  In this work, we introduce the harmonic generalization of the $m$-tuple
weight enumerators of codes over finite Frobenius rings. A harmonic version of
the MacWilliams-type identity for $m$-tuple weight enumerators of codes over
finite Frobenius ring is also given. Moreover, we define the demi-matroid
analogue of well-known polynomials from matroid theory, namely Tutte
polynomials and coboundary polynomials, and associate them with a harmonic
function. We also prove the Greene-type identity relating these polynomials to
the harmonic $m$-tuple weight enumerators of codes over finite Frobenius rings.
As an application of this Greene-type identity, we provide a simple
combinatorial proof of the MacWilliams-type identity for harmonic $m$-tuple
weight enumerators over finite Frobenius rings. Finally, we provide the
structure of the relative invariant spaces containing the harmonic $m$-tuple
weight enumerators of self-dual codes over finite fields.
