### Related Work  
The audio source separation problem can be formulated in a variety of settings. (See [3, 4, 5] for a thorough overview.) For the purpose of this work, we only regard the case that the input signal is single-channel and the separation is therefore always underdetermined. Different algorithms can be used if multiple channels are available (typically corresponding to microphones simultaneously recording the audio scene). Moreover, we always assume melodic instruments rather than speech or percussive instruments. While different kinds of prior information can be considered (such as specific characteristics of the sounds of the instruments, training data, or the musical score), we concentrate on the blind case with no prior information and instead rely on a learned parametric model for the sounds of the individual instruments. Many algorithms for this problem are based on non-negative matrix factorization (NMF) of the spectrogram. In the simplest form, each tone of an instrument at a particular pitch has its own representation as a dictionary atom [6, 7]. To make the representation of the sound of a particular instrument applicable at arbitrary pitch (_pitch-invariance_), one often employs _tensor factorization_, cf. [5]. In this case, the use of a log-frequency spectrogram (such as the constant-Q transform or the mel spectrogram) can be helpful, since it is also pitch-invariant in the sense that changing the fundamental frequency of a tone merely causes a shift in the representation [8, 9, 10, 11]. A separation approach that is mathematically equivalent to NMF is _probabilistic latent component analysis_ (PLCA) [12, 13] which also exists in variants that use a pitch-invariant model on top of a log-frequency spectrogram [14, 15]. The next step in abstraction is to model the individual harmonics separately while enforcing sparsity in the spectral representation [16]. Decreasing the variability, e.g., the number of parameters, in a model is generally beneficial since it reduces the risk of overfitting. However, we can go one step further by employing an explicit physical model for the tones of the instruments involving a minimum number of parameters. This has the additional advantage that while the previously discussed approaches with a log-frequency spectrogram can only be pitch-invariant if the instruments are tuned to the same log-frequency scale (_equal-temperament tuning_, cf. [17], defined as the frequency ratio corresponding to a musical interval being constant regardless of pitch), a physical model can be evaluated at any fundamental frequencies and sampled arbitrarily. This is crucial when dealing with acoustic instruments that either deliberately deviate from equal temperament or might simply be slightly out of tune. With a continuous model, it is thus possible to use a pitch-invariant representation on a linear-frequency spectrogram. However, the challenge then is to identify the fundamental frequencies on a continuous domain. [PERSON] et al. [18] use a peak detection and clustering algorithm to reduce the problem to a combinatorial one that can be approached via appropriate heuristics. [PERSON] et al. [19], in a polyphonic single-instrument setting, consider the fundamental frequency as an optimizable parameter and use an NMF-type update rule, but they remark that this is only possible on a local scale due to the high number of local minima. Even with a physical model, it turns out that the log-frequency spectrogram can still be helpful: [PERSON] and [PERSON] [20] use its pitch-invariance property to obtain the approximate fundamental frequency of a tone via simple cross-correlation. After that, numerical optimization is performed to improve the estimate on a local scale. The optimization procedure also incorporates inharmonicity, which violates pitch-invariance, and variable width of the peaks in the spectrum which can occur, for instance, at tone boundaries.