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We present a unifying framework that allows us to study the mixed crystalline-electromagnetic responses of topological semimetals in spatial dimensions up to $D=3$through dimensional augmentation and reduction procedures. We show how this framework illuminates relations between the previously known topological semimetals and use it to identify a new class of quadrupolar nodal line semimetals for which we construct a lattice tight-binding Hamiltonian. We further utilize this framework to quantify a variety of mixed crystalline-electromagnetic responses, including several that have not previously been explored in existing literature, and show that the corresponding coefficients are universally proportional to weighted momentum-energy multipole moments of the nodal points (or lines) of the semimetal. We introduce lattice gauge fields that couple to the crystal momentum and describe how tools including the gradient expansion procedure, dimensional reduction, compactification, and the Kubo formula can be used to systematically derive these responses and their coefficients. We further substantiate these findings through analytical physical arguments, microscopic calculations, and explicit numerical simulations employing tight-binding models. Published by the American Physical Society2024