Linear algebra formalism: Difference between revisions

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~~Aspects of tuning theory are often described in the language of '''linear algebra.''' This is because the space of [[just intonation~~|~~just intervals]] (and as it turns out, the space of [[radical interval]]s) constitutes a vector space. This can be determined by checking that intervals follow the axioms of linear~~ algebra:

Aspects of tuning theory are often described in the language of '''linear algebra'''. This is because the space of [[just intonation|just intervals]] (and as it turns out, the space of [[radical interval]]s) constitutes a [[lattice]], technically a {{w|free abelian group}} but with properties very similar to a {{w|vector space}} – for example, they share the property of having a [[basis]]. The main difference is that vector spaces are defined on real numbers whereas free abelian groups are defined on integers; it is thus customary to use linear algebra as a simplification. Besides, various metrics are defined by {{w|embedding}} the lattice in a {{w|normed vector space}}, further necessitating its use.

It can be verified that intervals follow the axioms of linear algebra:

* Because [[stacking]] corresponds to multiplication of rational numbers:

** Stacking intervals is associative. For example, ([[3/2]] * [[5/4]]) * [[2/1]] is the same as 3/2 * (5/4 * 2/1).

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== Mappings and matrices ==

{{Todo|complete section|inline=1}}A temperament mapping, in linear algebra, is represented by a matrix. A matrix is a grid of numbers, written like so:

A temperament mapping, in linear algebra, is represented by a matrix. A matrix is a grid of numbers, written like so:

<math>

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== Vals and tuning maps ==

== See also ==

* [[Matrix echelon forms]]

[[Category:Math]]

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-Aspects of tuning theory are often described in the language of '''linear algebra.''' This is because the space of [[just intonation|just intervals]] (and as it turns out, the space of [[radical interval]]s) constitutes a vector space. This can be determined by checking that intervals follow the axioms of linear algebra:
+{{Wikipedia|Linear algebra}}
+Aspects of tuning theory are often described in the language of '''linear algebra'''. This is because the space of [[just intonation|just intervals]] (and as it turns out, the space of [[radical interval]]s) constitutes a [[lattice]], technically a {{w|free abelian group}} but with properties very similar to a {{w|vector space}} – for example, they share the property of having a [[basis]]. The main difference is that vector spaces are defined on real numbers whereas free abelian groups are defined on integers; it is thus customary to use linear algebra as a simplification. Besides, various metrics are defined by {{w|embedding}} the lattice in a {{w|normed vector space}}, further necessitating its use.
+It can be verified that intervals follow the axioms of linear algebra:
 * Because [[stacking]] corresponds to multiplication of rational numbers:
 ** Stacking intervals is associative. For example, ([[3/2]] * [[5/4]]) * [[2/1]] is the same as 3/2 * (5/4 * 2/1).
@@ Line 20: / Line 23: @@
 == Mappings and matrices ==
-{{Todo|complete section|inline=1}}A temperament mapping, in linear algebra, is represented by a matrix. A matrix is a grid of numbers, written like so:
+{{Todo|complete section|inline=1}}
+A temperament mapping, in linear algebra, is represented by a matrix. A matrix is a grid of numbers, written like so:
 <math>
@@ Line 133: / Line 138: @@
 == Vals and tuning maps ==
 {{Todo|complete section|inline=1}}
+== See also ==
+* [[Matrix echelon forms]]
+[[Category:Math]]

Linear algebra formalism: Difference between revisions

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