Extended bra–ket notation: Difference between revisions

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== Basics ==

The name ''bra–ket notation'' refers to its two basic structures: the '''bra''' and the '''ket'''. A bra is a list which has a pointed bracket on its ''left'' side, and a ket is a list which has a pointed bracket on its ''right'' side. Bras are used for covectors (such as [[~~Val|vals~~]]), and kets are used for vectors (such as [[~~monzos~~]]). In a basic sense, vectors can be thought of as tangible objects (like intervals), and covectors are used for things you do to those objects (like temperaments).

The name ''bra–ket notation'' refers to its two basic structures: the '''bra''' and the '''ket'''. A bra is a list which has a pointed bracket on its ''left'' side, and a ket is a list which has a pointed bracket on its ''right'' side. Bras are used for covectors (such as [[val]]s), and kets are used for vectors (such as [[monzo]]s). In a basic sense, vectors can be thought of as tangible objects (like intervals), and covectors are used for things you do to those objects (like temperaments).

The usage patterns described in this section are the same as the standard usage of bra–ket notation in quantum mechanics.

=== Common applications ===

In RTT, covectors are used most frequently for ~~[[Val|~~vals]] (including [[tuning map]]s). For example, the val for 7-ET is notated as {{bra| 7 11 16 }}, and a ~~(tempered-prime)~~ tuning map for it might be {{bra|1209.682 1900.930 2764.988}}.

In RTT, covectors are used most frequently for vals (including [[tuning map]]s). For example, the val for 7-ET is notated as {{bra| 7 11 16 }}, and a tuning map for it might be {{bra| 1209.682 1900.930 2764.988 }}.

The most common type of vectors used in RTT is the ~~[[Monzos|~~monzo]] (both [[~~Prime~~-count vector~~|prime-count vectors~~]] and [[~~Generator~~-count vector~~|generator-count vectors~~]]). For example, the monzo for 45/32 is {{ket| -5 2 1 }}, and the monzo for ~45/32 in porcupine temperament is {{ket| 2 -11 }} (for the generators ~2/1 and ~11/10).

The most common type of vectors used in RTT is the monzo (both [[prime-count vector]]s and [[generator-count vector]]s). For example, the monzo for 45/32 is {{ket| -5 2 1 }}, and the tempered monzo for ~45/32 in porcupine temperament is {{ket| 2 -11 }} (for the generators ~2/1 and ~11/10).

=== Combining ===

When a bra and a ket are put together, the dot product is taken, and the result is a scalar (a single number). In order to be compatible in this way, the bra and ket must at least have the same number of entries, so that each of their entries will pair up for a product. The dot product corresponds to "applying" the covector to the vector.

When a bra and a ket are put together, the dot product is taken, and the result is a scalar (a single number). In order to be compatible in this way, the bra and ket must at least have the same number of entries, so that each of their entries will pair up for a product. The {{w|dot product}} corresponds to "applying" the covector to the vector.

Mapping 45/32 with 7-ET would be notated as {{nowrap|{{vmp| 7 11 16 | -5 2 1 }} {{=}} {{nowrap|7 ⋅ −5}} + {{nowrap|11 ⋅ 2}} + {{nowrap|16 ⋅ 1}}}} {{nowrap|{{=}} −35 + 22 + 16}} = 3. This means that 45/32 can be found at 3 steps of 7-ET.

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Dave Keenan and Douglas Blumeyer propose that it may be helpful to distinguish objects with distinctive shapes, such as [[generator tuning map]]s and generator-count vectors, by using curly brackets {…] […} in place of angle brackets ⟨…] […⟩, wherever the height or width of a vector or matrix is equal to the [[rank]] of the temperament, <math>r</math>. A mnemonic for the curly bracket is that it resembles the tilde (~) which is commonly used to mark approximated or tempered intervals, e.g. ~3/2 is an approximation of 3/2.

For example, while the monzo representing 5/4 in just intonation would be written {{vector| -2 0 1 }}, the mapped version of this in meantone could be written {{rket| -2 4 }}. And while the ~~tempered-prime~~ tuning map for quarter-comma meantone might be written {{bra| 1200.000 1896.578 2786.314 }}, the generator tuning map could be written {{rbra| 1200.000 696.578 }}.

For example, while the monzo representing 5/4 in just intonation would be written {{vector| -2 0 1 }}, the mapped version of this in meantone could be written {{rket| -2 4 }}. And while the ordinary tuning map for quarter-comma meantone might be written {{bra| 1200.000 1896.578 2786.314 }}, the generator tuning map could be written {{rbra| 1200.000 696.578 }}.

They further propose that the use of the normal angle bracket could be restricted to matrix widths and heights equal only to the [[dimensionality]] of the temperament, <math>d</math>, and any other width or height besides <math>d</math> and <math>r</math> would be given with plain square brackets […]. So, for example, a comma basis could be written [{{ket| 4 -4 1 }} {{ket| 7 0 -3 }}] because its width is equal to the [[nullity]] of the temperament, <math>n</math>. This is consistent with the fact that it is common for linear algebra texts to treat a nullspace basis not as a matrix but as a mere list of vectors.

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 == Basics ==
-The name ''bra–ket notation'' refers to its two basic structures: the '''bra''' and the '''ket'''. A bra is a list which has a pointed bracket on its ''left'' side, and a ket is a list which has a pointed bracket on its ''right'' side. Bras are used for covectors (such as [[Val|vals]]), and kets are used for vectors (such as [[monzos]]). In a basic sense, vectors can be thought of as tangible objects (like intervals), and covectors are used for things you do to those objects (like temperaments).
+The name ''bra–ket notation'' refers to its two basic structures: the '''bra''' and the '''ket'''. A bra is a list which has a pointed bracket on its ''left'' side, and a ket is a list which has a pointed bracket on its ''right'' side. Bras are used for covectors (such as [[val]]s), and kets are used for vectors (such as [[monzo]]s). In a basic sense, vectors can be thought of as tangible objects (like intervals), and covectors are used for things you do to those objects (like temperaments).
 The usage patterns described in this section are the same as the standard usage of bra–ket notation in quantum mechanics.
 === Common applications ===
-In RTT, covectors are used most frequently for [[Val|vals]] (including [[tuning map]]s). For example, the val for 7-ET is notated as {{bra| 7 11 16 }}, and a (tempered-prime) tuning map for it might be {{bra|1209.682 1900.930 2764.988}}.
+In RTT, covectors are used most frequently for vals (including [[tuning map]]s). For example, the val for 7-ET is notated as {{bra| 7 11 16 }}, and a tuning map for it might be {{bra| 1209.682 1900.930 2764.988 }}.
-The most common type of vectors used in RTT is the [[Monzos|monzo]] (both [[Prime-count vector|prime-count vectors]] and [[Generator-count vector|generator-count vectors]]). For example, the monzo for 45/32 is {{ket| -5 2 1 }}, and the monzo for ~45/32 in porcupine temperament is {{ket| 2 -11 }} (for the generators ~2/1 and ~11/10).
+The most common type of vectors used in RTT is the monzo (both [[prime-count vector]]s and [[generator-count vector]]s). For example, the monzo for 45/32 is {{ket| -5 2 1 }}, and the tempered monzo for ~45/32 in porcupine temperament is {{ket| 2 -11 }} (for the generators ~2/1 and ~11/10).
 === Combining ===
-When a bra and a ket are put together, the dot product is taken, and the result is a scalar (a single number). In order to be compatible in this way, the bra and ket must at least have the same number of entries, so that each of their entries will pair up for a product. The dot product corresponds to "applying" the covector to the vector.
+When a bra and a ket are put together, the dot product is taken, and the result is a scalar (a single number). In order to be compatible in this way, the bra and ket must at least have the same number of entries, so that each of their entries will pair up for a product. The {{w|dot product}} corresponds to "applying" the covector to the vector.
 Mapping 45/32 with 7-ET would be notated as {{nowrap|{{vmp| 7 11 16 | -5 2 1 }} {{=}} {{nowrap|7 ⋅ −5}} + {{nowrap|11 ⋅ 2}} + {{nowrap|16 ⋅ 1}}}} {{nowrap|{{=}} −35 + 22 + 16}} =&nbsp;3. This means that 45/32 can be found at 3 steps of 7-ET.
@@ Line 123: / Line 123: @@
 Dave Keenan and Douglas Blumeyer propose that it may be helpful to distinguish objects with distinctive shapes, such as [[generator tuning map]]s and generator-count vectors, by using curly brackets {…] […} in place of angle brackets ⟨…] […⟩, wherever the height or width of a vector or matrix is equal to the [[rank]] of the temperament, <math>r</math>. A mnemonic for the curly bracket is that it resembles the tilde (~) which is commonly used to mark approximated or tempered intervals, e.g. ~3/2 is an approximation of 3/2.
-For example, while the monzo representing 5/4 in just intonation would be written {{vector| -2 0 1 }}, the mapped version of this in meantone could be written {{rket| -2 4 }}. And while the tempered-prime tuning map for quarter-comma meantone might be written {{bra| 1200.000 1896.578 2786.314 }}, the generator tuning map could be written {{rbra| 1200.000 696.578 }}.
+For example, while the monzo representing 5/4 in just intonation would be written {{vector| -2 0 1 }}, the mapped version of this in meantone could be written {{rket| -2 4 }}. And while the ordinary tuning map for quarter-comma meantone might be written {{bra| 1200.000 1896.578 2786.314 }}, the generator tuning map could be written {{rbra| 1200.000 696.578 }}.
 They further propose that the use of the normal angle bracket could be restricted to matrix widths and heights equal only to the [[dimensionality]] of the temperament, <math>d</math>, and any other width or height besides <math>d</math> and <math>r</math> would be given with plain square brackets […]. So, for example, a comma basis could be written [{{ket| 4 -4 1 }} {{ket| 7 0 -3 }}] because its width is equal to the [[nullity]] of the temperament, <math>n</math>. This is consistent with the fact that it is common for linear algebra texts to treat a nullspace basis not as a matrix but as a mere list of vectors.