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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Note that this is the fundamental definition of what it means for something to be "a vector"; vectors are defined as objects in spaces where these axioms apply.

~~Additionally, the axioms of linear algebra contain the axioms of group theory, so that the just intervals under stacking can be considered a group.~~

Note that what we've described as multiplication is actually vector addition, and what we've described as exponentiation is actually multiplication of a vector (the interval) by a scalar (the exponent). Additionally, the unison is actually a zero vector. This makes sense if we think of intervals logarithmically, where multiplication of ratios becomes addition of [[cent]] values, the unison is 0 cents, and exponents become scale factors.

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 {{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 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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 Note that this is the fundamental definition of what it means for something to be "a vector"; vectors are defined as objects in spaces where these axioms apply.
-Additionally, the axioms of linear algebra contain the axioms of group theory, so that the just intervals under stacking can be considered a group.
 Note that what we've described as multiplication is actually vector addition, and what we've described as exponentiation is actually multiplication of a vector (the interval) by a scalar (the exponent). Additionally, the unison is actually a zero vector. This makes sense if we think of intervals logarithmically, where multiplication of ratios becomes addition of [[cent]] values, the unison is 0 cents, and exponents become scale factors.