False Vocal Fold Surface Waves During Sygyt Singing: a
theoretical study
by Chen-Gia Tsai
1. Introduction
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
Kargyraa song
"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. Theory
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).
3. Discussion
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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