## 2 Proposed Method
### Acoustic Model  
The acoustic model is a variant of the spectrogram factorisation-based model proposed in [21]. The model's primary goal is to explore the factorization of an input log-frequency spectrogram into components that have a close connection with singing characteristics such as voice type and the vocalization of different vowel sounds. We formulate the model dictionary templates into a six-dimensional tensor, representing log-frequency index, singer source, pitch, tuning deviation with 20 cent resolution, vowel type and voice type. Similarly to [9], the singer source and vowel type parameters constrain the search space into a mixture-of-subspaces, clustering a large variety of singers into a small number of categories. In this model, the voice type parameter corresponds to the vocal part (SATB), where each vocal part is linked to a distinct set of singers (the singer source). For details on the dictionary construction, see Section 2.1.2. As time-frequency representation, we use a normalised variable-Q transform (VQT) spectrogram [39] with a hop size of 20 ms and 20-cent frequency resolution. For convenience, we have chosen a pitch resolution that produces an integer number of bins per semitone (five in this case) and is also close to the range of just noticeable differences in musical intervals [40]. The input VQT spectrogram is denoted as \(X_{\omega,t}\in\mathbb{R}^{\Omega\times T}\), where \(\omega\) denotes log-frequency and \(t\) time. In the model, \(X_{\omega,t}\) is approximated by a bivariate probability distribution \(P(\omega,t)\), which is in turn decomposed as:  
\[P(\omega,t)=P(t)\sum_{s,p,f,\rho,\rho}P(\omega|s,p,f,o,v)P_{t}(s|p)P_{t}(f|p)P _{t}(o|p)P(v)P_{t}(p|v) \tag{1}\]  
where \(P(t)\) is the spectrogram energy (known quantity) and \(P(\omega|s,p,f,o,v)\) is the fixed pre-extracted spectral template dictionary. The variable \(s\) denotes the singer index (out of the collection of singer subjects used to construct the input dictionary); \(p\in\{21,\ldots,108\}\) denotes pitch in Musical Instrument Digital Interface (MIDI) scale; \(f\) denotes tuning deviation from 12-tone equal temperament in 20-cent resolution (\(f\in\{1,\ldots,5\}\), with \(f=3\) denoting ideal tuning); \(o\) denotes the vowel type; and \(v\) denotes  
Figure 1: Proposed system diagram. the voice type (e.g., soprano, alto, tenor, bass). The contribution of specific singer subjects from the training dictionary is modelled by \(P_{t}(s|p)\), i.e., the singer contribution per pitch over time. \(P_{t}(f|p)\) is the tuning deviation per pitch over time, and finally, \(P_{t}(o|p)\) is the time-varying vowel contribution per pitch. (Although \(P_{t}(o|p)\) is not explicitly used in this proposed approach, it is kept to ensure consistency with the Real World Computing (RWC) audio dataset [41] structure (see Section 2.1.2).) Unlike in [21] (which uses \(P_{t}(v|p)\)), this model decomposes the probabilities of pitch and voice type as \(P(v)P_{t}(p|v)\). That is, \(P(v)\) can be viewed as a mixture weight that denotes the overall contribution of each voice type to the whole input recording, and \(P_{t}(p|v)\) denotes the pitch activation for a specific voice type (e.g., SATB) over time. The factorization can be achieved by the expectation-maximization (EM) algorithm [42], where the unknown model parameters \(P_{t}(s|p)\), \(P_{t}(f|p)\), \(P_{t}(o|p)\), \(P_{t}(p|v)\) and \(P(v)\) are iteratively estimated.