False vocal fold surface waves during
Sygyt singing: A hypothesis
Chen-Gia Tsai, Yio-Wha Shau, and Tzu-Yu
Hsiao
Abstract
Overtone singing is a vocal
technique found in Central Asian cultures, by which one singer produces a high
pitch of nF0 along with a low drone pitch of F0. The pitch of nF0 arises from a
very sharp formant. Current physical modelling of overtone singing asserts that
the harmonic at nF0 is emphasized by a resonance of the vocal tract. However,
this approach could not explain the extraordinarily small bandwidth of this
formant.
This
paper offers a hypothesis that surface waves (Rayleigh waves) of the false vocal
folds might actively amplify the harmonic at nF0 in a specific technique of
overtone singing: Sygyt. We propose a loop for harmonic amplification, which is
composed of (1) the vocal tract with resonance nF0, (2) surface waves of the
false vocal folds, and (3) a varicose jet separating from the false folds. This
model receives indirect support from an experimental study on a novel human
vocalization, which is characterized by a prominent component at 4 kHz. During
this pure tonal vocalization, false fold surface vibrations were detected by
ultrasound colour Doppler imaging. High-frequency false fold surface waves may
also occur during Sygyt singing.
1. Introduction
Overtone singing (or throat
singing, biphonic 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 centered at nF0, as can be seen in Figs. 1 and 2.
Traditional techniques of overtone 
singing include Khoomei, Sygyt, Kargyraa and others.
There
are two approaches of physical modelling of overtone singing: (1) the
double-source theory [1], 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 a 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 [2].
Recent
attempts of physical modleling of Sygyt were concerned with calculation of the
transfer function of the vocal tract using one-dimensional models, successfully
predicting the formant frequency [2,3]. From a theoretical standpoint, however,
this approach may not be suitable for the tract with a rapidly flaring bell
section. A Sygyt singer raises the tongue so that the tract shape changes
abruptly at the narrowing of the tongue (marked with a red dot in Fig. 1b),
where the assumption of planar wave fronts breaks down, and evanescent
cross-modes can be excited in this flaring section even at low frequencies [4].
This may leads to errors in transfer function calculation using one-dimensional
models. An alternative approach of Matched Asymptotic Expansions for modelling
a Sygyt singer’s vocal tract was proposed in [5].
In
a two-resonator theory, a Sygyt singer’s vocal tract was modelled as a coupled
system of a longitudinal resonator that was from the glottis to the narrowing
of the tongue, and a Helmholtz resonator that was from the articulation by the
tongue to the mouth exit. Experiments showed that for some Sygyt voices with a
sharp formant two resonances were matched, while a melody pitch can be
perceived even in the case of not exactly matched resonances [6]. Although the
formant magnitude was shown to be increased by resonance matching [3], it is
unclear whether resonance-matching will reduce the formant bandwidth.
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 [5]. The spectrum of the Sygyt voice
shown in Fig. 1a has the 12th harmonic approximately 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.
The
insufficiency of the resonance theory is even more notable in another technique
of overtone singing: Kargyraa. A
Kargyraa singer uses his
false vocal folds to produce low pitched drone, manipulating his mouth opening
to change the vocal tract resonance. Spectra in Fig. 2 show that the centre
frequencies of the first and second formants of Kargyraa voices always stand in
the ratio of 1:2. This strange phenomenon suggests an unknown glottal source
that produces the outstanding component at F1, and its second harmonic.
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. We first discuss the interactions between the
false vocal fold surface waves (FVFSWs), the glottal flow and acoustic waves. A
preliminary experiment that provided indirect evidence of this model is then
addressed.
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. [7]). 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. [8]).
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 [9]. The trajectories of 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. Physical modelling of
Sygyt
Here we propose a physical
model that describes how FVFSWs absorb the energy of the glottal flow and
acoustic waves.
The false folds are
significantly adducted during Sygyt singing. Hence, the volume flow through
them (UF) is sensitive to FVFSWs. FVFSWs are supposed to be triggered by the
acoustic pressure, which is predominated by the resonance of the vocal tract
nF0. So we assume a FVFSW with the frequency of nF0.
Based
on the assumption of elliptic movements of fleshpoints on the false folds,
snapshots of this wave can be obtained. The ellipses in Figs. 3b and 
3c represent the trajectory of fleshpoints. We estimate the energy exchange
between the flow and the tissue occurs at one point. In Fig. 3b the work done
by the viscous flow at this point is positive. In Fig. 3c 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 induces varicose oscillations of UF, which produce the
harmonic at nF0 in the source signal. This harmonic is in turn reinforced by
the strong vocal tract resonance at nF0.
The
net work done by the sinusoidal acoustic wave with frequency nF0 at a point on
the false fold over a period can be positive or negative, depending on the
phase relationship between the FVFSW and the acoustic pressure. We suppose that
within a half wavelength of the FVFSW in the vicinity of the flow separation
point, the FVFSW absorbs the acoustic energy of the harmonic at nF0. Away from
this flow separation point, the FVFSW is expected to decay rapidly because of
large viscous losses in the tissue during high frequency vibrations. We thus
conclude that the total work done by the acoustic wave on the FVFSW is
positive.
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: a flow separating from
curved walls with mucosal layers receiving acoustic feedback. 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.
2.3. 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 supply the energy for sustaining
FVFSWs. It is interesting to note that FVFSWs have been observed in patients
suffering from ventricular dysphonia [10], 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.
The
loop described in our model tends to “unify” the double-source theory and the
resonance theory 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.
The
present model for Sygyt might also shed new light on the production of
high-frequency, whistle-like voice type of birds, dolphins, whales, and
groaning dogs. In this regard, our model is an updated version of the double-source
theory [1], which already drew parallels between the sounding mechanisms of
overtone singing and the whistle-like voice type, which is produced with the
false folds adducted.
It is interesting to compare
the harmonic-amplification loop with the sounding mechanism of flute-type
instruments, which is based on a loop composed of a vibrating jet and acoustic
waves filtered by a resonator. In the case of flutes the jet separates from the
musician’s lips, travelling along the mouth of the resonator towards a sharp
edge. When the instrument produces a tone, the jet oscillates at one of the
resonances of the pipe. The acoustic flow field near the flow separation point
excites sinuous oscillations of the jet. At the sharp edge, the jet is directed
alternately toward the inside and the outside of the resonator. This pulsing
injection induces an equivalent pressure difference across the mouth that
excites and maintains acoustic waves in the pipe [11]. The jet, like the false
fold mucosa, does not vibrate at any intrinsic resonance. It should be noted
that the acoustic flow induces sinuous oscillations of the jet at the mouth
hole of a flute, whereas the acoustic pressure excites FVFSWs that induce
varicose oscillations of the glottal flow.
While
a varicose jet is essential for whistle-like sound production, the role of wall
vibration is not fully understood. It has been suggested that the sounding
mechanism of human whistling is a loop composed of the jet and the oral cavity
with a prominent resonance. The pressure fluctuations due to the acoustic wave
at the flow separation point could induce varicose oscillations of the jet
without any wall vibration. This model is in an interesting contrast to our
model of Sygyt, which assumes vibrations of the compliant walls. To examine the
assumption of FVFSWs in our model of Sygyt, we measure surface vibrations
during whistle-like singing in vivo.
3. Experimental Study
3.1. Whistle-like voice
type
The present model of
“varicose jet oscillations induced by surface waves of curved walls in the
vicinity of the flow separation point” may provide 
insight into the production of the whistle-like voice type in birds and
mammals. It has been suggested that the production mechanism of bird whistled
song might be related to a retraction of the syringeal membranes while in
oscillation so that they no longer completely close, leading to a great
reduction in the harmonic content of the flow. An alternative explanation of
whistled song is that it is produced by pure aerodynamic means without any
vibrating surfaces [12]. However, recent experimental studies favour the
sounding mechanism of vibrating surface [13,14].
After
some practice, human can imitate dog’s groaning to produce high-frequency
whistle-like voices, which have a prominent component approximately at 4 kHz,
as shown in Fig. 4c. We hypothesize that the mechanism underlying this
vocalization is a varicose jet induced by FVFSWs.
Medical
ultrasound (
3.2. Methods
A commercially available,
high resolution
3.3. Results
CDI colour artefacts detected
surface vibrations of the right false vocal fold during pure tonal singing
(Fig. 4d). During warming up of this vocalization, surface vibrations of the right
vocal fold and the false fold were observed (Fig. 4b).
The
frequency of pure tonal singing was found to range from 3.7 kHz to 4.6 kHz. Out
of this range the voice lose the pure tonal characteristic, with breathy noises
accumulating at the prominent resonance.
4. Concluding Remarks
The observation of false fold
surface vibrations during pure tonal singing provides indirect support of our
model for Sygyt. As FVFSWs may generate 4 kHz pure tonal voices with the second
harmonic 30 dB (or more) weaker than the fundamental, it should be possible
that a Sygyt singer amplifies a selected harmonic of the voice produced by the
true vocal folds through FVFSWs.
The
role of acoustic feedback in FVFSW generation is not fully understood. When the
acoustic wave filtered by the resonator is strong enough to trigger FVFSWs, a
loop for pure tonal vocalization may be established. If not, periodic FVFSWs
may not occur. The laryngeal ventricle may be the Helmholtz resonator that is
responsible for the prominent resonance at 3.7-4.6 kHz. However, this
“resonance” model appears against experimental results about bird’s pure tonal
vocalization [13,14]. If the frequency of surface waves is not determined by
the tract resonance, it should be determined by the tissue curvature, elastic
properties, and the flow speed. In the case of Sygyt singing, however, it has
not been reported that a singer manipulates the false folds to change the
melody pitch. Further research is needed to compare the sounding mechanisms of
Sygyt singing and the pure tonal vocalization.
One
implication of our surface wave model is that the vertical motion of
fleshpoints on the true/false vocal folds may be critical to their
self-sustained oscillation. The two-mass and three-mass models of the vocal
folds [17,18] do not take into account the ellipse-like motion of vocal fold
fleshpoints, which is consistent with Rayleigh’s theory of surface waves and
has been demonstrated in excised canine larynx experiments [9]. We suggest that
the vertical motion of fleshpoints near the flow separation point can absorb
the kinetic energy of the glottal flow through viscous shear force.
The
effect of surface viscous shear stress exerted by a flow also plays a central
role in the system of a pair of fluttering flags in wind. This system shows
some notable similarities of the glottis. When the inter-flag distance lies in
a definite range the flags flutter in an out-of-phase state and generate a
pulsating flow, with striking similarities of the vocal fold vibration in the
chest register. Flow visualizations showed significant shear stress on the
flags exerted by the flow [19]. This finding suggests that viscous shear stress
on the vocal fold mucosa should not be ignored, especially in the vocalizations
with a large open quotient.
Next
to the viscosity effect, the surface shear stress may be attributed to the
carrying-along of the varicose flow. It was observed in a pair of flags that
the flag wave propagates along with the flow, while the wave of an isolated
flag propagates in the direction opposite to the flow. Note that the surface
shear stress dominates the system of a pair of flags but not an isolated flag
[19]. It is likely that the surface shear stress is due to the effect that a
varicose or sinuous flow carries along the flag wave. This approach may shed
new light on the mechanism of the self-sustained oscillation of the vocal
folds.
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