Rank and codimension: Difference between revisions

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The '''rank''' of a [[Regular_Temperaments|regular temperament]] is simply its dimension. ~~Mathematically~~, ~~it is the number of independent intervals~~, ~~called ''generators''~~, ~~which can be combined together to obtain any interval of~~ the ~~temperament~~. For instance, every interval of [[Meantone|meantone]] can be obtained as a combination of a certain number of octaves up or down, plus a certain number of flattened meantone fifths up or down. The terminology originally comes from group theory and linear algebra, although we are using the term "co-rank" slightly differently here.

The '''rank''' of a [[Regular_Temperaments|regular temperament]] is simply its dimension. For example, [[edo]]s are rank-1 because their pitches can be arranged in a 1-dimensional way, and MOSes and temperaments based on them are rank-2, because the two dimensions are the [[period]] and the [[generator]]. For instance, every interval of [[Meantone|meantone]] can be obtained as a combination of a certain number of octaves (the period) up or down, plus a certain number of flattened meantone fifths (the generator) up or down.

== Mathematical description ==

Mathematically, it is the number of independent intervals, called ''generators'', which can be combined together to obtain any interval of the temperament. The terminology originally comes from group theory and linear algebra, although we are using the term "co-rank" slightly differently here.

In the parlance of group theory, the intervals of a regular temperament comprise a [http://en.wikipedia.org/wiki/Free_abelian_group#Rank finitely generated free abelian group] with a rank equal to the number of generators. In the parlance of linear algebra, the rank of the temperament is also the rank of any mapping matrix defining the temperament.

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-The '''rank''' of a [[Regular_Temperaments|regular temperament]] is simply its dimension. Mathematically, it is the number of independent intervals, called ''generators'', which can be combined together to obtain any interval of the temperament. For instance, every interval of [[Meantone|meantone]] can be obtained as a combination of a certain number of octaves up or down, plus a certain number of flattened meantone fifths up or down. The terminology originally comes from group theory and linear algebra, although we are using the term "co-rank" slightly differently here.
+The '''rank''' of a [[Regular_Temperaments|regular temperament]] is simply its dimension. For example, [[edo]]s are rank-1 because their pitches can be arranged in a 1-dimensional way, and MOSes and temperaments based on them are rank-2, because the two dimensions are the [[period]] and the [[generator]]. For instance, every interval of [[Meantone|meantone]] can be obtained as a combination of a certain number of octaves (the period) up or down, plus a certain number of flattened meantone fifths (the generator) up or down.
+== Mathematical description ==
+Mathematically, it is the number of independent intervals, called ''generators'', which can be combined together to obtain any interval of the temperament. The terminology originally comes from group theory and linear algebra, although we are using the term "co-rank" slightly differently here.
 In the parlance of group theory, the intervals of a regular temperament comprise a [http://en.wikipedia.org/wiki/Free_abelian_group#Rank finitely generated free abelian group] with a rank equal to the number of generators. In the parlance of linear algebra, the rank of the temperament is also the rank of any mapping matrix defining the temperament.