Extended bra–ket notation: Difference between revisions

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====Examples====

For example, the [[mapping]] for meantone temperament is a matrix that looks like this:

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

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RTT mappings are typically thought of in terms of their ''rows''. This mapping has two rows; let's call them <math>~~𝒓_1~~</math> and <math>~~𝒓_2~~</math>. And so to notate this mapping in EBK, we can first imagine capturing the rows as bras like we would normally: <math>~~𝒓_1~~</math> = {{map|1 0 -4}} and <math>~~𝒓_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>~~𝒓_1~~</math> <math>~~𝒓_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, {{ket|{{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, we'd see this as:

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>C</math>, we'd see this as:

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RTT comma bases, as opposed to mappings, are more naturally thought of in terms of their ''columns''. This comma basis has two columns; let's call them <math>\textbf{c}_1</math> and <math>\textbf{c}_2</math>. And so to notate this comma basis in EBK, we can first imagine capturing the columns as kets like we would normally: <math>\textbf{c}_1</math> = {{ket|-11 7 0}} and <math>\textbf{c}_2</math> = {{ket|-7 3 1}}. Then, to put them together, we can think of this matrix as a single row containing these two columns, or in other words, a bra containing the two kets: {{map|<math>\textbf{c}_1</math> <math>\textbf{c}_2</math>}}, or fully written out, {{map|{{ket|-11 7 0}} {{ket|-7 3 1}}}}.

RTT comma bases, as opposed to mappings, are more naturally thought of in terms of their ''columns''. This comma basis has two columns; let's call them <math>\textbf{c}_1</math> and <math>\textbf{c}_2</math> (in this case, that conveniently works out as both 'c' for "column" and 'c' for "comma"!). And so to notate this comma basis in EBK, we can first imagine capturing the columns as kets like we would normally: <math>\textbf{c}_1</math> = {{ket|-11 7 0}} and <math>\textbf{c}_2</math> = {{ket|-7 3 1}}. Then, to put them together, we can think of this matrix as a single row containing these two columns, or in other words, a bra containing the two kets: {{map|<math>\textbf{c}_1</math> <math>\textbf{c}_2</math>}}, or fully written out, {{map|{{ket|-11 7 0}} {{ket|-7 3 1}}}}.

====History====

@@ Line 47: / Line 47: @@
 ====Examples====
-For example, the [[mapping]] for meantone temperament is a matrix that looks like this:
+For example, the [[mapping]] for meantone temperament is a matrix <math>M</math> that looks like this:
@@ Line 58: / Line 58: @@
-RTT mappings are typically thought of in terms of their ''rows''. This mapping has two rows; let's call them <math>𝒓_1</math> and <math>𝒓_2</math>. And so to notate this mapping in EBK, we can first imagine capturing the rows as bras like we would normally: <math>𝒓_1</math> = {{map|1 0 -4}} and <math>𝒓_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>𝒓_1</math> <math>𝒓_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, {{ket|{{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, we'd see this as:
+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>C</math>, we'd see this as:
@@ Line 72: / Line 72: @@
-RTT comma bases, as opposed to mappings, are more naturally thought of in terms of their ''columns''. This comma basis has two columns; let's call them <math>\textbf{c}_1</math> and <math>\textbf{c}_2</math>. And so to notate this comma basis in EBK, we can first imagine capturing the columns as kets like we would normally: <math>\textbf{c}_1</math> = {{ket|-11 7 0}} and <math>\textbf{c}_2</math> = {{ket|-7 3 1}}. Then, to put them together, we can think of this matrix as a single row containing these two columns, or in other words, a bra containing the two kets: {{map|<math>\textbf{c}_1</math> <math>\textbf{c}_2</math>}}, or fully written out, {{map|{{ket|-11 7 0}} {{ket|-7 3 1}}}}.
+RTT comma bases, as opposed to mappings, are more naturally thought of in terms of their ''columns''. This comma basis has two columns; let's call them <math>\textbf{c}_1</math> and <math>\textbf{c}_2</math> (in this case, that conveniently works out as both 'c' for "column" and 'c' for "comma"!). And so to notate this comma basis in EBK, we can first imagine capturing the columns as kets like we would normally: <math>\textbf{c}_1</math> = {{ket|-11 7 0}} and <math>\textbf{c}_2</math> = {{ket|-7 3 1}}. Then, to put them together, we can think of this matrix as a single row containing these two columns, or in other words, a bra containing the two kets: {{map|<math>\textbf{c}_1</math> <math>\textbf{c}_2</math>}}, or fully written out, {{map|{{ket|-11 7 0}} {{ket|-7 3 1}}}}.
 ====History====