  We study the map from conductances to edge energies for harmonic functions on
finite graphs with Dirichlet boundary conditions. We prove that for any
compatible acyclic orientation and choice of energies there is a unique choice
of conductances such that the associated harmonic function realizes those
orientations and energies. We call the associated function enharmonic. For
rational energies and boundary data the Galois group of ${\mathbb Q}^{tr}$ (the
totally real algebraic numbers) over ${\mathbb Q}$ permutes the enharmonic
functions, acting on the set of compatible acyclic orientations. A consequence
is the non-tileability of certain polygons by rational-area rectangles.
  For planar graphs there is an enharmonic conjugate function, together these
form the real and imaginary parts of a "fixed energy" analytic function. In the
planar scaling limit for ${\mathbb Z}^2$ (and the fixed south/west
orientation), these functions satisfy a nonlinear analog of the Cauchy-Riemann
equations, namely \begin{eqnarray*}u_xv_y &=& 1\\u_yv_x&=&-1.\end{eqnarray*} We
give an analog of the Riemann mapping theorem for these functions, as well as a
variational approach to finding solutions in both the discrete and continuous
settings.
