Rank-3 temperament: Difference between revisions

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A '''rank three temperament''' is a [[regular temperament]] with three generators. If one of the generators can be an octave, it is called a '''planar temperament''', though the word is sometimes applied to any rank three temperament. ~~The~~ octave classes of notes of a planar temperament can be embedded in a plane as a [[Wikipedia:Lattice|lattice]], ~~hence~~ the ~~name~~. The most elegant way to put a Euclidean metric, and hence a lattice structure, on the pitch classes of a planar temperament is to orthogonally project onto the subspace perpendicular to the space determined by 2 and the commas of the temperament. To do this we need a Euclidean metric on the space in which p-limit intervals reside as a lattice, and the most expeditious and theoretically justifiable choice of such a metric seems to be [[Euclidean interval space]].

A '''rank three temperament''' is a [[regular temperament]] with three generators. If one of the generators can be an octave, it is called a '''planar temperament''', though the word is sometimes applied to any rank three 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 [[Wikipedia:Lattice|lattice]]; and second, the set of all possible tunings of such a temperament is represented by a plane in a [[projective tuning space]] of 3 or higher dimensions. The most elegant way to put a Euclidean metric, and hence a lattice structure, on the pitch classes of a planar temperament is to orthogonally project onto the subspace perpendicular to the space determined by 2 and the commas of the temperament. To do this we need a Euclidean metric on the space in which p-limit intervals reside as a lattice, and the most expeditious and theoretically justifiable choice of such a metric seems to be [[Euclidean interval space]].

== Example ==

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	A '''rank three temperament''' is a [[regular temperament]] with three generators. If one of the generators can be an octave, it is called a '''planar temperament''', though the word is sometimes applied to any rank three temperament. ~~The~~ octave classes of notes of a planar temperament can be embedded in a plane as a [[Wikipedia:Lattice\|lattice]], ~~hence~~ the ~~name~~. The most elegant way to put a Euclidean metric, and hence a lattice structure, on the pitch classes of a planar temperament is to orthogonally project onto the subspace perpendicular to the space determined by 2 and the commas of the temperament. To do this we need a Euclidean metric on the space in which p-limit intervals reside as a lattice, and the most expeditious and theoretically justifiable choice of such a metric seems to be [[Euclidean interval space]].		A '''rank three temperament''' is a [[regular temperament]] with three generators. If one of the generators can be an octave, it is called a '''planar temperament''', though the word is sometimes applied to any rank three 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 [[Wikipedia:Lattice\|lattice]]; and second, the set of all possible tunings of such a temperament is represented by a plane in a [[projective tuning space]] of 3 or higher dimensions. The most elegant way to put a Euclidean metric, and hence a lattice structure, on the pitch classes of a planar temperament is to orthogonally project onto the subspace perpendicular to the space determined by 2 and the commas of the temperament. To do this we need a Euclidean metric on the space in which p-limit intervals reside as a lattice, and the most expeditious and theoretically justifiable choice of such a metric seems to be [[Euclidean interval space]].

	== Example ==		== Example ==