## 2 Proposed Method
### Acoustic Model
#### 2.1.2 Dictionary Extraction  
Dictionary \(P(\omega|s,p,f,o,v)\) with spectral templates from multiple singers is built based on English pure vowels (monophthongs), such as those used in the solfge system of learning music: Do, Re, Mi, Fa, Sol, La, Ti and Do. The dictionaries use spectral templates extracted from solo singing recordings in the Musical Instrument Sound subset of the Real World Computing (RWC) database (RWC-MDB-I-2001 Nos. 45-49) [41]. The recordings contain sequences of notes following a chromatic scale, where the range of notes varies according to the tessitura of distinct vocal parts. Each singer sings a scale in five distinct English vowels (/a/, /ae/, /i/, /b/, /u/). In total, we have used 15 distinct singers: 9 male and 6 female, consisting of 3 human subjects for each voice type (bass, baritone, tenor, alto, soprano).  
Although the aim of this work is the transcription of vocal quartets, we keep the spectral templates from all five voice types in the dictionary because we do not know in advance the voice types present in each audio recording. This decision allows the dictionary to cover a wider variety of vocal timbres during the spectral decomposition, although not all of the resulting voice assignment probabilities will be used during its integration with the music language model for a single song. Rather, our model dynamically aligns one of the dictionary's voice types to each vocal part in a song. This dynamic dictionary alignment is based on the music language model's voice assignments and is discussed further in Section 2.3.  
The fundamental frequency (\(f_{0}\)) sequence from each monophonic recording is estimated using the Probabilistic YIN (PYIN) algorithm [43]. Afterwards, the time-frequency representation is extracted using the VQT, with 60 bins per octave. A spectral template is extracted for each frame, regarding the singer source, vowel type and voice type. In order to incorporate multiple estimates from a common pitch, the set of estimates that fall inside the same pitch bin are replaced by its metrically-trimmed mean, discarding 20% of the samples as possible outliers. The use of the metrically-trimmed mean aims to reduce the influence of possible pitch inaccuracies obtained from the automatic application of the PYIN algorithm. However, there is no guarantee that the final estimate will be free of eventual outliers. The set of spectral templates is then pre-shifted across log-frequency in order to support tuning deviations for \(\pm 20\) and \(\pm 40\) cent and are stored into a six-dimensional tensor matrix \(P(\omega|s,p,f,o,v)\). Due to the available data from the chromatic scales, the resulting dictionary \(P(\omega|s,p,f,o,v)\) has some pitch templates missing, as shown in Figure 2a.  
To address the aforementioned issue, we have investigated alternative ways to fill out the missing templates in the dictionary, including spectrum estimation by replication [14,44], linear and nonlinear interpolation and a generative process based on Gaussian mixture models (inspired by [45,46]). Following experimentation, we have chosen the replication approach, where existing templates belonging to the same dictionary are used to fill in the missing parts of the pitch scale, as this has been shown to achieve the best performance [47]. In this approach, the spectral shape of a given pitch \(p_{n}\) is repeated (with the appropriate log-frequency shift) over all subsequent pitches\(p\in[p_{n+1},p_{m-1}]\) until another template is found (the pitch template \(p_{m}\)). Figure 2b illustrates the resulting dictionary templates of one singer example (vowel /a/) from our audio dataset, following the above replication process.