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 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:

** 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.

** 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.

* 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 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~~.

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

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}}

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 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 ==

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== Vals and tuning maps ==

*

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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 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:
-** 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.
-** 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.
 * Because exponentiation by integers is well-defined for rational numbers:
@@ Line 15: / Line 15: @@
 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.
+Additionally, the axioms of linear algebra contain the axioms of group theory, so that the just intervals under stacking can be considered a group.
-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}}
+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 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 ==
@@ Line 133: / Line 135: @@
 == Vals and tuning maps ==
 {{Todo|complete section|inline=1}}
+*