We will discuss a common generalization of the Taylor spectrum and the pseudospectrum, called the multivariate pseudospectrum. Applied to three or more hermitian matrices, this variation on the spectrum leads to spaces with interesting topology. When these matrices are almost commuting, their multivariate pseudospectrum that can be computed by numerical methods. Since the Taylor spectrum can be infinite, some form of $K$-theory is needed to make rigorous inferences from the numerics.