Estimation of driving impedance at the vocal folds

It is not straightforward to consider the acoustic load acting on the vocal folds. Vertical motion of the vocal folds, something like an outward swinging door (Fletcher, 1993;

Helmholtz and Ellis, 1875), would be driven by the pressure difference P – P across it.

1985). However, it is also possible that some of the force separating the vocal folds is provided by the average pressure in the glottis, like the sliding door motion described by Helmholtz and Fletcher, so the vocal folds might be affected by Z_sg and Z_vt in parallel, as suggested by Stevens (2000), Chi and Sonderegger (2007), and Lulich (2010).

Figure 4-7 shows the impedance and transfer functions from the glottis loaded by either:

Z_vt (pale), the series addition of Zsg and Zvt (dark); and the parallel addition of Z_sg and Zvt

(dashed). Both the series and parallel load impedances show additional peaks associated with the minima of the Z_sg. Such features have been observed as formants in the spectra of in vivo phonation (Fant et al., 1972) and swept sine measurements with excitation close to the open glottis in vivo (Fujimura and Lindqvist, 1971). These spectral features may influence the characteristics of breathy phonation, where the average r_g is large.

In Figure 4-4 and Figure 4-7, the impedance of the vocal tract is considered as seen from the lower part of the glottis, i.e. through its entire length. Since flow separation is expected to occur at the upper part of the glottis, it may be more useful to consider the impedance at the top of the glottis, i.e. looking through the end effect δ from equation ( 4-3 ), as the impedance seen by the vocal folds. Figure 4-8 is comparable with Figure 4-7 except that only the end effect of the glottal length is considered. This has a very similar effect on the impedance curve to increasing the glottal radius, as shown in Figure 4-4, i.e. the impedance extrema become more symmetric, and the frequencies of the pressure function maxima increase.

Figure 4-8 Calculations of impedance magnitude at the lips (top), impedance magnitude through the glottal end effect (middle), and pressure transfer function (Plip/Pglottis) (bottom) for a large glottal radius rg = 3 mm.

c.f. Figure 4-7. The solid pale line shows the impedance calculated independently of the subglottal tract. The solid dark line shows the impedance of the glottis considered as loaded with the series impedances of Zvt and Zsg. The dashed line shows the glottis with these impedances providing the load in parallel. In both cases, an extra peak in the transfer function occurs close to the minimum in Zsg (see Figure 4-6). The maxima in Zsg do not affect the transfer functions.

4.3. Summary

Low pitched, M1 phonation increases the bandwidths of the acoustic vocal tract resonances, but does not significantly change the resonance and formant frequencies, compared to closed glottis measurements. Some of this increase is accounted for by the inertive acoustic load that a partially open glottis adds to the vocal tract but it is expected that higher visco-thermal and/or turbulent losses due to the airflow through the glottis are also involved.

Considering the impedance of the glottis, two different behaviours are observed with high

For high glottal impedance (rg < 1 mm): f_A0 increases as Zg is lowered. The frequencies of the acoustic resonances Ri do not change significantly at the 1% level, and Ai change little.

The transfer function is highly asymmetric so that f_Fi and f_Ri of the measured impedance at the lips are close to the maximum in the output impedance at the glottis.

Resonances of the subglottal tract are not evident in the impedance measured through the lips during low pitched, modal phonation due to the small average glottal apertures (and possibly additional glottal energy loss factors).

For vowel production in speech and singing, R0 can be neglected since it lies below achievable f₀ even with period doubling phonation. However, A0 may influence the relative prominence of f₀ or its harmonics particularly when the mouth opening is small.

Furthermore, fA0 imposes a lower limit to f_F1, since at low frequencies the wall inertance is in series with the inertance of the mouth opening. So it is not possible to reduce f_F1 below f_A0. This also causes B_F1 to increase as f_F1 approaches this value in agreement with the literature. This is a possible reason for the limit of the vowel plane in the low F1 direction for all languages, which is an interesting observation about speech in general.

For low glottal impedance (rg > 1 mm): the impedance at the glottis looking out becomes more symmetric, so that f_Fi and B_Fi increase, as observed when changing from normal phonation to whispering in both the resonances and the formants.

Since some types of vocal fold motion may be driven by the series combination of Z_vt and Z_sg and other types possibly by the parallel combination, depending on the model, pressure transfer functions under these two conditions were calculated and both show peaks close to the minima of Z_sg, which may be observable in breathy phonation.

This chapter explores the P_sg, f₀ relationship of human excised larynges as a function of the type of laryngeal control⁷. The vibratory behaviour was controlled aerodynamically, by adjusting the air supply, and mechanically, by adjusting the laryngeal geometry corresponding to a combination of laryngeal muscle movements. No vocal tract was present in the experiments, so the acoustic load downstream of the vocal folds is small and inertive. In this configuration the impedance of the subglottal tract may be expected to influence the motion of the vocal folds, so the upstream impedance of the air supply was estimated and its influence on the vocal folds investigated.