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| Each temperament has two names: a traditional name and a [[color notation|color name]]. The traditional names are diverse in [[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]]. | | Each temperament has two names: a traditional name and a [[color notation|color name]]. The traditional names are diverse in [[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]]. |
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| Yet another recent development is the concept of a [[pergen]], appearing in our [[tour of regular temperaments]] 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 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 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 in our [[tour of regular temperaments]] as (P8, P5/2) or somesuch, which classifies temperaments by their period and the generator(s), giving ideas of how to notate these temperaments. For rank-2 temperaments we have developed a similar classification system called [[ploidacot]]. |
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| 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.
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| == Further reading == | | == Further reading == |