Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 624813333 - Original comment: ** |
Wikispaces>TallKite **Imported revision 624814191 - 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>2018-01-13 | : This revision was by author [[User:TallKite|TallKite]] and made on <tt>2018-01-13 03:54:52 UTC</tt>.<br> | ||
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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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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. | 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. | ||
The pentatonic MOS scales don't include fifth-split pergens such as fifth-4th. This is because a pentatonic genchain has only 4 steps, and can only divide a multigen into quarters. It would be possible to include pergens with a multigen which isn't actually generated. For example, 3L 2s using the sensei generator would be (P8, WWP5/7) [5]. The rationale would be that two sensei generators = 5/3, in effect a (P8, (5/3)/2) pseudo-pergen. | |||
||||||~ Tetratonic MOS scales ||~ secondary examples || | ||||||~ Tetratonic MOS scales ||~ secondary examples || | ||
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==Supplemental materials== | ==Supplemental materials== | ||
This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third- | This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block. | ||
This app lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper. | This app lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper. | ||
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__**Extra stuff:**__ | __**Extra stuff, not sure if it should be included:**__ | ||
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]. | 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]. We can use 64/63 to expand to the 7-limit. For (a,b,c,d) we get [k,s,g,r]: | ||
k = 12a + 19b + 28c + 34d | k = 12a + 19b + 28c + 34d | ||
s = 7a + 11b + 14c + 20d | s = 7a + 11b + 14c + 20d | ||
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c = -g | c = -g | ||
d = -r | d = -r | ||
The LCM of the pergen's two splitting fractions is called the **height** of the pergen. For example, {P8, P5} has height 1, and {P8/2, M2/4} has height 4. In single-pair notation, the enharmonic interval's number of ups or downs is equal to the height. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.</pre></div> | The LCM of the pergen's two splitting fractions is called the **height** of the pergen. For example, {P8, P5} has height 1, and {P8/2, M2/4} has height 4. In single-pair notation, the enharmonic interval's number of ups or downs is equal to the height. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.</pre></div> | ||
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<br /> | <br /> | ||
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.<br /> | 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.<br /> | ||
<br /> | |||
The pentatonic MOS scales don't include fifth-split pergens such as fifth-4th. This is because a pentatonic genchain has only 4 steps, and can only divide a multigen into quarters. It would be possible to include pergens with a multigen which isn't actually generated. For example, 3L 2s using the sensei generator would be (P8, WWP5/7) [5]. The rationale would be that two sensei generators = 5/3, in effect a (P8, (5/3)/2) pseudo-pergen.<br /> | |||
<br /> | <br /> | ||
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<!-- ws:start:WikiTextHeadingRule:71:&lt;h2&gt; --><h2 id="toc15"><a name="Further Discussion-Supplemental materials"></a><!-- ws:end:WikiTextHeadingRule:71 -->Supplemental materials</h2> | <!-- ws:start:WikiTextHeadingRule:71:&lt;h2&gt; --><h2 id="toc15"><a name="Further Discussion-Supplemental materials"></a><!-- ws:end:WikiTextHeadingRule:71 -->Supplemental materials</h2> | ||
<br /> | <br /> | ||
This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third- | This PDF is a rank-2 notation guide that shows the full lattice for the first 15 pergens, up through the third-splits block.<br /> | ||
<br /> | <br /> | ||
This app lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.<br /> | This app lists out thousands of pergens, and suggests periods, generators and enharmonics for each one. It can also list only those pergens supported by a specific edo. Written in Jesusonic, runs inside Reaper.<br /> | ||
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<br /> | <br /> | ||
<br /> | <br /> | ||
<u><strong>Extra stuff:</strong></u><br /> | <u><strong>Extra stuff, not sure if it should be included:</strong></u><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]. | 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]. We can use 64/63 to expand to the 7-limit. For (a,b,c,d) we get [k,s,g,r]:<br /> | ||
k = 12a + 19b + 28c + 34d<br /> | k = 12a + 19b + 28c + 34d<br /> | ||
s = 7a + 11b + 14c + 20d<br /> | s = 7a + 11b + 14c + 20d<br /> | ||
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c = -g<br /> | c = -g<br /> | ||
d = -r<br /> | d = -r<br /> | ||
<br /> | <br /> | ||
The LCM of the pergen's two splitting fractions is called the <strong>height</strong> of the pergen. For example, {P8, P5} has height 1, and {P8/2, M2/4} has height 4. In single-pair notation, the enharmonic interval's number of ups or downs is equal to the height. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.</body></html></pre></div> | The LCM of the pergen's two splitting fractions is called the <strong>height</strong> of the pergen. For example, {P8, P5} has height 1, and {P8/2, M2/4} has height 4. In single-pair notation, the enharmonic interval's number of ups or downs is equal to the height. The minimum number of ups or downs needed to notate the temperament is half the height, rounded down. If the height is 4 or 5, double-ups and double-downs will be needed.</body></html></pre></div> | ||