  Inspired by the idea of blurring the exponential function, we define blurred
variants of the $j$-function and its derivatives, where blurring is given by
the action of a subgroup of $\rm{GL}_2(\mathbb{C})$. For a dense subgroup (in
the complex topology) we prove an Existential Closedness theorem which states
that all systems of equations in terms of the corresponding blurred $j$ with
derivatives have complex solutions, except where there is a functional
transcendence reason why they should not. For the $j$-function without
derivatives we prove a stronger theorem, namely, Existential Closedness for $j$
blurred by the action of a subgroup which is dense in
$\rm{GL}_2^+(\mathbb{R})$, but not necessarily in $\rm{GL}_2(\mathbb{C})$.
  We also show that for a suitably chosen countable algebraically closed
subfield $C \subseteq \mathbb{C}$, the complex field augmented with a predicate
for the blurring of the $j$-function by $\rm{GL}_2(C)$ is model theoretically
tame, in particular, $\omega$-stable and quasiminimal.
