We propose an algorithm for the blind separation of single-channel audio signals. It is based on a parametric model that describes the spectral properties of the sounds of musical instruments independently of pitch. We develop a novel sparse pursuit algorithm that can match the discrete frequency spectra from the recorded signal with the continuous spectra delivered by the model. We first use this algorithm to convert an STFT spectrogram from the recording into a novel form of log-frequency spectrogram whose resolution exceeds that of the mel spectrogram. We then make use of the pitch-invariant properties of that representation in order to identify the sounds of the instruments via the same sparse pursuit method. As the model parameters which characterize the musical instruments are not known beforehand, we train a dictionary that contains them, using a modified version of Adam. Applying the algorithm on various audio samples, we find that it is capable of producing high-quality separation results when the model assumptions are satisfied and the instruments are clearly distinguishable, but combinations of instruments with similar spectral characteristics pose a conceptual difficulty. While a key feature of the model is that it explicitly models inharmonicity, its presence can also still impede performance of the sparse pursuit algorithm. In general, due to its pitch-invariance, our method is especially suitable for dealing with spectra from acoustic instruments, requiring only a minimal number of hyperparameters to be preset. Additionally, we demonstrate that the dictionary that is constructed for one recording can be applied to a different recording with similar instruments without additional training.