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 rank-3 [[regular temperament]], i.e. a 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]].

== Example ==

Revision as of 04:49, 6 January 2021 view source FloraC (talk \| contribs) autopatrolled, Bureaucrats, Interface administrators, Sysops 18,384 edits Readability improvements; recategory ← Older edit		Revision as of 20:58, 20 March 2021 view source Inthar (talk \| contribs) autopatrolled, emailconfirmed 15,004 edits m statement that a planar temperament is rank 3 Newer edit →
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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 rank-3 [[regular temperament]], i.e. a 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]].

	== Example ==		== Example ==