35. AIR COLUMN, REED, AND PLAYER'S WINDWAY INTERACTION IN MUSICAL INSTRUMENTS
Arthur H. Benade
Reprinted from Vocal Fold Physiology: Biomechanics, Acoustics
and Phonatory Control, Denver Center For Performing Arts (1985), ed Titze
and Scherer.
Traduction
en français
ABSTRACT
Musicians have always insisted on the importance of getting the
proper shapes in a windplayer's air passages. For this reason, the apparent
success of the current oscillation theory of reeds and musical air columns
without inclusion of the player's windway effects became increasingly mysterious
as the subject matured. Since this theory to date has been useful for guiding
the construction of fine instruments, confidence in its techniques is sufficient
to support a serious attack on the problem of extending it to include the
player's windway. Major energy production occurs at frequencies where A[(Z_{u} +
Z_{d})//Z_{r}]≈>l. Here A and Z_{r} are
the transconductance and impedance of the reed while Z_{u} and Z_{d} are
the input impedances of the air columns looking upstream and downstream
from the reed. Nonlinear effects couple these energy sources via heterodyne
action, whether or not Z_{u} appears in the accounting. Meaningful
extension of theory has been aided by the development of convenient pulseecho/FFTmeasurement
techniques for the Z's of both the instrument air column and the player's
windway. Most vowel (supraglottal) configurations give rise to Z_{u} peaks
in the range of 450 to 1500 Hz that are able to play a significant role
for instruments. The fact that these peaks do not coincide with speech formant
frequencies has helped to confuse the situation, as has the fact that some
players unconsciously exploit windway resonances, while many do not use
them at all.
I. INTRODUCTION
This report is intended to provide an introductory account of how the player's
own windway interacts with the reed and air column of a musical wind instrument.
Our formal understanding of the reed/aircolumn interaction is extremely
good today, to the extent that it is possible not only to describe the
acoustical nature of the interaction but also to use it as an effective
guide to the instrument maker in his labors to build a good instrument
or to improve an already existing one. For this reason, our task is relatively
simple: we need only to show how the additional complexities associated
with the player's windway modify the mathematical physics of the simpler
reed/aircolumn system and then examine the ways in which the modified
system differs in its behavior from the one that has been well studied.
The reader's first reaction to the preceding paragraph may well be a remark
such as, "Musicians for hundreds of years have insisted on the importance
of the mouth and throat configuration of anyone who is serious about playing
a wind instrument. How then can anyone claim to have understood a wind
instrument to a useful extent without taking this fact into account?
Furthermore, how can he then go on to announce the importance of the player's
windway as a new discovery?" It is my hope that the answers to these important
questions will themselves clarify the nature of just what it is that has
become newly understood.
For many years I have stoutly told my musician friends (and myself, in
my incarnation as a serious amateur player) that the role of the player's
windway could only be clarified after the other, more easily visible contributors
to the musical oscillation were properly elucidated.
As a matter of fact, today the question has inverted itself, taking the
form: "How did such a largely influential part of the dynamical system
remain incognito during the course of investigations covering many years
in which changes of only two or three parts in a thousand of many other acoustical
parameters could readily be associated with their dynamical and musical
consequences?"
It will perhaps be useful to outline the preceding remarks in the following
way before we look at the physics itself.
DOES THE LUNGTHROUGHMOUTH AIRWAY SIGNIFICANTLY INFLUENCE
THE PLAYING OF MUSICAL WIND INSTRUMENTS?
1. MUSICIANS ARE UNANIMOUSLY OF THE OPINION
THAT IT DOES.
2. THE MUSICAL ACOUSTICIAN HAS TENDED TO IGNORE
THE QUESTION, OR TO SET IT ASIDE AS A RELATIVELY SMALL INFLUENCE.
Item (2) above is a deliberate oversimplification. Measurements and speculations
of an acoustical nature have been made over the span of many decades, but
for various reasons no clear consensus has developed. The detailed recounting
of this branch of history will not contribute appreciably to our present
purpose, which is to give a compact description of what is known today,
in a form that will (hopefully) be intelligible to a readership whose major
concerns are with the biophysics of the player himself rather than with
the details of his interaction with a musical wind instrument.
I wish to make clear at this point in my introductory remarks that the
present report is intended to be little more than an announcement of some
of the recent results obtained in Cleveland. For the sake of brevity, I
will therefore run the risk of frustrating my readers and annoying workers
elsewhere whose results are not properly acknowledged. I shall, however,
mention here by name those of my coworkers past and present who have made
particularly large contributions (beyond the limits of any published work
of theirs) to the insights reported here; these are Walter Worman, George
Jameson, Stephen Thompson, and Peter Hoekje. The present report would not
have become possible without their direct collaboration. This is true of
George Jameson and Peter Hoekje in particular. Beyond this I will present
only those bibliographical details that can directly aid the reader in
his comprehension of the present discussion. A formal research report with
proper acknowledgement and documentation is being prepared by Hoekje and
myself for submission to the Journal of the Acoustical Society of America.
II. FORMULATION OF THE PROBLEM
Figure
351 shows the general nature of the dynamical system with which we are
concerned. The system may be considered as being the concatenation
of four major segments: the sublaryngeal airway (terminated at its lower
end by the player's lungs), the larynx (which in the present case is either
wide open or partially closed in a manner that does not permit it to oscillate),
the vocal tract (which is extensively adjustable via motions of the soft
palate, tongue, jaws, etc.), the reed of the musical instrument (whose operating
point, damping, etc., are controlled by the player's lip position and pressure),
and the musical air column (whose acoustical properties are controlled via
the player's fingers on the various keys and/or tone holes).
In any musical wind instrument, whether woodwind, brass, voice (or even
harmonica!), we find three interacting subsystems: an air passage from
the wind supply (the player's windway or the organ pipe's foot and the
wind chest below it), a flowcontrol device (the cane or lip reed of the
orchestral wind instrument, the singer's larynx, the free reed of the harmonica,
or the air reed of the flute player), and finally some sort of resonator
and radiating system that ultimately couples with the room into which the
sound is to be fed. Setting aside the flute family of instruments, the
flow control device is a valve whose degree of closure is determined by
the pressure difference across an operating surface.
Figure 352 shows two versions of the basic pressurecontrolled system.
One of the controlling pressures is maintained in part by the player's
lungs and in part is produced by the acoustic disturbances taking place
in the player's windway (PWW). The other pressure acting upon the
reedvalve is found within the instrument's mouthpiece as a manifestation
of the acoustical activity taking place in the instrument's air column
(IAC). In Figure 352a the valve action is arranged so that an increase
in the downstream pressure p_{d} leads to a greater flow. This
arrangement is typical of all the orchestral reed woodwinds and of the
organ reed pipes. Figure 352b shows, on the other hand, a system in which
the valve action is reversed so that an increase of p_{d} decreases
the flow u, as is typical in the orchestral brasses.
We
will find it convenient to define directions in this essentially onedimensional
waveguide system with the help of the words "upstream" and "downstream," these
being directly related to the direction of the normal flow of air from
the player's lungs out into the room. Thus one of the flowcontrolling
pressures acts on the upstream side of the reed, while the other is exerted
on its downstream surface. Terminology based on this convention prevents
ambiguities of the sort that arise if one simply uses the words "up" and "down." For
a clarinetist the airflow runs upward within the player, and then downward
through his instrument. The problem of similar description of what
goes on in a bassoon or tuba defies the imagination!.
We find it useful to characterize the PWW and the IAC via their impedances
as seen by the flow controller. We will refer to the impedance looking
upstream into the PWW as Z_{u}, while the impedance of the IAC
will be denoted by Z_{d}. The reed itself requires two characterizations,
since it plays two roles in the complete oscillating system. We define
its acoustic impedance Z_{r} in
terms of its displacement volume velocity when it moves in response to
a driving pressure exerted on either one of its surfaces (see Figure 352);
the other and perhaps more basic property of the reed is its flowcontrol
characteristic, which is in general a nonlinear function. This flowcontrol
characteristic is most conveniently specified by expressing the flow u
as a Taylor expansion in the pressure difference p between the two sides
of the reed, as given in Equation 351.
u(t) = u_{0} + A p(t) + B p^{2}(t) + C p^{3}(t)
+ + + + + (351)
Because the reed assembly functions as a springmassdamper system, we
recognize at once that Z_{r} shows a resonance property that makes
it inversely proportional to the factor D(ω)
that is set down as Equation 352.
(352)
Here ω_{r} is the natural frequency of
the reed and g_{r} is its half power bandwidth. We notice, further,
that since the actual flow rate of air through the reed depends on its
position (and so only indirectly on the activating pressure), the flowcontrol
coefficients are themselves resonant in their nature. That is, these coefficients
may be written as the product of their lowfrequency "steadystate" values
(A_{0}, B_{0}, C_{0}, . . .) and the factor D(ω)
defined above. This fact proves to be very important to our understanding
of the way musical instruments are played. We can usefully remark here
that A_{0} is positive for the woodwind valve system of Figure
352a and negative for that belonging to the brasses, as in Figure 352b.
Let us now write down the pressure and flow relationships on the upstream
and downstream sides of the reed, in terms of the impedances Z_{u},
Z_{d}, and Z_{r}. The positive direction of acoustic
flow is defined to be the downstream direction of the DC flow from the
player's lungs.
u = p_{d}/Z_{d} + (p_{d}  p_{u})/Z_{r} (353a)
u = p_{u}/Z_{u} + (p_{u}  p_{d})/Z_{r} (353b)
The first term on the right side of each of these equations simply expresses
the ordinary relation between pressure at the entryway of a waveguide and
the flow that goes into that waveguide. The second term gives a measure
of the flow that goes into the region vacated by the reed itself when it
moves under the influence of the pressure difference that acts upon its
two sides.
Equations 353a and 353b can be combined in an interesting and useful
way: the flow u through the reed aperture can be expressed very simply
in terms of the pressure difference p across the reed, as shown
in Equation 354.
p = u(Z_{u} + Z_{d})//Z_{r} (354)
That is to say, the pressure difference across the reed is proportional
to the sum of the upstream and downstream impedances, in parallel with
the reed impedance (which tends to be very large compared with the other
impedances, so that it has a secondary, though nontrivial, role in the
oscillation process). We will use the unadorned symbol Z to represent this
combined impedance.
As a preparation for the next step in the discussion, we should recapitulate
the nature of the problem whose solution we are trying to outline. When
a wind instrument is played, the upstream and downstream impedances (together
with the reed's own impedance) are coupled via a flowcontrolling valve
to the player's lungs, which serve as the primary source of compressed
air. The system is kept in oscillation by a feedback loop in which the
net acoustical disturbance at the reed (i.e., the pressure difference across
it) operates the flow controller, and the resulting flow serves as the
excitory stimulus for the upstream and downstream waves.
Equation 351 provides us with a formal representation of the pressureoperated
flowcontrol property u(p) of the reed, while Equation 354 provides in
a very compact form the pressure response property of the entire airway
system (PWW + IAC + REED) to a flow stimulus. We should notice that both
of these equations relate the flow u, which is the same on both sides of
the reed valve, to the pressure difference p across it. In other words,
our analysis can be carried out in terms of p and u via the net Z and the "control
polynomial" u(p), without our having to worry about the complications of
the individual responses of our three subsystems to the flow which they
jointly engender via a nonlinear coupling.
From the point of view of mathematical physics we have here an initial
explanation of why effects produced by the PWW did not automatically destroy
our ability to make meaningful calculations guided by, and checked
against, experiments with reeds and various types of IAC—all that
was necessary was that the PWW would not produce confusing and distracting
effects. We were fortunate, indeed, over a period of many years that such
was the case for long enough for us to get a firm grasp of the essential
physics.
We turn now in the briefest way to a sketch of how the essential behavior
of the system can be understood. Confining our attention for the moment
to the case of strictly periodic oscillations in the system, we write the
flow u(t) as a Fourier series:
u(t) = Σu_{n}cos(nω_{0}t
+ ψ_{n}) (355)
Here ω_{0} represents the frequency of
the tone being produced. Term by term this series represents the flow excitation
spectrum being applied to the (PWW + IAC + REED) system. Given the (net)
impedance Z(ω) of this system, we may write Z_{n} for
its magnitude at the frequency nω_{0} and Φ_{n} for
its phase. The pressure signal corresponding to u(t) can then be written
down.
p(t) = ΣZ_{n}u_{n}cos(nω_{0}t
+ ψ_{n} + Φ_{n}) (356)
As a matter of formal mathematics, Equations 351, 355, and 356 can be
solved simultaneously to give the pressure spectrum across the reed for
a given blowing pressure. While the detailed calculations are very tedious,
it proves possible to extract a great deal of useful information about
the system. This information, which can be readily checked against the
behavior of real systems, depends much more on the overall mathematical
structure of the problem than it does on the numerical values of the various
parameters. That is, the salient features of the solution can be summarized
very simply in a form that depends only on the systematic behavior of nonlinear
trigonometric equations. Furthermore, when the complete story is in, we
find (surprisingly enough) that the results show almost no sensitivity
to the phases of the impedances, or of the reed resonance factor (Equation
352)! This is not to say that the phases are irrelevant or that they have
random values—merely that the spectrum amplitudes are not sensitive
to the phases of the Z_{n}'s and the D_{n}'s.
Equations 357 and 358 will suffice here to indicate the nature of the
playing pressure spectrum as measured across the reed. In particular, the
fundamental component p_{l}, which is the pressure amplitude of
the disturbance at the playing frequency, obeys an equation of the form
(357)
Similarly, the higher components have amplitudes that can all be written
in the form
(358)
I want to point out that in these equations there is no explicit appearance
of the phase shifts associated with the flowcontrol parameters or the
impedances. Only the magnitudes are important when the oscillation is of
periodic type.
We will postpone discussion of these results until we have sketched out
a linear cousin to the analysis, in which we can see what happens to the
n'th component of the pressure when looked at by itself, the inescapable
nonlinear coupling between spectral components being represented by a flow
source U_{n} that is "external" to the component in question.
III. A LINEAR COUSIN TO THE PROBLEM
Suppose that our system is running in a steady oscillation at the frequency ω_{0},
with a part u(t) of the flow being produced through the linear term Ap
of the control polynomial, and part of it U(t) being externally imposed
by an asyetunspecified source having the same periodicity. If we use
the Fourier representation, the imposed flow may be written
U(t) = ΣU_{n}e^{jnω0t} (35.9)
and the pressure signal across the reed is
p(t) = ΣZ_{n}[u_{n} +
U_{n}]e^{jnω0t} (35.10)
Equation 3510 may be solved term by term for the flow component amplitudes
in terms of the combined impedance Z_{n} and the corresponding
transconductance A_{n} (evaluated at the frequencies ω_{n} of
interest):
u_{n} = A_{n}p_{n} = Z_{n}A_{n}[u_{n} +
U_{n}] (3511)
whence
u_{n} = U_{n}[(Z_{n}A_{n})/(1  Z_{n}A_{n})] (3512)
Here and in the discussion that follows through Equation 3513, the symbols
A_{n} and Z_{n} have their ordinary complex representation;
i.e., account is taken of both magnitude and phase. Equation 3512 has
the familiar form that represents the current gain u_{n}/U_{n} of
a feedback amplifier for which the openloop gain is Z_{n}A_{n}.
It is at once apparent, therefore, that each spectral component of the
flow is selfsustaining as an independent oscillator if the real part of
the openloop gain is exactly unity. That is, an energy input provided
by the external drive signal U_{n} is not needed to keep the oscillation
going.
However, if the openloop gain Z_{n}A_{n} is less than
unity, the amplitude of the flow component u_{n} is proportional
to U_{n}. Furthermore, the magnitude of u_{n} will die
away exponentially in time if U_{n} is abruptly shut off,
with a decay rate that is proportional to the discrepancy between
unity and the magnitude of the real part of Z_{n}A_{n}.
If, on the other hand, the openloop gain is greater than unity, an exponentially
growing oscillation can take place with a growth rate that is once
again proportional to the difference between unity and the magnitude
of the real part of the openloop gain. Under these conditions the feedback
system is able (for the component in question) to generate more energy
than it can dissipate, without need for an additional input via U_{n}.
So far as our present (oversimplified) model is concerned, we may summarize
by saying that the oscillation of each spectral component is independent
of all the others, and that it is inherently unstable. We are of course
very much accustomed to this sort of instability, which is shared by all
ordinary oscillators, and it is quite customary to recall the presence
of some amplitudedependent (nonlinear) additional damping which comes
into play to stabilize the amplitude of a real oscillator.
In the multicomponent musical oscillator there are, to be sure, several
amplitudedependent sources of damping beyond that implied by Z_{u},
Z_{d}, and Z_{r} (turbulent damping, for example). There
is, however, another way in which energy can be transferred in and out
of each spectral component, a way that not only assures the stability of
each component amplitude under muchlessstringent requirements on the
openloop gain, but also guarantees that the various amplitudes have a
welldefined relationship to one another. This is of course an absolute
requirement for a musical sound source whose tone color needs to be defined
for each condition of playing chosen by its user. The fundamentally nonlinear
nature of the control polynomial defined in Equation 351 shows (in simplest
terms) that whatever pressure signal components p_{n} might be
generated via the operations of the linear term in this polynomial, they
will immediately breed contributions to the entire collection of flow components
at all other harmonic frequencies according to the heterodyne (intermodulation)
arithmetic that may be generalized for arbitrary exponents from the trigonometric
relation
(McosP) (NcosQ) = (MN/2)[cos(P+Q) + cos(PQ)] (3513)
That is, the "externally imposed" flow components U_{n} that were
introduced in Equation 359 may now be understood to represent in a very
simple way (computationally useless but heuristically helpful) the transfer
of energy from each modal oscillator to its brothers. It is no longer required
that each component be precisely selfsustaining when looked at by itself;
all that is required is that as a group the spectral components can jointly
produce enough energy to supply their total energy expenditure to the outside
world.
Our quasilinear model provides us one more insight into the nature of
the realworld nonlinear system: every spectral component is connected
directly or indirectly to every other one, so that its phase is the resultant
of many influences. The nature of the oscillation is such that there are many ways
in which the actual phase of a given component can be reconciled with those
of its conferers. Proper analysis shows that, as a result, the spectral
amplitudes are determined almost exclusively by the magnitudes of the relevant
Z and A, B, C parameters and not by their phase angles (Thompson 1978).
The discussion so far in this section has shown that energy production
is favored at maxima of the A(ω)Z(ω)
product. In the woodwinds, A is very nearly A_{0} over much of
the spectral range because the reed's own natural frequency ω_{r} is
relatively high (e.g., 20003000 Hz for a clarinet). This being so, energy
production is favored at the impedance maxima of the PWWIACREED system.
This says (if for a moment we ignore Z_{u} and Z_{r}) that
oscillation is favored at the normalmode frequencies of the IAC taken
with its reed end closed, as has been recognized for at least 200 years
("the clarinet plays as a stopped pipe").
Another implication of our discussion is that the overall energy
production is largest if the impedance maxima are harmonically related
to one another. This assures that each of the heterodyne frequency components
generated from the harmonics of the played note finds itself matching one
of the energyproducing impedance maxima and thus transferring energy to
a productive place in the regenerative scheme. Let us put this in more
obviously musicrelated words of the sort used prior to the explicit inclusion
of PWW effects: a musical instrument whose impedance maxima (as modified
by the parallel but large Z_{r}) are harmonically related
is one that starts its tones well, produces a clear sound, provides controllable
dynamics and stable pitches, and is otherwise a most attractive instrument
in the hands of the player and in the ears of the listener. I have given
a very extensive discussion of these matters in chapters 20 through 22
of my book (Benade 1976). Conscious recognition of the usefulness of accurate
harmonic "alignments" of the air column resonances led (beginning around
1964) to a continuing evolution of laboratory and workshop techniques for
the measurement and correction of the positions of the resonances
belonging to essentially all the notes of an instrument's scale. The behavior
of instruments adjusted by means of these techniques has been much admired
by wellknown musicians, and the techniques themselves are beginning
to have a significant effect on the making of (at least artistgrade) instruments
of all sorts today.
We had temporarily set aside the possibility that the ZA product could
become large near the reed frequency ω_{r},
so that the harmonic for which nω_{0} ≈ ω_{r} might
contribute to the net energy production even though Z itself might not
be large. While the book contains numerous qualitative remarks concerning
the musical usefulness of this possibility in woodwinds, the detailed
physics of it was not elucidated till later (Thompson 1979). For present
purposes it will suffice to say that all really skilled woodwind players
exploit the possibility of an extra energy source at ω_{r} by
setting the reed frequency at some (any!) harmonic of the playing frequency
in order to further stabilize and purify their sound production via the
inclusion of an extra, accurately aligned participant in the "regime
of oscillation." For brass instruments, the player must pay attention
to ω_{r}, since the note he wishes to
play is selected directly by arranging the lipreed natural frequency to
lie just below the fundamental of the desired tone. Further discussion
of the curious dynamics of the brass instrument, with its reversedsign
value for the reed transconductance A(ω), would
take us too far from the goals of this report. It will suffice for us to
notice that the adjustability of the reed resonance frequency is a
musically important resource for the woodwind player and an unavoidable
necessity for the brass player. In both cases we find that a physiological
adjustment is used as an adjunct to the mechanical controls provided
by the player's hands on the keys, valves, and slides of his instrument.
We close this part of our thumbnail sketch of the (inherently nonlinear
and therefore very stable) sound production mechanism of the orchestral
wind instrument by pointing out once more that our understanding of it
reached a highly developed state without any account being taken of
the possibility that the player's windway could itself play a significant
role. Our present analysis has shown that Z_{u} enters the dynamical
equations in a manner that is entirely symmetrical with that of Z_{d}.
To the scientist this means that he does not need to rework all his equations
when he adds consideration of Z_{u} to his analysis of Z_{d} and
Z_{r}: the symbol Z merely takes on a slightly different meaning.
From the point of view of the musician it means that the player has one
additional physiological adjustmentresource at his disposal (whose dynamical
nature we now can see in a general way). For all of us, we have yet the
question of how the dynamical effect of this resource could remain scientifically
incognito for so long, a question to which a partial answer will be given
below.
IV. SPECTRAL IMPLICATIONS
Now that we have sketched out the general nature of the nonlinear multicomponent
regeneration process that functions in the orchestral wind instruments,
we are in a position to examine the spectrum of the control pressure signal
p(t), as given in Equations 357 and 358 above. Recall that in these equations
we need only the magnitudes of the Z, A, B, C parameters! The first thing
that we notice is that the denominators of these equations are almost exactly
like the denominator of Equation 3512, from which we learned of the crucial
importance of the Z_{n}A_{n} product in controlling
the amount of energy generation that can take place at the n'th harmonic.
The only unfamiliar feature is the presence of other spectral components
whose influence is added to the direct effect of the component in question.
In Equations 357 and 358 these extra p_{j}'s are the explicit
representations (in an essentially exact formulation) of the "imposed flow" contributions
that were introduced heuristically in Equation
359. Aside from this, the denominators have almost exactly the same meaning
in the exact formulation that they did in our introductory version. We
can see this explicitly in Equation 357, which gives information about
the fundamental component of the spectrum. We begin by considering the
form taken by this equation in the lowamplitude limit, where the quadratic
and higherorder terms in the flow polynomial (Equation 351) have no role
to play. Under these conditions, the fact that p_{2} and other
higherorder components are zero means that if there is to be any oscillation
at all at the fundamental frequency, then (1  Z_{l}A) must
vanish, exactly as we have come to expect.
We turn now to a consideration of the numerator of Equation 358. This
shows a remarkably simple pair of overall relationships (that are well
substantiated by experiment under suitable conditions), as we can see from
the abridged version set down as Equation 3514.
p_{n} = Z_{n}p_{l}^{n} . (other, slowmoving
terms) (3514)
The first of these relationships is that the general shape of the reeddrive
pressure spectrum is well caricatured by the envelope of the controlling
aggregate impedance, and the second is that the n'th pressure amplitude
component is proportional to the n'th power of the fundamental component
amplitude as this changes with the player's blowing pressure. In other
words, as one plays a crescendo, keeping his embouchure and PWW constant,
the oscillation "blossoms" from a nearly pure sinusoid into a waveform
whose components grow progressively to the fully developed mezzoforte
distribution implied by Equation 358. Playing louder yet causes the reed
to close fully for a growing fraction of each cycle, giving rise to an
entirely new type of spectral development that has its envelope determined
by the dutycycle of the puffs of air through the reed. Beyond this we
need only to notice the exact parallelism of mathematical form in the denominators
of Equations 357 and 358.
It is only a brief step now to a description of the two spectra (which
can be measured) on each side of the reed: that is, the spectrum measured
in the instrument's mouthpiece (as has been done for many years during
the development of the basic theory outlined here) and the spectrum measured
in the player's mouth. If we write (p_{n})_{u} and (p_{n})_{d} for
these two pressurespectrum components and recall that
Z_{n} = ((Z_{u} + Z_{d})//Z_{r})_{n} ,
then
(p_{n})_{u} = u_{n}(Z_{u})_{n} =
p_{n}(Z_{u} /Z)_{n} (3515a)
(p_{n})_{d} = u_{n}(Z_{d})_{n} =
p_{n}(Z_{d} /Z)_{n} (3515b)
If (as has been known for many years), Z_{r} is large enough to
have only a small influence on the magnitude of Z, and if (as was presumed
for almost as many years) Z_{u} is relatively small and featureless,
equations like 357 and 358 appear to apply directly to the mouthpiece
spectrum, calculated using Z_{d} obtained from
measurements of the IAC. Experiments of this sort in fact have been done
and have provided a significant fraction of the evidence that has to date
supported our confidence in the theory as outlined. Notice once again our
debt to the curious but fortunate accident that the influence of the PWW
did not intrude upon our consciousness until we were ready to cope with
it!
It has been a truism of the subject that changes in the mouthpiece pressurespectrum
amplitudes should directly reflect changes in the corresponding impedance
peak heights, as is made explicit by the numerator of Equation 358 and
the leading factor in Equation 3514. It is but a short step from this
for us to invoke the upstream/downstream symmetry of the system as justification
for the idea that changes in Z_{u} produced by tongue and mouth
movements by the player will produce exactly parallel changes in the pressure
spectrum as measured in the player's mouth. However, it is not at once
obvious what happens to the spectrum on one side of the reed as the
result of changes in impedance on the other side.
Differentiation of the writtenout form of Equation 3515a with respect
to Z_{u}, and of Equation 3515b with respect to Z_{d},
gives us an explicit representation of these crossinfluences. When this
is done, a very surprising result is obtained:
TO FIRST APPROXIMATION, CHANGING Z ON ONE SIDE OF THE REED MAKES NO CHANGE
IN THE SPECTRUM ON THE OTHER SIDE!
On closer examination we find that there are indeed small changes, especially
if the perturbed spectral component is one of those for which the
ZA product is nearly unity—if, in other words it is very nearly able
to balance its own energy budget, and so support itself without feeding
energy to, or absorbing it from, the other components.
We close this discussion of the overall theoretical formulation of the
wind instrument regeneration process with a short summary of the major
points, leaving the broader implications till after the presentation
of some experimental data on the influence of the PWW on the playing regimes
of real instruments. The first point which should be made is that the upstream
and downstream impedances appear symmetrically in the theory. The second
point is that everything about the oscillation is directly determined by
aggregate Z as defined in Equation 354. The third point is that if the
magnitude peaks of the aggregate Z function are harmonically related, the
oscillation is stabilized, made clean and noisefree, and given a controllable
nature that is favorable to good musical performance. The fourth point
is that while changes in Z_{u} and Z_{d} alter the spectrum
as observable on the same side as the changes are made, there is generally
little or no change on the other side of the reed.
We may take item three above as giving an analytical indication of why
a player might find it advantageous to manipulate his PWW. Similarly, item
four can give us a hint as to why these effects were not immediately detectable
in the course of ordinary researchmeasurements were made only on the
downstream side of the reed!
V. IMPEDANCE MEASUREMENTS ON THE PLAYER'S WINDWAY
As
has already been remarked, one of the reasons why many of us took it for
granted that the PWW would have little effect on the basic regeneration
processes of a musical wind instrument was the assumption that the multibranched,
softwalled air passages the player's lungs acted as an essentially reflectionless
termination of the sub and supraglottal airway. We were further encouraged
in the belief that the upstream airway was unlikely to have an important
role by the fact that the pipe foot and wind chest of a pipe organ have
a relatively small physical (but not musically negligible!) influence
on the sound and the stability of tone production. Twentyfive years ago
this gave sufficient reason to move forward boldly, under the guidance
of the writings of Henri Bouasse (Bouasse, 192930) and with the stimulation
shortly afterwards of the accurate pioneering measurements of the clarinet
reed's flowcontrol transconductance (A_{0}) carried out by John
Backus (Backus 1963).
While precision measurements of IAC input impedances could be made from
the earliest part of this active period (see the examples of measurement
technique in Benade 1973), the necessarily slow frequencysweep techniques
then available could not be adapted to measurements on the highly variable
PWW. The more recent arrival of convenient FFT procedures has led
many of us to devise flowimpulse excitation methods, where the impedance
is deduced from the Fourier transform of the pressure response signal.
Members of my audience are far better acquainted with the history of this
subject than I, so the present listing of references is only intended to
indicate some of the earlier influences on my own thinking about this sort
of procedure (Oliver 1964; Rosenberg and Gordon 1966; Fransson 1975; Dawson
1976; Kruger 1980). The remaining paragraphs of this section will be devoted
first to an indication of the nature of the apparatus we have begun to
use, then to the display of the PWW input impedance (Z_{u}) measured
for various vocal tract configurations, and finally a description of some
of the information that can be gained from them.
The impedance head used in our present experiments is of the sort shown
in Figure 353 (Ibisi and Benade 1982). The primary sound source is a 27mm
diameter piezoelectric "beeper" disc bonded to the end of a short piece
of 20mm ID, 32mm OD heavywall phenolic tubing by a bead of RTV rubber.
The pressure signal is detected by an electret microphone whose 3mm aperture
looks into the tube only 12 mm from the face of the piezoelectric driver.
If the piezoelectric transducer is considered to be a lossless singlemode
harmonic oscillator, then a linearly rising ramp drive voltage will produce
a single velocity pulse of the form
v(t) = V[1  cos(2πt/T)] (3516)
for 0 < t < T (zero otherwise), provided that the ramp duration T
is exactly equal to the natural period of oscillation of the transducer.
Only a slight modification of the drive voltage waveform is required to
assure a very similar excitation velocity signal when account is taken
of the fact that the transducer is a damped oscillator (a detailed report
of these and other matters is in preparation for submission to JASA). Suffice
it to say that our excitation pulse has a FWHM of about 0.083 milliseconds,
so that FFT measurement of Z_{u} is possible without correction
up to well beyond the 2500Hz limit of our present major concern.

