Qualitative and Quantitative Dynamics of .Vowels 
Thomas C. Bourgeois 
Institute for Neurogenic Communication Disorders 
University of Arizona 
Tucson, Arizona 85721 
bourg@cnet, shs. arizona, edu 
Richard T. Oehrle 
Department of Linguistics 
University of Arizona 
Tucson, Arizona 85721 
oehrle@convxl, ccit. arizona, edu 
Introduction 
A pervasive and fundamental property of 
spoken language is the nesting of quasi-periodic 
structures, ranging ti'om the vibration of the vocal 
cords to the iteration of syllables, accents, and 
higher-order prosodic objects. The long-range 
goal of the research reported here is to bring to 
bear on the study of this phenomenon the methods 
and insights of the study of dynamical systems, 
in the hope that this will increase our 
understanding of the computation of spoken 
language. In this paper, we describe this point of 
view and illustrate the results we have obtained to 
date in a study of English vowels, within and 
across individual speakers. 
Two perspeeU'ves on speech 
Speech is a physical event: it is produced by the 
mechanical actions of the human articulators 
and propagates itself through physical media. At 
the same time, speech is the carrier of richly- 
structured linguistic information. From this 
latter point of view, speech events constitute 
tokens of a symbolic system. The fundamental 
question prompting the research reported here is: 
what makes this duality possible? 
* This research was funded in part by Research 
and Training Center Grant P60 DC-01409 from 
the National Instutute on Deafness and other 
Communication Disorders. The present state of 
this research has benefitted from comments made 
by the participants of NWAVE 22 and the Forum 
on Information-Based Linguistics held at the 
University of Arizona during the spring of 1993. 
The authors wish to acknowledge the support of 
Fred Richards, who first brought the phase portrait 
to our attention and who wrote the reconstruction 
and Poincar6 section code, and John Coleman, 
Kerry Green, and two anonymous reviewers for 
their helpful suggeslSons. 
Pretheoretically, an adequate answer to this 
question should address the interaction in spoken 
language of variation and stability. Speech is 
symbolically stable across an impressive range 
of variation of the physical signal, variation 
observable within a given speaker, across 
speakers with a common language, and across 
the range of dialects and languages. In spite of the 
scope and pervasiveness of this variation, speech 
is not entirely arbitrary. For example, the 
phonological adaptation of speech is not arbitrary: 
vowel spaces do not cross-cut each other in 
random fashion. This suggests that the symbolic 
properties of speech are not the result of a purely 
conventionalist association between the space of 
speech sounds and their phonological 
interpretation. A deeper analysis can be found in 
the work of Stevens (1972) and Liljencrants & 
Lindblom (1972). Stevens notes that the vocal tract 
is constructed in such a way that there are regions 
of articulatory variation which produce relatively 
little acoustic variation. Working from the 
perspective of human action theory, Tuller & 
Kelso (1991) interpreted Stevens' notion as 
implying the existence of regions of dynamical 
stability in the speech production mechanism, 
and not as implying the existence of invariant 
acoustic properties within the signal. 
Liljencrants & Lindblom explore the hypothesis 
that phonological systems are solutions to the 
problems presented by ease of articulation, on the 
one hand, and ease of perception, on the other. 
Our approach to the problem focuses on how the 
intrinsic physical properties of speech are 
adaptable to the demands of symbolic 
phonological representation. To explore this 
question, we base our research on the concepts and 
methods of dynamical systems (Abraham & 
Shaw, 1992). Our work investigates the 
trajectories of vowels in a so-called 'phase space 
representation' of the associated acoustic wave- 
71 
tbrm regarded as a function A mapping time t to 
amplitude A(t). This reconstruction transforms a 
point A(t O) in the wave form to an ordered triple in 
the phase space of the form <A(t0) , A'(t0) , A"(t0)> 
where the y- and z-coordinates correspond to the 
first and second time derivative; temporally 
successive values of the function A(t) become 
successive points in the phase space. Thus time is 
'parameterized' in the sense that it is not directly 
represented in the space, although it can always be 
recovered by considering only the behavior of the 
x-axis, which mirrors the original function A(t). 
The resulting trajectory is a closed (or nearly 
closed) curve in three dimensions which repeats 
(or nearly repeats) itself with each glottal cycle. 
Pitch is also indirectly represented, encoded 
within the representation as the distance along the 
trajectory between successive samples of the 
speech wave (at fixed temporal intervals): higher 
pitches correspond to more distant successive 
measurements. In other words, the phase-space 
reconstruction includes all the information found 
in spectrographic representations of speech, but 
normalizes across fundamental frequency 
variation. An example follows. Consider below a 
fragment of the waveform produced during one 
male talker's production of the vowel \[u\] in the 
context of the word who'd. 
The wave form can be characterized qualitatively 
as having two large peaks, one smaller than the 
other, which repeat with each period. The phase 
portrait reconstructed from this wave form 
appears below in the form of a stereogram (join 
the two center dots by crossing your eyes to see the 
three-dimensionality of the resulting image): 
.f' 
~: {, • 
• i ". .), 
... :: '..... ~ ...- 
.'2.:.:.:.:...... ,~,~?..,.""° : ' 
t 
~:'2,2.:.:.:,. : .., 
.:.ii::: 
_:. . 
If. 
In the reconstruction, the largest peak of the wave 
form constitutes the largest loop, and the smaller 
peak corresponds to the smaller loop. The 
trajectory winds twice around the center in the xy 
plane and twice around the the center in the xz 
plane. We provide an interpretation of these 
windings below. 
What makes this point of view attractive is, 
first of all, its physical realism: speech events in 
fact constitute a dynamical system, and as such, 
the dynamics of the articulators and the 
acoustical dynamics they produce in the ambient 
media around them are directly characterizable 
as dynamic systems. Equally attractive is that the 
study of dynamical systems brings together, in a 
single integrated framework, quantitative and 
qualitative methods, a feature which has been 
exploited in the study of physical systems since 
the pioneering insights of Poincar~. That is, we 
can study the dynamic aspects of speech to any 
desired degree of quantitative detail, in the same 
space that accommodates a non-quantitative, 
qualitative investigation of behavior. In 
particular, then, one may identify the phonetic 
properties of a speech event with the quantitative 
aspects of its behavior, and ask whether or to what 
extent the qualitative aspects of this dynamic 
system support phonological analysis. If these 
qualitative physical aspects of speech do in fact 
support phonological analysis, then the 
simultaneous co-existence of 'phonetic' and 
'phonological' properties in the same space 
provides an interesting alternative to the view 
that phonological properties are modeled in a 
discrete space of 'distinctive features' and 
phonetic realization corresponds to a map from 
this discrete space to a corresponding space of 
continuous phonetic parameters. Thinking of 
phonological properties as the natural qualitative 
distinctions that exist in the continuous phase 
spaces of particular speech events makes it 
'f2 
possible to reconcile the apparent abstractness of 
phonological properties with their intrinsic 
dependence on such physical parameters as 
duration, amplitude, and frequency. 
Phase space reconstruetiorm of vowels 
As we have said, the phase-space 
reconstruction makes it possible to study 
quantitative and qualitative aspects of vowels (in 
particular, and the full range of speech sounds, in 
general) in the same space. The qualitative 
aspects of dynamic behavior correspond to 
fundamental properties of attractors within the 
phase-space. The presence of such attractors is 
revealed by stability in the phase-space trajectory. 
A continuous phase space can support discretely- 
structured forms of stable behavior. Thus, one 
and the same trajectory may be studied from the 
point of view of the continuous space or from the 
point of view of the discrete parameters which 
control the shape of that trajectory. It is this basic 
duality which we seek to exploit. 
An attractor represents a natural limit of a 
phase-portrait. For example, consider the 
behavior of a damped pendulum which swings 
through a series of decreasing arcs until it 
eventually comes to rest. A phase-space 
reconstruction of its behavior consists of the set of 
points (x, y) in the Plane, where the x-coordinate 
represents the displacement---positive or 
negative---of the pendulum at any given point in 
time, and the y-coordinate represents its velocity. 
Since the pendulum swings with decreasing 
displacement and correspondingly decreasing 
velocity, its phase-portrait consists of an arc 
spiraling through the phase space and ending in 
the origin---a point attractor. ~ ~xlc3t 
Other kinds of attractors are possible. The 
attractor for a bowed string, for example, is 
periodic: a closed curve in the planar phase space. 
A fundamental question in investigating speech 
as a dynamic system is the character of the 
attractors in the phase space. 
The double-Helmholz resonator model of 
the vocal tract provides a convenient and 
straightforward means to introduce the geometry 
of the vowel trajectory in phase space, the torus. 
The torus is the product S 1 × S 1 of two circles. 
Thus, the two dimensions model the oscillatory 
properties of the two chambers, while the trajectory 
in the product of the dimensions models their 
coupling for a given value of their controlling 
parameters (that is, by hypothesis, for the phonetic 
value of a particular vowel) 1. Consider the double 
resonant cavity schematized below: 
Here, the two chambers A and B are coupled to 
each other by a connecting tube. For the moment, 
let us imagine that there is no coupling between 
the two chambers. Let chamber A have a single 
resonant frequency ¢o and chamber B have a 
different resonant frequency Q. We can then 
reconstruct the phase space trajectory as a circle 
in a plane whose points are determined by the 
ordered triple 
1 Because the trajectory is derived from the actual 
acoustic waveform, our interpretation of such a 
trajectory is not in ~ principle restricted by 
limitations due to the simplicity of the double- 
resonator model. By changing the parameters of 
reconstruction we can easily embed the resulting 
trajectories in a state space of arbitrarily higher 
dimension. That is, we can relate the trajectory 
not to a line on the surface of a torus but rather to a 
rope on that surface, or we can think of this rope as 
inhabiting the space enclosed by the toms, rather 
than constraining it to occupy the surface of that 
torus. Such extensions are straightforward; 
whether they would be required in a more 
adequate model remains an open question at this 
time. 
73 
<sin z, cos z, -sin T>; z = cot. Similarly, we can 
reconstruct the phase-space trajectory for B as a 
circle in a plane whose points are determined by 
<sin T, cos T, -sin T>; T = ~2t. Because the 
resonant frequencies co and f~ are disparate, we 
can consider the two planar phase spaces to be 
orthogonal to each other, as below: 
A 
If we now translate the two phase space 
trajectories and adjust the scale appropriately, it 
is easy to visualize that the space traced out by a 
point simultaneously constrained to move along 
the curve described by A and the curve described 
by B will be the surface of a torus. 
An example of such a trajectory is shown below. 
~.~--.--====-~=,.... 
im 
The fundamental hypothesis of this paper 
is that while the vowel space is acoustically 
continuous, the shape of the trajectories within the 
phase space representation corresponding to 
vowels of different quality are topologically 
distinguishable, and that the trajectories 
corresponding to vowels of the same quality 
across talkers are homeomorphic---that is, 
topologically indistinguishable. We are 
particularly interested in trajectories which are 
periodic with respect to both dimensions of the 
torus and their coupling. Topologically, these 
trajectories are torus knots of type (re,n), where Ill 
and n are relatively prime and m represents the 
period of the trajectory with regard to one of the 
circular dimensions of the torus and n represents 
the period of the trajectory with regard to the other 
circular dimension (see Crowell & Fox, 1993). 
These two parameters may be coupled in 
distinguishable ways as well. Even with m and n 
quite small, this space of possibilities gives rise to 
complex varieties of behavior which can be 
distinguished on simple, discrete grounds. 
Considering speech within the phase space 
representation, then, might provide insight into 
the continuous/symbolic duality which exists in 
both the production and perception of natural 
language. 
Data 
We have collected the vowels \[i I E ae ^ u U o al 
within the context hid from four adult males, four 
adult females, and two children (a boy, 11, and a 
girl, 9) during separate recording sessions, and 
stored the productions on a digital audio tape 
sampling at 44 kHz with 16 bits quantization. We 
then resampled these tokens onto a PC using a 
separate AID converter at 22kHz with 8 bits 
quantization. In order to study the dynamics of 
these vowels both within the glottal cycle as well 
as within the syllable, we extracted pitch-periods 
(one iteration of the closed curve in the phase 
space) from three regions of the syllable: the first 
quarter, the middle, and the third quarter. Each of 
the resulting arrays was transformed into the 
phase space using the method described in Gibson 
et al. (1992). At the time of writing, we present 
results obtained from analyzing one adult male 
(D.B.) and two adult female talkers. 
For reference to other kinds of speech 
analysis, the table below compiles the 
fundamental frequency and formant 
measurements for the three speakers we have 
i 
iilialyzed ,'is li liinction or vowel quality, averagl;d 
liCl'llss l,hc three positions in tilt; syllabic. 
\[Insert table 1 here.\] 
These measurements' are in general accord with 
those presented in Peterson & Barney (1952), and 
suggest that the voWels within our corpus are 
r phonetically unremarkable. 
The phase space trajectories of nine vowels 
from the male talker (D.B.) are given below: 
Insert figure 1 here. 
As our topology suggests, we discuss two 
qualitative parameters which serve to distinguish 
the phase portraits from each other: the 
smoothness of the trajectory, which we can take to 
lie |'elated to the wind4ng number around the 'tube' 
of the torus, and the number of trips each 
trajectory makes about the origin, the 'circle' of 
the torus. Descending in height across the 
inventory, the trajectories of those vowels 
produced with central or posterior articulation \[u 
U o ^ a\] orbit the origin in a characteristically 
smooth fashion relative to their anterior 
counterparts \[i I E ae whose portraits show a great 
deal more local activity. We can interpret this 
local activity as an increase in the number of 
loops around the smaller diameter of the torus. In 
other words, the winding number m is large for 
front vowels, and small for back vowels. 
Comparing now across the trajectories of vowels, 
the number of rotations around the origin 
increases as tongue height is lowered. That is, 
high vowels such as \[i I u U\] show fewer trips 
around the origin than low vowels \[ae a\]. In terms 
of the state space, we can say that the winding 
number for the large diameter of the torus n 
increases as vowel height decreases. 
Consider now the following phase space 
reconstructions of the vowel /u/ spoken by two 
Jbmale talkers (upper two phase portraits) and two 
nmle talkers (lower two phase portraits): 
\[I-nsertfigure 2 here. I 
With regard to the two qualitative parameters we 
discussed above, certain similarities are 
preserved within the vowel trajectories across 
talkers. The winding number of the small ring 
(i.e. around tim 'tube') is small, as is the winding 
nillnbl;r o1" the large ring (i.e. llrOillid the 'circh;'). 
These qualitative similarities of the winding 
number can be made quantitative by counting 
them, using a technique developed by Poincar6 
(we will consider only the winding number of the 
'circle' here). 
The Poincar6 section provides a means of 
simplifying the dynamics of a phase portrait by 
considering not the whole path within the space, as 
we have done above, but rather a plane which 
intersects the phase space such that all of the 
trajectories pass through it. Consistent with 
common practice, we choose the plane associated 
with the phase-zero point of the (large) oscillator 
and, for a given glottal cycle, count how many 
times the trajectory passes through the plane in a 
single direction. By this method, we obtain for the 
continuous trajectory a discrete observation of its 
winding number. For the example below, the 
Poincar6 section contains a single point, so the 
winding number would be 1. 
?:! i I :: :? 
The following table summarizes the results 
we have obtained for Poincar6 sections of phase 
space reconstructions of pitch periods exerpted 
from 1/4, 1/2, and 3/4 of the way through the vowel 
portion of the syllable, (T. 
i I E ae u U o ^ a 
¢d4 D.B. 1 2 3 4 2 3 3 4 5 
S.O. 1 2 3 4 1 2 3 5 5 
L.W. 1 2 3 3 1 2 3 3 5 
(~/2 D.B. 2 2 3 3 2 3 3 4 6 
S.O. 2 3 3 4 1 3 3 3 4 
L.W. 1 2 3 4 2 2 2 4 4 
3~d4 D.B. 2 3 4 5 2 3 3 4 6 
S.O. 2 3 3 4 2 2 2 3 6 
L.W. 1 3 4 4 2 3 2 4 4 
As the data illustrate, the winding number 
increases with a decrease in vowel height, 
consistent with our qualitative observations for a 
single speaker. Because the winding number is a 
measure of trips around the torus and therefore an 
integer, it provides us with a means of 
discretizing the vowel space in a way which is not 
completely arbitrary, but rather reflects the 
internal structure of the trajectories through the 
state space itself. Within this space, the 
trajectories can be grouped together as members of 
an equivalence class which itself is a function of 
the controlling parameter of vowel height. 
Specifically, high vowels \[i I u U\] can be thought of 
as being associated with trajectories of winding 
number n ~ 2, mid vowels are associated with 
trajectories of winding number 3 ~ n ~ 4, and low 
vowels are associated with trajectories of winding 
number n ~ 5. Much of our data conform to this 
generalization with only a few outliers. As for 
those data which do fall outside of this grouping, it 
is important to remember that the data given are 
based on a single Poincar~ section for a single 
pitch period of the relevant vowel. A more 
thorough analysis would undoubtedly include 
both P-sections for the phase angles (0, ~/2, ~, 3~/2) 
as a means of distinguishing local behavior near 
the phase plane from the global properties of the 
trajectory, and for additional pitch-periods in the 
signal. 
We predict then, that a change in the winding 
number n for a vowel reconstructed as a phase 
space trajectory will correspond to a change in the 
perceived phonetic category of the vowel; 
successively larger values of the winding 
number n correspond to succesively lower vowel 
height categories. In the feature system of SPE, a 
change from n = 2 to n = 3 corresponds to a change 
from \[+high\] to \[-high\], a change from n = 4 to n = 5 
corresponds to a change from \[-low\] to \[+low\]. 
A reviewer has correctly pointed out to us that 
the winding number around the large diameter of 
the torus appears to be correlated with the number 
of harmonics between the fundamental frequency 
and the first vowel formant (although it remains 
to be confirmed, by extension it is most likely the 
case that the winding number around the 'tube' of 
the torus is correlated with the number of 
harmonics between the fundamental frequency 
and the second vowel formant). Because we have 
described this number as a means for evaluating 
the perceived articulatory height of the vowel, it 
seems appropriate to determine whether or not 
there is a precedent in the literature for an 
interaction between the fundamental frequency 
and the first formant either in vowel production or 
vowel perception. The following bricf chronology 
features the highlights of our investigation into 
this question. 
The interaction between F 0 and F! 
Since the early 1950's researchers have 
observed an interaction between fhndamental 
frequency and vowel perception. Potter and 
Steinberg (1950), who measured the vowels of 
male, female, and child speakers, found that an 
increase in fundamental frequency across 
talkers was correlated with an increase in the 
absolute frequency values of the formants within 
a particular vowel category. While they suggested 
that fundamental frequency variation might 
offer a means for normalizing formant 
frequency values, they decided it was "a dubious 
possibility" since, among other things, formants 
are a product of the physical aspects of the vocal 
tract and have little dependence on fundamental 
frequency. However, they also found an effect of a 
change in fundamental frequency on the 
perception of synthetic vowels whose formant 
structure remained constant: as fundamental 
frequency was increased, the perceived frequency 
of the first formant decreased. That is, a vowel 
whose formant structure corresponded to a male 
\[a\] was perceived as an \[a\] when synthesized with 
the fundamental frequency of a male, but as a 
(child's) \[O\] when synthesized with the 
fundamental frequency of a child. Similarly, a 
vowel whose formant structure corresponded to a 
male \[ae\] was perceived as an \[ae\] when 
synthesized with the fundamental frequency of a 
male, but as a (child's) vowel somewhere between 
\[ae\] and \[El when synthesized with the 
fundamental frequency of a child. They report 
further evidence, albeit anecdotal, that when 
helium was used as a propagation medium for 
adult male vowels or an artificial larynx was 
used to excite the vowel formants of a child (thus 
raising or lowing, respectively, the fundamental 
frequency of the subject while leaving the vocal 
tract constant), that a speaker will "make 
adjustments in his formant frequencies in order 
to maintain a given vowel sound." Similar 
findings are also to be found in Peterson (1961), 
who reports, again anecdotally, that "if a man 
raises his fundamental voice frequency to 
correspond to that of n child (falsetto), and the 
higher tbrmants are removed by filtering, the 
acoustical result corresponds very closely to the 
\[Ol of a child with low-pass filtering and. may be 
so interpreted by a listener." More systematic 
studies of the effects of F 0 on perception of vowels 
were conducted by Miller (1953), Fujisaki & 
Kawashima (1968), and Carlson et al. (1975). 
Each of these studies reported a similar shift in 
the perceptual boundary between vowel categories 
as fundamental frequency was changed: an 
increase in fundamental frequency leads to a 
decrease in the perceived value of the first 
\[brmant, and thus an increase in the perceived 
articulation height of the vowel. In sum, several 
studies have indlicated that a person's 
Ihndamental fi'equency interacts with both vowel 
production and vowel perception, and that the 
product of this interaction appears to be under the 
control ot' the speaker to some degree. 
The observations reported in Scott (1976) 
provide some insight into predicting the impact on 
w)wel perception of the interaction between F 0 and 
F 1. Scott explored the perceptual consequences of 
manipulating the temporal fine-structure of 
w~wel waveforms, and found that the perceptual 
boundary dividing a synthesized continuum 
whose endpoints were /i/ and /I/ was correlated 
with a change in the number of positive-going 
slopes in the wavef0rm: those stimuli with two 
positive-going slopes were categorized as/i/while 
those with three positive-going slopes were 
c~ltcgorizcd as /l/. In a fi)llow-up experiment, F 0 
and F 1 were manipulated in three synthetic 
continua to produce a series with a waveform 
change from three cycles of F 1 per fundamental 
period to four cycles Of F 1 per fundamental period 
at different points along the seven-step 
coutinuum. For those continua whose waveform 
changes occurred near the category boundary (the 
region where the tokens became ambiguous), the 
position of the boundary shifted to the stimulus 
where the waveform change occurred. This 
research suggests that, at least for ambiguous 
vowels (i.e., those :near the boundary of two 
distinct phonetic categories), category 
membership can be decided on the basis of the 
temporal fine-structure of the wave form. 
The Scott study bears close similarity with the 
dynamic approach discussed here. The temporal 
line-structure that Scott manipulated has a direct 
correlation to the winding number of the phase 
space trajectory. Specifically, those tokens which 
contain an extra cycle of F 1 per fundamental 
period are also those whose winding number n 
increases by 1. As Scott observed, such a change 
was detectable perceptually, and its detection 
corresponds to a change in the mapping of the 
acoustic stimulus from one phonetic category to 
another. Relating this to the table of values from 
the Poincar6 analysis, one can see that in natural 
speech, the change between/i/and/I/for a given 
speaker is consistent with an integral increase in 
the winding number. Hence, Scott's prediction 
that temporal fine structure is correlated with the 
perceptual phonetic category of a synthetic vowel 
is borne out in our natural speech data as well. 
Put in phonological terms, the perceptual 
distinction between /i/ and /I/ which Scott 
investigated is typically characterized not as a 
change in vowel height, but rather as a change in 
the value of the \[tense\] or \[ATR\] feature. Because 
we have limited our analysis to distinctions of 
vowel height and net other dimensions which 
delimit the vowel space, there are clear 
differences between that study and the predictions 
we make here; we cannot comment on the extent 
of those differences at this stage in our research. 
More work will certainly be required to verify the 
connection between the parameters 
distinguishing the vowel space and those aspects 
of vowel (production) dynamics represented 
within the wave form. However, the connection 
between our approach and Scott's results (and the 
legacy of research which precedes it) is 
compelling. It suggests the otherwise 
unanticipated result that the oscillator driving 
vowel production (the glottal source which 
produces F 0) and the resonant cavity which 
determines vowel quality are entrained (coupled) 
in frequency. 
Conclusion 
In this discussion we have provided only a 
very cursory analysis of a small set of talkers, but 
it nevertheless illustrates the potential power 
which this theoretical perspective can have as a 
tool for resolving the continuous/discrete duality 
we mentioned above. It is important also to note 
that this technique of phase space reconstruction 
and subsequent P-section analysis can be 
obtained without any specialized hardware 
beyond that needed to discretize the wave form 
77 
itself and does not rely on the Fourier transform. 
As an analysis toolkit, then, this approach offers 
an augmentation to current spectral analysis 
techniques by reducing some of the cross-talker 
variation that such techniques cannot abstract 
away from via a 'vocal-tract internal' means of 
normalizing across differences in talkers and 
situations. 
As a final consideration along these lines, we 
point out two additional curiosities about speech 
that may also succumb to analysis under the 
dynamical perspective. First, as early as 1947 
French and Steinberg showed that speech could be 
either low-pass filtered or high-pass filtered at 1.9 
kHz while retaining around 68% of its 
intelligibility. This suggests that the global 
structure of the vowel's dynamics may in fact be 
retained in spite of the filtering process at this 
'magic' frequency. If this proves to be true, then 
our approach offers a unique perspective from 
which a straightforward account of this 
phenomenon can be obtained. Second, Licklider 
and Pollack (1948) showed that speech subjected to 
differentiation followed by infinite peak clipping 
(which preserves only the zero crossings of the 
wave form) was also highly intelligible--in fact, 
about 90% intelligible. Although we have not 
explored this fully, such a transformation seems 
intuitively related to the Poincar~ section 
analysis which we have provided above. 

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