Physical Modeling of the vocal tract of a Sygyt singer
Source theory v.s. Resonance theory
Two types of overtone-singing should be distinguished: Sygyt and Kargyraa. In Sygyt performances, the rising tongue divides the vocal tract into two cavities, which are connected by a narrow channel, whereas the tongue does not rise in Kargyraa performances.
Up until now, two major theories have been proposed on the production of the melody pitch: (1) The 'double-source' theory (Chernov & Maslov 1987), which asserts the existence of a second sound source such as a whistle-like mechanism formed by the narrowing of the false vocal folds (ventricular folds) in addition to the true vocal fold vibration; and (2) the 'resonance' theory, which asserts that only a glottal sound source exists, but that an upper harmonic is so emphasized by an extreme resonance of the vocal tract that it is segregated from the other components and heard as another pitch. The fact that the melody pitches producible by the singer are limited to the harmonic series of the drone supports the resonance theory (Adachi & Yamada 1999).
Physical modeling of the resonance of the vocal tract of Sygyt singers includes: (1) rear cavity theory, (2) front cavity theory, and (3) resonance-matching theory. The glottal sound source of Sygyt voices is rich in harmonics. This has been attributed to the short open duration of the glottis (Bloothooft et al. 1992, Adachi & Yamada 1999).
Rear cavity theory
Based on vocal tract shape measurements by MRI, Adachi and Yamada (1999) reported that the resonance of the rear cavity, that was from the glottis to the narrowing of the tongue, produced the sharp formant Fk. The resonance of the front cavity, that was from the articulation by the tongue to the mouth exit, was not critical to the production of the melody pitch. The length of the rear cavity decreases as fk increases.
Adachi and Yamada (1999) synthesized tones from transfer functions calculated with and without the front cavity, finding that the front cavity did not affect the formant frequencies, although the magnitude of Fk decreased due to the lack of the front cavity resonance. It is important to note that Adachi and Yamada calculated the transfer functions of a Sygyt singer's vocal tract using a one-dimensional model, in which the tract shape was approximated as a succession of cones. While such models are widely used in speech research, I argue that the change in the tract shape at the articulation point is so abrupt that the assumption of planar-wave fronts clearly breaks down. Theoretically, one-dimensional models are unsuitable for a Sygyt singer's vocal tract.
In practice, the rear cavity theory is not supported by a non-traditional technique of ovetone-singing used by Tran Quang Hai, who calls it 'one-cavity technique' because the tongue does not rise to divide the vocal tract into two cavities. However, there is an articulation point at the soft palate, as to pronounce the velar /ng/. The melody of fk is produced by manipulating the opening of the front cavity, while the rear cavity, that is from the glottis to the soft palate, remains unchanged. This technique suggests that the front cavity may be more important for the production of fk.
Front cavity theory
Based on preliminary impedance measurements of vocal tract by a Jew's harp, Tsai (2001) reported that the resonance of the front cavity determined fk. The author modeled the front cavity as a Helmholtz resonator driven by a flow source U1 at the articulation point. The transfer function can be calculated according to Eq. (6.65) in [Fletcher & Rossing 1991].
Owing to the tract shape at the articulation point, the flow U1 is presumed to be incompressible. It is known that in regions of fast change in pipe geometry, such as a tone hole or the pipe termination, the Helmholtz number He1 implies that the wave equation can locally be approximated by the Laplace equation, which describes an incompressible potential flow (Hirschberg & Kergomard 1995). In overtone-singing, the acoustic flow at the articulation point is therefore incompressible (compact region). This is not true for normal phonations.
The front cavity theory failed to explain the small bandwidth of Fk. Fig. 2 compares the matched theoretical spectral envelops and recorded spectra of a Sygyt voice and a Jew's harp tone, which were produced by me with the same front cavity. It can be seen that the Fk bandwidth of the voice is smaller than that of the Jew's harp tone. The latter was produced without the rear cavity because the rising tongue completely closed the channel between the front and the rear cavities. This discrepancy suggests that the rear cavity may play a role in sharpening Fk.
Figure 2: Spectra of a Sygyt voice (left) and a Jew's harp tone (right) produced with the same front cavity.
The resonance-matching theory takes into account the contributions of both the front and the rear cavities, whose resonances are more or less matched to produce a sharp Fk. Kob (2002) reported that an improvement of the second resonance by about 15 dB was achieved by matching two resonance frequencies, which was fulfilled by manipulating the mouth opening. Although this theory appears to 'unified' the theories of rear/front cavity, it should be noted that according to Table 6.1 in [Kob 2002], the resonance of the front cavity was just close to the second resonance of the rear cavity; Fk could be sharp enough for pitch production without an exact resonance-matching.
Kob (2002) calculated the transfer functions of a Sygyt singer's vocal tract using an improved method of continuous-time interpolated multiconvolution (Barjau et al. 1999), which was originally developed to calculate the impulse response of wind instruments with tone-hole discontinuities. However, this approach does not predict the flow field at the articulation point. Fig. 3 displays the shape of a Sygyt singer's vocal tract and the potential field at the articulation point. As can be seen from the isobar (equal-potential) lines, the acoustic flow has a higher velocity near the tongue. This contradicts the assumption of planar-wave fronts in Kob's calculation.
Figure 3: Shape of a Sygyt singer's vocal tract (left) and the isobar lines at the articulation point (right).
The limitations of one-dimensional models of the vocal tract or the bore of wind instruments should be borne in mind: even at low frequencies evanescent cross-modes will be excited in the rapidly flaring bell section because of strong mode coupling (e.g., Pagneux et al. 1996). In a Sygyt singer's vocal tract, one-dimensional models are suitable only for the rear cavity.
The vocal tract sould be divided into four regions, in which the wave equations have different forms for approximation. In light of Matched Asymptotic Expansions, the global solution can be obtained by 'gluing' the local solutions together (Hirschberg & Kergomard 1995). The four regions are (1) the rear cavity, (2) the compact region at the articulation point, (3) the front cavity as a Helmholtz resonator, and (4) the compact region at the mouth opening. The rear cavity is approximated as a succession of cones, where the acoustic field is governed by the Webster equation for He1. At the articulation point and at the mouth opening, the incompressible air is approximated as a piston. The front cavity is a Helmholtz resonator with a short neck.
If the transfer function of a Sygyt singer's vocal tract does not predict the small bandwidth of the second formant, one should consider the possible effect of acoustic feedback to the glottal source (Levin and Edgerton 1999). This may be related to the nonlinear effect of the adducted ventricular folds.