The task of bandwidth extension addresses the generation of missing high frequencies of audio signals based on knowledge of the low-frequency part of the sound. This task applies to various problems, such as audio coding or audio restoration. In this article, we focus on efficient bandwidth extension of monophonic and polyphonic musical signals using a differentiable digital signal processing (DDSP) model. Such a model is composed of a neural network part with relatively few parameters trained to infer the parameters of a differentiable digital signal processing model, which efficiently generates the output full-band audio signal. We first address bandwidth extension of monophonic signals, and then propose two methods to explicitely handle polyphonic signals. The benefits of the proposed models are first demonstrated on monophonic and polyphonic synthetic data against a baseline and a deep-learning-based resnet model. The models are next evaluated on recorded monophonic and polyphonic data, for a wide variety of instruments and musical genres. We show that all proposed models surpass a higher complexity deep learning model for an objective metric computed in the frequency domain. A MUSHRA listening test confirms the superiority of the proposed approach in terms of perceptual quality.