Kite's thoughts on pergens: Difference between revisions
Wikispaces>TallKite **Imported revision 624814715 - Original comment: ** |
Wikispaces>TallKite **Imported revision 624815117 - Original comment: ** |
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=__Applications__= | =__Applications__= | ||
One obvious application is to name regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names. Many temperaments have pergen-like names: Hemififths is (P8, P5/2), semihemi is (P8/2, P4/2), triforce is (P8/3, P4/2), both tetracot and semihemififths are (P8, P5/4), fourfives is (P8/4, P5/5), pental is (P8/5, P5), and fifive is (P8/2, P5/5). Pergen names are an improvement over these because they specify more exactly what is split. Some temperament names are what might be called pseudo- | One obvious application is to name regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names. Many temperaments have pergen-like names: Hemififths is (P8, P5/2), semihemi is (P8/2, P4/2), triforce is (P8/3, P4/2), both tetracot and semihemififths are (P8, P5/4), fourfives is (P8/4, P5/5), pental is (P8/5, P5), and fifive is (P8/2, P5/5). Pergen names are an improvement over these because they specify more exactly what is split. Some temperament names are what might be called a pseudo-pergen, because either it contains more than 2 primes, or because the multigen isn't actually a generator. For example, sensei, or semisixth, implies a pseudo-pergen (P8, (5/3)/2) that contains 3 primes. Meantone (mean = average, tone = major 2nd) implies a pseudo-pergen of (P8, (5/4)/2), but the tone isn't a generator. | ||
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 higher-primes-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 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 higher-primes-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. | ||
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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. | ||
||||||~ Tetratonic MOS scales ||~ secondary examples || | ||||||~ Tetratonic MOS scales ||~ secondary examples || | ||
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||= 9L 1s ||= (P8, P4/2) [10] ||= quarter-4th decatonic ||< || | ||= 9L 1s ||= (P8, P4/2) [10] ||= quarter-4th decatonic ||< || | ||
The pentatonic MOS scales don't include fifth-split pergens. 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. | |||
Nonstandard prime subgroups may provide a better understanding of some MOS scales. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, (5/4)/2), where 5·G = 7/4. | |||
==Combining pergens== | ==Combining pergens== | ||
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<br /> | <br /> | ||
One obvious application is to name regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names. Many temperaments have pergen-like names: Hemififths is (P8, P5/2), semihemi is (P8/2, P4/2), triforce is (P8/3, P4/2), both tetracot and semihemififths are (P8, P5/4), fourfives is (P8/4, P5/5), pental is (P8/5, P5), and fifive is (P8/2, P5/5). Pergen names are an improvement over these because they specify more exactly what is split. Some temperament names are what might be called pseudo- | One obvious application is to name regular temperaments in a logical, consistent manner, avoiding the need to memorize many arbitrary names. Many temperaments have pergen-like names: Hemififths is (P8, P5/2), semihemi is (P8/2, P4/2), triforce is (P8/3, P4/2), both tetracot and semihemififths are (P8, P5/4), fourfives is (P8/4, P5/5), pental is (P8/5, P5), and fifive is (P8/2, P5/5). Pergen names are an improvement over these because they specify more exactly what is split. Some temperament names are what might be called a pseudo-pergen, because either it contains more than 2 primes, or because the multigen isn't actually a generator. For example, sensei, or semisixth, implies a pseudo-pergen (P8, (5/3)/2) that contains 3 primes. Meantone (mean = average, tone = major 2nd) implies a pseudo-pergen of (P8, (5/4)/2), but the tone isn't a generator.<br /> | ||
<br /> | <br /> | ||
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 higher-primes-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.<br /> | 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 higher-primes-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.<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.<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 /> | ||
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The pentatonic MOS scales don't include fifth-split pergens. 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 /> | |||
Nonstandard prime subgroups may provide a better understanding of some MOS scales. For example, 6L 1s can be roulette [7], with a 2.5.7 pergen (P8, (5/4)/2), where 5·G = 7/4.<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:67:&lt;h2&gt; --><h2 id="toc13"><a name="Further Discussion-Combining pergens"></a><!-- ws:end:WikiTextHeadingRule:67 -->Combining pergens</h2> | <!-- ws:start:WikiTextHeadingRule:67:&lt;h2&gt; --><h2 id="toc13"><a name="Further Discussion-Combining pergens"></a><!-- ws:end:WikiTextHeadingRule:67 -->Combining pergens</h2> | ||