Kite's thoughts on pergens: Difference between revisions

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~~Overwhelmed? See~~

A '''pergen''' (pronounced "peer-gen") is a way of classifying a [[regular temperament]] solely by its [[periods and generators|period and generator(s)]]. For any temperament, there are many possible periods and generators. For the pergen, they are chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible. Every rank-2, rank-3, rank-4, etc. temperament has a pergen. Assuming the prime [[subgroup]] includes both 2 and 3, a rank-2 temperament's period is either an octave or some fraction of it, and its generator is either a fifth or some fraction of some 3-limit interval. Since both period and generator are conventional musical intervals or some fractions of them, the pergen gives great insight into notating a temperament. Several temperaments may share the same pergen, in fact, every [[strong extension]] of a temperament has the same pergen as the original temperament. Thus pergens classify temperaments but do not uniquely identify them. "c" in a pergen means compound (widened by one octave), e.g. ccP5 is a 5th plus two 8ves, or 6/1.

[~~http://tallkite~~.~~com/misc_files/notation%20guide%20for%20rank~~-2~~%20pergens~~.~~pdf '''TallKite~~.~~com/misc_files/notation guide for~~ rank-2 pergens.~~pdf'''] for practical notation examples~~.

~~''See~~ also~~: [[Rank~~-2 temperaments by ~~mapping~~ of 3]]''

Pergens also provide a way to name precise tunings of any rank-2 temperament. Meantone tunings are named third-comma, quarter-comma, two-fifths-comma, etc. for the fraction of an 81/80 comma that the 5th is flattened by. (The octave is assumed to be just.) This can be generalized to all temperaments. For example, fifth-comma [[Porcupine|Porcupine aka Triyo]] has the 5th sharpened by one-fifth of [[250/243]] ({{monzo| 1 -5 3 }}). Sharpened not flattened because the comma is fourthwards not fifthwards, i.e. it has prime 3 in the denominator not the numerator. Given the comma fraction, the generator's exact size can be deduced from the pergen. Here the pergen is (P8, P4/3). Because the 5th is sharpened, the 4th is flattened. Because the generator is 1/3 of a 4th, the generator is flattened by 1/3 of 1/5 of a comma, or 1/15 comma. If the temperament's comma doesn't contain prime 3, the next larger prime is used. For example, Augmented aka Trigu tempers out 128/125. The third-comma tuning sharpens 5/4 by just enough to equate it to a third of an 8ve. If a temperament has multiple commas, the comma fraction refers to the first comma in the color name.

~~__FORCETOC__~~

Overwhelmed? See [http://tallkite.com/misc_files/notation%20guide%20for%20rank-2%20pergens.pdf ''Notation guide for rank-2 pergens''] for practical notation examples.

~~==Definition==~~

~~A '''pergen''' (pronounced "peer~~-~~gen") is a way~~ of classifying a regular temperament solely by its period and generator(s). For any temperament, there are many possible periods and generators. For the pergen, they are chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible.

= Definition =

If a rank-2 temperament uses the primes 2 and 3 in its comma(s), or in its prime subgroup (i.e. doesn't explicitly exclude the octave or the fifth), then the period can be expressed as the octave 2/1, or some fraction of an octave. Furthermore, the generator can usually be expressed as some 3-limit interval, or some fraction of such an interval. Both fractions are always of the form 1/N, thus the octave and/or the 3-limit interval is '''split''' into N parts. The interval which is split into multiple generators is the '''multigen'''. The 3-limit multigen is referred to not by its ratio but by its conventional name, e.g. P5, M6, m7, etc.

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-Overwhelmed? See
+A '''pergen''' (pronounced "peer-gen") is a way of classifying a [[regular temperament]] solely by its [[periods and generators|period and generator(s)]]. For any temperament, there are many possible periods and generators. For the pergen, they are chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible. Every rank-2, rank-3, rank-4, etc. temperament has a pergen. Assuming the prime [[subgroup]] includes both 2 and 3, a rank-2 temperament's period is either an octave or some fraction of it, and its generator is either a fifth or some fraction of some 3-limit interval. Since both period and generator are conventional musical intervals or some fractions of them, the pergen gives great insight into notating a temperament. Several temperaments may share the same pergen, in fact, every [[strong extension]] of a temperament has the same pergen as the original temperament. Thus pergens classify temperaments but do not uniquely identify them. "c" in a pergen means compound (widened by one octave), e.g. ccP5 is a 5th plus two 8ves, or 6/1.
-[http://tallkite.com/misc_files/notation%20guide%20for%20rank-2%20pergens.pdf '''TallKite.com/misc_files/notation guide for rank-2 pergens.pdf'''] for practical notation examples.
-''See also: [[Rank-2 temperaments by mapping of 3]]''
+Pergens also provide a way to name precise tunings of any rank-2 temperament. Meantone tunings are named third-comma, quarter-comma, two-fifths-comma, etc. for the fraction of an 81/80 comma that the 5th is flattened by. (The octave is assumed to be just.) This can be generalized to all temperaments. For example, fifth-comma [[Porcupine|Porcupine aka Triyo]] has the 5th sharpened by one-fifth of [[250/243]] ({{monzo| 1 -5 3 }}). Sharpened not flattened because the comma is fourthwards not fifthwards, i.e. it has prime 3 in the denominator not the numerator. Given the comma fraction, the generator's exact size can be deduced from the pergen. Here the pergen is (P8, P4/3). Because the 5th is sharpened, the 4th is flattened. Because the generator is 1/3 of a 4th, the generator is flattened by 1/3 of 1/5 of a comma, or 1/15 comma. If the temperament's comma doesn't contain prime 3, the next larger prime is used. For example, Augmented aka Trigu tempers out 128/125. The third-comma tuning sharpens 5/4 by just enough to equate it to a third of an 8ve. If a temperament has multiple commas, the comma fraction refers to the first comma in the color name.
-__FORCETOC__
+Overwhelmed? See [http://tallkite.com/misc_files/notation%20guide%20for%20rank-2%20pergens.pdf ''Notation guide for rank-2 pergens''] for practical notation examples.
-==Definition==
-A '''pergen''' (pronounced "peer-gen") is a way of classifying a regular temperament solely by its period and generator(s). For any temperament, there are many possible periods and generators. For the pergen, they are chosen to use the fewest, and smallest, prime factors possible. Fractions are allowed, e.g. half-octave, but avoided if possible.
+{{See also| Rank-2 temperaments by mapping of 3 }}
+= Definition =
 If a rank-2 temperament uses the primes 2 and 3 in its comma(s), or in its prime subgroup (i.e. doesn't explicitly exclude the octave or the fifth), then the period can be expressed as the octave 2/1, or some fraction of an octave. Furthermore, the generator can usually be expressed as some 3-limit interval, or some fraction of such an interval. Both fractions are always of the form 1/N, thus the octave and/or the 3-limit interval is '''split''' into N parts. The interval which is split into multiple generators is the '''multigen'''. The 3-limit multigen is referred to not by its ratio but by its conventional name, e.g. P5, M6, m7, etc.