## 2 Proposed Method
### Model Integration  
In this section, we describe the integration of the acoustic model and the music language model into a single system that jointly performs multi-pitch detection and voice assignment from audio. The pitch activations \(P_{t}(p|v)\) for each voice type from the PLCA dictionary (bass, tenor, baritone, alto and soprano) are quite noisy, resulting in very low accuracy for voice assignment, as can be seen from our results (Table 1, row [PERSON] and [PERSON] [21]). However, we have found that a good prior distribution for \(P_{t}(p|v)\) can drive the spectrogram factorisation towards a more meaningful voice  
Figure 4: An example of the music language model being run on the detected pitches from Figure 3 with a beam size of two and two voices. Notes are represented as “[pitch,frame]” and are colour-coded based on their ground truth voice assignment. (Notes belonging to the upper voice are shown in red and notes belonging to the lower voice are shown in blue.) The observed note sets are listed beneath each \(O_{t}\). Notice the false positive pitch detection \([41,3]\) in \(O_{3}\). The two most likely state hypotheses at each step are listed in the large rectangles above each state \(S_{t}\), where the voices are notated with braces. The most likely state hypothesis at each step appears on the bottom row, and each state has an incoming arrow indicating which prior state hypothesis was used to transition into that state. Those state hypotheses with an entirely correct voice assignment are represented by a thick border. assignment. This prior is given by the music language model, and its integration into the system pipeline is performed in two stages. Since multi-pitch detections from the acoustic model are the input for the music language model, spurious detections can result in errors during the voice separation process. Therefore, in the first stage, we run the EM algorithm using only the acoustic model from Section 2.1 for 15 iterations to allow for convergence to stable multi-pitch detections. Next, the system runs for 15 more EM iterations, this time also using the music language model from Section 2.2. During each EM iteration in this second stage, the acoustic model is run first, and then, the language model is run on the resulting multi-pitch detections. To integrate the two models, we apply a fusion mechanism inspired by the one used in [52] to improve the acoustic model's pitch activations based on the resulting voice assignments. The output of the language model is introduced into the acoustic model as a prior to \(P_{t}(p|v)\). During the acoustic model's EM updates, Equation (6) is modified as:  
\[P_{t}^{new}(p|v)=\alpha P_{t}(p|v)+(1-\alpha)\phi_{t}(p|v), \tag{17}\]  
where \(\alpha\) is a weight parameter controlling the effect of the acoustic and language model and \(\phi\) is a hyperparameter defined as:  
\[\phi_{t}(p|v)\propto P_{t}^{a}(p|v)P_{t}(p|v). \tag{18}\]  
\(P_{t}^{a}(p|v)\) is calculated from the most probable final HMM state \(S_{t_{max}}\) using the pitch score \(\Delta_{p}(V,n)\) from the HMM transition function of Equation (14). For \(V\), we use the voice \(V_{v}\in S_{t_{max}}\) as it was at frame \(t-1\), and for \(n\), we use a note at pitch \(p\). The probability values are then normalised over all pitches per voice. The pitch score returns a value of one when the \(V\) is an empty voice (thus becoming