The most salient feature of this hierarchical scheme is the absence of FFTs. While FFT-driven algorithms have proven to be highly efficient on a wide range of hardware, the global transposes required severely hamper parallel scalability. The short-ranged, direct interactions can be efficiently parallelized using neutral territory methods, while the long-ranged interactions in this rudimentary algorithm require a single one-sided import from all appropriate remote nodes; this communication pattern is vastly different to the all-to-all patterns required for FFTs and related to the locally essential tree (LET) concept of the FMM. A more elaborate variant avoids the explicit list of remote nodes in step 4 by introducing a downwards pass, in which the potentials from a parent's nearest neighbors subtracted from the accumulated potentials, which are then propagated down to the children of that parent node; we have made a C++ code to demonstrate this scheme freely available. As well as considering different potential accumulation schemes, the independent nature of the cubature points permits distributing the work over these points, instead of by partitioning over particles; in this case a reduction operation would be required to combine each processor's partial contribution to the energy and forces.