False Vocal Fold Surface Waves During Sygyt Singing: a theoretical study by Chen-Gia Tsai
Overtone singing is a vocal technique found in Central Asian cultures such as Tuva and Mongolia, by which one singer produces a high pitch of nF0 along with a low drone pitch of F0 (F0 is the fundamental frequency, n = 6, 7, ...13 in typical performances). The voice of overtone singing is characterized by a sharp formant centred at nF0.
There are two approaches of physical modelling of overtone singing: (1) the double-source theory (Chernov and Maslov 1987), which asserts the existence of a second sound source that is responsible for the melody pitch; and (2) the resonance theory, which asserts that a harmonic is emphasized by an extreme resonance of the vocal tract. The fact that the melody pitches producible by the singer are limited to the harmonic series of the drone was regarded as robust support of the resonance theory (Adachi and Yamada 1999).
From a psychoacoustic point of view, a small bandwidth of the prominent formant is critical to a clear melody in Sygyt singing. A preliminary study using an autocorrelation model for pitch extraction suggested that the pitch strength of nF0 increased along with the Q value of this formant, with the formant magnitude playing a secondary role (see Perception of overtone singing). The amplified harmonic in a Sygyt voice can be 15 dB stronger than its flanking components. If the amplification of this harmonic cannot be explained in terms of vocal tract impedance, it should be attributed to the source signal.
Figure 1: Spectrum of a Sygyt voice produced by a singer from Tuva. The 18th harmonic is 15 dB stronger than its flanking components. It is likely that the false vocal folds generate the 9th, 18th, and even 27th harmonics.
The insufficiency of the resonance theory is notable in the spectra shown in Figs 1 and 2. The formant at 3 kHz of the Sygyt voice (Fig. 1) is so sharp that it may not be explained by tract filtering. On the other hand, the centre frequencies of the first and second formants of Kargyraa voices always stand in the ratio of 1:2 (Fig.2). This strange phenomenon suggests the existence of an unknown glottal source that produces the outstanding component at F1, and its second harmonic.
Figure 2: Two snapshot spectra of a
"The far side of a dry riverbed"
The goal of this study is to offer a physical model based on a nonlinear loop that explains the harmonic amplification in Sygyt. This model asserts that surface waves (Rayleigh waves) of the adducted false vocal folds can actively amplify a harmonic. In this theoretical study I discuss the interactions between the false vocal fold surface waves (FVFSWs), the glottal flow and acoustic waves.
2.1 Rayleigh surface waves
The Rayleigh surface wave is a specific superposition of a transverse wave and a longitudinal wave of an elastic solid (see, e.g. Achenbach 1984). Its amplitude is significant only near the surface and attenuates exponentially with the depth. The trajectories of material particles are ellipses. At the surface the normal displacement is about 1.5 times the tangential displacement. The velocity of Rayleigh waves, independent on the wavelength, is about 0.9 times the transverse wave velocity. Rayleigh's theory of surface waves has been generalized to viscoelastic solids (see, e.g. Romeo 2001).
The assumption of Rayleigh surface wave on the false vocal folds is supported, although indirectly, by recent measurements of the medial surface dynamics of the vocal folds (Berry et al. 2001). The trajectories of surface fleshpoints were approximately ellipses, with the length ratio of the two axes varying in the range of 1.5-2.0. This value is in remarkable agreement with Rayleigh's theory of surface waves.
2.2 Surface wave instability
The mucosal wave grows in amplitude when propagating in the same direction as the glottal flow. It is a phenomenon of wave instability with similarity of a fluttering flag in the wind. Mathematically, it can easily be shown that the mucosal wave and the flag wave absorb the kinetic energy of the flow through the effects of the Coriolis force. Other effects contributing to wave instability are (1) the centrifugal force and (2) the viscous force at the separation point. Unfortunately, these effects have not been taken into account in two-mass or three-mass models of the vocal folds.
Fluid-structure interaction (Paidoussis 1998) is important in biomechanics (Carpenter et al. 2000, Huber 2000, Fenlon and David 2001). In the field of voice research, the fluid-structure interactions occurring around the true/false glottis are poorly understood. It is instructive to compare them to the system of fluttering flags in the wind ( Chang et al. 1991, Chang and Moretti 1991, Tang et al. 2003, Watanabe et al. 2002, Zhang et al. 2000, Zhu and Peskin 2002, Zhu and Peskin 2003 ).
It has been proposed that flag flutter is caused either by vortex-shedding from the flagpole, or else by pressure-feedback from the vortex-street in the wake of a flat plate or sheet. However, observed flutter does not match either Strouhal frequency (Zhang et al. 2000). Hence, one should look for an instability phenomenon.
The pressure difference across the flag generated by a potential flow field can be described by aerodynamic mass terms resembling the "gyroscopic" inertia, Coriolis, and centrifugal coefficients: where w is the displacement of the flag, U the far-field flow velocity. This equation can be dated back to Bourrieres (1939) in a paper on the dynamics of pipes conveying fluid. This paper, published in the year of the outbreak of the Second World War, was effectively 'lost', and researchers re-derived this equation in 1950s and 1960s (see Paidoussis 1998, page 59).
I suggest that the second term, which corresponds to the effect of the Coriolis force, contributes to the surface wave (dynamic) instability, which has been shown in the measurement of the medial surface dynamics of the vocal folds (Berry et al. 2001). This is consistent with the vocal fold model proposed by Horáček and vec (2002), who regard the term of the Coriolis force as the aerodynamic damping. The surface wave instability can be attributed to a negative aerodynamic damping. Moreover, the centrifugal force may also play a role in wave (static) instability (Moretti 2003). Further investigations are needed to quantify the glottic fluid-structure interactions.
2.3 Physical modelling of Sygyt
We suppose that the surface wave is triggered at the narrowing of the false vocal folds where the flow velocity is high enough to induce significant surface wave instability. The FVFSW grows in amplitude while travelling upward, significantly modulating the flow at the point of flow separation.
Based on the assumption of elliptic movements of fleshpoints on the false folds, snapshots of this wave can be obtained. The ellipses in Figs. 3a and 3b represent the trajectory of fleshpoints. We estimate the energy exchange between the flow and the tissue occurs at one point. In Fig. 3a the work done by the viscous flow at this point is positive. In Fig. 3b the flow separates upstream, performing no work (or positive work, if back-flow appears) at this point. It can easily be seen that over a period the FVFSW absorbs energy from the flow in the vicinity of the flow separation point, which moves back and forth at a crest of the FVFSW, modulating the flow through the false folds at frequency of nF0. This leads to a varicose jet producing the harmonic at nF0 in the source signal. This harmonic is in turn reinforced by the strong vocal tract resonance at nF0.
Figure 3a and b: Snapshots of the surface wave on the left false fold. The dashed curve represents the rest position of the surface.
To sum up, a loop for Sygyt is established in terms of (1) linear resonator: the vocal tract with resonance at nF0, (2) energy source: pressure difference across the false glottis, and (3) nonlinear amplifier: the fluid-structure interaction around the false glottis. This self-sustained oscillator differs from the true vocal folds in that the false fold mucosa does not vibrate at any intrinsic resonance, but rather respond to the acoustic pressure.
Figure 4: Feedback loop for harmonic amplification in Sygyt.
The present model of "varicose jet oscillations induced by surface waves of curved walls in the vicinity of the flow separation point" could be regarded as a counter-part of the jet-resonator model discussed by Meissner (2002). It should be noted that both the jet blown by a flute player and the false fold mucosa do not vibrate at their intrinsic resonance, but respond to the acoustic field. That is why their vibration frequency can be changed rapidly by manipulating the resonators (the fingering for flute playing and the tongue position for Sygyt singing).
Helmholtz resonator - Sinuous jet
Helmholtz resonator - Varicose jet - Surface wave
Acoustic flow acting on the free jet
Acoustic pressure acting on the surface
Flow separation - Jet instability
Surface wave instability - Flow separation
Coriolis force: acoustic flow/jet interaction
Coriolis force: surface wave/jet interaction
Table 1: A comparison of the feedback loop of a flute-like system (left) and Sygyt (right).
The present model explains the crucial role of the adduction of the false folds in Sygyt technique. Because of this adduction the flow velocity over their mucosal layers is high enough to induce FVFSW instability. It is interesting to note that FVFSWs have been observed in patients suffering from ventricular dysphonia (Nasri et al. 1996), although their frequencies appeared to be much lower than those during Sygyt singing.
From an empirical standpoint, learning Sygyt is much more difficult than it is implicated by the resonance theory. In workshops of overtone singing, it has been repeatedly observed that only very few people are able to produce voices with a clear melody pitch. The present model predicts that one cannot sing Sygyt well even when manipulating the tract shape perfectly, because his false folds are not correctly adducted, or their mucosal layers do not have a proper shape, thickness, and viscoelastic properties.
During a 4 kHz pure tonal vocalization, significant surface waves of the false vocal folds have been detected (Tsai et al. 2004). This provides indirect evidence supporting my Sygyt model.
Figure 5: Spectrum of a pure tonal voice produced by me. During this vocalization, strong surface vibrations of the false vocal fold were detected by colour Doppler imaging (Tsai et al. 2004).
4. Concluding Remarks
The surface wave of the false folds may appear in some Sygyt singers. However, a general conclusion could not be given because there are different types of Sygyt technique.
The resonance theory and the double-source theory are not exclusive. The loop described in our model tends to "unify" these two theories of overtone singing. Whereas the true vocal folds and the vocal tract are, as usual, viewed as the independent source and filter, the false fold mucosa plays a key role in introducing acoustic feedback into the loop for harmonic amplification. This loop may also occur for other constrictions in the vocal tract, such as the soft palate (see velar-like voice with a sharp singer's formant).
Our model may also shed new light on the physical modelling of the vocal folds and the possible effect of acoustic feedback, especially for the phonations with large open quotient values. The model of Rayleigh waves and the effects of Coriolis force/centrifugal force in the glottic fluid-structure interaction demand further research.
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