Kite's thoughts on pergens: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-12 09:16:10 UTC</tt>.

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-12 18:32:44 UTC</tt>.

: The original revision id was <tt>~~624782391~~</tt>.

: The original revision id was <tt>624808115</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

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Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first porcupine is triple yellow, and the second one is triple yellow and red. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a JI-agnostic manner, and the color name indicates the harmonic possibilities: the prime subgroup, and what types of chord progressions it supports. Both names indicate the rank, the pergen name more directly.

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, ~~the MOS scales of~~ any two temperaments that have the same pergen tend to be the ~~same~~.

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.

||||~ pentatonic MOS scales ||~ ||~ secondary examples ||

||= 1L 4s ||= (P8, P5/3) [5] ||= third-5th pentatonic || third-4th, quarter-4th, quarter-5th ||

||= 2L 3s ||= (P8, P5) [5] ||= unsplit pentatonic || third-11th ||

||= 3L 2s ||= (P8, P12/5) [5] ||= quarter-12th pentatonic || quarter-11th ||

||= 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 ||

||= 2L 4s ||= (P8/2, P5) [6] ||= half-8ve hexatonic || ||

||= 3L 3s ||= (P8/3, P5) [6] ||= third-8ve hexatonic || ||

||= 4L 2s ||= (P8/2, P4/2) [6] ||= half-everything hexatonic || ||

||= 5L 1s ||= (P8, P5/3) [6] ||= third-5th hexatonic || ||

||||~ heptatonic MOS scales ||~ ||~ ||

||= ||= ||= || ||

The third main application, which the rest of this article will focus on, is that pergens allow a systematic exploration of notations for regular temperaments, without having to examine each of the thousands of individual temperaments. The discussion focusses on rank-2 temperaments that include primes 2 and 3.

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Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first porcupine is triple yellow, and the second one is triple yellow and red. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a JI-agnostic manner, and the color name indicates the harmonic possibilities: the prime subgroup, and what types of chord progressions it supports. Both names indicate the rank, the pergen name more directly.

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, ~~the MOS scales of~~ any two temperaments that have the same pergen tend to be the ~~same~~.

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.

<tr>

<th colspan="2">pentatonic MOS scales

</th>

<th>

</th>

<th>secondary examples

</th>

</tr>

<tr>

<td style="text-align: center;">1L 4s

</td>

<td style="text-align: center;">(P8, P5/3) [5]

</td>

<td style="text-align: center;">third-5th pentatonic

</td>

<td>third-4th, quarter-4th, quarter-5th

</td>

</tr>

<tr>

<td style="text-align: center;">2L 3s

</td>

<td style="text-align: center;">(P8, P5) [5]

</td>

<td style="text-align: center;">unsplit pentatonic

</td>

<td>third-11th

</td>

</tr>

<tr>

<td style="text-align: center;">3L 2s

</td>

<td style="text-align: center;">(P8, P12/5) [5]

</td>

<td style="text-align: center;">quarter-12th pentatonic

</td>

<td>quarter-11th

</td>

</tr>

<tr>

<td style="text-align: center;">4L 1s

</td>

<td style="text-align: center;">(P8, P4/2) [5]

</td>

<td style="text-align: center;">half-4th pentatonic

</td>

<td>

</td>

</tr>

<tr>

<th colspan="2">hexatonic MOS scales

</th>

<th>

</th>

<th>

</th>

</tr>

<tr>

<td style="text-align: center;">1L 5s

</td>

<td style="text-align: center;">(P8, P4/3) [6]

</td>

<td style="text-align: center;">third-4th hexatonic

</td>

<td>quarter-4th, quarter-5th, fifth-4th, fifth-5th

</td>

</tr>

<tr>

<td style="text-align: center;">2L 4s

</td>

<td style="text-align: center;">(P8/2, P5) [6]

</td>

<td style="text-align: center;">half-8ve hexatonic

</td>

<td>

</td>

</tr>

<tr>

<td style="text-align: center;">3L 3s

</td>

<td style="text-align: center;">(P8/3, P5) [6]

</td>

<td style="text-align: center;">third-8ve hexatonic

</td>

<td>

</td>

</tr>

<tr>

<td style="text-align: center;">4L 2s

</td>

<td style="text-align: center;">(P8/2, P4/2) [6]

</td>

<td style="text-align: center;">half-everything hexatonic

</td>

<td>

</td>

</tr>

<tr>

<td style="text-align: center;">5L 1s

</td>

<td style="text-align: center;">(P8, P5/3) [6]

</td>

<td style="text-align: center;">third-5th hexatonic

</td>

<td>

</td>

</tr>

<tr>

<th colspan="2">heptatonic MOS scales

</th>

<th>

</th>

<th>

</th>

</tr>

<tr>

</td>

</td>

</td>

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</table>

The third main application, which the rest of this article will focus on, is that pergens allow a systematic exploration of notations for regular temperaments, without having to examine each of the thousands of individual temperaments. The discussion focusses on rank-2 temperaments that include primes 2 and 3.

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<h2 id="toc16"><a name="Further Discussion-Misc notes"></a>Misc notes</h2>

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 <a href="http://xenharmonic.wikispaces.com/pergen+names">http://xenharmonic.wikispaces.com/pergen+names</a>

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 <a href="http://xenharmonic.wikispaces.com/pergen+names">http://xenharmonic.wikispaces.com/pergen+names</a>

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]:

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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 09:16:10 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-12 18:32:44 UTC</tt>.<br>
-: The original revision id was <tt>624782391</tt>.<br>
+: The original revision id was <tt>624808115</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>
@@ Line 117: / Line 117: @@
 Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first porcupine is triple yellow, and the second one is triple yellow and red. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a JI-agnostic manner, and the color name indicates the harmonic possibilities: the prime subgroup, and what types of chord progressions it supports. Both names indicate the rank, the pergen name more directly.
-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, the MOS scales of any two temperaments that have the same pergen tend to be the same.
+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.
+||||~ pentatonic MOS scales ||~   ||~ secondary examples ||
+||= 1L 4s ||= (P8, P5/3) [5] ||= third-5th pentatonic || third-4th, quarter-4th, quarter-5th ||
+||= 2L 3s ||= (P8, P5) [5] ||= unsplit pentatonic || third-11th ||
+||= 3L 2s ||= (P8, P12/5) [5] ||= quarter-12th pentatonic || quarter-11th ||
+||= 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 ||
+||= 2L 4s ||= (P8/2, P5) [6] ||= half-8ve hexatonic ||   ||
+||= 3L 3s ||= (P8/3, P5) [6] ||= third-8ve hexatonic ||   ||
+||= 4L 2s ||= (P8/2, P4/2) [6] ||= half-everything hexatonic ||   ||
+||= 5L 1s ||= (P8, P5/3) [6] ||= third-5th hexatonic ||   ||
+||||~ heptatonic MOS scales ||~   ||~   ||
+||=   ||=   ||=   ||   ||
+||=   ||=   ||=   ||   ||
+||=   ||=   ||=   ||   ||
+||=   ||=   ||=   ||   ||
+||=   ||=   ||=   ||   ||
+||=   ||=   ||=   ||   ||
+||=   ||=   ||=   ||   ||
+||=   ||=   ||=   ||   ||
+||=   ||=   ||=   ||   ||
+||=   ||=   ||=   ||   ||
+||=   ||=   ||=   ||   ||
+||=   ||=   ||=   ||   ||
+||=   ||=   ||=   ||   ||
+||=   ||=   ||=   ||   ||
+||=   ||=   ||=   ||   ||
+||=   ||=   ||=   ||   ||
+||=   ||=   ||=   ||   ||
+||=   ||=   ||=   ||   ||
 The third main application, which the rest of this article will focus on, is that pergens allow a systematic exploration of notations for regular temperaments, without having to examine each of the thousands of individual temperaments. The discussion focusses on rank-2 temperaments that include primes 2 and 3.
@@ Line 1,088: / Line 1,120: @@
 Pergens group many temperaments into one category, which has its advantages and its disadvantages. Some temperament names also do this, for example porcupine refers to not only 2.3.5 with 250/243, but also 2.3.5.7 with 250/243 and 64/63. Color names are the only type of name that never does this. The first porcupine is triple yellow, and the second one is triple yellow and red. Together, the pergen name and the color name supply a lot of information. The pergen name indicates the melodic possibilities in a JI-agnostic manner, and the color name indicates the harmonic possibilities: the prime subgroup, and what types of chord progressions it supports. Both names indicate the rank, the pergen name more directly.&lt;br /&gt;
 &lt;br /&gt;
-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, the MOS scales of any two temperaments that have the same pergen tend to be the same.&lt;br /&gt;
+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.&lt;br /&gt;
+&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;th colspan="2"&gt;pentatonic MOS scales&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;secondary examples&lt;br /&gt;
+&lt;/th&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;1L 4s&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, P5/3) [5]&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;third-5th pentatonic&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;third-4th, quarter-4th, quarter-5th&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;2L 3s&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, P5) [5]&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;unsplit pentatonic&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;third-11th&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;3L 2s&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, P12/5) [5]&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;quarter-12th pentatonic&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;quarter-11th&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;4L 1s&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, P4/2) [5]&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;half-4th pentatonic&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;th colspan="2"&gt;hexatonic MOS scales&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;&lt;br /&gt;
+&lt;/th&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;1L 5s&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, P4/3) [6]&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;third-4th hexatonic&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;quarter-4th, quarter-5th, fifth-4th, fifth-5th&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;2L 4s&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/2, P5) [6]&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;half-8ve hexatonic&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;3L 3s&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/3, P5) [6]&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;third-8ve hexatonic&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;4L 2s&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8/2, P4/2) [6]&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;half-everything hexatonic&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;5L 1s&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;(P8, P5/3) [6]&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;third-5th hexatonic&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;th colspan="2"&gt;heptatonic MOS scales&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;&lt;br /&gt;
+&lt;/th&gt;
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+&lt;br /&gt;
 &lt;br /&gt;
 The third main application, which the rest of this article will focus on, is that pergens allow a systematic exploration of notations for regular temperaments, without having to examine each of the thousands of individual temperaments. The discussion focusses on rank-2 temperaments that include primes 2 and 3.&lt;br /&gt;
@@ Line 2,490: / Line 2,823: @@
 &lt;!-- ws:start:WikiTextHeadingRule:62:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc16"&gt;&lt;a name="Further Discussion-Misc notes"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:62 --&gt;Misc notes&lt;/h2&gt;
   &lt;br /&gt;
-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 &lt;!-- ws:start:WikiTextUrlRule:3103:http://xenharmonic.wikispaces.com/pergen+names --&gt;&lt;a href="http://xenharmonic.wikispaces.com/pergen+names"&gt;http://xenharmonic.wikispaces.com/pergen+names&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:3103 --&gt;&lt;br /&gt;
+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 &lt;!-- ws:start:WikiTextUrlRule:3518:http://xenharmonic.wikispaces.com/pergen+names --&gt;&lt;a href="http://xenharmonic.wikispaces.com/pergen+names"&gt;http://xenharmonic.wikispaces.com/pergen+names&lt;/a&gt;&lt;!-- ws:end:WikiTextUrlRule:3518 --&gt;&lt;br /&gt;
 &lt;br /&gt;
 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]:&lt;br /&gt;