Extended bra–ket notation: Difference between revisions

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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.

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 ====

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==== 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 ===

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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.

== Equivalent in matrix notation ==

In ordinary matrix notation, a matrix can be written in terms of its rows or columns using horizontal or vertical bars.<ref>L.N. Trefethen and D. Bau, III, [https://www.stat.uchicago.edu/~lekheng/courses/309/books/Trefethen-Bau.pdf Numerical linear algebra]. Society for industrial and applied mathematics, 1997.</ref>

For example, the meantone mapping above can be written in terms of its rows as

<math>

\begin{bmatrix}

1 & 0 & -4 \\

0 & 1 & 4 \\

\end{bmatrix}

=

\begin{bmatrix}

1 & 0 & -4 \\

\hline

0 & 1 & 4 \\

\end{bmatrix}

</math>

or symbolically, following the notation above, as

<math>

\begin{bmatrix}

\hspace{1.45em} 𝒎_1 \hspace{1.45em} \\

\hline

𝒎_2

\end{bmatrix}

</math>

Likewise, the comma basis above can be written in terms of its columns as

<math>

\left[

\begin{array}{cc}

-11 & -7 \\

7 & 3 \\

0 & 1

\end{array}

\right]

=

\left[

\begin{array}{c|c}

-11 & -7 \\

7 & 3 \\

0 & 1

\end{array}

\right]

</math>

or symbolically as

<math>

\left[

\begin{array}{c|c}

& \\

\textbf{c}_1 & \textbf{c}_2 \\

& \\

\end{array}

\right]

</math>

This notation is widely understood and can also express matrices with block structure by combining horizontal and vertical bars.

== Notes ==

@@ Line 28: / Line 28: @@
 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 ====
@@ Line 68: / Line 68: @@
 ==== 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 ===
@@ Line 130: / Line 130: @@
 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.
+== Equivalent in matrix notation ==
+In ordinary matrix notation, a matrix can be written in terms of its rows or columns using horizontal or vertical bars.<ref>L.N. Trefethen and D. Bau, III, [https://www.stat.uchicago.edu/~lekheng/courses/309/books/Trefethen-Bau.pdf Numerical linear algebra]. Society for industrial and applied mathematics, 1997.</ref>
+For example, the meantone mapping above can be written in terms of its rows as
+<math>
+\begin{bmatrix}
+& 0 & -4 \\
+& 1 & 4 \\
+\end{bmatrix}
+=
+\begin{bmatrix}
+& 0 & -4 \\
+\hline
+& 1 & 4 \\
+\end{bmatrix}
+</math>
+or symbolically, following the notation above, as
+<math>
+\begin{bmatrix}
+\hspace{1.45em} 𝒎_1 \hspace{1.45em} \\
+\hline
+𝒎_2
+\end{bmatrix}
+</math>
+Likewise, the comma basis above can be written in terms of its columns as
+<math>
+\left[
+\begin{array}{cc}
+-11 & -7 \\
+& 3 \\
+& 1
+\end{array}
+\right]
+=
+\left[
+\begin{array}{c|c}
+-11 & -7 \\
+& 3 \\
+& 1
+\end{array}
+\right]
+</math>
+or symbolically as
+<math>
+\left[
+\begin{array}{c|c}
+ & \\
+\textbf{c}_1 & \textbf{c}_2 \\
+ & \\
+\end{array}
+\right]
+</math>
+This notation is widely understood and can also express matrices with block structure by combining horizontal and vertical bars.
 == Notes ==