Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 624808115 - Original comment: ** |
Wikispaces>TallKite **Imported revision 624808533 - 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>2018-01-12 | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-12 19:08:55 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>624808533</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
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Pergens can also be used to notate rank-2 scales, e.g. (P8, P4/3) [7] = third-fourth heptatonic is a JI-agnostic name for the Porcupine [7] scale. As long as the 5th is tuned fairly accurately, any two temperaments that have the same pergen tend to have the same MOS scales. Third-fourth heptatonic is the 1L6s scale. All MOS scales can be named after a pergen. There are multiple pergens that can generate the MOS scale, preference is given to the simpler one, and the one that makes a reasonable L/s ratio. A ratio of 3 or more makes a scale that's too lopsided. For example, 3L2s (anti-pentatonic) has a generator in the 400-480¢ range, suggesting both P11/4 and P12/4. But the former, with a just 11th, makes L = 351¢ and s = 73.5¢, and L/s = 4.76, quite large. The latter with a just 12th makes L = 249¢, s = 226.5¢, and L/s = 1.10, much better. | Pergens can also be used to notate rank-2 scales, e.g. (P8, P4/3) [7] = third-fourth heptatonic is a JI-agnostic name for the Porcupine [7] scale. As long as the 5th is tuned fairly accurately, any two temperaments that have the same pergen tend to have the same MOS scales. Third-fourth heptatonic is the 1L6s scale. All MOS scales can be named after a pergen. There are multiple pergens that can generate the MOS scale, preference is given to the simpler one, and the one that makes a reasonable L/s ratio. A ratio of 3 or more makes a scale that's too lopsided. For example, 3L2s (anti-pentatonic) has a generator in the 400-480¢ range, suggesting both P11/4 and P12/4. But the former, with a just 11th, makes L = 351¢ and s = 73.5¢, and L/s = 4.76, quite large. The latter with a just 12th makes L = 249¢, s = 226.5¢, and L/s = 1.10, much better. | ||
||||~ | ||||||~ Tetratonic MOS scales ||~ secondary examples || | ||
||= 1L 3s ||= (P8, P4/2) [4] ||= half-4th tetratonic || third-4th, third-5th || | |||
||= 2L 2s ||= (P8/2, P5) [4] ||= half-8ve tetratonic || || | |||
||= 3L 1s ||= (P8, P5/2) [4] ||= half-fifth tetratonic || || | |||
||||||~ Pentatonic MOS scales ||~ || | |||
||= 1L 4s ||= (P8, P5/3) [5] ||= third-5th pentatonic || third-4th, quarter-4th, quarter-5th || | ||= 1L 4s ||= (P8, P5/3) [5] ||= third-5th pentatonic || third-4th, quarter-4th, quarter-5th || | ||
||= 2L 3s ||= (P8, P5) [5] ||= unsplit pentatonic || third-11th || | ||= 2L 3s ||= (P8, P5) [5] ||= unsplit pentatonic || third-11th || | ||
||= 3L 2s ||= (P8, P12/5) [5] ||= quarter-12th pentatonic || quarter-11th || | ||= 3L 2s ||= (P8, P12/5) [5] ||= quarter-12th pentatonic || quarter-11th || | ||
||= 4L 1s ||= (P8, P4/2) [5] ||= half-4th pentatonic || || | ||= 4L 1s ||= (P8, P4/2) [5] ||= half-4th pentatonic || || | ||
||||~ | ||||||~ Hexatonic MOS scales ||~ || | ||
||= 1L 5s ||= (P8, P4/3) [6] ||= third-4th hexatonic || quarter-4th, quarter-5th, fifth-4th, fifth-5th || | ||= 1L 5s ||= (P8, P4/3) [6] ||= third-4th hexatonic || quarter-4th, quarter-5th, fifth-4th, fifth-5th || | ||
||= 2L 4s ||= (P8/2, P5) [6] ||= half-8ve hexatonic || || | ||= 2L 4s ||= (P8/2, P5) [6] ||= half-8ve hexatonic || || | ||
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||= 4L 2s ||= (P8/2, P4/2) [6] ||= half-everything hexatonic || || | ||= 4L 2s ||= (P8/2, P4/2) [6] ||= half-everything hexatonic || || | ||
||= 5L 1s ||= (P8, P5/3) [6] ||= third-5th hexatonic || || | ||= 5L 1s ||= (P8, P5/3) [6] ||= third-5th hexatonic || || | ||
||||~ | ||||||~ Heptatonic MOS scales ||~ || | ||
||= | ||= 1L 6s ||= (P8, P4/3) [7] ||= third-4th heptatonic || quarter-4th, fifth-4th, fifth-5th, sixth-4th, sixth-5th || | ||
||= | ||= 2L 5s ||= (P8, P11/3) [7] ||= third-11th heptatonic || fifth-WW4th, sixth-WW5th || | ||
||= | ||= 3L 4s ||= (P8, P5/2) [7] ||= half-5th heptatonic || fifth-12th || | ||
||= | ||= 4L 3s ||= (P8, P11/5) [7] ||= fifth-11th heptatonic || sixth-12th || | ||
||= | ||= 5L 2s ||= (P8, P5) [7] ||= unsplit heptatonic || sixth-WW4th || | ||
||= | ||= 6L 1s ||= (P8, P5/4) [7] ||= quarter-5th heptatonic || || | ||
|| | ||||||~ Octotonic MOS scales ||~ || | ||
||= ||= ||= || || | ||= ||= ||= || || | ||
||= ||= ||= || || | ||= ||= ||= || || | ||
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<table class="wiki_table"> | <table class="wiki_table"> | ||
<tr> | <tr> | ||
<th colspan="2"> | <th colspan="3">Tetratonic MOS scales<br /> | ||
</th> | |||
<th>secondary examples<br /> | |||
</th> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">1L 3s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8, P4/2) [4]<br /> | |||
</td> | |||
<td style="text-align: center;">half-4th tetratonic<br /> | |||
</td> | |||
<td>third-4th, third-5th<br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">2L 2s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8/2, P5) [4]<br /> | |||
</td> | |||
<td style="text-align: center;">half-8ve tetratonic<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<td style="text-align: center;">3L 1s<br /> | |||
</td> | |||
<td style="text-align: center;">(P8, P5/2) [4]<br /> | |||
</td> | |||
<td style="text-align: center;">half-fifth tetratonic<br /> | |||
</td> | |||
<td><br /> | |||
</td> | |||
</tr> | |||
<tr> | |||
<th colspan="3">Pentatonic MOS scales<br /> | |||
</th> | </th> | ||
<th><br /> | <th><br /> | ||
</th> | </th> | ||
</tr> | </tr> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
<th colspan=" | <th colspan="3">Hexatonic MOS scales<br /> | ||
</th> | </th> | ||
<th><br /> | <th><br /> | ||
</th> | </th> | ||
</tr> | |||
</tr> | |||
<tr> | <tr> | ||
<td style="text-align: center;">1L 5s<br /> | <td style="text-align: center;">1L 5s<br /> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
<th colspan=" | <th colspan="3">Heptatonic MOS scales<br /> | ||
</th> | </th> | ||
<th><br /> | <th><br /> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">1L 6s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">(P8, P4/3) [7]<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">third-4th heptatonic<br /> | ||
</td> | </td> | ||
<td><br /> | <td>quarter-4th, fifth-4th, fifth-5th, sixth-4th, sixth-5th<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">2L 5s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">(P8, P11/3) [7]<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">third-11th heptatonic<br /> | ||
</td> | </td> | ||
<td><br /> | <td>fifth-WW4th, sixth-WW5th<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">3L 4s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">(P8, P5/2) [7]<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">half-5th heptatonic<br /> | ||
</td> | </td> | ||
<td><br /> | <td>fifth-12th<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">4L 3s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">(P8, P11/5) [7]<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">fifth-11th heptatonic<br /> | ||
</td> | </td> | ||
<td><br /> | <td>sixth-12th<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">5L 2s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">(P8, P5) [7]<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">unsplit heptatonic<br /> | ||
</td> | </td> | ||
<td><br /> | <td>sixth-WW4th<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">6L 1s<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">(P8, P5/4) [7]<br /> | ||
</td> | </td> | ||
<td style="text-align: center;"><br /> | <td style="text-align: center;">quarter-5th heptatonic<br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
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</tr> | </tr> | ||
<tr> | <tr> | ||
< | <th colspan="3">Octotonic MOS scales<br /> | ||
</th> | |||
<th><br /> | |||
</th> | |||
</ | |||
< | |||
</ | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
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<!-- ws:start:WikiTextHeadingRule:62:&lt;h2&gt; --><h2 id="toc16"><a name="Further Discussion-Misc notes"></a><!-- ws:end:WikiTextHeadingRule:62 -->Misc notes</h2> | <!-- ws:start:WikiTextHeadingRule:62:&lt;h2&gt; --><h2 id="toc16"><a name="Further Discussion-Misc notes"></a><!-- ws:end:WikiTextHeadingRule:62 -->Misc notes</h2> | ||
<br /> | <br /> | ||
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:3553:http://xenharmonic.wikispaces.com/pergen+names --><a href="http://xenharmonic.wikispaces.com/pergen+names">http://xenharmonic.wikispaces.com/pergen+names</a><!-- ws:end:WikiTextUrlRule:3553 --><br /> | ||
<br /> | <br /> | ||
Gedras can be expanded to 5-limit two ways: one, by including another keyspan that is compatible with 7 and 12, such as 9 or 16. Two, the third number can be the comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. If 64/63 is 7's notational comma, for (a,b,c,d) we get [k,s,g,r]:<br /> | Gedras can be expanded to 5-limit two ways: one, by including another keyspan that is compatible with 7 and 12, such as 9 or 16. Two, the third number can be the comma 81/80. Thus 5/4 would be a M3 minus a comma, [4, 2, -1]. If 64/63 is 7's notational comma, for (a,b,c,d) we get [k,s,g,r]:<br /> | ||