Tour of regular temperaments: Difference between revisions
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Yet another recent development is the concept of a [[pergen]], appearing here as (P8, P5/2) or somesuch. Every rank-2, rank-3, rank-4, etc. temperament has a pergen, which specifies the period and the generator. Assuming the prime subgroup includes both 2 and 3, the period is either an octave or some fraction of it, and the generator is either some 3-limit interval or some fraction of one. 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 don't uniquely identify them. "W" in a pergen means "widened by one 8ve", e.g. WWP5 is a 5th plus two 8ves, or 6/1. | Yet another recent development is the concept of a [[pergen]], appearing here as (P8, P5/2) or somesuch. Every rank-2, rank-3, rank-4, etc. temperament has a pergen, which specifies the period and the generator. Assuming the prime subgroup includes both 2 and 3, the period is either an octave or some fraction of it, and the generator is either some 3-limit interval or some fraction of one. 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 don't uniquely identify them. "W" in a pergen means "widened by one 8ve", e.g. WWP5 is a 5th plus two 8ves, or 6/1. | ||
Each temperament has two names: a traditional name and a [[Color notation|color name]]. The traditional names are [[Temperament Names|arbitrary | Each temperament has two names: a traditional name and a [[Color notation|color name]]. The traditional names are [[Temperament Names|arbitrary]], but the color names are systematic and rigorous, and the comma can be deduced from the color name. Wa = 3-limit, yo = 5-over, gu = 5-under, zo = 7-over, and ru = 7-under. See also [[Color notation/Temperament Names|Color Notation/Temperament Names]]. | ||
=Equal temperaments (Rank-1 temperaments)= | =Equal temperaments (Rank-1 temperaments)= | ||
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===[[Apotome family|Apotome or Lawa family]] (P8/7, ^1)=== | ===[[Apotome family|Apotome or Lawa family]] (P8/7, ^1)=== | ||
This family tempers out the apotome, [-11 7 0> = 2187/2048, which is a 3-limit comma. | This family tempers out the apotome, [-11 7 0> = 2187/2048, which is a 3-limit comma. | ||
===[[Blackwood|Blackwood or Sawa family]] (P8/5, ^1)=== | |||
This family tempers out the limma, [8 -5 0> = 256/243, which is a 3-limit comma. | |||
===[[Gammic family|Gammic or Laquinquadyo family]] (P8, P5/20)=== | ===[[Gammic family|Gammic or Laquinquadyo family]] (P8, P5/20)=== | ||