Both at low and higher cyclotron harmonics, properly accounting for finite Larmor radius effects is crucial in many ion cyclotron resonance frequency heating scenario's creating high energy tails. This paper discusses an extension TOMCAT-U of the 1D TOMCAT tokamak plasma wave equation solver (Van Eester and Koch 1998 Plasma Phys. Control. Fusion40 1949) to arbitrary harmonics and arbitrary wavelengths while only keeping leading order terms in equilibrium variation terms. Rather than adopting the particle position, the guiding center position is used as the independent variable when writing down an expression for the dielectric response that is suitable for numerical application. This choice of independent variable yields intuitive expressions involving the Kennel–Engelmann operator which can directly be linked to the corresponding expressions in the RF diffusion operator appearing in the Fokker–Planck equation. It also guarantees that a positive definite power transfer from waves to particles is ensured for any of the wave modes in a plasma in which all populations have a Maxwellian distribution, as is expected from first principles. Rather than relying on a truncated Taylor series expansion of the dielectric response, an integrodifferential approach that retains all finite Larmor radius effects is proposed. To keep the required computation time for this generalized description reasonable, tabulation of integrals is intensively used. Although the accent is on the presentation of the upgraded formalism as well as the adopted recursions and tabulations, a few examples are provided to illustrate the potential of the new wave code that relies on these tabulations.