Extended bra–ket notation: Difference between revisions

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===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 ~~GC-vectors and~~ [[generator tuning map]]s, by using ~~a curly bracket~~ [...} in place of an angle ~~bracket~~ [..⟩, wherever the height or width of a vector or matrix is equal to the [[rank]] of the temperament, <math>r</math>.

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 urly 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 {{map|1200.000 1896.578 2786.314}}, the generator tuning map could be written {{rbra|1200.000 696.578}}.

We ~~considered using a round~~ bracket ~~for this~~, [...), ~~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~~. ~~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~~.

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 [{{vector|4 -4 1}} {{vector|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.

~~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 [{{vector|4 -4 1}} {{vector|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.

We also considered using curved angle brackets <span style="font-family: 'Cambria Math', 'STIX Two Math'">⧼</span>...] [...<span style="font-family: 'Cambria Math', 'STIX Two Math'">⧽</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 example, it is only because we specified that they be shown in the font '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.

==Notes==

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 ===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 GC-vectors and [[generator tuning map]]s, by using a curly bracket [...} in place of an angle bracket [..⟩, wherever the height or width of a vector or matrix is equal to the [[rank]] of the temperament, <math>r</math>.
+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 urly 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 {{map|1200.000 1896.578 2786.314}}, the generator tuning map could be written {{rbra|1200.000 696.578}}.
-We considered using a round bracket for this, [...), 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. 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.
+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 [{{vector|4 -4 1}} {{vector|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.
-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 [{{vector|4 -4 1}} {{vector|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.
+We also considered using curved angle brackets <span style="font-family: 'Cambria Math', 'STIX Two Math'">⧼</span>...] [...<span style="font-family: 'Cambria Math', 'STIX Two Math'">⧽</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 example, it is only because we specified that they be shown in the font '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.
 ==Notes==