3.3. Summary
4.2.1. High glottal impedance
4.2.1.4. The impedance at the vocal folds from the glottis
4.2.1.4.1. The impedance seen by the glottis
The next step is to use the measurements made at the lips and a vocal tract model to infer properties of the tract when excited at the glottis. 'Looking in' from the lips, the vocal tract looks like a duct closed or almost closed at the glottis. 'Looking out' from the glottis, the vocal tract looks like a duct open at the lips. Thus, at lowest order approximation, the resonances (impedance minima) measured at the lips are anti-resonances (impedance maxima) at a closed glottis, and vice versa.
For low frequencies and a compact vocal tract, the parameters associated with the mechanical resonances of the tract do not depend upon which end of the tract is used for measurements. For higher frequencies, and provided that the simple cylindrical model is retained, the values estimated for the attenuation constant α would be the same for excitation at both ends. However, bandwidth measurements made with the subjects’ lips sealed around the impedance head could differ from bandwidths that would apply for excitation at the glottis when the subject has an open mouth. Thus the bandwidth measurements of the resonances in the preceding section are not directly comparable with formant bandwidths reported in the literature. For example, the transmission loss due to the resistive term of the radiation impedance of an open cylinder causes the magnitudes of the impedance extrema calculated at the glottis to decrease more rapidly with frequency than those of a cylinder closed at the lips.
So, to consider the impedance at the glottis 'looking out' into the vocal tract: the tract is a non-rigid cylindrical duct, as modelled previously, but is open at the lips with a large flange (the face) as in equation ( 4-2 ). This impedance is an estimate of the impedance as ‘seen’
by the glottis looking toward the open mouth. In Figure 4-4 the impedance of the vocal tract is shown through glottal radii rg = 7 mm and 1 mm. Decreasing the radius of the glottis (increasing the acoustic inertance) causes the extrema in the impedance spectrum to become asymmetric. Thus, for small rg the minima in |Z| decrease in frequency to approach the almost stationary maxima in |Z| as described in Wolfe et al. (2009b).
Figure 4-4 Calculated impedance magnitude of the vocal tract from the glottis (top), pressure transfer function (Plip/Pglottis) (middle), and triangles showing the bandwidth of the transfer function maxima (bottom) for a vocal tract with glottal radius rg = 5 mm (dark) and 1 mm (pale). The solid lines in the lower plot show two formant bandwidth estimations for f0 = 80 Hz and 300 Hz using the estimation equation from Hawks and Miller (1995).
4.2.1.4.2. Transfer function estimation
The top graph in Figure 4-4 shows that, for a glottis with effective radius 1 mm, the inertance dominates the load impedance. For a radius of 5 mm, however, the resonances of the vocal tract dominate. It is expected that the maxima in the pressure transfer function from glottis to lips largely determine the formants Fi. The frequencies and bandwidths of the maxima are shown as triangles in the lower plot of Figure 4-4. These BFi can then be directly compared to the values in the literature determined using external measurements.
Hawks and Miller (1995) determined an empirical f0 dependent relationship between fFi and BFi based on published closed glottis swept sine excitation measurements with a closed glottis (Fant, 1961; Fujimura and Lindqvist, 1971). This relationship is shown in Figure 4-4 along with the values of BFi determined in this fashion. In this context, one should note that estimates of BFi during phonation are greater than the estimates based on closed glottis data. For example, BFi determined from spectral examination of recorded utterances has been reported between 67 and 333 Hz, with median values of BF1, BF2, and BF3 of 130, 150
and 185 Hz respectively (Bogert, 1953). Pham Thi Ngoc and Badin (1994) reported an increase in BF1 of 40-120% from closed glottis to phonation for the French vowel [a]
(where fF1 is close to the fR1 measured in this study) and up to 140% for higher resonances.
The difference in the model values presented here is not so large since the extra losses at the glottis have not been incorporated into the model. Nevertheless, the differences in BRi for the model calculations of the transfer function show increases of 20-100% (see Figure 4-2).
Figure 4-4 also illustrates that increasing the glottal radius increases fF1, which agrees with the experimental evidence on comparisons between phonated and whispered vowels (Jovicic, 1998; Kallail and Emanuel, 1984; Swerdlin et al., 2010), the latter presumably having a larger effective rg.
4.2.1.4.3. Implications for closed vowels
The value of BFi indicated in Figure 4-4 depends strongly on the boundary conditions of the tract at the lips. The impedances from which the parameters of the vocal tract model are derived are measured just inside the lips. So a more accurate model of the impedance at the glottis would include an additional 10 mm in length to account for the position of the measurement head inside the lips, plus the end effect to simulate radiation impedance.
Furthermore, the vowel gesture used in this study, when spoken under normal conditions, would have an effective lip aperture smaller than that enforced by the experiment. In Figure 4-5 the pale curve shows the impedance and transfer function from the glottis where the effective lip radius is 7 mm. Compared to the 10 mm radius and shorter vocal tract in Figure 4-4 the determined values of BFi with the slight lip closure, shown with pale triangles in Figure 4-5, are slightly decreased, and occur at lower frequency due to the longer vocal tract.
Figure 4-5 Calculated impedance magnitude at the glottis (top), pressure transfer function (Plip/Pglottis) (middle), and bandwidth of the transfer function maxima (bottom) for a vocal tract with additional lip length of 10 mm (cf. no lip in Figure 4-4). The pale lines and triangles show the relatively open effective lip radius = 7 mm, and dark lines and triangles show an almost closed effective lip radius = 1 mm for the same non-rigid vocal tract model. The dashed lines show a rigid vocal tract with almost closed lips radius = 1 mm.
The solid lines in the lower plot show formant bandwidth estimations for f0 = 80 Hz and 300 Hz (Hawks and Miller, 1995).
Figure 4-5 also includes the impedance calculated for almost closed lips to demonstrate that BF1 increases as fF1 decreases below ~500 Hz, as observed by Fujimura and Lindqvist (1971). Hawks and Miller (1995) considered this behaviour by using two different sets of parameters for low and high frequency in their BFi estimation. However, their calculation allows for unrealistically low fF1 since it does not consider how the non-rigid vocal tract influences fF1. If the vocal tract were rigid, decreasing the effective lip radius would decrease fF1 until the impedance curve reached an infinitely high impedance at low frequency. However, the inertance of the non-rigid wall (discussed in 3.2.2) acts in parallel with the increasing inertance of the shrinking lip aperture, so the lower limit of fF1 is determined by the non-rigid wall anti-resonance at fA0. Figure 4-5 shows examples of these two situations, with a dashed line for the rigid vocal tract and a solid dark line for the non-rigid vocal tract. The lower limit of fF1 occurs at ~300 Hz for men and ~320 Hz for women
using the average data from the vocal tract configurations in this study. It may be possible to make changes to the area of the tract, i.e. to deviate from cylindrical geometry, to lower this limit. However, it should not be possible for fF1 to occur lower than the closed glottis fA0 reported in Table 3-1 (210 and 245 Hz for men and women respectively). This finding also has implications to the acoustic load on the vocal folds with the large inertive loads used in straw phonation, which is discussed in Chapter 5.
Finally, the combination of a low fF1 due to lip closure and the mechanical properties of the wall may be important for the production of bilabial plosives such as [b], in which the initial broadband sound burst produced is observed to have a characteristic low frequency prominence and downward spectral tilt (Delebecque et al., 2013; Kewley-Port, 1983;
Stevens, 2000).