3-limit interval with a monzo (a,b), there is a unique gedra [k,s], and vice versa:<br /> | Every 3-limit interval can be uniquely expressed as the combination of a keyspan and a stepspan. This combination is called a <strong>gedra</strong>, analogous to a monzo, but written in brackets not parentheses: 3/2 = (-1,1) is a 7-semitone 5th, thus (-1,1) = [7,4]. 9/8 = (-3,2) = [2,1] = a 2-semitone 1-step interval. The octave 2/1 = [12,7]. For any 3-limit interval with a monzo (a,b), there is a unique gedra [k,s], and vice versa:<br /> | ||
<span style="display: block; text-align: | <span style="display: block; text-align: left;">&gt; k = 12a + 19b</span><span style="display: block; text-align: left;">&gt; s = 7a + 11b</span><br /> | ||
The matrix ((12,19) (7,11)) is unimodular, and can be inverted, and (a,b) can be derived from [k,s]:<br /> | The matrix ((12,19) (7,11)) is unimodular, and can be inverted, and (a,b) can be derived from [k,s]:<br /> | ||
<span style="display: block; text-align: center;">a = -11k + 19b</span><span style="display: block; text-align: center;">b = 7a - 12b</span><br /> | <span style="display: block; text-align: center;">a = -11k + 19b</span><span style="display: block; text-align: center;">b = 7a - 12b</span><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:3877: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:3877 --><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 /> | |||
<!-- ws:start:WikiTextUrlRule:3878: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:3878 --><br /> | |||
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Screenshot of the first 38 pergens:<br /> | |||
<!-- ws:start:WikiTextLocalImageRule:2215:&lt;img src=&quot;/file/view/alt-pergenLister.png/624838213/800x526/alt-pergenLister.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 526px; width: 800px;&quot; /&gt; --><img src="/file/view/alt-pergenLister.png/624838213/800x526/alt-pergenLister.png" alt="alt-pergenLister.png" title="alt-pergenLister.png" style="height: 526px; width: 800px;" /><!-- ws:end:WikiTextLocalImageRule:2215 --><br /> | |||
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<!-- ws:start:WikiTextHeadingRule:75:&lt;h2&gt; --><h2 id="toc17"><a name="Further Discussion-Misc notes"></a><!-- ws:end:WikiTextHeadingRule:75 -->Misc notes</h2> | <!-- ws:start:WikiTextHeadingRule:75:&lt;h2&gt; --><h2 id="toc17"><a name="Further Discussion-Misc notes"></a><!-- ws:end:WikiTextHeadingRule:75 -->Misc notes</h2> | ||
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Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. Earlier drafts of this article can be found at <!-- ws:start:WikiTextUrlRule: | Pergens were discovered by Kite Giedraitis in 2017, and developed with the help of Praveen Venkataramana. Earlier drafts of this article can be found at <!-- ws:start:WikiTextUrlRule:3879:http://xenharmonic.wikispaces.com/pergen+names --><a href="http://xenharmonic.wikispaces.com/pergen+names">http://xenharmonic.wikispaces.com/pergen+names</a><!-- ws:end:WikiTextUrlRule:3879 --><br /> | ||
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