Kite's thoughts on pergens: Difference between revisions
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==Miscellaneous Notes== | ==Miscellaneous Notes== | ||
=== Combining pergens === | |||
Tempering out 250/243 creates third-4th, and 49/48 creates half-4th, and tempering out both commas creates sixth-4th. Therefore (P8, P4/3) + (P8, P4/2) = (P8, P4/6). If adding a comma to a temperament doesn't change the pergen, it's a strong extension, otherwise it's a weak extension. | Tempering out 250/243 creates third-4th, and 49/48 creates half-4th, and tempering out both commas creates sixth-4th. Therefore (P8, P4/3) + (P8, P4/2) = (P8, P4/6). If adding a comma to a temperament doesn't change the pergen, it's a strong extension, otherwise it's a weak extension. | ||
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If a false double pergen can be broken down into two simpler ones, that may help with finding a double-pair notation with an EU of a 2nd or less. For example, sixth-4th's single pair notation has an EU of a 4th. But since sixth-4th is half-4th plus third-4th, and those two have a good EU, sixth-4th can be notated with one pair from half-4th and another from third-4th. | If a false double pergen can be broken down into two simpler ones, that may help with finding a double-pair notation with an EU of a 2nd or less. For example, sixth-4th's single pair notation has an EU of a 4th. But since sixth-4th is half-4th plus third-4th, and those two have a good EU, sixth-4th can be notated with one pair from half-4th and another from third-4th. | ||
=== Expanding gedras === | |||
Gedras can be expanded to 5-limit or higher by including another keyspan that is compatible with 7 and 12, such as 9 or 16. But a more useful approach is for the third number to 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]: | Gedras can be expanded to 5-limit or higher by including another keyspan that is compatible with 7 and 12, such as 9 or 16. But a more useful approach is for the third number to 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]: | ||
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Gedras can also be expanded by adding an entry for ups/downs. The entry shows the '''upspan''', which is the number of ups the interval has. Downed intervals have a negative upspan. An entry for '''liftspan''' can also be added. For example, vM3 = [4,2,-1], and ^^\4 = [5,3,2,-1]. | Gedras can also be expanded by adding an entry for ups/downs. The entry shows the '''upspan''', which is the number of ups the interval has. Downed intervals have a negative upspan. An entry for '''liftspan''' can also be added. For example, vM3 = [4,2,-1], and ^^\4 = [5,3,2,-1]. | ||
=== Height of a pergen === | |||
The LCM of the pergen's two splitting fractions could be 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 EU's number of ups or downs is equal to the height. The <u>minimum</u> 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. | The LCM of the pergen's two splitting fractions could be 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 EU's number of ups or downs is equal to the height. The <u>minimum</u> 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. | ||
=== Generalizing the pergen === | |||
See [[User:AthiTrydhen/Abstract pergens]] | |||
Pergens were discovered by [[KiteGiedraitis|Kite Giedraitis]] in 2017, and developed with the help of [[ | === Credits === | ||
Pergens were discovered by [[KiteGiedraitis|Kite Giedraitis]] in 2017, and developed with the help of [[Praveen Venkataramana]]. | |||
== Addenda (late 2023) == | == Addenda (late 2023) == | ||