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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== 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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== ~~Wedgies and multivals~~ ==

== See also ==

{{Wikipedia|Exterior algebra}}'''Exterior algebra''' is a branch of linear {{w|Algebra over a field|algebra}} which focuses on combining vectors into structures called "multivectors". Such a combination is called the "exterior product" or "wedge product" and is denoted with <math>\wedge</math>.

* [[Matrix echelon forms]]

In many cases, the same things exterior algebra is used for in music theory can be accomplished using normal linear algebra. The matrix approach is usually preferred for pedagogical reasons (more people are familiar with matrices compared to exterior products) and computational reasons, (most common numerical libraries are primarily intended for matrix operations). Still, in some more abstract or advanced applications, the exterior algebra may still be used if it is more natural.

~~=== Wedge product ===~~

The wedge product is the n-dimensional generalization of the cross product. It produces not a vector, but a structure called a ''multivector'' with entries corresponding to combinations of entries in the starting vectors. For two vectors of the same length, [~~a1 a2 a3 ... an] and~~ [~~b1 b2 b3 ... bn~~], we go through every pair of indices ''i, j'' up to ''n'' where ''j'' > ''i,'' and the entry ci,j of the wedge product is aibj - biaj. cj,i is equal to -(ci,j), and ci,i where the two indices are the same is 0.

~~The wedge product is used in regular temperament theory to combine [[vals~~]] into [[Wedgie|multivals]], hence why multivals are called "wedgies". For example, wedging ⟨5 8 12] and ⟨7 11 16] (the patent vals for 5edo and 7edo) yields ⟨⟨(5*11-8*7) (5*16-12*7) (8*16-12*11)]], which simplifies to ⟨⟨(55-56) (80-84) (128-132)]] and thus to ⟨⟨-1 -4 -4]], which is the wedgie for 5 & 7, a.k.a. meantone.

The wedge product can be generalized to combine ''n'' vals together, where instead of every pair of indices, we have every combination of ''n'' indices. This results in wedgies for rank-3 temperaments and beyond.

~~For a higher-dimensional multivector with elements ci, j, k..., the entries at all permutations of the same indices have the same absolute value, and swapping two indices flips the sign.~~

~~=== See also ===~~

* [[~~Wedgie]]~~

[[Category:Math]]

* [[Hodge dual]]

* [[Dave Keenan & Douglas Blumeyer's guide to EA for RTT]]

* [[Interior product]]

* [[Recoverability]]

* [[User:~~Mike Battaglia/Exterior Norm Conjecture Table~~]]~~~~

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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 14: / Line 17: @@
 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.
@@ Line 22: / 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 136: / Line 139: @@
 {{Todo|complete section|inline=1}}
-== Wedgies and multivals ==
+== See also ==
-{{Wikipedia|Exterior algebra}}'''Exterior algebra''' is a branch of linear {{w|Algebra over a field|algebra}} which focuses on combining vectors into structures called "multivectors". Such a combination is called the "exterior product" or "wedge product" and is denoted with <math>\wedge</math>.
+* [[Matrix echelon forms]]
-In many cases, the same things exterior algebra is used for in music theory can be accomplished using normal linear algebra. The matrix approach is usually preferred for pedagogical reasons (more people are familiar with matrices compared to exterior products) and computational reasons, (most common numerical libraries are primarily intended for matrix operations). Still, in some more abstract or advanced applications, the exterior algebra may still be used if it is more natural.
-=== Wedge product ===
-The wedge product is the n-dimensional generalization of the cross product. It produces not a vector, but a structure called a ''multivector'' with entries corresponding to combinations of entries in the starting vectors. For two vectors of the same length, [a<sub>1</sub> a<sub>2</sub> a<sub>3</sub> ... a<sub>n</sub>] and [b<sub>1</sub> b<sub>2</sub> b<sub>3</sub> ... b<sub>n</sub>], we go through every pair of indices ''i, j'' up to ''n'' where ''j'' > ''i,'' and the entry c<sub>i,j</sub> of the wedge product is a<sub>i</sub>b<sub>j</sub> - b<sub>i</sub>a<sub>j</sub>. c<sub>j,i</sub> is equal to -(c<sub>i,j</sub>), and c<sub>i,i</sub> where the two indices are the same is 0.
-The wedge product is used in regular temperament theory to combine [[vals]] into [[Wedgie|multivals]], hence why multivals are called "wedgies". For example, wedging ⟨5 8 12] and ⟨7 11 16] (the patent vals for 5edo and 7edo) yields ⟨⟨(5*11-8*7) (5*16-12*7) (8*16-12*11)]], which simplifies to ⟨⟨(55-56) (80-84) (128-132)]] and thus to ⟨⟨-1 -4 -4]], which is the wedgie for 5 & 7, a.k.a. meantone.
-The wedge product can be generalized to combine ''n'' vals together, where instead of every pair of indices, we have every combination of ''n'' indices. This results in wedgies for rank-3 temperaments and beyond.
-For a higher-dimensional multivector with elements c<sub>i, j, k...</sub>, the entries at all permutations of the same indices have the same absolute value, and swapping two indices flips the sign.
-=== See also ===
-* [[Wedgie]]
+[[Category:Math]]
-* [[Hodge dual]]
-* [[Dave Keenan & Douglas Blumeyer's guide to EA for RTT]]
-* [[Interior product]]
-* [[Recoverability]]
-* [[User:Mike Battaglia/Exterior Norm Conjecture Table]]<!-- main article -->

Linear algebra formalism: Difference between revisions

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