Regular temperament: Difference between revisions
→Further reading: Somewhat major restructure to draw more attention to the monzo/val/etc pages |
ArrowHead294 (talk | contribs) |
||
| Line 45: | Line 45: | ||
Another recent contribution to the field of temperament is the concept of [[optimization]], which can take many forms. The point of optimization is to minimize the difference between a temperament and JI by finding an optimal tuning for the generator. The two most frequently used forms of optimization are [[POTE tuning|POTE]] ("Pure-Octave Tenney-Euclidean"), [[TOP tuning|TOP]] ("Tenney OPtimal", or "Tempered Octaves, Please") and more recently [[CTE]] ("Constained Tenney-Euclidean"), which has become the new standard instead of POTE since POTE is meant to be an approximation. Optimization is rather intensive mathematically, but it is seldom left as an exercise to the reader; most temperaments are presented here in their optimal forms in terms of POTE and CTE generators. In addition, for each temperament there is a [[Optimal ET sequence|sequence of equal temperaments]] showing possible [[equal-step tuning]]s in the order of better absolute accuracy to JI. The most common browser tools used for finding optimal tunings (useful for investigating new temperaments) are [[Graham Breed]]'s [http://x31eq.com/temper/ Temperament Finder] and [[User:Sintel|sintel]]'s [https://sintel.pythonanywhere.com/ Temperament Calculator]; the former gives temperament names (usually consistent with the wiki) and implements a wide variety of features like finding related temperaments while the latter implements [[CTE]] and more complex types of subgroups (like allowing ratios as generators) and supports an alternative notation to [[warts]] that is more convenient for arbitrary subgroups. | Another recent contribution to the field of temperament is the concept of [[optimization]], which can take many forms. The point of optimization is to minimize the difference between a temperament and JI by finding an optimal tuning for the generator. The two most frequently used forms of optimization are [[POTE tuning|POTE]] ("Pure-Octave Tenney-Euclidean"), [[TOP tuning|TOP]] ("Tenney OPtimal", or "Tempered Octaves, Please") and more recently [[CTE]] ("Constained Tenney-Euclidean"), which has become the new standard instead of POTE since POTE is meant to be an approximation. Optimization is rather intensive mathematically, but it is seldom left as an exercise to the reader; most temperaments are presented here in their optimal forms in terms of POTE and CTE generators. In addition, for each temperament there is a [[Optimal ET sequence|sequence of equal temperaments]] showing possible [[equal-step tuning]]s in the order of better absolute accuracy to JI. The most common browser tools used for finding optimal tunings (useful for investigating new temperaments) are [[Graham Breed]]'s [http://x31eq.com/temper/ Temperament Finder] and [[User:Sintel|sintel]]'s [https://sintel.pythonanywhere.com/ Temperament Calculator]; the former gives temperament names (usually consistent with the wiki) and implements a wide variety of features like finding related temperaments while the latter implements [[CTE]] and more complex types of subgroups (like allowing ratios as generators) and supports an alternative notation to [[warts]] that is more convenient for arbitrary subgroups. | ||
Each temperament has two names: a traditional name and a [[Color notation|color name]]. The traditional names are diverse in terms of [[Temperament names|sources]], whereas the color names are systematic and rigorous, and the comma(s) can be deduced from the color name. Wa = 3-limit | Each temperament has two names: a traditional name and a [[Color notation|color name]]. The traditional names are diverse in terms of [[Temperament names|sources]], whereas the color names are systematic and rigorous, and the comma(s) can be deduced from the color name. {{nowrap|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]]. | ||
Yet another recent development is the concept of a [[pergen]], appearing [[Tour of Regular Temperaments|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(s). 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 5th 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 don't 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. | Yet another recent development is the concept of a [[pergen]], appearing [[Tour of Regular Temperaments|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(s). 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 5th 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 don't 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. | ||