Extended bra–ket notation: Difference between revisions

Line 139:

The stylistic reasoning is that it resembles the use commas in large numbers to separate every third digit (e.g. 1,000,000). The first comma being after prime 3 rather than after the third prime (prime 5) is due to the notational specialness of primes 2 and 3, those being the primes capable of being encoded by standard sheet music (prime 2 by Pythagorean nominals A,B,C,D,E,F,G and prime 3 by sharp and flat symbols).

===~~Curly~~ brackets ~~for tempered objects~~===

===Variant including curly and square brackets===

Dave Keenan and Douglas Blumeyer propose that it may be helpful to distinguish ~~tempered~~ objects, such as GC-vectors and [[generator tuning map]]s, by using curly brackets in place of angle 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 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 ~~tempered~~ 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 [-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].

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.

Furthermore, 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}}].

==Notes==

@@ Line 139: / Line 139: @@
 The stylistic reasoning is that it resembles the use commas in large numbers to separate every third digit (e.g. 1,000,000). The first comma being after prime 3 rather than after the third prime (prime 5) is due to the notational specialness of primes 2 and 3, those being the primes capable of being encoded by standard sheet music (prime 2 by Pythagorean nominals A,B,C,D,E,F,G and prime 3 by sharp and flat symbols).
-===Curly brackets for tempered objects===
+===Variant including curly and square brackets===
-Dave Keenan and Douglas Blumeyer propose that it may be helpful to distinguish tempered objects, such as GC-vectors and [[generator tuning map]]s, by using curly brackets in place of angle 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 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 tempered 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 [-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].
 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.
+Furthermore, 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}}].
 ==Notes==