Rank-3 temperament: Difference between revisions
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A '''rank-3 temperament''' is a [[regular temperament]] with three [[generator]]s. If one of the generators can be an [[2/1|octave]], it is called a '''planar temperament''', though the word is sometimes applied to any rank-3 temperament. There are two interpretations for the name ''planar temperament'': first, the octave classes of notes of a planar temperament can be embedded in a plane as a [[lattice]]; and second, the set of all possible tunings of such a temperament is represented by a plane in a [[projective tuning space]] of three or more dimensions. | A '''rank-3 temperament''' is a [[regular temperament]] with three [[generator]]s. If one of the generators can be an [[2/1|octave]], it is called a '''planar temperament''', though the word is sometimes applied to any rank-3 temperament. There are two interpretations for the name ''planar temperament'': first, the octave classes of notes of a planar temperament can be embedded in a plane as a [[lattice]]; and second, the set of all possible tunings of such a temperament is represented by a plane in a [[projective tuning space]] of three or more dimensions. | ||
See [[Tour of regular temperaments #Rank-3 temperaments]] for a list of families, clans, and collections of rank-3 temperaments. | |||
== Euclidean metric on the lattice == | == Euclidean metric on the lattice == | ||
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If we list 2 first in the list of commas, the matrix ''P'' for any planar temperament will always have a first row and first column with coefficients of 0. We may also change coordinates for ''P'', by monzo-weighting the columns of ''P'', which is to say, scalar multiplying the successive rows by log<sub>2</sub>(''q'') for each of the primes ''q'' up to ''p'', which allows us to project unweighted monzos without first transforming coordinates. | If we list 2 first in the list of commas, the matrix ''P'' for any planar temperament will always have a first row and first column with coefficients of 0. We may also change coordinates for ''P'', by monzo-weighting the columns of ''P'', which is to say, scalar multiplying the successive rows by log<sub>2</sub>(''q'') for each of the primes ''q'' up to ''p'', which allows us to project unweighted monzos without first transforming coordinates. | ||
== | == See also == | ||
== | * [[:Category: Rank-3 temperaments]] | ||
* [[ | |||
== External links == | == External links == | ||