Extended bra–ket notation: Difference between revisions

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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 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.

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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}}.

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 ===

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==== 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.

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 }}.

@@ 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.
@@ Line 13: / Line 13: @@
 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 ===
@@ Line 40: / Line 40: @@
 ==== 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:
@@ Line 79: / Line 79: @@
 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 }}.