Extended bra–ket notation: Difference between revisions

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:

==== 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 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 ===