Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 624838239 - Original comment: ** |
Wikispaces>TallKite **Imported revision 624838431 - 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>2018-01-14 08: | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-14 08:32:05 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>624838431</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: | ||
> k = 12a + 19b | |||
The matrix | > s = 7a + 11b | ||
The matrix [(12,19) (7,11)] is unimodular, and can be inverted, and (a,b) can be derived from [k,s]: | |||
> a = -11k + 19b | |||
> b = 7a - 12b | |||
Gedras can be manipulated exactly like monzos. Just as adding two intervals (a,b) and (a',b') gives us (a+a',b+b'), likewise [k,s] added to [k',s'] equals [k+k',s+s']. If the GCD of a and b is n, then (a,b) is a stack of n identical intervals, with (a,b) = (na', nb') = n(a',b'), and if (a,b) is converted to [k,s], then the GCD of k and s is also n, and [k,s] = [nk',ns'] = n[k',s']. | Gedras can be manipulated exactly like monzos. Just as adding two intervals (a,b) and (a',b') gives us (a+a',b+b'), likewise [k,s] added to [k',s'] equals [k+k',s+s']. If the GCD of a and b is n, then (a,b) is a stack of n identical intervals, with (a,b) = (na', nb') = n(a',b'), and if (a,b) is converted to [k,s], then the GCD of k and s is also n, and [k,s] = [nk',ns'] = n[k',s']. | ||
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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 | http://www.tallkite.com/misc_files/alt-pergensLister.zip | ||
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<br /> | <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 /> | 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 /> | ||
< | <ul class="quotelist"><li>k = 12a + 19b</li><li>s = 7a + 11b</li></ul>The matrix [(12,19) (7,11)] is unimodular, and can be inverted, and (a,b) can be derived from [k,s]:<br /> | ||
The matrix | <ul class="quotelist"><li>a = -11k + 19b</li><li>b = 7a - 12b</li></ul>Gedras can be manipulated exactly like monzos. Just as adding two intervals (a,b) and (a',b') gives us (a+a',b+b'), likewise [k,s] added to [k',s'] equals [k+k',s+s']. If the GCD of a and b is n, then (a,b) is a stack of n identical intervals, with (a,b) = (na', nb') = n(a',b'), and if (a,b) is converted to [k,s], then the GCD of k and s is also n, and [k,s] = [nk',ns'] = n[k',s'].<br /> | ||
< | |||
Gedras can be manipulated exactly like monzos. Just as adding two intervals (a,b) and (a',b') gives us (a+a',b+b'), likewise [k,s] added to [k',s'] equals [k+k',s+s']. If the GCD of a and b is n, then (a,b) is a stack of n identical intervals, with (a,b) = (na', nb') = n(a',b'), and if (a,b) is converted to [k,s], then the GCD of k and s is also n, and [k,s] = [nk',ns'] = n[k',s'].<br /> | |||
<br /> | <br /> | ||
Gedras greatly facilitate finding a pergen's period, generator and enharmonic(s). A given fraction of a given 3-limit interval can be approximated by simply dividing the keyspan and stepspan directly, and rounding off. This approximation will usually produce an enharmonic interval with the smallest possible keyspan and stepspan, which is the best enharmonic for notational purposes. As noted above, the smaller of two equivalent periods or generators is preferred, so fractions of the form N/2 should be rounded down, not up. For example, consider the half-5th pergen. P5 = [7,4], and half a 5th is approximately [round(7/2), round (4/2)] = [3,2] = (5,-3) = m3. Here xE = M - n<span class="nowrap">⋅</span>G = P5 - 2<span class="nowrap">⋅</span>m3 = [7,4] - 2<span class="nowrap">⋅</span>[3,2] = [7,4] - [6,4] = [1,0] = A1.<br /> | Gedras greatly facilitate finding a pergen's period, generator and enharmonic(s). A given fraction of a given 3-limit interval can be approximated by simply dividing the keyspan and stepspan directly, and rounding off. This approximation will usually produce an enharmonic interval with the smallest possible keyspan and stepspan, which is the best enharmonic for notational purposes. As noted above, the smaller of two equivalent periods or generators is preferred, so fractions of the form N/2 should be rounded down, not up. For example, consider the half-5th pergen. P5 = [7,4], and half a 5th is approximately [round(7/2), round (4/2)] = [3,2] = (5,-3) = m3. Here xE = M - n<span class="nowrap">⋅</span>G = P5 - 2<span class="nowrap">⋅</span>m3 = [7,4] - 2<span class="nowrap">⋅</span>[3,2] = [7,4] - [6,4] = [1,0] = A1.<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:3879: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:3879 --><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: | <!-- ws:start:WikiTextUrlRule:3880: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:3880 --><br /> | ||
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Screenshot of the first 38 pergens:<br /> | Screenshot of the first 38 pergens:<br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:2227:&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:2227 --><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:3881:http://xenharmonic.wikispaces.com/pergen+names --><a href="http://xenharmonic.wikispaces.com/pergen+names">http://xenharmonic.wikispaces.com/pergen+names</a><!-- ws:end:WikiTextUrlRule:3881 --><br /> | ||
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