Extended bra–ket notation: Difference between revisions

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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> = {{map|1 0 -4}} and <math>𝒎_2</math> = {{map|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~~|{{map|1 0 -4}} {{map|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> = {{map|1 0 -4}} and <math>𝒎_2</math> = {{map|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, {{rket|{{map|1 0 -4}} {{map|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 {{vector|-11 7}} and {{vector|-7 3 1}}, respectively. As a matrix <math>\mathrm{C}</math>, we'd see this as:

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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 mapping would look like [{{map|1 0 -4}}, {{map|0 1 4}}], and 7-ET~~'s comma basis~~ 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 mapping would look like [{{map|1 0 -4}}, {{map|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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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 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>.

For examples, while the PC-vector representing 5/4 would be written {{vector|-2 0 1}}, the mapped version of this in meantone could be written [-2 4}. And while the tuning map for quarter-comma meantone might be written {{map|1200.000 1896.578 2786.314}}, the ~~generators~~ tuning map could be written {1200.000 696.578].

For examples, 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}}.

The stylistic reasoning is that the curly bracket resembles the tilde (~) which is commonly used to mark approximated or tempered intervals, e.g. ~3/2 is an approximation of 3/2.

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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> = {{map|1 0 -4}} and <math>𝒎_2</math> = {{map|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|{{map|1 0 -4}} {{map|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> = {{map|1 0 -4}} and <math>𝒎_2</math> = {{map|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, {{rket|{{map|1 0 -4}} {{map|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 {{vector|-11 7}} and {{vector|-7 3 1}}, respectively. As a matrix <math>\mathrm{C}</math>, we'd see this as:
@@ Line 84: / Line 84: @@
 ====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 mapping would look like [{{map|1 0 -4}}, {{map|0 1 4}}], and 7-ET's comma basis 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 mapping would look like [{{map|1 0 -4}}, {{map|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 143: / Line 143: @@
 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 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>.
-For examples, while the PC-vector representing 5/4 would be written {{vector|-2 0 1}}, the mapped version of this in meantone could be written [-2 4}. And while the tuning map for quarter-comma meantone might be written {{map|1200.000 1896.578 2786.314}}, the generators tuning map could be written {1200.000 696.578].
+For examples, 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}}.
 The stylistic reasoning is that the curly bracket resembles the tilde (~) which is commonly used to mark approximated or tempered intervals, e.g. ~3/2 is an approximation of 3/2.