Extended bra–ket notation - Revision history

FloraC: Style

2025-11-04T09:14:03Z

Style

← Older revision		Revision as of 09:14, 4 November 2025
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	Vectors (including covectors) are used to represent tuning theory objects of various dimensionality. For example, the monzo {{ket\| 1 -2 1 }} represents the interval 10/9, which would be found as a point in the ''three''-dimensional space of 5-limit JI lattice, while the monzo {{ket\| 0 -1 1 1 -1 }} represents the interval 35/33, which would be found as a point in the ''five''-dimensional space of the 11-limit JI lattice. However, regardless of the dimensionality of the musical object represented, a vector itself will always be a ''one''-dimensional structure, in the sense that it is a simple list of numbers. Due to this, vectors are always fairly easy to embed in similarly one-dimensional strings of text or data cells of tables.		Vectors (including covectors) are used to represent tuning theory objects of various dimensionality. For example, the monzo {{ket\| 1 -2 1 }} represents the interval 10/9, which would be found as a point in the ''three''-dimensional space of 5-limit JI lattice, while the monzo {{ket\| 0 -1 1 1 -1 }} represents the interval 35/33, which would be found as a point in the ''five''-dimensional space of the 11-limit JI lattice. However, regardless of the dimensionality of the musical object represented, a vector itself will always be a ''one''-dimensional structure, in the sense that it is a simple list of numbers. Due to this, vectors are always fairly easy to embed in similarly one-dimensional strings of text or data cells of tables.

	But RTT uses a number of ''two''-dimensional structures as well — i.e. numbers arranged in a grid of rows and columns — which are called matrices. Having the ability to present these objects one-dimensionally can be quite helpful too, and so the first way in which EBK extends bra–ket notation is designed to provide that.		But RTT uses a number of ''two''-dimensional structures as well – i.e. numbers arranged in a grid of rows and columns – which are called matrices. Having the ability to present these objects one-dimensionally can be quite helpful too, and so the first way in which EBK extends bra–ket notation is designed to provide that.

	==== Advantage ====		==== Advantage ====
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	==== Alternatives ====		==== Alternatives ====
	In many wiki writings, mappings and comma bases are provided as ''lists'' of vectors, notated using square brackets on both sides and commas between entries, like this: [''a'', ''b'', ''c'', …]. So meantone's mapping would look like [{{bra\| 1 0 -4 }}, {{bra\| 0 1 4 }}], and a comma basis for 7-ET would look like [{{ket\| -11 7 0 }}, {{ket\| -7 3 1 }}]. This notation is completely sufficient and unambiguous, but — for better or worse — does not emphasize the matrix-like structure of the data quite as strongly.		In many wiki writings, mappings and comma bases are provided as ''lists'' of vectors, notated using square brackets on both sides and commas between entries, like this: [''a'', ''b'', ''c'', …]. So meantone's mapping would look like [{{bra\| 1 0 -4 }}, {{bra\| 0 1 4 }}], and a comma basis for 7-ET would look like [{{ket\| -11 7 0 }}, {{ket\| -7 3 1 }}]. This notation is completely sufficient and unambiguous, but – for better or worse – does not emphasize the matrix-like structure of the data quite as strongly.

	=== Repetition, for multivectors ===		=== Repetition, for multivectors ===

FloraC: Review on the last edit

2025-11-04T09:12:16Z

Review on the last edit

VectorGraphics at 04:17, 4 November 2025

2025-11-04T04:17:31Z

← Older revision		Revision as of 04:17, 4 November 2025
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	'''Extended bra–ket notation''', or '''EBK''' for short, is ~~a potential~~ name for the notation system for [[regular temperament theory]] (RTT) objects that has gradually developed over time, with contributions from various theoreticians.		'''Extended bra–ket notation''', or '''EBK''' for short, is the name for the notation system for [[regular temperament theory]] (RTT) objects in the linear algebra formalism that has gradually developed over time, with contributions from various theoreticians. EBK involves enclosing lists of values in sets of brackets, with pointed brackets used to distinguish different types of lists.

	~~Several stylistic variations are possible, and no formal style has yet been established, but it can be safely said that~~ EBK ~~always~~ involves enclosing lists of values in sets of brackets, with pointed brackets used to distinguish different types of lists.

	Per the name, EBK extends {{w\|Bra–ket notation}}, which is used in quantum mechanics. The use of bra–ket notation for RTT was originally proposed by [[Gene Ward Smith]] in February 2002<ref>https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3355#3356</ref>.		Per the name, EBK extends {{w\|Bra–ket notation}}, which is used in quantum mechanics. The use of bra–ket notation for RTT was originally proposed by [[Gene Ward Smith]] in February 2002<ref>https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3355#3356</ref>.

	== Basics ==		== 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, and kets are used for vectors (~~for more information on the difference between covectors and vectors, see~~ [[~~variance~~]]).		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 usage patterns described in this section are the same as the standard usage of bra–ket notation in quantum mechanics.		The usage patterns described in this section are the same as the standard usage of bra–ket notation in quantum mechanics.

	=== Common applications ===		=== Common applications ===
	In RTT, covectors are used most frequently for [[~~map~~]]~~s and~~ [[tuning map]]s. For example, the ~~map~~ 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}}.		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}}.

	The most common ~~types~~ of vectors used in RTT ~~are~~ [[prime-count ~~vector~~]]~~s (PC-vectors)~~ and [[generator-count ~~vector~~]]~~s (GC-vectors~~). For example, the ~~PC-vector~~ for 45/32 is {{ket\| -5 2 1 }}, and the ~~GC-vector~~ for ~45/32 in porcupine temperament is {{ket\| 2 -11 }}.		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).

	=== Combining ===		=== 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.		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.

	~~So 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.		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.

	~~Or mapping the GC-vector for~~ porcupine's ~45/32 with a generator tuning map like {{bra\| 1200 162.737 }} would look like {{nowrap\|{{vmp\| 1200 162.737 \| 2 -11 }} {{=}} 1200 ⋅ 2 + 162.737 ⋅ −11}} {{nowrap\|{{=}} 2400 + −1790.107}} = 609.893.		Mapping porcupine's ~45/32 with a generator tuning map like {{bra\| 1200 162.737 }} (reasonable tunings for ~2/1 and ~11/10 in porcupine) would look like {{nowrap\|{{vmp\| 1200 162.737 \| 2 -11 }} {{=}} 1200 ⋅ 2 + 162.737 ⋅ −11}} {{nowrap\|{{=}} 2400 + −1790.107}} = 609.893.

	== Extensions ==		== Extensions ==
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	=== Alternating, for matrices ===		=== Alternating, for matrices ===
	Vectors (including covectors) are used to represent ~~musical~~ objects of various dimensionality. For example, the ~~PC-vector~~ {{ket\| 1 -2 1 }} represents the interval 10/9, which would be found as a point in the ''three''-dimensional space of 5-limit JI lattice, while the ~~PC-vector~~ {{ket\| 0 -1 1 1 -1 }} represents the interval 35/33, which would be found as a point in the ''five''-dimensional space of the 11-limit JI lattice. However, regardless of the dimensionality of the musical object represented, a vector itself will always be a ''one''-dimensional structure, in the sense that it is a simple list of numbers. Due to this, vectors are always fairly easy to embed in similarly one-dimensional strings of text or data cells of tables.		Vectors (including covectors) are used to represent tuning theory objects of various dimensionality. For example, the monzo {{ket\| 1 -2 1 }} represents the interval 10/9, which would be found as a point in the ''three''-dimensional space of 5-limit JI lattice, while the monzo {{ket\| 0 -1 1 1 -1 }} represents the interval 35/33, which would be found as a point in the ''five''-dimensional space of the 11-limit JI lattice. However, regardless of the dimensionality of the musical object represented, a vector itself will always be a ''one''-dimensional structure, in the sense that it is a simple list of numbers. Due to this, vectors are always fairly easy to embed in similarly one-dimensional strings of text or data cells of tables.

	But RTT uses a number of ''two''-dimensional structures as well — i.e. numbers arranged in a grid of rows and columns — which are called matrices. Having the ability to present these objects one-dimensionally can be quite helpful too, and so the first way in which EBK extends bra–ket notation is designed to provide that.		But RTT uses a number of ''two''-dimensional structures as well — i.e. numbers arranged in a grid of rows and columns — which are called matrices. Having the ability to present these objects one-dimensionally can be quite helpful too, and so the first way in which EBK extends bra–ket notation is designed to provide that.
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	RTT mappings are typically thought of in terms of their ''rows''. This mapping <math>M</math> has two rows; following mathematical conventions, let's call them <math>𝒎_1</math> and <math>𝒎_2</math>. And so to notate this mapping in EBK, we can first imagine capturing the rows as bras like we would normally: <math>𝒎_1</math> = {{bra\| 1 0 -4 }} and <math>𝒎_2</math> = {{bra\| 0 1 4 }}. Then, to put them together, we can think of this matrix as a single column containing these two rows, or in other words, a ket containing the two bras: {{ket\| <math>𝒎_1</math> <math>𝒎_2</math> }}, or fully written out, {{ket\| {{bra\| 1 0 -4 }} {{bra\| 0 1 4 }} }}.		RTT mappings are typically thought of in terms of their ''rows''. This mapping <math>M</math> has two rows; following mathematical conventions, let's call them <math>𝒎_1</math> and <math>𝒎_2</math>. And so to notate this mapping in EBK, we can first imagine capturing the rows as bras like we would normally: <math>𝒎_1</math> = {{bra\| 1 0 -4 }} and <math>𝒎_2</math> = {{bra\| 0 1 4 }}. Then, to put them together, we can think of this matrix as a single column containing these two rows, or in other words, a ket containing the two bras: {{ket\| <math>𝒎_1</math> <math>𝒎_2</math> }}, or fully written out, {{ket\| {{bra\| 1 0 -4 }} {{bra\| 0 1 4 }} }}.

	For another example, the canonical [[comma basis]] for 7-ET consists of the two commas 2187/2048 and 135/128, with ~~PC-vectors~~ {{ket\|-11 7}} and {{ket\| -7 3 1 }}, respectively. As a matrix <math>\mathrm{C}</math>, we'd see this as:		For another example, the canonical [[comma basis]] for 7-ET in the 5-limit consists of the two commas 2187/2048 and 135/128, with monzos {{ket\|-11 7}} and {{ket\| -7 3 1 }}, respectively. As a matrix <math>\mathrm{C}</math>, we'd see this as:

	<math>		<math>
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	The second extension of EBK from standard bra–ket notation is the repetition of brackets, allowing for the representation of [[multivector]]s.		The second extension of EBK from standard bra–ket notation is the repetition of brackets, allowing for the representation of [[multivector]]s.

	~~Some advanced practitioners of RTT~~ use multivectors as an alternative way to represent temperaments (besides mappings and comma bases). These multivectors come in various grades, such as 2-vectors and 3-vectors. In fact, ordinary vectors are simply 1-vectors. In order to distinguish a <math>g</math>-vector from a 1-vector, the brackets that would normally be used can be repeated <math>g</math> times, where <math>g</math> is the grade.		It is sometimes useful to use multivectors as an alternative way to represent temperaments (usually in the form of [[wedgies]]), besides mappings and comma bases. These multivectors come in various grades, such as 2-vectors and 3-vectors. In fact, ordinary vectors are simply 1-vectors. In order to distinguish a <math>g</math>-vector from a 1-vector, the brackets that would normally be used can be repeated <math>g</math> times, where <math>g</math> is the grade.

	For example, the 2-vector (bivector) representing meantone temperament uses two sets of brackets: {{multivector\| 1 4 4 }}. ~~The~~ 3-~~covector representing 7-limit 31-ET {{multicovector\|rank=~~3~~\| -87 72 -49 31 }}~~.		For example, the 2-vector (bivector) representing meantone temperament (written in standard math notation as <math>
			\left[ \begin{matrix}
			0 & 1 & 4 \\
			-1 & 0 & 4 \\
			-4 & -4 & 0 \\
			\end{matrix} \right]
			</math>) uses two sets of brackets: {{multivector\| 1 4 4 }}. Note that only 3 entries are written down, but in fact the entire value of the 2-vector can be determined from the value of these 3 entries.

	The repetition-for-multivectors extension was developed in November of 2003 by Dave Keenan<ref>https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_7525#7749</ref>.		The repetition-for-multivectors extension was developed in November of 2003 by Dave Keenan<ref>https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_7525#7749</ref>.
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	The stylistic reasoning is that it resembles the use of commas in large numbers to separate every third digit (e.g. 1,000,000). The first comma being after prime 3 rather than after the third prime (prime 5) is due to the notational specialness of primes 2 and 3, those being the primes capable of being encoded by standard sheet music (prime 2 by clefs and staff positions and prime 3 by Pythagorean nominals A, B, C, D, E, F, G and sharp and flat symbols).		The stylistic reasoning is that it resembles the use of commas in large numbers to separate every third digit (e.g. 1,000,000). The first comma being after prime 3 rather than after the third prime (prime 5) is due to the notational specialness of primes 2 and 3, those being the primes capable of being encoded by standard sheet music (prime 2 by clefs and staff positions and prime 3 by Pythagorean nominals A, B, C, D, E, F, G and sharp and flat symbols).

			This may be useful to retain for wedgies, to hint at the multi-dimensional structure. For example: {{Multival\|1 4, 4}}.

	=== Variant including curly and square brackets ===		=== Variant including curly and square brackets ===
	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.		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 ~~PC-vector~~ representing 5/4 would be written {{vector\| -2 0 1 }}, the mapped version of this in meantone could be written {{rket\| -2 4 }}. And while the 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 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 }}.

	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.		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.

FloraC: FloraC moved page Extended bra-ket notation to Extended bra–ket notation: Follow Wikipedia

2025-08-10T18:45:53Z

FloraC moved page Extended bra-ket notation to Extended bra–ket notation: Follow Wikipedia

← Older revision	Revision as of 18:45, 10 August 2025
(No difference)

FloraC: I see no reason to capitalize the term bra-ket. It also seems Wikipedia endorses the use of an en dash here. Fix wiki markup

2025-08-10T18:45:25Z

I see no reason to capitalize the term bra-ket. It also seems Wikipedia endorses the use of an en dash here. Fix wiki markup

Show changes

FloraC: This is group theory, not linear algebra

2025-08-10T18:39:31Z

This is group theory, not linear algebra

← Older revision		Revision as of 18:39, 10 August 2025
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	'''Extended Bra-Ket notation''', or '''EBK''' for short, is a potential name for the notation system for [[regular temperament theory]] (RTT) objects ~~in the [[linear algebra formalism]]~~ that has gradually developed over time, with contributions from various theoreticians.		'''Extended Bra-Ket notation''', or '''EBK''' for short, is a potential name for the notation system for [[regular temperament theory]] (RTT) objects that has gradually developed over time, with contributions from various theoreticians.

	Several stylistic variations are possible, and no formal style has yet been established, but it can be safely said that EBK always involves enclosing lists of values in sets of brackets, with pointed brackets used to distinguish different types of lists.		Several stylistic variations are possible, and no formal style has yet been established, but it can be safely said that EBK always involves enclosing lists of values in sets of brackets, with pointed brackets used to distinguish different types of lists.
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	In RTT, covectors are used most frequently for [[map]]s and [[tuning map]]s. For example, the map 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}}.		In RTT, covectors are used most frequently for [[map]]s and [[tuning map]]s. For example, the map 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 types of vectors used in RTT are [[prime-count vector]]s (PC-vectors) and [[generator-count vector]]s (GC-vectors)~~, the linear algebra counterpart of [[monzos]]~~. For example, the PC-vector for 45/32 is {{ket\| -5 2 1 }}, and the GC-vector for ~45/32 in porcupine temperament is {{ket\| 2 -11 }}.		The most common types of vectors used in RTT are [[prime-count vector]]s (PC-vectors) and [[generator-count vector]]s (GC-vectors). For example, the PC-vector for 45/32 is {{ket\| -5 2 1 }}, and the GC-vector for ~45/32 in porcupine temperament is {{ket\| 2 -11 }}.

	=== Combining ===		=== Combining ===
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	==== Examples ====		==== Examples ====
	For example, the [[mapping]] for meantone temperament is ~~represented as~~ a matrix <math>M</math> that looks like this:		For example, the [[mapping]] for meantone temperament is a matrix <math>M</math> that looks like this:

VectorGraphics at 04:14, 25 June 2025

2025-06-25T04:14:45Z

← Older revision		Revision as of 04:14, 25 June 2025
Line 1:		Line 1:
	'''Extended Bra-Ket notation''', or '''EBK''' for short, is a potential name for the notation system for [[regular temperament theory]] (RTT) objects that has gradually developed over time, with contributions from various theoreticians.		'''Extended Bra-Ket notation''', or '''EBK''' for short, is a potential name for the notation system for [[regular temperament theory]] (RTT) objects in the [[linear algebra formalism]] that has gradually developed over time, with contributions from various theoreticians.

	Several stylistic variations are possible, and no formal style has yet been established, but it can be safely said that EBK always involves enclosing lists of values in sets of brackets, with pointed brackets used to distinguish different types of lists.		Several stylistic variations are possible, and no formal style has yet been established, but it can be safely said that EBK always involves enclosing lists of values in sets of brackets, with pointed brackets used to distinguish different types of lists.
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	In RTT, covectors are used most frequently for [[map]]s and [[tuning map]]s. For example, the map 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}}.		In RTT, covectors are used most frequently for [[map]]s and [[tuning map]]s. For example, the map 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 types of vectors used in RTT are [[prime-count vector]]s (PC-vectors) and [[generator-count vector]]s (GC-vectors). For example, the PC-vector for 45/32 is {{ket\| -5 2 1 }}, and the GC-vector for ~45/32 in porcupine temperament is {{ket\| 2 -11 }}.		The most common types of vectors used in RTT are [[prime-count vector]]s (PC-vectors) and [[generator-count vector]]s (GC-vectors), the linear algebra counterpart of [[monzos]]. For example, the PC-vector for 45/32 is {{ket\| -5 2 1 }}, and the GC-vector for ~45/32 in porcupine temperament is {{ket\| 2 -11 }}.

	=== Combining ===		=== Combining ===
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	==== Examples ====		==== Examples ====
	For example, the [[mapping]] for meantone temperament is a matrix <math>M</math> that looks like this:		For example, the [[mapping]] for meantone temperament is represented as a matrix <math>M</math> that looks like this:


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	The second extension of EBK from standard bra-ket notation is the repetition of brackets, allowing for the representation of [[multivector]]s.		The second extension of EBK from standard bra-ket notation is the repetition of brackets, allowing for the representation of [[multivector]]s.

	Some advanced practitioners of RTT use multivectors~~, such as [[wedgie]]s,~~ as an alternative way to represent temperaments (besides mappings and comma bases). These multivectors come in various grades, such as 2-vectors and 3-vectors. In fact, ordinary vectors are simply 1-vectors. In order to distinguish a <math>g</math>-vector from a 1-vector, the brackets that would normally be used can be repeated <math>g</math> times, where <math>g</math> is the grade.		Some advanced practitioners of RTT use multivectors as an alternative way to represent temperaments (besides mappings and comma bases). These multivectors come in various grades, such as 2-vectors and 3-vectors. In fact, ordinary vectors are simply 1-vectors. In order to distinguish a <math>g</math>-vector from a 1-vector, the brackets that would normally be used can be repeated <math>g</math> times, where <math>g</math> is the grade.

	For example, the 2-vector (bivector) representing meantone temperament uses two sets of brackets: {{multivector\| 1 4 4 }}. The 3-covector representing 7-limit 31-ET {{multicovector\|rank=3\| -87 72 -49 31 }}.		For example, the 2-vector (bivector) representing meantone temperament uses two sets of brackets: {{multivector\| 1 4 4 }}. The 3-covector representing 7-limit 31-ET {{multicovector\|rank=3\| -87 72 -49 31 }}.

FloraC: Fix pronouns

2025-06-14T07:06:38Z

Fix pronouns

← Older revision		Revision as of 07:06, 14 June 2025
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	For example, while the PC-vector representing 5/4 would be written {{vector\|-2 0 1}}, the mapped version of this in meantone could be written {{rket\| -2 4 }}. And while the 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 PC-vector representing 5/4 would be written {{vector\|-2 0 1}}, the mapped version of this in meantone could be written {{rket\| -2 4 }}. And while the 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 }}.

	We 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.		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.

	Instead of curly brackets, we considered using round brackets (parentheses), (…] […), however it is sometimes necessary to include parenthesized expressions as entries in the bra or ket, which would be difficult to parse if we had round bra or ket brackets.		Instead of curly brackets, they considered using round brackets (parentheses), (…] […), however it is sometimes necessary to include parenthesized expressions as entries in the bra or ket, which would be difficult to parse if we had round bra or ket brackets.

	We also considered using curved angle brackets <span style="font-family: STIXGeneral, 'Cambria Math', 'STIX Two Math', serif">⧼</span>…] […<span style="font-family: STIXGeneral, 'Cambria Math', 'STIX Two Math', serif">⧽</span>, (Unicode U+29FC and U+29FD), however in some fonts these are almost indistinguishable from ordinary angle brackets. If you see distinctly-curved angle brackets in the preceding sentence, it may only be because we specified ~~that they~~ be shown in one of the fonts 'STIXGeneral', 'Cambria Math' (Windows) or 'STIX Two Math' (Mac). Here's an ordinary angle bracket followed by a curved angle bracket without special treatment: ⟩⧽. If this situation should change in future, a mnemonic for the curved angle brackets is that they can also be called "rounded" angle brackets, and a rounded number is an approximate number in the same way that a tempered interval is an approximation of a just interval. Curly brackets would still be used as a substitute in ASCII-only environments.		They also considered using curved angle brackets <span style="font-family: STIXGeneral, 'Cambria Math', 'STIX Two Math', serif">⧼</span>…] […<span style="font-family: STIXGeneral, 'Cambria Math', 'STIX Two Math', serif">⧽</span>, (Unicode U+29FC and U+29FD), however in some fonts these are almost indistinguishable from ordinary angle brackets. If you see distinctly-curved angle brackets in the preceding sentence, it may only be because they have been specified to be shown in one of the fonts 'STIXGeneral', 'Cambria Math' (Windows) or 'STIX Two Math' (Mac). Here's an ordinary angle bracket followed by a curved angle bracket without special treatment: ⟩⧽. If this situation should change in future, a mnemonic for the curved angle brackets is that they can also be called "rounded" angle brackets, and a rounded number is an approximate number in the same way that a tempered interval is an approximation of a just interval. Curly brackets would still be used as a substitute in ASCII-only environments.

	== Notes ==		== Notes ==

ArrowHead294 at 13:45, 26 February 2025

2025-02-26T13:45:20Z

← Older revision		Revision as of 13:45, 26 February 2025
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	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.		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.

	So mapping 45/32 with 7-ET would be notated as {{~~vmprod~~\| 7 11 16 \| -5 2 1 }} = {{nowrap\|7 ~~⋅ -5~~}} + {{nowrap\|11 ~~⋅~~ 2}} + {{nowrap\|16 ~~⋅~~ 1}} = ~~-35~~ + 22 + 16 = 3.		So 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.

	Or mapping the GC-vector for porcupine's ~45/32 with a generator tuning map like {{bra\| 1200 162.737 }} would look like {{~~vmprod~~\| 1200 162.737 \| 2 -11 }} = {{~~nowrap\|~~1200 ~~⋅~~ 2}} + {{nowrap\|~~162.737 ⋅ -11}}~~ = 2400 + ~~-1790~~.107 = 609.893.		Or mapping the GC-vector for porcupine's ~45/32 with a generator tuning map like {{bra\| 1200 162.737 }} would look like {{nowrap\|{{vmp\| 1200 162.737 \| 2 -11 }} {{=}} 1200 ⋅ 2 + 162.737 ⋅ −11}} {{nowrap\|= 2400 + −1790.107}} = 609.893.

	== Extensions ==		== Extensions ==

ArrowHead294 at 18:28, 11 February 2025

2025-02-11T18:28:38Z

← Older revision		Revision as of 18:28, 11 February 2025
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	<references />		<references />

			{{Navbox notation}}
	{{~~Notation navbox~~}}

	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]
	[[Category:Notation]]		[[Category:Notation]]

← Older revision		Revision as of 09:12, 4 November 2025
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	== Basics ==		== 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.		The usage patterns described in this section are the same as the standard usage of bra–ket notation in quantum mechanics.

	=== Common applications ===		=== 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 ===		=== 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.		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.		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.		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.