This paper proposes a new 3D generative model that learns to synthesize shape variations based on a single example. While generative methods for 3D objects have recently attracted much attention, current techniques often lack geometric details and/or require long training times and large resources. Our approach remedies these issues by combining sparse voxel grids and multiscale point, normal, and color sampling within an encoder-free neural architecture that can be trained efficiently and in parallel. We show that our resulting variations better capture the fine details of their original input and can capture more general types of surfaces than previous SDF-based methods. Moreover, we offer interactive generation of 3D shape variants, allowing more human control in the design loop if needed.