This paper introduces MR-CQTdiff, a novel neural-network architecture for diffusion-based audio generation that leverages a multi-resolution Constant-$Q$ Transform (C$Q$T). The proposed architecture employs an efficient, invertible CQT framework that adjusts the time-frequency resolution on an octave-by-octave basis. This design addresses the issue of low temporal resolution at lower frequencies, enabling more flexible and expressive audio generation. We conduct an evaluation using the Fréchet Audio Distance (FAD) metric across various architectures and two datasets. Experimental results demonstrate that MR-CQTdiff achieves state-of-the-art audio quality, outperforming competing architectures.