Kite's thoughts on pergens: Difference between revisions
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A rank-4 temperament has a pergen of four intervals, rank-5 has five intervals, etc. A rank-1 temperament could have a pergen of one, such as (P8/12) for 12-edo or (P12/13) for 13-ed3, but there's no particular reason to do so. In fact, edos and edonois are simply rank-1 pergens, and what the concept of edos does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher. | A rank-4 temperament has a pergen of four intervals, rank-5 has five intervals, etc. A rank-1 temperament could have a pergen of one, such as (P8/12) for 12-edo or (P12/13) for 13-ed3, but there's no particular reason to do so. In fact, edos and edonois are simply rank-1 pergens, and what the concept of edos does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher. | ||
In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime: (P8, P5, 81/80, 64/63, ...). The higher prime's exponent in the comma's monzo must be 1 or -1. The commas can be replaced with microtonal accidentals: (P8, P5, ^1, /1,...). | |||
=__Derivation__= | =__Derivation__= | ||
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||= 9L 1s ||= (P8, P4/2) [10] ||= quarter-4th decatonic ||< || | ||= 9L 1s ||= (P8, P4/2) [10] ||= quarter-4th decatonic ||< || | ||
The | The tetratonic MOS scales don't include quarter-split pergens, because a tetratonic genchain has only 3 steps, and can only divide a multigen into thirds. It would be possible to include pergens with a multigen which isn't actually generated. For example, 3L 2s using the sensei generator would be (P8, WWP5/7) [5]. The rationale would be that two sensei generators = 5/3, in effect a (P8, (5/3)/2) pseudo-pergen. | ||
Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, (5/4)/2), where 5·G = 7/4. | |||
==Combining pergens== | ==Combining pergens== | ||
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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. | This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block. | ||
http://www.tallkite.com/misc_files/pergens.pdf | |||
This app 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. | This app 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. | ||
(link coming soon) | |||
( | |||
==Misc notes== | ==Misc notes== | ||
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A rank-4 temperament has a pergen of four intervals, rank-5 has five intervals, etc. A rank-1 temperament could have a pergen of one, such as (P8/12) for 12-edo or (P12/13) for 13-ed3, but there's no particular reason to do so. In fact, edos and edonois are simply rank-1 pergens, and what the concept of edos does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher.<br /> | A rank-4 temperament has a pergen of four intervals, rank-5 has five intervals, etc. A rank-1 temperament could have a pergen of one, such as (P8/12) for 12-edo or (P12/13) for 13-ed3, but there's no particular reason to do so. In fact, edos and edonois are simply rank-1 pergens, and what the concept of edos does for rank-1 temperaments, the concept of pergens does for temperaments of rank 2 or higher.<br /> | ||
<br /> | <br /> | ||
In keeping with the higher-prime-agnostic nature of pergens, untempered just intonation has a pergen of the octave, the fifth, and a list of commas, each containing only one higher prime: (P8, P5, 81/80, 64/63, ...). The higher prime's exponent in the comma's monzo must be 1 or -1. The commas can be replaced with microtonal accidentals: (P8, P5, ^1, /1,...).<br /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:45:&lt;h1&gt; --><h1 id="toc2"><a name="Derivation"></a><!-- ws:end:WikiTextHeadingRule:45 --><u>Derivation</u></h1> | <!-- ws:start:WikiTextHeadingRule:45:&lt;h1&gt; --><h1 id="toc2"><a name="Derivation"></a><!-- ws:end:WikiTextHeadingRule:45 --><u>Derivation</u></h1> | ||
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The | The tetratonic MOS scales don't include quarter-split pergens, because a tetratonic genchain has only 3 steps, and can only divide a multigen into thirds. It would be possible to include pergens with a multigen which isn't actually generated. For example, 3L 2s using the sensei generator would be (P8, WWP5/7) [5]. The rationale would be that two sensei generators = 5/3, in effect a (P8, (5/3)/2) pseudo-pergen.<br /> | ||
<br /> | <br /> | ||
Some MOS scales are better understood using a pergen with a nonstandard prime subgroup. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, (5/4)/2), where 5·G = 7/4.<br /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:67:&lt;h2&gt; --><h2 id="toc13"><a name="Further Discussion-Combining pergens"></a><!-- ws:end:WikiTextHeadingRule:67 -->Combining pergens</h2> | <!-- ws:start:WikiTextHeadingRule:67:&lt;h2&gt; --><h2 id="toc13"><a name="Further Discussion-Combining pergens"></a><!-- ws:end:WikiTextHeadingRule:67 -->Combining pergens</h2> | ||
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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:3833: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:3833 --><br /> | |||
<br /> | <br /> | ||
This app 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 /> | This app 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 /> | ||
(link coming soon)<br /> | |||
( | |||
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<!-- ws:start:WikiTextHeadingRule:73:&lt;h2&gt; --><h2 id="toc16"><a name="Further Discussion-Misc notes"></a><!-- ws:end:WikiTextHeadingRule:73 -->Misc notes</h2> | <!-- ws:start:WikiTextHeadingRule:73:&lt;h2&gt; --><h2 id="toc16"><a name="Further Discussion-Misc notes"></a><!-- ws:end:WikiTextHeadingRule:73 -->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:3834:http://xenharmonic.wikispaces.com/pergen+names --><a href="http://xenharmonic.wikispaces.com/pergen+names">http://xenharmonic.wikispaces.com/pergen+names</a><!-- ws:end:WikiTextUrlRule:3834 --><br /> | ||
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