Linear algebra formalism - Revision history

VectorGraphics: VectorGraphics moved page Mathematical formalism to Linear algebra formalism over redirect

2025-06-29T20:45:54Z

VectorGraphics moved page Mathematical formalism to Linear algebra formalism over redirect

← Older revision	Revision as of 20:45, 29 June 2025
(No difference)

VectorGraphics at 20:45, 29 June 2025

2025-06-29T20:45:00Z

← Older revision		Revision as of 20:45, 29 June 2025
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	{{Todo\|complete section\|inline=1}}		{{Todo\|complete section\|inline=1}}

	~~== Wedgies and multivals ==~~		*
	{{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>.


	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~~11-87) (516-127) (816-1211)]], 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]]
	* [[Hodge dual]]
	* [[Dave Keenan & Douglas Blumeyer's guide to EA for RTT]]
	* [[Interior product]]
	* [[Recoverability]]
	* [[User:Mike Battaglia/Exterior Norm Conjecture Table]]<!-- main article -->

VectorGraphics: Added exterior algebra page

2025-06-28T23:39:40Z

Added exterior algebra page

@@ Line 136: / Line 136: @@
 {{Todo|complete section|inline=1}}
-== Exterior algebra ==
+== Wedgies and multivals ==
 === Wedge product ===
@@ Line 146: / Line 150: @@
 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.

VectorGraphics: VectorGraphics moved page Linear algebra formalism to Mathematical formalism

2025-06-28T23:29:46Z

VectorGraphics moved page Linear algebra formalism to Mathematical formalism

← Older revision	Revision as of 23:29, 28 June 2025
(No difference)

VectorGraphics at 21:12, 28 June 2025

2025-06-28T21:12:39Z

← Older revision		Revision as of 21:12, 28 June 2025
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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.		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.		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.

VectorGraphics at 21:08, 28 June 2025

2025-06-28T21:08:09Z

← Older revision		Revision as of 21:08, 28 June 2025
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	=== Wedge product ===		=== Wedge product ===
	The wedge product is the n-dimensional generalization of the cross product. It produces not a vector, but a structure 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 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 ⟨⟨(511-87) (516-127) (816-1211)]], 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 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 ⟨⟨(511-87) (516-127) (816-1211)]], 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.		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.

VectorGraphics at 20:56, 28 June 2025

2025-06-28T20:56:09Z

← Older revision		Revision as of 20:56, 28 June 2025
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	== Vals and tuning maps ==		== Vals and tuning maps ==
	{{Todo\|complete section\|inline=1}}		{{Todo\|complete section\|inline=1}}

			== Exterior algebra ==

			=== Wedge product ===
			The wedge product is the n-dimensional generalization of the cross product. It produces not a vector, but a structure 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 ⟨⟨(511-87) (516-127) (816-1211)]], 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.

Fredg999: MOS:PIPESTYLE

2025-06-27T01:30:25Z

MOS:PIPESTYLE

← Older revision		Revision as of 01:30, 27 June 2025
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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 ~~intervals~~]]) 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 vector space. This can be determined by checking that intervals follow the axioms of linear algebra:

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

Lériendil at 17:07, 26 June 2025

2025-06-26T17:07:50Z

← Older revision		Revision as of 17:07, 26 June 2025
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	Aspects of tuning theory are often described in the language of '''linear algebra.''' This is because the space of just intervals (and as it turns out, the space of radical intervals) 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 intervals]]) constitutes a vector space. This can be determined by checking that intervals follow the axioms of linear algebra:

	* Because stacking corresponds to multiplication of rational numbers:		* 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).		** Stacking intervals is associative. For example, ([[3/2]] * [[5/4]]) * [[2/1]] is the same as 3/2 * (5/4 * 2/1).
	** Stacking intervals is commutative. For example, 3/2 * 5/4 is the same as 5/4 * 3/2.		** Stacking intervals is commutative. For example, 3/2 * 5/4 is the same as 5/4 * 3/2.
	** The unison, 1/1, is the identity for stacking, as for an interval ''v'', 1/1 * ''v'' = ''v'' itself.		** The unison, [[1/1]], is the identity for stacking, as for an interval ''v'', 1/1 * ''v'' = ''v'' itself.
	** Every interval has a descending counterpart, which when stacked with that interval produces the unison (1/1). For example, 5/4 * 4/5 = 1/1.		** Every interval has a descending counterpart, which when stacked with that interval produces the unison (1/1). For example, 5/4 * 4/5 = 1/1.
	* Because exponentiation by integers is well-defined for rational numbers:		* Because exponentiation by integers is well-defined for rational numbers:
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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.		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.

	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.		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.

	Conventionally, vectors are notated as lists of numbers representing coordinates in space. In linear algebra, these coordinates are interpreted as scale factors on several arbitrarily chosen "basis vectors" representing the space, which are scaled and added together to produce the vector in question. The standard and most intuitive way of notating intervals as vectors is to take the entries of that interval's monzo and interpret them as vector coordinates, where each basis vector represents a different generator in the monzo's subgroup (usually a prime ~~number~~). Because the entries of a monzo represent exponents on a prime, the intuition of stacking as addition carries over. For example, the interval 5/4, which has a monzo of 2.3.5 [-2 0 1], may be interpreted as the vector [-2 0 1⟩, where we use an [[Extended bra-ket notation\|angle bracket ⟩]] on the right to indicate that it is a vector. This notation is called the ''generator-count vector'', and may in this case be called the ''prime-count vector'', since the generators are the primes. However, it is also common to call the vector itself a monzo (and hence, terms like "eigenmonzo"). {{Todo\|complete intro\|inline=1}}		Conventionally, vectors are notated as lists of numbers representing coordinates in space. In linear algebra, these coordinates are interpreted as scale factors on several arbitrarily chosen "basis vectors" representing the space, which are scaled and added together to produce the vector in question. The standard and most intuitive way of notating intervals as vectors is to take the entries of that interval's [[monzo]] and interpret them as vector coordinates, where each basis vector represents a different generator in the monzo's [[subgroup]] (usually a [[prime harmonic]]). Because the entries of a monzo represent exponents on a prime, the intuition of stacking as addition carries over. For example, the interval 5/4, which has a monzo of 2.3.5 [-2 0 1], may be interpreted as the vector [-2 0 1⟩, where we use an [[Extended bra-ket notation\|angle bracket ⟩]] on the right to indicate that it is a vector. This notation is called the ''generator-count vector'', and may in this case be called the ''prime-count vector'', since the generators are the primes. However, it is also common to call the vector itself a monzo (and hence, terms like "[[eigenmonzo]]"). {{Todo\|complete intro\|inline=1}}

	== Mappings and matrices ==		== Mappings and matrices ==

VectorGraphics at 01:04, 26 June 2025

2025-06-26T01:04:31Z

@@ Line 1: / Line 1: @@
-Aspects of tuning theory are often described in the language of '''linear algebra.'''
+Aspects of tuning theory are often described in the language of '''linear algebra.''' This is because the space of just intervals (and as it turns out, the space of radical intervals) constitutes a vector space. This can be determined by checking that intervals follow the axioms of linear algebra:
-{{Todo|complete intro|inline=1}}
+* Because stacking corresponds to multiplication of rational numbers:
-== Monzos and vectors ==
+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.
-{{Todo|complete section|inline=1}}A vector is a list of numbers, written like so: <math> \begin{pmatrix}
--2\\
+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.
-== Vals and covectors ==
+Conventionally, vectors are notated as lists of numbers representing coordinates in space. In linear algebra, these coordinates are interpreted as scale factors on several arbitrarily chosen "basis vectors" representing the space, which are scaled and added together to produce the vector in question. The standard and most intuitive way of notating intervals as vectors is to take the entries of that interval's monzo and interpret them as vector coordinates, where each basis vector represents a different generator in the monzo's subgroup (usually a prime number). Because the entries of a monzo represent exponents on a prime, the intuition of stacking as addition carries over. For example, the interval 5/4, which has a monzo of 2.3.5 [-2 0 1], may be interpreted as the vector [-2 0 1⟩, where we use an [[Extended bra-ket notation|angle bracket ⟩]] on the right to indicate that it is a vector. This notation is called the ''generator-count vector'', and may in this case be called the ''prime-count vector'', since the generators are the primes. However, it is also common to call the vector itself a monzo (and hence, terms like "eigenmonzo"). {{Todo|complete intro|inline=1}}
 == Mappings and matrices ==
-{{Todo|complete section|inline=1}}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 23: / Line 29: @@
 </math>
-This matrix can be thought of as a "function" that you apply to a vector to get out another vector. This matrix has 3 columns, meaning the vector it takes as an "input" will have 3 elements, and it has 2 rows, meaning the vector you get out will have 2 elements. So, this is a "function" down from 3-dimensional space to 2-dimensional space.
+This matrix can be thought of as a "function" that you apply to a vector (in this case, a generator-count vector) to get out another vector (here, another generator-count vector in the tempered space). This matrix has 3 columns, meaning the vector it takes as an "input" will have 3 elements (representing a rank-3 subgroup of JI), and it has 2 rows, meaning the vector you get out will have 2 elements (representing a rank-2 temperament). So, this is a "function" down from 3-dimensional space to 2-dimensional space. The columns tell us where to find the generators of our original subgroup in the space of tempered intervals. One of the key properties of linear algebra is that knowing that alone allows us to tell where every other interval ends up, and is why regular temperaments are covered so much in xenharmonic theory as opposed to irregular temperaments.
 == Matrix operations ==
@@ Line 123: / Line 131: @@
 {{Todo|complete section|inline=1}}
-=== Practical usage ===
+== Vals and tuning maps ==
-Let's say we want to determine the tuning of 6/5 in quarter-comma meantone.
+{{Todo|complete section|inline=1}}