Extended bra–ket notation: Difference between revisions

So mapping 45/32 with 7-ET would be notated as {{nowrap|{{vmprod|7 11 16|-5 2 1}} {{=}} 7·(−5) + 11·2 + 16·1 {{=}} −35 + 22 + 16 {{=}} 3}}.  

Or mapping the GC-vector for porcupine's ~45/32 with a generator tuning map like {{map|1200 162.737}} would look like {{nowrap|{{vmprod|1200 162.737|2 -11}} {{=}} 1200·2 + 162.737·(−11) {{=}} 2400 + (−1790.107) {{=}} 609.893}}.   

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.

\left[ \begin{matrix}

\end{matrix} \right]

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 = \val{1 & 0 & -4}</math> and <math>𝒎_2 = \val{0 & 1 & 4}</math>. 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: <math>\monzo{𝒎_1 & 𝒎_2}</math>, or fully written out, <math>\monzo{\monzo{1 & 0 & -4}\,\monzo{0 & 1 & 4}}</math>.

For another example, one possible canonical [[comma basis]] for [[7edo|7-tone equal temperament]] consists of the two intervals [[81/80]] and [[135/128]], with PC-vectors {{vector|-4 4 -1}} and {{vector|-7 3 1}}, respectively. As a matrix <math>\mathrm{C}</math>, we'd see this as:

\left[ \begin{matrix}

-4 & -7 \\

4 & 3 \\

-1 & 1 \\

\end{matrix} \right]

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 = \monzo{-4 & 4 & -1}</math> and <math>\textbf{c}_2 = \monzo{-7 & 3 & 1}</math>. 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: <math>\val{\textbf{c}_1 & \textbf{c}_2}</math>, or fully written out, <math>\val{\monzo{-4 & 4 & -1}\,\monzo{-7 & 3 & 1}}</math>.

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: <math>\left[a, b, c, \ldots\right]</math>. So meantone's mapping would look like <math>\left[\val{1 & 0 & -4}, \val{0 & 1 & 4}\right]</math>, and a comma basis for 7-ET would look like <math>\left[\monzo{-4 & 4 & -1}\, \monzo{-7 & 3 & 1}\right]</math>. This notation is completely sufficient and unambiguous, but&mdash;for better or worse&mdash;does not emphasize the matrix-like structure of the data quite as strongly.

For example, the 2-vector (bivector) representing meantone temperament uses two sets of brackets: {{multivector|1 4 4}}. The 3-covector representing 7-limit 31-ET {{multicovector|rank=3| -87 72 -49 31 }}.

! Symbol !! Unicode !! description

| &#x27E8; || [http://unicode-table.com/en/27E8/ U+27E8] || MATHEMATICAL LEFT ANGLE BRACKET

| &#x27E9; || [http://unicode-table.com/en/27E9/ U+27E9] || MATHEMATICAL RIGHT ANGLE BRACKET

The stylistic reasoning is that it resembles the use of 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 clefs and staff positions and prime 3 by Pythagorean nominals {{nowrap|A, B, C, D, E, F, G,}} along with sharps and flats).

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 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>. 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.

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', serif;">⧼</span>...] [...<span style="font-family: 'Cambria Math', 'STIX Two Math', serif">⧽</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 sentence, 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. Curly brackets would still be used as a substitute in ASCII-only environments.