Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 626601993 - Original comment: ** |
Wikispaces>TallKite **Imported revision 626602309 - 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>2018-02-19 06: | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-02-19 06:59:13 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>626602309</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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For a pergen (P8/m, M/n), let x be any number that divides m, R be any 3-limit ratio, and z be any integer. The interval z·P8 ± x·R splits into x parts. Let y be any number that divides n, the interval z·M ± y·R splits into y parts. If x divides both m and n, the interval z·P8 + z'·M ± x·R splits into x parts (z' is any integer). | For a pergen (P8/m, M/n), let x be any number that divides m, R be any 3-limit ratio, and z be any integer. The interval z·P8 ± x·R splits into x parts. Let y be any number that divides n, the interval z·M ± y·R splits into y parts. If x divides both m and n, the interval z·P8 + z'·M ± x·R splits into x parts (z' is any integer). | ||
||~ pergen ||~ ||~ secondary splits <= P12 || | |||
|| (P8/2, P5) || half-8ve || M2/2, m6/2, A4/2, d12/2 || | |||
|| (P8, P5/2) || half-5th || P11/4, || | |||
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==Singles and doubles== | ==Singles and doubles== | ||
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For a pergen (P8/m, M/n), let x be any number that divides m, R be any 3-limit ratio, and z be any integer. The interval z·P8 ± x·R splits into x parts. Let y be any number that divides n, the interval z·M ± y·R splits into y parts. If x divides both m and n, the interval z·P8 + z'·M ± x·R splits into x parts (z' is any integer).<br /> | For a pergen (P8/m, M/n), let x be any number that divides m, R be any 3-limit ratio, and z be any integer. The interval z·P8 ± x·R splits into x parts. Let y be any number that divides n, the interval z·M ± y·R splits into y parts. If x divides both m and n, the interval z·P8 + z'·M ± x·R splits into x parts (z' is any integer).<br /> | ||
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<th>pergen<br /> | |||
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<th>secondary splits &lt;= P12<br /> | |||
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<td>(P8/2, P5)<br /> | |||
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<td>half-8ve<br /> | |||
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<td>M2/2, m6/2, A4/2, d12/2<br /> | |||
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<td>(P8, P5/2)<br /> | |||
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<td>half-5th<br /> | |||
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<td>P11/4,<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:73:&lt;h2&gt; --><h2 id="toc8"><a name="Further Discussion-Singles and doubles"></a><!-- ws:end:WikiTextHeadingRule:73 -->Singles and doubles</h2> | <!-- ws:start:WikiTextHeadingRule:73:&lt;h2&gt; --><h2 id="toc8"><a name="Further Discussion-Singles and doubles"></a><!-- ws:end:WikiTextHeadingRule:73 -->Singles and doubles</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. It includes alternate enharmonics for many pergens.<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. It includes alternate enharmonics for many pergens.<br /> | ||
<!-- ws:start:WikiTextUrlRule: | <!-- ws:start:WikiTextUrlRule:7041: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:7041 --><br /> | ||
(screenshot)<br /> | (screenshot)<br /> | ||
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Alt-pergenLister lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. 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. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. 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:7042:http://www.tallkite.com/misc_files/alt-pergenLister.zip --><a class="wiki_link_ext" href="http://www.tallkite.com/misc_files/alt-pergenLister.zip" rel="nofollow">http://www.tallkite.com/misc_files/alt-pergenLister.zip</a><!-- ws:end:WikiTextUrlRule:7042 --><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:4240:&lt;img src=&quot;/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 460px; width: 704px;&quot; /&gt; --><img src="/file/view/alt-pergenLister.png/624838213/704x460/alt-pergenLister.png" alt="alt-pergenLister.png" title="alt-pergenLister.png" style="height: 460px; width: 704px;" /><!-- ws:end:WikiTextLocalImageRule:4240 --><br /> | ||
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Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:<br /> | Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:<br /> | ||