Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 626602309 - Original comment: ** |
Wikispaces>TallKite **Imported revision 626613657 - 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 | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-02-19 11:49:21 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>626613657</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). | ||
||~ | ||~ category ||||~ __pergen__ ||~ secondary splits <= P12 || | ||
|| (P8/2, P5) || half-8ve || M2/2, m6/2, | ||~ halves ||||= all pergens ||= M3/2, d5/2, A5/2, m7/2, M9/2, A11/2 || | ||
|| (P8, | || ||= (P8/2, P5) ||= half-8ve ||= M2/2, A4/2, m6/2, M10/2, d12/2 || | ||
|| || | || ||= (P8, P4/2) ||= half-4th ||= m2/2, M6/2, A8/2, m10/2, P12/2 || | ||
|| || | || ||= (P8, P5/2) ||= half-5th ||= A1/2, m3/2, M7/2, m9/2, P11/2 || | ||
|| | || ||= (P8/2, P4/2) ||= half-everything ||= (every 3-limit interval)/2 || | ||
|| || | ||~ thirds ||||= all pergens ||= A4/3, m10/3 || | ||
|| || | || ||= (P8/3, P5) ||= third-8ve ||= m3/3, M6/3, A11/3, d12/3 || | ||
|| || | || ||= (P8, P4/3) ||= third-4th ||= A1/3, m7/3, M7/3, M10/3 || | ||
|| || | || ||= (P8, P5/3) ||= third-5th ||= m2/3, m6/3, M9/3, A8/3 || | ||
|| || | || ||= (P8, P11/3) ||= third-11th ||= M2/3, M3/3, m9/3, P12/3 || | ||
|| || || || | || ||= (P8/3, P4/2) ||= third-8ve half-4th ||= || | ||
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<table class="wiki_table"> | <table class="wiki_table"> | ||
<tr> | <tr> | ||
<th> | <th>category<br /> | ||
</th> | </th> | ||
<th><br /> | <th colspan="2"><u>pergen</u><br /> | ||
</th> | </th> | ||
<th>secondary splits &lt;= P12<br /> | <th>secondary splits &lt;= P12<br /> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
< | <th>halves<br /> | ||
</ | </th> | ||
<td> | <td colspan="2" style="text-align: center;">all pergens<br /> | ||
</td> | </td> | ||
<td> | <td style="text-align: center;">M3/2, d5/2, A5/2, m7/2, M9/2, A11/2<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td>(P8, P5 | <td><br /> | ||
</td> | |||
<td style="text-align: center;">(P8/2, P5)<br /> | |||
</td> | </td> | ||
<td>half- | <td style="text-align: center;">half-8ve<br /> | ||
</td> | </td> | ||
<td> | <td style="text-align: center;">M2/2, A4/2, m6/2, M10/2, d12/2<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;">(P8, P4/2)<br /> | ||
</td> | |||
<td style="text-align: center;">half-4th<br /> | |||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;">m2/2, M6/2, A8/2, m10/2, P12/2<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;">(P8, P5/2)<br /> | ||
</td> | |||
<td style="text-align: center;">half-5th<br /> | |||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;">A1/2, m3/2, M7/2, m9/2, P11/2<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;">(P8/2, P4/2)<br /> | ||
</td> | |||
<td style="text-align: center;">half-everything<br /> | |||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;">(every 3-limit interval)/2<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
< | <th>thirds<br /> | ||
</ | </th> | ||
<td><br /> | <td colspan="2" style="text-align: center;">all pergens<br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;">A4/3, m10/3<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;">(P8/3, P5)<br /> | ||
</td> | |||
<td style="text-align: center;">third-8ve<br /> | |||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;">m3/3, M6/3, A11/3, d12/3<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;">(P8, P4/3)<br /> | ||
</td> | |||
<td style="text-align: center;">third-4th<br /> | |||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;">A1/3, m7/3, M7/3, M10/3<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;">(P8, P5/3)<br /> | ||
</td> | |||
<td style="text-align: center;">third-5th<br /> | |||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;">m2/3, m6/3, M9/3, A8/3<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
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<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;">(P8, P11/3)<br /> | ||
</td> | </td> | ||
<td style="text-align: center;">third-11th<br /> | |||
</td> | |||
<td style="text-align: center;">M2/3, M3/3, m9/3, P12/3<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td><br /> | <td><br /> | ||
</td> | |||
<td style="text-align: center;">(P8/3, P4/2)<br /> | |||
</td> | |||
<td style="text-align: center;">third-8ve half-4th<br /> | |||
</td> | |||
<td style="text-align: center;"><br /> | |||
</td> | </td> | ||
</tr> | </tr> | ||
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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:7106: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:7106 --><br /> | ||
(screenshot)<br /> | (screenshot)<br /> | ||
<br /> | <br /> | ||
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<br /> | <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 /> | 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:7107: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:7107 --><br /> | ||
<br /> | <br /> | ||
Screenshot of the first 38 pergens:<br /> | Screenshot of the first 38 pergens:<br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:4284:&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:4284 --><br /> | ||
<br /> | <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 /> | 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 /> | ||