Linear algebra formalism
Aspects of tuning theory are often described in the language of linear algebra.
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Monzos and vectors
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A vector is a list of numbers, written like so: [math]\displaystyle{ \begin{pmatrix} -2\\ 0\\ 1 \end{pmatrix} }[/math].
Vals and covectors
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Mappings and matrices
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A matrix is a grid of numbers, written like so:
[math]\displaystyle{ \begin{bmatrix} 1 & 0 & -4\\ 0 & 1 & 4 \end{bmatrix} }[/math]
This matrix can be thought of as a "function" that you apply to a vector to get out another vector. This matrix has 3 columns, meaning the vector it takes as an "input" will have 3 elements, and it has 2 rows, meaning the vector you get out will have 2 elements. So, this is a "function" down from 3-dimensional space to 2-dimensional space.
Matrix operations
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Dot product
The dot product is a way to combine two vectors to get out a single number. Say we want to take the dot product of the vectors [math]\displaystyle{ \begin{pmatrix}12\\19\\28\end{pmatrix} }[/math] and [math]\displaystyle{ \begin{pmatrix}-2\\0\\1\end{pmatrix} }[/math]. To do so, follow these steps:
- Write the vectors separated by a dot to denote the dot product: [math]\displaystyle{
\begin{pmatrix}
12\\
19\\
28\\
\end{pmatrix}
\cdot
\begin{pmatrix}
-2\\
0\\
1
\end{pmatrix}
}[/math]
- This may also be notated [math]\displaystyle{ \langle 12, 19, 28 \vert -2, 0, 1\rangle }[/math]; from this derives the notation for vals and monzos.
- Multiply the corresponding elements, and add the results together: [math]\displaystyle{ \left(12\cdot-2\right)+\left(19\cdot0\right)+\left(28\cdot1\right) = -24 + 0 + 28 = 4 }[/math]
Multiply matrix by vector
We write the "application" of a matrix like so:
[math]\displaystyle{ \begin{bmatrix} 1 & 0 & -4\\ 0 & 1 & 4 \end{bmatrix} \begin{bmatrix} -2\\ 0\\ 1 \end{bmatrix} }[/math]
where the second object is the vector.
To write the first element of our output, we take the dot product of the first row of our matrix with our vector: [math]\displaystyle{ \begin{pmatrix} 1\\ 0\\ -4\\ \end{pmatrix} \cdot \begin{pmatrix} -2\\ 0\\ 1 \end{pmatrix} = \left(1\cdot-2\right)+\left(0\cdot0\right)+\left(-4\cdot1\right) = -2 + 0 + -4 = -6 }[/math]
We do the same thing for the second element of our output, computing [math]\displaystyle{ \begin{pmatrix} 0\\ 1\\ 4\\ \end{pmatrix} \cdot \begin{pmatrix} -2\\ 0\\ 1 \end{pmatrix} = 4 }[/math].
Thus, our output is [math]\displaystyle{ \begin{bmatrix} -6\\ 4\\ \end{bmatrix} }[/math] .
Multiply matrix by matrix
A matrix can act on another matrix, as well. In this case, the matrix on the right can simply be treated as several vectors next to each other.
[math]\displaystyle{ \begin{bmatrix} 1 & 0 & -4\\ 0 & 1 & 4 \end{bmatrix} \begin{bmatrix} 1 & -1 & -2\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & -1 & -6\\ 0 & 1 & 4 \end{bmatrix} }[/math]
Multiply row vector by matrix
This is to taking the dot product as matrix*matrix multiplication is to matrix*vector multiplication. You take the dot product of the row vector with each successive column of the matrix, and write the result as another row vector. Any matrix*vector operation can be rewritten in this format by swapping rows and columns; the reason these are distinguished is because it is conventional to represent certain things as column vectors and different things as row vectors (i.e. monzos and vals); in this case, vectors represented as rows are called "covectors".
Determinant
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Practical usage
Let's say we want to determine the tuning of 6/5 in quarter-comma meantone.
Stage 1: JI to temperament
[math]\displaystyle{ \begin{bmatrix} 1 & 0 & -4\\ 0 & 1 & 4 \end{bmatrix} }[/math]
This matrix I've been using as an example is actually a "function" that converts a 5-limit interval in monzo format (with the entries corresponding to powers of 2, 3, and 5) into a corresponding meantone interval in an analogous format (with the entries representing powers of meantone's tempered 2 and 3) called "tempered monzos" or "tmonzos".
So, let's take the monzo for 6/5, [1 1 -1⟩, and apply this matrix to it: [math]\displaystyle{ \begin{bmatrix} 1 & 0 & -4\\ 0 & 1 & 4 \end{bmatrix} \begin{bmatrix} 1\\ 1\\ -1 \end{bmatrix} = \begin{bmatrix} 5\\ -3 \end{bmatrix} }[/math].
The result is the meantone tmonzo representing a tempered 6/5.
Stage 2: Temperament to tuning
Now, we have the tmonzo. We'll be introducing something called a tval, which gives us a specific tuning of our temperament the same way a regular val gives us a specific tuning of just intonation. The quarter-comma meantone tval for this meantone mapping is ⟨1200 ~1896.5784] in cents. This is where the dot product comes in: [math]\displaystyle{ \langle 1200, ~1896.5784 \vert 5, -3\rangle }[/math].
Computing this dot product yields ~310.265, which is exactly the size of the QCM minor third in cents!