# Spectral Envelope Transformation in Singing Voice for Advanced Pitch Shifting
## 3 Analysis of Parameters and Their Variation along Pitch
### Formant-Based Model for Spectral Envelope Transformation  
At the sight of the relationships unveiled, we propose a model of spectral envelope transformation for pitch shifting based on the displacement of the location of the formants \(F1\), \(F2\) and \(F3\), with the pitch. The objective is to attain a pitch shifting of any frequency gap. Since we are considering singing voice,  
Figure 4: \(F2\) evolution with pitch, harmonic component.  
Figure 5: \(F3\) evolution with pitch, harmonic component.  
the frequency shift is conveniently expressed in semitones. In an equally-tempered scale, the frequency \(f0^{\prime}\) of the target pitch can be defined by the following expression:  
\[f0^{\prime}=f0\cdot 2^{\left(\frac{N}{2}\right)^{\pm 1}} \tag{10}\]  
where \(f0\) is the original pitch before the shifting and \(N\) is the number of semitones corresponding to the desired frequency shift. The \(\pm\) symbol refers to increasing or decreasing the pitch. With all this, the frequency displacement \(\Delta f0\) is defined as:  
\[\Delta f0=f0^{\prime}-f0 \tag{11}\]  
Then, after the analysis described before, the following generic expression is used to modify the parameters \(F_{i}\):  
\[F_{i}^{\prime}=F_{i}+g_{i}(\Delta f0) \tag{12}\]  
where \(F_{i}^{\prime}\) is the objective value of central frequency of formant \(i\), \(F_{i}\) stands for the original value and \(g_{i}(\cdot)\) is a function dependent on the format number \(i\).  
By means of linear regression, we obtained the following expressions for the model transformation functions:  
\[F_{1}^{\prime}=F_{1}+g_{1}(\Delta f0)=F_{1}+A\cdot\Delta f0 \tag{13}\]  
\[F_{2}^{\prime}=F_{2}+g_{2}(\Delta f0)=F_{2}+10^{(2\cdot F_{1}^{\prime}\cdot B \cdot c^{\prime}\cdot\Delta f0)} \tag{14}\]  
\[F_{3}^{\prime}=F_{3}+g_{3}(\Delta f0)=F_{3}+10^{(3\cdot F_{1}^{\prime}\cdot D \cdot c^{\prime}\cdot\Delta f0)} \tag{15}\]  
where the constants \(A,B,C,D\) and \(E\) are defined in Table 2. Note the similar, but different values found for the harmonic and the residual component of the signal, as described in Section 3.2.  
Observe that the values in Table 2 imply that medium-large frequency displacements are required in order to perceive the effect of the envelope transformation. Generally, our observations indicated that increasing or decreasing the pitch by a fifth is enough to perceive the changes. Smaller shifts do not produce significant changes in the spectral envelope, which made the difference between our model and a conventional pitch shifting algorithm unnoticeable/hardly noticeable.  
Finally, note that a change of \(f0\) produces the same effect for lower and higher pitches since the relationship between formants is already defined by linear or logarithmic expressions (13)-(15). However, it must be kept in mind that there is a huge difference in the perceptual influence of this transformations in lower or higher frequencies due to the relationship between \(f0\) and the perceived note. Higher frequencies need larger changes of \(f0\) for changing a note (see Equation (10)).