Plücker coordinates
Plücker coordinates (also known as the wedgie), are a way to assign coordinates to temperaments, by viewing them as elements of some projective space.
Definition
We have a Grassmannian variety [math]\displaystyle{ \mathrm{Gr} (k, n) }[/math] consisting of the k-dimensional subspaces of [math]\displaystyle{ \mathbb{R}^n }[/math]. The rational points on this variety can be identified with rank-k temperaments on a JI space with n primes.
Let [math]\displaystyle{ M }[/math] be an element of [math]\displaystyle{ \mathrm{Gr} (k, n) }[/math], spanned by basis vectors [math]\displaystyle{ m_1, \ldots, m_k }[/math]. We can embed the Grassmannian into a projective space using the Plücker map: $$ \begin{align} \iota: \mathrm{Gr} (k, n) & \to \mathbf{P}\left(\Lambda^{k} \mathbb{R}^n \right) \\ \operatorname {span} (m_1, \ldots, m_k) & \mapsto \left[ m_1 \wedge \ldots \wedge m_k \right] \, . \end{align} $$
Here, [math]\displaystyle{ \Lambda^{k} \mathbb{R}^n }[/math] is the k-th exterior power of our original space [math]\displaystyle{ \mathbb{R}^n }[/math]. The dimension of [math]\displaystyle{ \mathrm{Gr} (k, n) }[/math] is [math]\displaystyle{ k(n-k) }[/math], while the dimension of [math]\displaystyle{ \Lambda^{k} \mathbb{R}^n }[/math] is [math]\displaystyle{ \binom{n}{k} }[/math], which is typically much larger.
Examples
The space of lines through the origin is exactly projective space, so [math]\displaystyle{ \mathrm{Gr} (1, n) \cong \mathbf{P} (\mathbb{R}^n) }[/math]. In 3 dimensions, a plane through the origin is completely defined by its normal, so we get that [math]\displaystyle{ \mathrm{Gr} (2, 3) \cong \mathrm{Gr} (1, 3) \cong \mathbf{P} (\mathbb{R}^3) }[/math], the projective plane.
The simplest non-trivial case is [math]\displaystyle{ \mathrm{Gr} (2, 4) }[/math]. An element [math]\displaystyle{ M }[/math] spanned by two lines [math]\displaystyle{ x, y }[/math], can be represented as the matrix $$ \begin{equation} \begin{bmatrix} x_{1} & x_{2} & x_{3} & x_{4} \\ y_{1} & y_{2} & y_{3} & y_{4} \end{bmatrix} \, . \end{equation} $$
These are not 'proper' coordinates, as doing row operations on this matrix preserves the row-span. Put another way, we can always multiply by some [math]\displaystyle{ g \in GL_k (\mathbb{R}) }[/math].
The projective coordinates can be calculated by taking the determinants of all [math]\displaystyle{ 2 \times 2 }[/math] sub-matrices
$$ p_{ij} = \begin{vmatrix} x_i & x_j \\ y_i & y_j \end{vmatrix} \, , $$
which finally gives us
$$ \begin{equation} \iota (M) = \left[ x \wedge y \right] = \left[ p_{12} : p_{13} : p_{14} : p_{34} : p_{42} : p_{23} \right] \, . \end{equation} $$
Note the use of colons to signify that these coordinates are homogeneous.
Plücker relations
The coordinates must satisfy some algebraic relations called Plücker relations. Generally, the projective space is much 'larger' than the Grassmannian, and the image in the projective space is some quadric surface.
For the example above on [math]\displaystyle{ \mathrm{Gr} (2, 4) }[/math], the Plücker relation is
$$ p_{12} p_{34} - p_{13} p_{24} + p_{14} p_{23} = 0 \, . $$
Note that in this case, there is only one such relation, but in higher dimensions there will be many.
Rational points
A rational point [math]\displaystyle{ P }[/math] on [math]\displaystyle{ \mathrm{Gr}(k, n) }[/math] is a k-dimensional subspace such that [math]\displaystyle{ P \cap \mathbb{Z}^n }[/math] is a rank k sublattice of [math]\displaystyle{ \mathbb{Z}^n }[/math].
The same relations as above can be derived, where the homogeneous coordinates have entries in [math]\displaystyle{ \mathbb{Z} }[/math] instead. Note that in the corresponding equivalence relation, the matrices [math]\displaystyle{ g \in GL_k (\mathbb{Z}) }[/math] have [math]\displaystyle{ |\det (g)| = 1 }[/math].
Because the Plücker coordinates are homogeneous, we can always put them in a 'canonical' form by dividing all entries by their GCD and ensuring the first element is non-negative.