Acoustic Interaction Between the Glottal Source and the Vocal Tract

Martin Rothenberg

Syracuse University

Syracuse, NY, USA

*Vocal Fold Physiology, *K, N. Stevens and M. Hirano,
Eds., University of Tokyo Press, pp. 305-328, 1981

*(Proceedings of the Vocal Fold Physiology Conference held in Kurume, Japan January
15-19, 1980.)*

Though the glottal sound source is often considered to have a volume velocity waveform independent of the supralaryngeal configuration during vowel-like sounds, it has long been suspected that the separation of sound source and vocal tract can lead to a significant error in the estimation of voice quality (Fant, 1960; Flanagan 1968). It is generally realized that there can be appreciable first formant energy absorbed by the glottis during the open phase of the glottal cycle, and that this energy can cause oscillations on the glottal flow (volume velocity) waveform and a change in the frequency and damping of the formant. However, it is not generally recognized that this interaction can also have a strong effect on the overall waveshape of the glottal pulse, and, in particular, on the amount of high frequency energy generated at the instant of vocal fold closure.

High speed and stroboscopic motion pictures of the glottis during chest voice have generally yielded a rather symmetrical, triangular waveform for the projected glottal area, as, for example, in the samples shown by Dr. Hiroto in Chapter 1 of this book. On the other hand, measurements of the glottal flow waveform by inverse-filtering the sound pressure or the flow at the mouth have often shown a markedly unsymmetrical waveform, with a slowly-rising glottal opening phase and a sharply terminating glottal closing phase (for example, see Miller, 1959; Holmes, 1962; and Rothenberg, 1973). The glottal flow waveforms in simulation studies have usually shown these same characteristics, as we have seen in the contribution to this conference by Dr. Ishizaka and Dr. Titze (Chapters 17 and 18 of this book). This dissymmetry of the glottal flow waveform can be an important determinant of voice quality in that it increases the high frequency energy of the waveform, as compared to the projected area waveform, and concentrates the energy in the glottal closed phase, during which the vocal tract is most efficient (Fant, 1979).

Of the possible causes for this flow dissymmetry, the following
three appear to me to be the most likely to be significant:

(1) a non-invariant relationship between projected glottal area and glottal
flow conductance (the reciprocal of flow resistance) due to the different vocal
fold configurations that exist during the opening and closing phases,

(2) the small Component of air flow which is due to the air displaced by vocal
fold motion (Rothenberg, 1973; Rothenberg and Zahorian, 1977),

(3) the effect of the supraglottal acoustic impedance.

The relationship between projected glottal area and flow conductance
is likely to be significant in determining voice quality under some conditions.
We will not consider this factor further in this paper, however, and we make
the common first-approximation assumption that the relation between glottal
conductance and projected glottal area is invariant.

Since the air displaced by vocal fold motion tends to decrease the glottal flow
as the folds separate, and increase the flow as the folds come together (as
in a hand clap), this component will tend to cause a dissymmetry of the type
we are discussing. A simple calculation shows that the component will be small,
but not necessarily negligible (Rothenberg, 1973. See also the simulation result
by Flanagan and Ishizaka (1978).) However, in this paper we will not consider
this component further, and we will study, instead, the effect of the third
factor. We will consider the effect of supraglottal loading on the glottal waveform,
using a model which is valid at the fundamental frequency and lowest order harmonics,
since it is these components which most strongly influence the overall waveshape
of the glottal pulse. Our comments will be restricted to acoustic interaction
and not include the effect that the supraglottal pressure variations might have
on the motion of the vocal folds.

**A MODEL FOR SOURCE-TRACT ACOUSTIC INTERACTION**

Fig. 1 shows a simple linear. lumped-parameter model for the glottis and the
supraglottal vocal tract. The elements in the model are in standard electrical
circuit form, but are defined acoustically as follows:

Zg = glottal impedance (The dissipative part of the glottal
impedance is termed the glottal resistance.)

Yg = glottal admittance (The dissipative part of the glottal
admittance is termed the glottal conductance.)

Ug = glottal volume velocity

Psg = subglottal pressure

Zt = impedance of the supraglottal vocal tract as seen
by the glottis

Pt = supraglottal pressure

In this paper. we will consider the subglottal pressure to be constant and the
glottal inertance Lg to be zero. The glottal admittance
will then be a pure conductance that is the inverse of the glottal resistance
Rg and is equal to Ug/(Psg-Pt).
In the following theoretical development, we will initially consider this admittance
to have the symmetrical, triangular shape shown at the upper right in Fig. 2
as the vocal folds open and close during the glottal air pulse. Though this
representation does not properly reflect the flow dependency of the glottal
resistance (which for larger glottal areas causes the differential or small-signal
flow resistance to be somewhat higher than the resistance defined by pressure/flow,
and causes both measures of resistance to increase with volume velocity for
a given glottal area), it should yield a good approximation to the actual glottal
flow for small values of Zt, if the variation in projected
glottal area is approximately triangular (Flanagan, 1958). In the common assumption
of independence between the glottal source and the vocal tract, the supraglottal
impedance Zt is considered negligible compared to Zg,
and therefore, as shown in the curve for Rt = 0 in Fig.
2, the glottal air flow would have the same shape as the glottal admittance
function Yg. In the figure, the pressure, admittance and
time scales are normalized to arbitrary units (Psg = 1,
Yg maximum = 1, pulsewidth = 2).

In looking at the effect of a non-negligible Zt,
let us first assume Zt to be purely dissipative, as shown
in Fig. 2, for some representative values of Rt. In this
case, the air flow Ug is given by

Ug(t) = Psg/Rg(t) + Rt = Psg/(1/Yg(t)) + Rt | (1) |

At the initiation and termination of the air flow pulset Rg is
high and dominates Rtt and the resulting flow is similar to that which would
occur with no Rt. However, as Rg decreases (and Yg increases) the flow becomes
limited by Rt. For high values of Rtt the flow Ug approaches a rectangular pulse
of amplitude Psg/Rt, as it is "switched" on and off by the onset and
offset of Yg. This rectangular pulse will have a much higher ratio of high-frequency
energy to low-frequency energy than would be the case with Rt equal to zero
(the non-interactive model).

A reactive supraglottal impedance will also cause an increase in the proportion
of high frequency energy, The way in which this change occurs, however, is different
from that for the dissipative case. If the lowest supra-glottal resonance (the
first formant Fl) is higher than the voice fundamental frequency F0,
as is usually the case, the supraglottal loading will be inertive at frequencies
between F0 and F1. In addition,
in a more accurate representation the inertive component of the glottal impedance
would be added to the load impedance in determining the flow, thus increasing
the importance of the effect of inertive loading. In this paper, we will present
a mathematical analysis for only the case of an inertive reactance, assuming,
for simplicity, that Zt is due to a pure inertance Lt, so that Zt(w) = jwLt.
Fig. 3 shows our simplified model with an inertive supraglottal loading of magnitude
Lt. The flow Ug for this model is governed by the following differential equations:

Lt(dUg/dt) + (Ug/t) = 1, for 0 £ t £ 1, | (2A) |

Lt(dUg/dt) + (Ug/t-2) = 1, for 1 £ t £ 2, | (2B) |

With the added condition that Ug(0) = 0.

As can be verified by substitution in the equations, the solution for this set
of differential equations is

Ug = t/(Lt + 1) for 0 £ t £ 1. | (3A) |

For 1 £ t £ 2 the solution has 2 forms, depending on the magnitude of Lt:

(2Lt/Lt^{2}
- 1)(2-t)^{(1/Lt)} - (2-t)/(Lt
- 1) for Lt ¹ 1 |
||

Ug = | (3B) | |

[1/2 - ln(2-t)](2-t) for Lt = 1 |

These functions are plotted in Fig. 3 for some representative
value of Lt. It can be seen that inertive loading would
tend to give the flow pulse its often-noted skew to the right and rapid termination.
Since the high frequency energy produced at a discontinuity of slope tends to
be proportional to the change of slope at the discontinuity. the high frequency
energy produced at the termination of the glottal pulse can be greatly increased
by inertive loading. The terminal slope increases rapidly with Lt.
even at values of Lt which do not drastically reduce the
peak amplitude of the flow waveform. This differs from the dissipative case,
in which a steep termination is obtained only by limiting the pulse height,
i.e., in which the proportion of high frequency energy is increased by reducing
the low frequency energy (which varies with the pulse height). In the inertive
case, there is an actual increase in high frequency energy caused by the loading.

It can also be seen in Fig. 3 that the response between t = 0
and t = 1 is a linear increase in flow with a smaller slope than for the non-interactive
case. Thus the high frequency energy produced at the onset of the glottal pulse
is reduced. In the limit. with high values of Lt, Ug
approaches a "sawtooth" or "ramp" waveform with a linear
increases from zero flow at t = 0 to a peak flow of 2/(Lt
+ l) at t = 2. followed by a sudden decrease to zero flow at t = 2. The spectrum
of such a pulse falls off at only 20 dB/decade compared to the 40 dB/decade
fall-off for the triangular pulse that occurs with Lt
= 0.

However, the increase of high frequency energy produced by the glottal closure
is not just an asymptotic affect that is present only with high values of Lt.
The increase of terminal slope occurs with relatively small values of Lt.
and in fact. the terminal slope becomes infinite for values of Lt
greater than unity. This result can be seen from the time derivative of Ug(t).
For 1 £ t £
2 the derivative is

(1/Lt - 1) - (2/Lt^{2}
- 1)(2-t)^{((1/Lt) - 1)} for Lt ¹
1 |
||

dUg/dt = | (4) | |

½ + ln(2-t) for Lt = 1 |

For 0 £ t £ 1 the slope is l/(L+l) for all values of L. Thus the slope at the onset of air flow decreases gradually with Lt. while the terminal slope increases rapidly for values of Lt approaching unity as the term l/(Lt-l) increases and dominates the second term. For Lt > 1 the second term becomes infinite as t approaches 2, since (1/Lt - 1) is negative. Inertive loading also delays the peak of the volume velocity flow. For the simple model of Fig. 3, the time at which Ug is maximum can be calculated by setting the time derivative equal to zero and solving for t. This procedure yields:

2 - ((Lt + 1)/(2))^{(Lt)/(1-Lt)} for Lt ¹
1 |
|||

T at Ug maximum |
= | (5) | |

2 - e^{-1/2} = 1.39 for Lt = 1 |

To show the effect of inertive loading with an admittance waveform
that is more rounded near the peak than is a triangle, Fig. 4 gives the flow
that would result if Yg were one-half cycle of a sinusoid.
Since we have not derived a closed-form solution for this case, the curves were
obtained by means of a digital simulation of the differential equation. With
the sinusoidal admittance, the peaks of the flow waveforms become more rounded
than for the triangle, especially for small values of Lt.
In addition, the peak of the flow waveform is delayed somewhat more, as a function
of Lt, than for the triangular admittance function, and
the value of Lt at which the derivative at t - 2 becomes infinite appears to
be somewhat less than unity. Otherwise, the responses for the sinusoidal and
triangular cases are very similar. The insensitivity of the general form of
the volume velocity to the details of the glottal admittance function with higher
values of Lt supports our assumption that a rough approximation
to the actual admittance can be informative.

Another simplification that we have made in modeling the glottal admittance waveform has been to assume that the admittance before and after the glottal pulse is zero. Inverse-filtering studies and motion pictures of the glottis during voicing have shown that there is often a patent air path between the arytenoid cartilages. This flow path would tend to show up in our model as a fixed impedance (or admittance) in parallel with the time-varying component. In the non-interactive model, the result would be primarily to offset the flow waveform from zero flow, without much change in the shape of the glottal pulse. However, with inertive loading assumed, the added admittance would alter the form of the differential equation and therefore affect the pulse waveshape, primarily near the onset and offset of the glottal pulse. To see in more detail that the effect of such glottal "leakage" might be, we have, in Fig. 5, inserted a fixed admittance, Yg - l, of 0.2 times the amplitude of the time-varying component of Yg, Yg-ac in parallel with the time varying component. In doing this, we have again assumed a linear model for the glottal admittance and neglected the inertive component. Offsets of this magnitude are commonly observed during inverse-filtering of oral air flow, and may be much larger in pathological voice, or, naturally, during breathy voice. The general result of this glottal leakage is to cause a gradual onset of the glottal flow pulse, and a more gradual offset, with less high frequency energy produced at both locations.

The reduction of high frequency energy at the instant of glottal
closure is of special interest because of its strong potential affect on voice
quality. Without "leakage," the high frequency energy is produced
by the sealed glottis forcing the flow to zero, in opposition to the influence
of the inertance, which acts to continue the flow. With leakage, the decrease
in Yg stops when Ug is still decreasing
slowly. At that instant the glottal flow pattern changes to a more gradual,
exponential decay, with a time constant (Yg-l)(Lt).
It may be significant that, according to this simple model, the stronger the
source-tract interaction (the higher the value of Lt)
the greater is the degradation of the high frequency energy caused by Yg-l.
In other words, according to this model, inertive loading of the glottal source
will cause an increase in high frequency energy on glottal closure only if there
is very little air leakage during the interval of vocal fold closure.

Before turning to the experimental corroboration of acoustic loading-effects
on the glottal waveform, we should note that the above theoretical development
has been for a symmetrical glottal admittance pulse. Though a symmetrical pulse
(whether triangular or sinusoidal) may be a good first approximation to admittance
functions derived from projected area measurements, there may be a good deal
of dissymmetry in individual cases. To see how such dissymmetry could interact
with loading effects, we solved the differential equation resulting from the
triangular admittance function pulse used for Fig. 3 (having a total width or
duration of two, a peak value of unity and no "leakage"), but with
the peak occurring at some arbitrary time Tm. instead
of at t = 1. If Tm is between zero and one, the admittance
function tilts to the left, and if Tm is between one and two, the tilt is to
the right with Tm = 1 being the case illustrated above
in Fig. 3. The differential equations for Ug would then
be:

Lt(dUg/dt) + (Tm/t)Ug = 1, for 0 £ t £ Tm | (6A) |

Lt(dUg/dt) + ((2-Tm)/(2-t))Ug = 1, for Tm £ t £ 2 | (6B) |

As long as Lt is not exactly equal to unity
(since Lt = 1 results in an indeterminacy in the solution
form) the solution to these equations is

Ug = t/(Lt + Tm) for 0 £ L £ Tm | (7A) |

Ug = K(2-t)((2-Tm)/L) + (2-t)/(L -2 + Tm) for Tm £ t £ 2, | (7B) |

where

K = [2L(2-Tm)^{((Tm-2)/L)}]
/ [L^{2} + 2(Tm-1)L
+ Tm^{2} - 2Tm] |
(7C) |

As for the special case of Tm = 1 (a symmetrical pulse), the onset of the pulse is linear, and the offset is steeper than the onset. However, the skew of the admittance pulse is now a function of both Lt and Tm. We illustrate this by showing that the "critical" value of Lt, i.e., the value above which the derivative becomes infinite at t - 2, will vary with Tm. The relationship can be derived by first differentiating Ug in the interval Tm £ t £ 2:

(dUg/dt) = (K(Tm-2)/(Lt)) * (2-t) * (2-Tm-Lt)/(Lt) + 1/(Lt-2+Tm) | (8) |

As t approaches two, this function becomes infinite for values
of Tm and Lt that make the exponent
in the first term negative. This occurs for values of Lt
greater than (2 - Tm). Thus, if the admittance pulse is
skewed to the right (Tm > 1) the flow termination will
become infinite at smaller values of Lt, and vice-versa.
For example, if Tm = 4/3, for a rather high but possibly
attainable opening time to closing time ratio of two, the critical value of
Lt is reduced to 2/3 as compared to the value of unity
that holds for the symmetrical admittance function.

**THE MEASUREMENT OF SOURCE-TRACT ACOUSTIC INTERACTION**

Though an inertive loading of the glottal source would produce flow waveforms
very much like those observed by using standard inverse-filtering techniques
given a symmetrical variation of glottal admittance it still remains to be shown
that this type of glottal-supraglottal interaction is significant in such activities
as speech or singing. There are a number of ways that we can study this during
actual vocalizations. The most direct method might be to measure Zt
somehow, but this requires the measurement of the pressure just above the glottis,
and is difficult to implement. (However, see Dr. Koike's Chapter 14 in this
book.) Using another approach, we have attempted to implement. a "nonlinear
inverse filter" in which some of the effects of the glottal-supraglottal
interaction are removed (Rothenberg and Zahorian, 1977). The resulting waveform,
shown in Fig. 6, is more directly related to Yg than is
the actual flow Ug. To aid in visualizing the assymetry
of each waveform, the connected lines above the waveform were drawn in to match
the maximum slopes of the rising and falling segments, neglecting the oscillations
at the frequency of the first formant in the case of the. linear filter. It
can be seen that in this case the supraglottal impedance did cause an alteration
of Ug similar to that produced by inertive loading in
our model.

Another approach in measuring the affect of Zt is to change Zt while keeping the glottal area function approximately invariant. One way this can be done is by changing the vowel value. In the example in Fig. 7, the glottal flow was measured by inverse filtering oral air flow, while vocal fold movements were monitored by simultaneously recording the vocal fold contact area (VFCA) waveform obtained from a Laryngograph (Fourcin, 1974). The VFCA waveform is shown inverted in the figure, since the Inverse. VFCA waveform tends to rise and fall with the glottal airflow. The samples shown in the figure were from the center of the vowel in the nonsense syllable /b Vowel p/ in the syllable sequence /b a p b ae p ---/. Each row in the figure represents one such sequence, with the first two sequences (top two rows) spoken slowly enough so that each vowel had a distinct steady-state segment, while the last sequence (bottom row) was spoken at a natural rate. Vocal effort was at a moderate conversational level. The stop consonants help assure a good velopharyngeal closure during the vowel, which is important for accurate inverse filtering. Filter parameters corresponding to the frequency and damping of the first three formants were adjusted manually during a repetitive playback of the vowel sample, using the VFCA waveform as an aid in defining the glottal closed and open periods (Rothenberg, 1979). The low-frequency response limitation of the standard automatic amplitude control and high-pass filtering in the Laryngograph probably caused a slight falling of the inverse VFCA trace during the glottal open phase, but this distortion should be similar for all samples.

Except perhaps for the first (topmost) repetition of /a/ and the second (middle) repetition of / ae /, a constancy of the VFCA waveform between samples suggests that the differences in the Ug waveform were not caused by different vibratory patterns of the vocal folds. We refer here to the general shape of the waveform and not the amplitude, since the waveform amplitude depends on the larynx position relative to the electrodes, and therefore can vary with the vowel articulation.

If the two samples having a non-representative VFCA waveform are
ignored, the figure shows that, of the vowels tested, those with a constriction
closer to the glottis and a higher first formant (at the left in the figure)
tended to have a glottal flow waveform which was more skewed to the right, with
a steeper flow termination, and I therefore might be expected to have more high
frequency energy generated by the termination of tile glottal closure. In fact,
for the /i/ samples shown, the sharpest discontinuity in waveform slope, and
therefore greatest production of high-frequency energy, appears to be at the
peak of the waveform, and not at the instant of glottal closure. The overall
shapes of the flow waveforms in Fig. 7 are roughly similar to those produced
by the model in Figs. 3 and 4, as inertive loading is varied, except for the
added oscillations at the first formant frequency. The oscillatory component
of the interaction is not included in our simple model. Though there was considerable
waveform variability between speakers, the general trend toward more dissymetry
with vowels closer to /a/ was also found in the samples from two other speakers
tested. one male and one female.

These results are at least roughly consistent with calculations
made from a simple model of the vocal tract. For a lossless supraglottal vocal
tract 17 cm long, with a uniform area of 5 cm^{2},
the impedance Zt will be inertive for frequencies below
the first formant frequency of 500 Hz (or. more precisely, below a frequency
just slightly under the first formant). The magnitude of this impedance will
be approximately 8 tan(w/2000) in either acoustic
ohms or units of cm H^{2}O per liter/sec. In order
to represent the actual vocal tract impedance below F1
by a pure inertance. it is necessary to find a linear approximation to the actual
impedance function. Assuming that in our example the frequencies between 125
Hz and 375 Hz. representing an F0 of 125 Hz and the second
and third harmonics are of most interest we can make a linear approximation
by estimating Lt as the derivative of |Zt|
with respect to w at some intermediate frequency,
say 250 Hz:

Lt » (d|Zt|)/(dw) = (d/dw)[8 tan(w/2000)] at w = 250(2p) = 0.008 at w = 250(2p) | (6) |

To normalize Lt to the scales used for Figs.
2, 3, and 4, it is necessary to multiply by the actual maximum glottal admittance
(or conductance, ignoring Lg) and divide by the actual
time for one-half the glottal pulse (approximately 2 ms at F0
= 125 Hz). In calculating interactive effects, the differential conductance
dUg/d(Psg - Pt)
should be used. For a flow of 500 ml/sec and a maximum glottal area of 0.16
mm^{2}, typical adult male values, the maximum
glottal conductance would be approximately .05 in cgs units, as computed from
A^{2}/rUg
(Flanagan, 1958; Fant, 1960). Defining Lt as the value
of Lt normalized to the scales used for Figs. 2, 3, and
4, we obtain:

Lt (normalized) = Lt
(actual) x (0.05/0.002) = 0.2. |
(7) |

Vowels with a higher F1 would be expected
to yield a value of __L__t that is valid over a wider
frequency range, since more harmonics would be included before Fl
is surpassed and the reactance becomes compliant. In addition, vocal tract configurations
in which the pharynx is more constricted than for a neutral vowel might be expected
to lead to a higher value of __L__t. Since the value
of __L__t also depends upon the maximum glottal admittance,
a vocal fold vibratory pattern that resulted in a wider than average glottal
opening would also increase __L__t, as would the inertive
component of an unusually pronounced constriction at the entrance to the larynx
or at the false vocal folds. Thus a value of Lt of at
least 0.5 for a back vowel such as /a/ in some speakers is not inconceivable.
Any glottal inertance would add to this figure.

Finally, we have also tested for the presence of supra-glottal
loading effects by comparing vocalizations made with air and with a large proportion
of helium mixed with the air. By reducing the acoustic inertance in the vocal
tract, the helium would be expected to reduce any supra-glottal loading effect,
if present. (Though, unlike the change of vowel value, the use of inspired helium
also affects the glottal inertance. To affect only the acoustic loading, it
would be necessary to introduce the helium only into the supraglottal pathway.)
Fig. 8 shows the general result we obtain. Both waveforms in the figure were
obtained by inverse filtering the oral volume velocity for a male adult holding
a vocal tract position for an /ae/ vowel. The fundamental frequency in each
case was about 110 Hz. It can be seen that the symmetry of the waveform increases
significantly with helium displacing some of the air.

**CONCLUSIONS**

It appears that the often-noted skewing of the glottal flow waveform which results in the primary vocal tract excitation being at the instant of glottal closure can be caused by a combination of factors, including dissymmetry in the glottal admittance function, air displaced by vocal fold motion, and acoustic interaction with the supraglottal impedance. The inertance of the glottal slit may also be a factor, at least at the smaller glottal openings, when this inertance is largest.

To study the effect of acoustic interaction, we have defined a
normalized supraglottal inertance __L__t, which approximates
the actual loading of the glottis at frequencies between F0
and F1, assuming F1 is appreciably
larger than F0. This model for the supraglottal loading
indicates that for a symmetrical triangular admittance pulse the effect of acoustic
interaction becomes large if __L__t is close to or
exceeds unity, since at __L__t = 1 the derivative of
the glottal flow becomes infinite at the instant of glottal closure. This critical
value of __L__t will vary with the symmetry and general
shape of the admittance pulse, but appears to remain in a range of roughly 0.5
to 2.0 for the type of admittance variation to be found in normal speech. The
model also indicates that at higher values of __L__t
the volume velocity waveform tends to have a characteristic shape which is rather
insensitive to the details of the glottal admittance function.

A rough analysis of the magnitude of the supraglottal vocal tract impedance shows that values of Lt of at least 0.5 are not implausible, with the value in any given case being a function of the vowel value. the maximum admittance of the glottis during the glottal cycle and the configuration of the larynx above the vocal folds.

A simple representation of the glottal admittance function that would result from an incomplete vocal fold closure during the glottal cycle indicates that this factor is also important. in that it can appreciably modify the effect of source-tract acoustic interaction. However, small leakage paths may not be as significant as our results in Fig. 5 suggest, since we have neglected the glottal inertance, a factor that could be relatively large with a small opening.

Our results demonstrate that acoustic interaction can cause the
glottal source waveform to vary widely as a function of vowel value and F0,
since the first formant must be high compared to F0 in
order for the supraglottal loading to be inertive at an appreciable number of
glottal harmonics, and because the magnitude of the impedance below F1
varies as a function of vowel value. It is also possible that different
pronunciations (allophonic variations) of the same vowel phoneme can result
in appreciably different source spectra because of differences in the impedance
seen by the glottis. The degree to which this phenomenon can explain differences
in voice quality in speech and singing would be an interesting subject for future
research.

**ACKNOWLEDGMENTS **

The work described here was supported by research grant NS08919
from the National Institutes of Health. James T. Mahshie assisted in the experimental
work, and the author was fortunate to have the help of Wilbur R. LePage in the
solution of the differential equations.

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