Same eigenvalue spectrum with different matrices

Question

There are two matrices $D$ and $T$ which provides eigenvalue spectrum (dispersion relation) according to $$ E(K) = \operatorname{eig}(D + T \exp(iK) + T' \exp(-iK)) $$ $$ K = 0:dK:\pi $$ Where K is a list of real numbers from 0 to $\pi$ spaced by $dK$. Is it possible to find 2 new matrices $d$ and $t$ with much smaller size, such that in some range of spectrum ($E_1 < Eigenvalues < E_2)$, we can reconstruct the old dispersion ($E(K)$)? $$ e(K) = \operatorname{eig}(d + t \exp(iK) + t' \exp(-iK)) \approx E(K) $$ $$ E_1 < E < E_2 $$ Note $T'$ is the conjugate transpose of $T$.