Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 630798459 - Original comment: ** |
Wikispaces>TallKite **Imported revision 630798507 - 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>2018-07-04 | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-07-04 04:00:04 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>630798507</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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To find the pergen, find the edo, then find the row that corresponds to the period, then find the column that corresponds to the generator. | To find the pergen, find the edo, then find the row that corresponds to the period, then find the column that corresponds to the generator. | ||
||~ EDO ||~ Period |||||||||||||||||| | ||~ EDO ||~ Period ||||||||||||||||||||||~ Generator in edosteps || | ||
||~ ||~ in edosteps ||~ 1 ||~ 2 ||~ 3 ||~ 4 ||~ 5 ||~ 6 ||~ 7 ||~ 8 ||~ 9 ||~ 10 ||~ | ||~ ||~ in edosteps ||~ 1 ||~ 2 ||~ 3 ||~ 4 ||~ 5 ||~ 6 ||~ 7 ||~ 8 ||~ 9 ||~ 10 ||~ 11 || | ||
||~ 5 ||~ 5 = P8 ||= P4/2 ||= P5 ||= ||= ||= ||= ||= ||= ||= ||= ||= || | |||
||~ 6 ||~ 6 = P8 ||= P4/2 ||= - ||= ||= ||= ||= ||= ||= ||= ||= ||= || | ||~ 6 ||~ 6 = P8 ||= P4/2 ||= - ||= ||= ||= ||= ||= ||= ||= ||= ||= || | ||
||~ " ||~ 3 = P8/2 ||= P5 ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= || | |||
||~ 7 ||~ 7 = P8 ||= P4/3 ||= P5/2 ||= P5 ||= ||= ||= ||= ||= ||= ||= ||= || | ||~ 7 ||~ 7 = P8 ||= P4/3 ||= P5/2 ||= P5 ||= ||= ||= ||= ||= ||= ||= ||= || | ||
||~ 8 ||~ 8 = P8 ||= P4/3 ||= - ||= P5 ||= ||= ||= ||= ||= ||= ||= ||= || | |||
||~ " ||~ 4 = P8/2 ||= P5 ||= - ||= ||= ||= ||= ||= ||= ||= ||= ||= || | ||~ " ||~ 4 = P8/2 ||= P5 ||= - ||= ||= ||= ||= ||= ||= ||= ||= ||= || | ||
||~ 9 ||~ 9 = P8 ||= P4/4 ||= P4/2 ||= - ||= P5 ||= ||= ||= ||= ||= ||= ||= || | |||
||~ " ||~ 3 = P8/3 ||= P5 ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= || | ||~ " ||~ 3 = P8/3 ||= P5 ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= || | ||
||~ 10 ||~ 10 = P8 ||= P4/4 ||= - ||= P5/2 ||= - ||= ||= ||= ||= ||= ||= ||= || | |||
||~ " ||~ 5 = P8/2 ||= P5 ||= P4/2 ||= ||= ||= ||= ||= ||= ||= ||= ||= || | ||~ " ||~ 5 = P8/2 ||= P5 ||= P4/2 ||= ||= ||= ||= ||= ||= ||= ||= ||= || | ||
||~ 11 ||~ 11 = P8 ||= P4/5 ||= P5/3 ||= P5/2 ||= P11/4 ||= P5 ||= ||= ||= ||= ||= ||= || | |||
||~ 12 ||~ 12 = P8 ||= P4/5 ||= - ||= - ||= - ||= P5 ||= ||= ||= ||= ||= ||= || | ||~ 12 ||~ 12 = P8 ||= P4/5 ||= - ||= - ||= - ||= P5 ||= ||= ||= ||= ||= ||= || | ||
||~ " ||~ 6 = P8/2 ||= P5 ||= - ||= - ||= ||= ||= ||= ||= ||= ||= ||= || | |||
||~ " ||~ 4 = P8/3 ||= P5 ||= - ||= ||= ||= ||= ||= ||= ||= ||= ||= || | ||~ " ||~ 4 = P8/3 ||= P5 ||= - ||= ||= ||= ||= ||= ||= ||= ||= ||= || | ||
||~ " ||~ 3 = P8/4 ||= P5 ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= || | |||
||~ 13b ||~ 13 = P8 ||= P4/6 ||= P4/3 ||= P4/2 ||= P12/5 ||= P12/4 ||= P5 ||= ||= ||= ||= ||= || | ||~ 13b ||~ 13 = P8 ||= P4/6 ||= P4/3 ||= P4/2 ||= P12/5 ||= P12/4 ||= P5 ||= ||= ||= ||= ||= || | ||
||~ 14 ||~ 14 = P8 ||= P4/6 ||= - ||= P4/2 ||= - ||= P11/4 ||= - ||= ||= ||= ||= ||= || | |||
||~ " ||~ 7 = P8/2 ||= P5 ||= P4/3 ||= P4/2 ||= ||= ||= ||= ||= ||= ||= ||= || | ||~ " ||~ 7 = P8/2 ||= P5 ||= P4/3 ||= P4/2 ||= ||= ||= ||= ||= ||= ||= ||= || | ||
||~ 15 ||~ 15 = P8 ||= P4/6 ||= P4/3 ||= - ||= P12/6 ||= - ||= - ||= P11/3 ||= ||= ||= ||= || | |||
||~ " ||~ 5 = P8/3 ||= P5 ||= P4/2 ||= ||= ||= ||= ||= ||= ||= ||= ||= || | ||~ " ||~ 5 = P8/3 ||= P5 ||= P4/2 ||= ||= ||= ||= ||= ||= ||= ||= ||= || | ||
||~ " ||~ 3 = P8/5 ||= P4/3 ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= || | |||
||~ 16 ||~ 16 = P8 ||= P4/7 ||= - ||= P5/3 ||= - ||= P12/5 ||= - ||= P5 ||= ||= ||= ||= || | ||~ 16 ||~ 16 = P8 ||= P4/7 ||= - ||= P5/3 ||= - ||= P12/5 ||= - ||= P5 ||= ||= ||= ||= || | ||
||~ " ||~ 8 = P8/2 ||= P5 ||= - ||= P5/3 ||= - ||= ||= ||= ||= ||= ||= ||= || | |||
||~ " ||~ 4 = P8/4 ||= P5 ||= - ||= ||= ||= ||= ||= ||= ||= ||= ||= || | ||~ " ||~ 4 = P8/4 ||= P5 ||= - ||= ||= ||= ||= ||= ||= ||= ||= ||= || | ||
|| | ||~ 17 ||~ 17 = P8 ||= P4/7 ||= P5/5 ||= P11/8 ||= P11/6 ||= P5/2 ||= P11/4 ||= P5 ||= P11/3 ||= ||= ||= || | ||
|| | ||~ 18b ||~ 18 = P8 ||= P4/8 ||= - ||= - ||= - ||= P5/2 ||= - ||= P12/4 ||= - ||= ||= ||= || | ||
||~ | ||~ " ||~ 9 = P8/2 ||= P5 ||= P4/4 ||= - ||= P4/2 ||= ||= ||= ||= ||= ||= ||= || | ||
||~ " ||~ | ||~ " ||~ 6 = P8/3 ||= P5/2 ||= - ||= - ||= ||= ||= ||= ||= ||= ||= ||= || | ||
|| | ||~ " ||~ 3 = P8/6 ||= P5 ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= || | ||
|| | ||~ 19 ||~ 19 = P8 ||= P4/8 ||= P4/4 ||= P11/9 ||= P4/2 ||= P12/6 ||= P12/5 ||= WWP5/7 ||= P5 ||= P11/3 ||= ||= || | ||
||~ | ||~ 20 ||~ 20 = P8 ||= P4/8 ||= - ||= P5/4 ||= - ||= - ||= - ||= P11/4 ||= - ||= W<span style="vertical-align: super;">3</span>P5/8 ||= ||= || | ||
||~ | ||~ " ||~ 10 = P8/2 ||= M2/4 ||= - ||= P5/4 ||= - ||= - ||= ||= ||= ||= ||= ||= || | ||
||~ " ||~ 5 = P8/4 ||= P4/2 ||= P5 ||= ||= ||= ||= ||= ||= ||= ||= ||= || | |||
||~ " ||~ 4 = P8/5 ||= P5/4 ||= - ||= ||= ||= ||= ||= ||= ||= ||= ||= || | |||
||~ 21 ||~ 21 = P8 ||= P4/9 ||= P5/6 ||= - ||= P5/3 ||= P11/6 ||= - ||= - ||= W<span style="vertical-align: super;">3</span>P4/9 ||= - ||= P11/3 ||= || | |||
||~ " ||~ 7 = P8/3 ||= P5/2 ||= P5 ||= P4/3 ||= ||= ||= ||= ||= ||= ||= ||= || | |||
||~ " ||~ 3 = P8/7 ||= P5/3 ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= || | |||
|| | ||~ 22 ||~ 22 = P8 ||= P4/9 ||= - ||= P4/3 ||= - ||= P12/7 ||= - ||= P12/5 ||= - ||= P5 ||= - ||= || | ||
||~ " ||~ 11 = P8/2 ||= M2/4 ||= P5 ||= P4/3 ||= P12/5 ||= P12/7 ||= ||= ||= ||= ||= ||= || | |||
||~ 23 ||~ 23 = P8 ||= P4/10 ||= P4/5 ||= P11/11 ||= P12/9 ||= P4/2 ||= P12/6 ||= WWP4/8 ||= WWP4/7 ||= P12/4 ||= P5 ||= P11/3 || | |||
||~ | ||~ 24 ||~ 24 = P8 ||= P4/10 ||= - ||= - ||= - ||= P4/2 ||= - ||= P5/2 ||= - ||= - ||= - ||= W<span style="vertical-align: super;">4</span>P5/10 || | ||
||~ | ||~ " ||~ 12 = P8/2 ||= M2/4 ||= - ||= - ||= - ||= P4/2 ||= ||= ||= ||= ||= ||= || | ||
|| | ||~ " ||~ 8 = P8/3 ||= P5/2 ||= - ||= P4/2 ||= ||= ||= ||= ||= ||= ||= ||= || | ||
||~ " ||~ 6 = P8/4 ||= P4/2 ||= - ||= - ||= ||= ||= ||= ||= ||= ||= ||= || | |||
||~ " ||~ 4 = P8/6 ||= P4/2 ||= - ||= ||= ||= ||= ||= ||= ||= ||= ||= || | |||
||~ " ||~ 3 = P8/8 ||= P5 ||= ||= ||= ||= ||= ||= ||= ||= ||= ||= || | |||
||~ ||~ ||~ 1 ||~ 2 ||~ 3 ||~ 4 ||~ 5 ||~ 6 ||~ 7 ||~ 8 ||~ 9 ||~ 10 ||~ 11 || | |||
||~ ||~ ||~ 1 ||~ 2 ||~ 3 ||~ 4 ||~ 5 ||~ 6 ||~ 7 ||~ 8 ||~ 9 ||~ 10 ||~ 11 | |||
| Line 3,113: | Line 3,109: | ||
<br /> | <br /> | ||
<u><span style="font-size: 110%;">Mizarian Porcupine Overture by Herman Miller (P8, P4/3)</span></u><br /> | <u><span style="font-size: 110%;">Mizarian Porcupine Overture by Herman Miller (P8, P4/3)</span></u><br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:6485:&lt;img src=&quot;/file/view/Mizarian%20Porcupine%20Overture.png/628798699/800x692/Mizarian%20Porcupine%20Overture.png&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 692px; width: 800px;&quot; /&gt; --><img src="/file/view/Mizarian%20Porcupine%20Overture.png/628798699/800x692/Mizarian%20Porcupine%20Overture.png" alt="Mizarian Porcupine Overture.png" title="Mizarian Porcupine Overture.png" style="height: 692px; width: 800px;" /><!-- ws:end:WikiTextLocalImageRule:6485 --><br /> | ||
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Pergen squares can be generalized to any prime subgroup by representing the notes as dots. Below are the first 32 rank-2 pergens in a completely JI-agnostic format. A is the interval of equivalence, the period of the unsplit pergen. B is the generator of the unsplit pergen. For 2.3 pergens, A = 8ve and B = 5th. The (A, (A-B)/2) square corresponds to (P8, P4/2). In the 2.5 subgroup, B = 5/4. In Bohlen-Peirce, A = 3/1 and B = 5/3. True doubles are in red. The true/false property of a pergen is independent of the prime subgroup. Imperfect multigens are in green. Imperfect is generalized to other subgroups as requiring multiples of B in the pergen.<br /> | Pergen squares can be generalized to any prime subgroup by representing the notes as dots. Below are the first 32 rank-2 pergens in a completely JI-agnostic format. A is the interval of equivalence, the period of the unsplit pergen. B is the generator of the unsplit pergen. For 2.3 pergens, A = 8ve and B = 5th. The (A, (A-B)/2) square corresponds to (P8, P4/2). In the 2.5 subgroup, B = 5/4. In Bohlen-Peirce, A = 3/1 and B = 5/3. True doubles are in red. The true/false property of a pergen is independent of the prime subgroup. Imperfect multigens are in green. Imperfect is generalized to other subgroups as requiring multiples of B in the pergen.<br /> | ||
<!-- ws:start:WikiTextLocalImageRule: | <!-- ws:start:WikiTextLocalImageRule:6486:&lt;img src=&quot;/file/view/pergen%20squares.png/627986281/pergen%20squares.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/pergen%20squares.png/627986281/pergen%20squares.png" alt="pergen squares.png" title="pergen squares.png" /><!-- ws:end:WikiTextLocalImageRule:6486 --><br /> | ||
A similar chart could be made for all rank-3 pergens, using pergen cubes.<br /> | A similar chart could be made for all rank-3 pergens, using pergen cubes.<br /> | ||
<br /> | <br /> | ||
| Line 6,414: | Line 6,410: | ||
<th>Period<br /> | <th>Period<br /> | ||
</th> | </th> | ||
<th colspan=" | <th colspan="11">Generator in edosteps<br /> | ||
</th> | </th> | ||
</tr> | </tr> | ||
| Line 6,454: | Line 6,438: | ||
<th>10<br /> | <th>10<br /> | ||
</th> | </th> | ||
<th> | <th>11<br /> | ||
</th> | </th> | ||
</tr> | </tr> | ||
| Line 6,473: | Line 6,449: | ||
</td> | </td> | ||
<td style="text-align: center;">P5<br /> | <td style="text-align: center;">P5<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 6,509: | Line 6,477: | ||
</td> | </td> | ||
<td style="text-align: center;">-<br /> | <td style="text-align: center;">-<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 6,543: | Line 6,503: | ||
</th> | </th> | ||
<td style="text-align: center;">P5<br /> | <td style="text-align: center;">P5<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 6,583: | Line 6,535: | ||
</td> | </td> | ||
<td style="text-align: center;">P5<br /> | <td style="text-align: center;">P5<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 6,636: | Line 6,580: | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | |||
<tr> | |||
</tr> | |||
<tr> | |||
<th>&quot;<br /> | <th>&quot;<br /> | ||
</th> | </th> | ||
| Line 6,653: | Line 6,589: | ||
</td> | </td> | ||
<td style="text-align: center;">-<br /> | <td style="text-align: center;">-<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 6,693: | Line 6,621: | ||
</td> | </td> | ||
<td style="text-align: center;">P5<br /> | <td style="text-align: center;">P5<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 6,723: | Line 6,643: | ||
</th> | </th> | ||
<td style="text-align: center;">P5<br /> | <td style="text-align: center;">P5<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 6,765: | Line 6,677: | ||
</td> | </td> | ||
<td style="text-align: center;">-<br /> | <td style="text-align: center;">-<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 6,797: | Line 6,701: | ||
</td> | </td> | ||
<td style="text-align: center;">P4/2<br /> | <td style="text-align: center;">P4/2<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 6,839: | Line 6,735: | ||
</td> | </td> | ||
<td style="text-align: center;">P5<br /> | <td style="text-align: center;">P5<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 6,888: | Line 6,776: | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | |||
</tr> | |||
<tr> | <tr> | ||
<th>&quot;<br /> | <th>&quot;<br /> | ||
| Line 6,907: | Line 6,787: | ||
</td> | </td> | ||
<td style="text-align: center;">-<br /> | <td style="text-align: center;">-<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 6,941: | Line 6,813: | ||
</td> | </td> | ||
<td style="text-align: center;">-<br /> | <td style="text-align: center;">-<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 6,975: | Line 6,839: | ||
</th> | </th> | ||
<td style="text-align: center;">P5<br /> | <td style="text-align: center;">P5<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 7,021: | Line 6,877: | ||
</td> | </td> | ||
<td style="text-align: center;">P5<br /> | <td style="text-align: center;">P5<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 7,057: | Line 6,905: | ||
</td> | </td> | ||
<td style="text-align: center;">-<br /> | <td style="text-align: center;">-<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 7,087: | Line 6,927: | ||
</td> | </td> | ||
<td style="text-align: center;">P4/2<br /> | <td style="text-align: center;">P4/2<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 7,140: | Line 6,972: | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | |||
</tr> | |||
<tr> | <tr> | ||
<th>&quot;<br /> | <th>&quot;<br /> | ||
| Line 7,157: | Line 6,981: | ||
</td> | </td> | ||
<td style="text-align: center;">P4/2<br /> | <td style="text-align: center;">P4/2<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 7,191: | Line 7,007: | ||
</th> | </th> | ||
<td style="text-align: center;">P4/3<br /> | <td style="text-align: center;">P4/3<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 7,239: | Line 7,047: | ||
</td> | </td> | ||
<td style="text-align: center;">P5<br /> | <td style="text-align: center;">P5<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 7,269: | Line 7,069: | ||
</td> | </td> | ||
<td style="text-align: center;">-<br /> | <td style="text-align: center;">-<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 7,301: | Line 7,093: | ||
</td> | </td> | ||
<td style="text-align: center;">-<br /> | <td style="text-align: center;">-<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 7,330: | Line 7,114: | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<th><br /> | <th>17<br /> | ||
</th> | </th> | ||
<th><br /> | <th>17 = P8<br /> | ||
</th> | </th> | ||
<td style="text-align: center;">P4/7<br /> | |||
</td> | |||
<td style="text-align: center;">P5/5<br /> | |||
</td> | |||
<td style="text-align: center;">P11/8<br /> | |||
</td> | |||
<td style="text-align: center;">P4/7<br /> | |||
</td> | |||
<td style="text-align: center;">P5/5<br /> | |||
</td> | |||
<td style="text-align: center;">P11/8<br /> | |||
</td> | |||
<td style="text-align: center;">P11/6<br /> | <td style="text-align: center;">P11/6<br /> | ||
</td> | </td> | ||
| Line 7,385: | Line 7,133: | ||
</td> | </td> | ||
<td style="text-align: center;">P11/3<br /> | <td style="text-align: center;">P11/3<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 7,421: | Line 7,161: | ||
</td> | </td> | ||
<td style="text-align: center;">-<br /> | <td style="text-align: center;">-<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 7,449: | Line 7,181: | ||
</td> | </td> | ||
<td style="text-align: center;">P4/2<br /> | <td style="text-align: center;">P4/2<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 7,483: | Line 7,207: | ||
</td> | </td> | ||
<td style="text-align: center;">-<br /> | <td style="text-align: center;">-<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 7,515: | Line 7,231: | ||
</th> | </th> | ||
<td style="text-align: center;">P5<br /> | <td style="text-align: center;">P5<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 7,567: | Line 7,275: | ||
</td> | </td> | ||
<td style="text-align: center;">P11/3<br /> | <td style="text-align: center;">P11/3<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 7,603: | Line 7,303: | ||
</td> | </td> | ||
<td style="text-align: center;">W<span style="vertical-align: super;">3</span>P5/8<br /> | <td style="text-align: center;">W<span style="vertical-align: super;">3</span>P5/8<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 7,631: | Line 7,323: | ||
</td> | </td> | ||
<td style="text-align: center;">-<br /> | <td style="text-align: center;">-<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 7,661: | Line 7,345: | ||
</td> | </td> | ||
<td style="text-align: center;">P5<br /> | <td style="text-align: center;">P5<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 7,697: | Line 7,373: | ||
</td> | </td> | ||
<td style="text-align: center;">-<br /> | <td style="text-align: center;">-<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 7,749: | Line 7,417: | ||
</td> | </td> | ||
<td style="text-align: center;">P11/3<br /> | <td style="text-align: center;">P11/3<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 7,771: | Line 7,431: | ||
</td> | </td> | ||
<td style="text-align: center;">P4/3<br /> | <td style="text-align: center;">P4/3<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 7,803: | Line 7,455: | ||
</th> | </th> | ||
<td style="text-align: center;">P5/3<br /> | <td style="text-align: center;">P5/3<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 7,857: | Line 7,501: | ||
</td> | </td> | ||
<td style="text-align: center;">-<br /> | <td style="text-align: center;">-<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 7,883: | Line 7,519: | ||
</td> | </td> | ||
<td style="text-align: center;">P12/7<br /> | <td style="text-align: center;">P12/7<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 7,931: | Line 7,559: | ||
</td> | </td> | ||
<td style="text-align: center;">P11/3<br /> | <td style="text-align: center;">P11/3<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 7,967: | Line 7,587: | ||
</td> | </td> | ||
<td style="text-align: center;">W<span style="vertical-align: super;">4</span>P5/10<br /> | <td style="text-align: center;">W<span style="vertical-align: super;">4</span>P5/10<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 7,991: | Line 7,603: | ||
</td> | </td> | ||
<td style="text-align: center;">P4/2<br /> | <td style="text-align: center;">P4/2<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 8,023: | Line 7,627: | ||
</td> | </td> | ||
<td style="text-align: center;">P4/2<br /> | <td style="text-align: center;">P4/2<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 8,059: | Line 7,655: | ||
</td> | </td> | ||
<td style="text-align: center;">-<br /> | <td style="text-align: center;">-<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 8,093: | Line 7,681: | ||
</td> | </td> | ||
<td style="text-align: center;">-<br /> | <td style="text-align: center;">-<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
| Line 8,147: | Line 7,727: | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
| Line 8,221: | Line 7,755: | ||
</th> | </th> | ||
<th>11<br /> | <th>11<br /> | ||
</th> | </th> | ||
</tr> | </tr> | ||
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Screenshots of the first 2 pages:<br /> | Screenshots of the first 2 pages:<br /> | ||
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Alt-pergenLister lists out thousands of rank-2 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 or edo pair. Written in Jesusonic, runs inside Reaper.<br /> | Alt-pergenLister lists out thousands of rank-2 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 or edo pair. Written in Jesusonic, runs inside Reaper.<br /> | ||
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The first section (PERGEN and Per/Gen cents) describes each pergen without regard to notational issues. The period and generator's cents are given, assuming a 5th of 700¢ + c. The generator is reduced, e.g. (P8/2, P5) has a generator of 100¢ + c, not 700¢ + c. The next two sections show a possible notation for P and G. The last section shows the unreduced pergen, and for false doubles, a possible single-pair notation. Horizontal lines group the pergens into blocks (half-splits, third-splits, etc). Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red.<br /> | The first section (PERGEN and Per/Gen cents) describes each pergen without regard to notational issues. The period and generator's cents are given, assuming a 5th of 700¢ + c. The generator is reduced, e.g. (P8/2, P5) has a generator of 100¢ + c, not 700¢ + c. The next two sections show a possible notation for P and G. The last section shows the unreduced pergen, and for false doubles, a possible single-pair notation. Horizontal lines group the pergens into blocks (half-splits, third-splits, etc). Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red.<br /> | ||
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Screenshots of the first 69 pergens:<br /> | Screenshots of the first 69 pergens:<br /> | ||
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The first 29 pergens supported by 12edo:<br /> | The first 29 pergens supported by 12edo:<br /> | ||
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Some of the pergens supported by 15edo. A red asterisk means partial support.<br /> | Some of the pergens supported by 15edo. A red asterisk means partial support.<br /> | ||
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Pergens supported by 19edo. Edos that are a prime number support only 1 pergen per block.<br /> | Pergens supported by 19edo. Edos that are a prime number support only 1 pergen per block.<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 /> | ||