cs.SDOct 19, 2021
SaveA mathematical model of the vowel space
Organizations: GIPSA-PCMD
Abstract
The articulatory-acoustic relationship is many-to-one and non linear and this is a great limitation for studying speech production. A simplification is proposed to set a bijection between the vowel space (f1, f2) and the parametric space of different vocal tract models. The generic area function model is based on mixtures of cosines allowing the generation of main vowels with two formulas. Then the mixture function is transformed into a coordination function able to deal with articulatory parameters. This is shown that the coordination function acts similarly with the Fant's model and with the 4-Tube DRM derived from the generic model.
Figures & tables
Figure 1: The vowel space of the generic vocal tract model and its 8 characteristic vowels (Mm. Multimedia and supplementary material ). The area functions (Mm. Multimedia and supplementary material ) are plotted for each vowel with, at the top, the pairs of cosine Fourier coefficients which are introduced in the vowel equation (Eq. 4a b). For [o,e], we have .
Figure 2: Plot of the functions expressed with the values of derived in Table 1 . This shows how are set at their respective phases and it focuses on for . The expression of is consistent with this given by Eq. 3b because and .
Figure 3: Simulation in condition together with the vowel space with obtained with the coordination function (Mm. Multimedia and supplementary material and Mm. Multimedia and supplementary material ). All (non figured) points of are inside of this. The spaces derived from the Schroeder-Ehrenfest relation and Fourier coefficients before and after rectification ("SE" from and ) and the biased relation ("est" from ) are plotted relative to the neutral reference . The bias transforms the circular shape "SE" from into the triangular shape "est".
Figure 4: (a) The design of the Fant model redrawn from ( Badin et al., 1990 , Fig. 1) and the definition of geometrical parameters used in Table 3 (b) Simulation in condition together with the vowel space with obtained with the coordination function (Mm. Multimedia and supplementary material and Mm. Multimedia and supplementary material ). All (non figured) points of are inside of this.
Figure 5: Simulations of in condition and : (a) DRM (b) Fant model. Abscissa: Fourier cosine coefficients of the area function. Ordinate: Deviations from neutral in relative values for the two first formants. The circled gray surfaces represent the configurations produced by the coordination function in and the dots represent which is a simulation of the 4 parameters of each model but uncorrelated in the same domain of variation .
Figure 6: Plot of the estimates of the deviations from neutral in relative values for DRM . For the first formant, this is close to the bisecting line thanks to the bias.
Figure 7: Overlay of DRM data with ( Boë et al., 2019 , Fig. 4 with ) . This shows that a 4-tube model driven by 7 parameters has a significantly larger space than one with only 2 parameters. The human vowel space is depicted by background ellipses and only the [i] area is not well covered by this model.