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


Musicians have always insisted on the importance of getting the proper shapes in a wind-player'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[(Zu + Zd)//Zr]≈>l. Here A and Zr are the transconductance and impedance of the reed while Zu and Zd 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 Zu appears in the accounting. Meaningful extension of theory has been aided by the development of convenient pulse-echo/FFT-measurement techniques for the Z's of both the instrument air column and the player's windway. Most vowel (supraglottal) configurations give rise to Zu 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.


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/air-column 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/air-column 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.

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.


Figure 35-1 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 flow-control 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 35-2 shows two versions of the basic pressure-controlled 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 reed-valve 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 35-2a the valve action is arranged so that an increase in the downstream pressure pd leads to a greater flow. This arrangement is typical of all the orchestral reed woodwinds and of the organ reed pipes. Figure 35-2b shows, on the other hand, a system in which the valve action is reversed so that an increase of pd decreases the flow u, as is typical in the orchestral brasses.

We will find it convenient to define directions in this essentially one-dimensional 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 flow-controlling 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 Zu, while the impedance of the IAC will be denoted by Zd. The reed itself requires two characterizations, since it plays two roles in the complete oscillating system. We define its acoustic impedance Zr 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 35-2); the other and perhaps more basic property of the reed is its flow-control characteristic, which is in general a nonlinear function. This flow-control 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 35-1.

u(t) = u0 + A p(t) + B p2(t) + C p3(t) + + + + +                               (35-1)

Because the reed assembly functions as a spring-mass-damper system, we recognize at once that Zr shows a resonance property that makes it inversely proportional to the factor D(ω) that is set down as Equation 35-2.


Here ωr is the natural frequency of the reed and gr 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 flow-control coefficients are themselves resonant in their nature. That is, these coefficients may be written as the product of their low-frequency "steady-state" values (A0, B0, C0, . . .) 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 A0 is positive for the woodwind valve system of Figure 35-2a and negative for that belonging to the brasses, as in Figure 35-2b.

Let us now write down the pressure and flow relationships on the upstream and downstream sides of the reed, in terms of the impedances Zu, Zd, and Zr. The positive direction of acoustic flow is defined to be the downstream direction of the DC flow from the player's lungs.

u = pd/Zd + (pd - pu)/Zr                                                                     (35-3a)

-u = pu/Zu + (pu - pd)/Zr                                                                    (35-3b)

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 35-3a and 35-3b 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 35-4.

p = u(Zu + Zd)//Zr                                                                               (35-4)

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 non-trivial, 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 flow-controlling 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 35-1 provides us with a formal representation of the pressure-operated flow-control property u(p) of the reed, while Equation 35-4 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) = Σuncos(nω0t + ψn)                                                                      (35-5)

Here ω0 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 Zn for its magnitude at the frequency nω0 and Φn for its phase. The pressure signal corresponding to u(t) can then be written down.

p(t) = ΣZnuncos(nω0t + ψn + Φn)                                                          (35-6)

As a matter of formal mathematics, Equations 35-1, 35-5, and 35-6 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 35-2)! 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 Zn's and the Dn's.

Equations 35-7 and 35-8 will suffice here to indicate the nature of the playing pressure spectrum as measured across the reed. In particular, the fundamental component pl, which is the pressure amplitude of the disturbance at the playing frequency, obeys an equation of the form


Similarly, the higher components have amplitudes that can all be written in the form


I want to point out that in these equations there is no explicit appearance of the phase shifts associated with the flow-control 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 Un that is "external" to the component in question.


Suppose that our system is running in a steady oscillation at the frequency ω0, 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 as-yet-unspecified source having the same periodicity. If we use the Fourier representation, the imposed flow may be written

U(t) = ΣUnejnω0t                                                                                     (35.9)

and the pressure signal across the reed is

p(t) = ΣZn[un + Un]ejnω0t                                                                       (35.10)

Equation 35-10 may be solved term by term for the flow component amplitudes in terms of the combined impedance Zn and the corresponding transconductance An (evaluated at the frequencies ωn of interest):

un = Anpn = ZnAn[un + Un]                                                                      (35-11)


un = Un[(ZnAn)/(1 - ZnAn)]                                                                      (35-12)

Here and in the discussion that follows through Equation 35-13, the symbols An and Zn have their ordinary complex representation; i.e., account is taken of both magnitude and phase. Equation 35-12 has the familiar form that represents the current gain un/Un of a feedback amplifier for which the open-loop gain is ZnAn. It is at once apparent, therefore, that each spectral component of the flow is self-sustaining as an independent oscillator if the real part of the open-loop gain is exactly unity. That is, an energy input provided by the external drive signal Un is not needed to keep the oscillation going.

However, if the open-loop gain ZnAn is less than unity, the amplitude of the flow component un is proportional to Un. Furthermore, the magnitude of un will die away exponentially in time if Un is abruptly shut off, with a decay rate that is proportional to the discrepancy between unity and the magnitude of the real part of ZnAn.

If, on the other hand, the open-loop 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 open-loop 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 Un.

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 amplitude-dependent (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 amplitude-dependent sources of damping beyond that implied by Zu, Zd, and Zr (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 much-less-stringent requirements on the open-loop gain, but also guarantees that the various amplitudes have a well-defined 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 35-1 shows (in simplest terms) that whatever pressure signal components pn 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(P-Q)]                            (35-13)

That is, the "externally imposed" flow components Un that were introduced in Equation 35-9 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 self-sustaining 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 quasi-linear model provides us one more insight into the nature of the real-world 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 A0 over much of the spectral range because the reed's own natural frequency ωr is relatively high (e.g., 2000-3000 Hz for a clarinet). This being so, energy production is favored at the impedance maxima of the PWW-IAC-REED system. This says (if for a moment we ignore Zu and Zr) that oscillation is favored at the normal-mode 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 energy-producing impedance maxima and thus transferring energy to a productive place in the regenerative scheme. Let us put this in more obviously music-related 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 Zr) 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 well-known musicians, and the techniques themselves are beginning to have a significant effect on the making of (at least artist-grade) instruments of all sorts today.

We had temporarily set aside the possibility that the ZA product could become large near the reed frequency ωr, so that the harmonic for which nω0 ≈ ωr 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 ωr 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 ωr, since the note he wishes to play is selected directly by arranging the lip-reed natural frequency to lie just below the fundamental of the desired tone. Further discussion of the curious dynamics of the brass instrument, with its reversed-sign 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 Zu enters the dynamical equations in a manner that is entirely symmetrical with that of Zd. To the scientist this means that he does not need to rework all his equations when he adds consideration of Zu to his analysis of Zd and Zr: 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 adjustment-resource 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.


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 35-7 and 35-8 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 35-12, from which we learned of the crucial importance of the ZnAn 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 35-7 and 35-8 these extra pj's are the explicit representations (in an essentially exact formulation) of the "imposed flow" contributions that were introduced heuristically in Equation 35-9. 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 35-7, which gives information about the fundamental component of the spectrum. We begin by considering the form taken by this equation in the low-amplitude limit, where the quadratic and higher-order terms in the flow polynomial (Equation 35-1) have no role to play. Under these conditions, the fact that p2 and other higher-order components are zero means that if there is to be any oscillation at all at the fundamental frequency, then (1 - ZlA) must vanish, exactly as we have come to expect.

We turn now to a consideration of the numerator of Equation 35-8. 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 35-14.

pn = Znpln . (other, slow-moving terms)                                                   (35-14)

The first of these relationships is that the general shape of the reed-drive 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 35-8. 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 duty-cycle 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 35-7 and 35-8.

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 (pn)u and (pn)d for these two pressure-spectrum components and recall that

Zn = ((Zu + Zd)//Zr)n ,


(pn)u = un(Zu)n = pn(Zu /Z)n                                                                  (35-15a)

(pn)d = un(Zd)n = pn(Zd /Z)n                                                                  (35-15b)  

If (as has been known for many years), Zr is large enough to have only a small influence on the magnitude of Z, and if (as was presumed for almost as many years) Zu is relatively small and featureless, equations like 35-7 and 35-8 appear to apply directly to the mouthpiece spectrum, calculated using Zd 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 pressure-spectrum amplitudes should directly reflect changes in the corresponding impedance peak heights, as is made explicit by the numerator of Equation 35-8 and the leading factor in Equation 35-14. 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 Zu 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 written-out form of Equation 35-15a with respect to Zu, and of Equation 35-15b with respect to Zd, gives us an explicit representation of these cross-influences. When this is done, a very surprising result is obtained:


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 35-4. The third point is that if the magnitude peaks of the aggregate Z function are harmonically related, the oscillation is stabilized, made clean and noise-free, and given a controllable nature that is favorable to good musical performance. The fourth point is that while changes in Zu and Zd 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 research-measurements were made only on the downstream side of the reed!


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 multi-branched, 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. Twenty-five years ago this gave sufficient reason to move forward boldly, under the guidance of the writings of Henri Bouasse (Bouasse, 1929-30) and with the stimulation shortly afterwards of the accurate pioneering measurements of the clarinet reed's flow-control transconductance (A0) 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 frequency-sweep 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 flow-impulse 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 (Zu) 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 35-3 (Ibisi and Benade 1982). The primary sound source is a 27-mm diameter piezoelectric "beeper" disc bonded to the end of a short piece of 20-mm ID, 32-mm OD heavy-wall phenolic tubing by a bead of RTV rubber. The pressure signal is detected by an electret microphone whose 3-mm 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 single-mode 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)]                                                                       (35-16)

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 Zu is possible without correction up to well beyond the 2500-Hz limit of our present major concern.

The upper part of Figure 35-4 shows the pulse-echo sequence observed when the driver is attached to a piece of 20-mm ID copper tubing open at the far end and 570 mm long. Notice that doubling the height of the initial pulse makes it a member of the alternating-polarity, exponentially decaying sequence of the later pulses, exactly as theory predicts. The lower part of the same figure shows the input impedance of this air column as calculated via FFT from the time waveform in the upper part of the figure. Both of these displays are plotted from data stored in a Hewlett-Packard 3582A real-time analyzer.

Figure 35-5 will orient us to the general magnitudes of the peak values of Zu and Zd in a more-or-less musical context. The display is linear in Z and in frequency. The tall peak visible in the neighborhood of 1300 Hz belongs to the measured Zu under the following conditions: the subject has formed his vocal tract in the manner customary for articulating the vowel [ah] and the end of the impedance head has been inserted between his lips, with his teeth propped apart by the "tooth rest" on the impedance head to approximate the spacing they have on a clarinet. The less-tall sequence of resonance peaks that cross the entire figure reproduces the impedance curve for the 20-mm ID pipe shown in the preceding figure. The small circles placed above the peaks of this impedance curve show where they would be seen if the tube ID had been 15 mm rather than 20 mm. This 15-mm reference ID is chosen because it matches closely the size of the bore on a normal Bb clarinet (whose resonance peaks tend in fact to be less than about half as tall because of the additional damping associated with the complexities of the open and closed tone holes). It is clear that PWW impedance peaks can be very significantly taller than any of those that we might find in a real IAC. We will postpone any of the implications of this remark until Sec. VI, after we have looked at a little more data.

Figures 35-6, 35-7, and 35-8 show the measured Zu curves (expressed logarithmically via the dB notation) as a linear function of frequency for the vowels [ah], [eh], and [ih]. In all cases the frequencies of the principal resonant peaks are marked, along with an indication of the wave impedance of a 20-mm ID tube, which we can use as a calibration value. It is not quite coincidental that in all three cases the low- and high-frequency limits of the measured curves match this reference value. We will return to a discussion of this phenomenon as soon as the figures have been described.

A dash-dot curved line will be seen in all three of these impedance diagrams. A separate curve of this type was originally drawn freehand on a copy of each resonance curve on the basis of criteria that will become clear very shortly. It was then verified that these three curves were all very similar. A single composite curve was then constructed by an informal averaging and smoothing procedure. It is this composite that is displayed.

In the theory of nonuniform horns it is convenient to define the wave impedance as seen at the input as the function Z0(ω) that would be measured if the horn were to be given a reflectionless termination at its far end. This wave impedance can be real or imaginary, even if damping does not exist in the body of the horn. It is not difficult to show, then, that when the horn is given an arbitrary termination (real, imaginary, or complex), the measured input impedance will have maximum and minimum magnitudes that bound it in the following way:

(Zin)max = [(1 + F)/(1 – F)]+1         (35-17a)

(Zin)min = [(1 + F)/(1 – F)]-1        (35-17b)

Here the symbol F represents the fraction of the downward signal amplitude that returns to the input end after reflection from the termination at the other end. The attenuation considered here is due to all losses undergone in one round trip down and back in the waveguide plus the losses that take place for whatever reason at the termination itself. It is not possible to give a similarly straightforward account of the frequencies at which these extrema are found.

It is clear from Equations 35-17a and 35-17b that in many cases measurement of the peak and dip values of Zin permits the determination of Z0(ω)--all one needs to do is to calculate the geometric means of adjacent pairs of Zmax and Zmin. The broken line in the figures is such a midline reconciled to give a curve for Z0(ω) that is consistent with all the vowel configurations studied. The fact that various such configurations share a nearly common wave-impedance behavior is an interesting and surprising fact, concerning which we can extract some further information.

Horn theory tells us that in the limit of high frequencies, Z0(ω) tends toward a value that is equal to the wave impedance R0 of a uniform waveguide whose entering cross section matches that of the horn. Figures 35-6, 35-7, and 35-8 suggest on the basis of this property that the PWW has an entry way cross section that is close to that of a 20-mm ID pipe (our impedance head is short enough that its own cross section does not produce complications in any of the interpretations that we are making on the basis of general horn theory).

The fact that the peaks and dips in the measured Zu's become less pronounced shows that the lungs (which serve as the termination of the PWW) are becoming less and less reflective at high frequencies. Thus, our original assumption that the PWW shows little or no resonant behavior is only justified in the limit of high frequencies.

We may usefully invoke horn theory to aid in the extraction of yet more information about the acoustical nature of the PWW. If the horn has a taper at its entryway (whether enlarging or contracting), then we find that the wave impedance rises proportionally to the frequency from a zero at zero frequency, and then that it ultimately levels out toward the value R0 when ωx0/c >> 1. Here x0 is the length of the apical cone implied by the horn's initial taper and its entryway cross section. The fact that our deduced curve for Z0(ω) remains level at the value R0 implies, then, that our PWW has very little taper in the region of the mouth (at least under the conditions of our measurements!).

Yet another piece of global information can be deduced from a study of our curves for the input impedance itself as measured for the PWW. If the PWW has an opening to the outer world anywhere along its length, then Zin has a first-order zero at zero frequency. If, on the other hand, the PWW is air-tight except at its input end, the impedance has a simple pole at zero frequency. It is of course a truism that if the horn is given a nonreflecting termination, Zin = Z0(ω) at zero frequency (which limits to R0). Inspection of the resonance curves of Figures 35-6, 35-7, and 35-8 therefore confirms that the player's lungs function as a closed and only somewhat reflecting termination for the PWW.

One final piece of information about the PWW can be gained from an examination of the pattern of its input impedance. It was, in fact, the feature that immediately called itself to my attention when the data were first in hand. All of the impedance curves showed a pattern of peaks and dips superposed on a broad hump centered in the neighborhood of 1000 Hz. Better put, there is a broad hump in the wave impedance itself, and this hump is fairly independent of the other details of PWW structure. This sort of overall pattern is reminiscent of the generic behavior of the input impedance for all valve settings of a brass wind instrument, where the explanation is well known (Benade 1976, sec. 20.4, p. 400). When account is taken of the other systematic implications of the PWW shape as outlined above, one is led to suspect that it possesses a constriction a short way upstream, such that (speaking very informally), at the input end, a sort of Helmholtz resonator is constructed whose natural frequency matches that of the broad hump.

Figure 35-9 provides the first evidence of the plausibility of this sort of deduction. Here we see the input impedance of a very long piece of 20-mm ID tubing having an essentially nonreflecting termination. That is, we have a direct measurement of the wave impedance Z0(ω). The horizontal line across the graph verifies that a uniform tube does in fact have a frequency-independent measured wave impedance. The upper curve shows the broad hump that was described in the preceding paragraph, as is produced by the introduction of a properly proportioned constriction at a suitable distance from the drive end. These proportions may be calculated from the position and the height of the hump, with no free parameters to take care of other features of the shape of the Z0(ω) curve. There is a gratifying similarity between the shape of this curve and the (presumably analogous) freehand curve that was drawn on the earlier impedance curves.

Figure 35-10 shows the input impedance of an open-ended tube 1900 mm long into which the same constriction is installed. The Z0(ω) curve from Figure 35-9 is superposed to show in summary form how precisely the general ideas of horn theory correspond with the experimental results. The high- and low-frequency limits of Zin and the wave impedance are illustrated, along with the fact that on a dB plot, the wave impedance peaks and dips. The fact that in the present case the horn is open to the air is indicated by the low value of the input impedance at low frequencies.

Figure 35-11 is a very similar illustration of the behavior of a nonuniform waveguide. The only essential feature in which this differs from its predecessor is the overall length of the tube, chosen this time to approximate the length of the PWW (as given in Figure 35-1). Despite the fact that the pipe is open at the far end, it is clear that the general pattern of peaks and dips is strongly reminiscent of those measured for various vowel configurations of the PWW. In particular, the mean spacing of the peaks is the same, as is expected for one-dimensional ducts of equal length.


We will consider first the spectra measured on the two sides of the reed of a clarinet-like air column having three accurately aligned impedance maxima (located at odd multiples of 240 Hz). Above this (i.e., above a tone-hole-lattice cutoff frequency of about 1250 Hz), the impedance lies close to the value characteristic of an infinite line of 15-mm ID. This means that the IAC itself can generate energy strongly at the three odd harmonics of a 240-Hz playing frequency. Heterodyne transfer of energy between these harmonics constitutes the major source of energy in the entire oscillation. Absorption of energy at the impedance minima located at 480 and 960 Hz (even multiples of 240 Hz) and at all harmonic frequencies that lie above the tone-hole-lattic cutoff frequency assures amplitude stability.

On the upstream side of the reed we have the PWW, whose impedance function can be readily modified to give a wide variety of resonance peak distributions of the sort illustrated in Figures 35-6, 35-7, and 35-8 and discussed in Sec V. For present purposes it will suffice to recall that the tallest peak of Zu(ω) tends to be two or three times as tall as the tallest peak of the IAC impedance curve.

The discussion in Sec. V showed that one obtains good, steady oscillation if the net impedance controlling the reed has a set of harmonically related peaks. We did not at that time face the question of what would happen to the system if (as in the present case) the net impedance has a set of harmonic peaks plus one or more inharmonic, "maverick" peaks as well. Our recent experiments show that (except under special circumstances to which we shall return shortly) a stable regime of oscillation normally sets itself up under the domination of the harmonic set of peaks. This "civilized" regime of oscillation involves energy production at a set of harmonic frequencies in the manner already outlined, WITH NO PRODUCTION AT THE FREQUENCY OF A MAVERICK PEAK! This seems a little surprising at first, but if a sinusoid at the frequency of the maverick peak were to be produced, its nonlinearly generated harmonics, and the intermodulation product frequencies generated by it via nonlinear coupling to the harmonic components otherwise generated, would in general lie at points of the overall impedance curve where they would function as a heavy drain on the energy budget of the complete oscillation. It turns out, then, that such oscillations are not normally possible. As a result we find that the IAC dominates the sound production of our experimental system, as it does in the case of a real clarinet, with the PWW playing an auxiliary role in the manner which we are about to demonstrate.

The top half of Figure 35-12 shows the sound spectrum (pn)d measured in the mouthpiece of our IAC under two experimental conditions. The lower half of the figure shows the corresponding two spectra (pn)u measured in the player's mouth. The solid lines in both graphs shows the spectra measured under the "normal" condition that no resonances in the PWW are anywhere near the resonances of the IAC. The mouthpiece spectrum under these conditions is of exactly the form that has become familiar to us in our work with woodwinds over many years. Its shape is controlled by the impedance of the IAC, as we have become accustomed to take for granted (see Equation 35-8). We recognize, now, that the relative invisibility of the PWW generally comes from the fact that its resonances do not normally lie in places that give rise to the special circumstances referred to above, but not yet elucidated. The solid line in the lower part of the figure, which shows the nature of the corresponding spectrum in the player's mouth, does not have any particular features that we need to dwell upon at the moment.

If one listens to the signal picked up by a microphone within the player's mouth as he alters the configuration of his tongue, etc., it is at once apparent that one or another of the harmonics that belongs to the played note is boosted very considerably as a resonance of the PWW is brought into tune with it. The dashed lines in the two parts of Figure 35-12 show the nature of the spectra observed when the PWW has been configured so that the frequency of its major peak is brought into coincidence (at 960 Hz) with the fourth harmonic of the played note. We notice first of all that there is a 40-dB (100-fold) increase in the strength of this harmonic in the player's mouth, along with a considerable strengthening of all the neighboring harmonics as a result of intermodulation between this strong component and its neighbors.

The spectrum measured within the mouthpiece of our "clarinet" under the special condition of harmonic tuning of the PWW (as given by the dotted lines in the upper diagram) shows no particularly noteworthy changes. The fundamental and second-harmonic amplitudes are essentially unchanged, while most of the other harmonics are strengthened by 5 or 6 dB (a doubling in amplitude). The fourth harmonic, which one might think should be greatly strengthened, has its level raised by only 12 dB (a four-fold increase in amplitude). These results illustrate and confirm the mathematical result described in Sec. IV to the effect that alterations in Zu and in Zd show their predominant effects on the same side of the reed as the changes are made, with only small changes being detectable on the other side.

One may confirm the perceptual smallness of the changes in the mouthpiece signal by listening to it while the PWW is being altered. One becomes aware only of a rather subtle change in the sound of the sort one is accustomed to hearing as a fine player makes an effort to achieve the best possible tone. In other words, the listener gets only a hint of the drastic changes that are taking place within the player's mouth. Our picture of what is going on as the clarinet reed collaborates with the IAC and PWW is at last becoming clear. Our understanding of the ways in which the PWW could be important in a general way without ever making its presence obvious is also beginning to develop. For example, proper alignment of a PWW resonance can improve a note by stabilizing the regime of oscillation and reducing the stray noise even though its influence on the externally perceived tone color may be quite small.

We now take our first step in describing the special circumstances under which the PWW can make its presence felt in an overt way. Consider a dynamical system in which the IAC is designed so as to give only a single strong resonance peak (an example of which is shown in the lower part of Figure 35-13, where the peak is at a frequency fa = 340 Hz). If then the reed is instructed on its downstream side by such an air column and on its upstream side by some version of the PWW, there is no harmonic collection of resonance peaks that can cooperate in setting up a well-defined regime of oscillation and its corresponding harmonic spectrum. If the player will, however, explore the possible variations in Zu(ω), it proves possible for him to find one or more cases where the tallest peak of the PWW resonance lies at such a frequency fb that energy generation at this frequency and at fa is sufficient to feed all of their intermodulation products, some of which may fall on top of one or more of the remaining (inharmonically positioned) resonance peaks of the PWW. The upper parts of Figure 35-13 show the enormously complicated mouthpiece and mouth spectra measured for such a special case, whose sound is recognizable by musicians to be of the complexly interwoven type that they are accustomed to calling a "multiphonic." Oscillatory energy is primarily produced near the frequency of the IAC impedance maximum and at one (or perhaps more) of the PWW resonance peaks. Because of the strong nonlinearity of the reed valve, these two or three primary spectral components have bred a whole host of intermodulation products. We have produced numerous versions of the IAC and PWW configuration described here, and are always able to produce multiphonics in exactly the same way. In all cases it has been possible to analyze the spectra in the manner already worked out for the analogous type of oscillation produced by an IAC whose resonances are inharmonic (Benade 1976, chap. 25).

There are two more special cases of IAC-PWW interaction that we need to consider. It proves readily possible for one to find configurations of the PWW in which the strongest of the resonance peaks is so placed that its coupling with the harmonically positioned resonances of the IAC can disrupt even a going regime of harmonic oscillation and replace it with a multiphonic of rather considerable complexity. The precise dynamical requirements for this sort of transition have not been worked out so far, but it turns out that a player can, with only a little practice, learn to provoke such a multiphonic starting from almost any note of his normal scale. We will say no more about this class of phenomena (which needs considerably more study), turning instead to a dynamically much simpler, but musically more spectacular special case. A solution has been found to the rather challenging acoustical problem of designing an IAC (for use with a clarinet reed and mouthpiece facing) for which the input impedance Zd seen by the reed is essentially resonance-free (basically real); the magnitude of this impedance is no more than about one tenth that of a normal 15-mm ID clarinet bore. Clearly, such an IAC can give no "instructions" to the downstream side of the reed, leaving it entirely under the influence of the PWW! Attempts to play on this air column are quickly rewarded with a variety of easily controllable tones whose pitches are determined directly either by the 450- to 1400-Hz range of easily available resonances of the PWW or by the 2000- to 3000-Hz range of reed resonance frequencies, where ZA can be large even when Z itself is no more than that provided at high frequencies by the PWW. In the first case, motions of the player's tongue, etc., vary the pitch, with very little change due to alterations of the embouchure tension that controls the resonance frequency of the reed itself. In the other case, the playing frequency depends on embouchure tension almost exclusively (via the reed frequency). It shows no influence from the (lower-frequency) resonances of the PWW that are controlled by the player's vocal tract configuration. The listener finds it very easy to recognize which one of the two ways of playing is in operation, and to recognize the transition of dominance from one to the other when the player manages to crowd one of them into the domain of easy playing of the other.

It is quickly apparent that a good reed-player has no trouble in learning to control the resonances of his PWW for quasimusical purposes. I possess a cassette tape on which microphone signals were recorded both from the upstream and the downstream sides of the reed while George Jameson performed (on demand, in quick succession, and with essentially no practice) a wide variety of familiar musical extracts, among which are themes from Haydn's Trumpet Concerto and "Surprise Symphony," Mozart's Clarinet Concerto, the Sextette from "Lucia di Lammermoor," and "Hearts and Flowers." In all cases the fluency of performance and accuracy of intonation was at least equal to that of any amateur whistler possessed of a good ear. It is essential to notice that I am NOT prepared to say that the ease with which this new mode of performance was learned gives evidence that the task is parallel to one that is familiar to a player in the normal course of his professional music-making! While it may turn out to be so after we have made a proper study of the question, we should remind ourselves at this point that the same player would probably have done equally well on a slide-whistle or a musical saw (which requires motor skills that are quite unfamiliar to him).


For the present we will content ourselves with the knowledge that the reed has shown itself to be perfectly willing to take instruction from resonances on either its upstream or downstream sides. We have also learned that harmonically related resonances on the two sides of the reed can work together to give a steady and clean tone; that a maverick (inharmonic) resonance will not usually upset the steady operations of the reed controlled by a harmonically aligned set of resonances; and that, under special circumstances, it can disrupt the oscillation, producing instead a multiphonic of considerable complexity.

To the extent that our musical investigations have progressed, we have good reason to believe that most (though by no means all) present-day woodwind players use the resonances of their PWWs to "fill out," "clarify," or "fine-tune" the notes they play. Among the brass players the analogous behavior seems to be much more common. In any event, connections are beginning to be recognized between the age-old urgings of music teachers and the positions of various resonances in the PWW. We are also forced to be very much aware that the nature of these connections is not easily foreseen by the physicist. In particular, we must remember that it is utterly absurd to expect, a priori, that the positions of the vowel formant frequencies have any direct connection with the impedance maxima of the PWW as we have been discussing them. This is because the PWW resonances of interest to us are associated with the normal modes of an air column having a high impedance (i.e., a "closed" boundary at the player's mouth and a complicated boundary at the other end provided by the lungs, whereas the vowel formant frequencies belong with a (much shorter) air column that has a low-Z boundary at the mouth end and a high-Z (essentially closed) one at the larynx end!

In conclusion I will summarize the whole paper by outlining the reasons why all of these player's windway effects managed to stay out of scientific sight for so long. First of all, every player learns almost at once the easy task of avoiding "pathological" configurations of the PWW that give rise to those multiphonics that can take place despite the presence of aligned resonances of the IAC; these sounds can be recognized as the shrieks often produced by beginners.

Secondly, the audible effects of resonance alignment in the PWW are rather subtle and not readily recognized in the instrument's mouthpiece spectrum or in its concert-hall descendant. As we have seen, their existence was really only demonstrable once the overall picture had begun to be worked out. Many other subtleties of tone color and response had to be recognized and rationalized (if not quantified) before the PWW effects could be meaningfully studied.

A third reason, while quite personal to the present author, is quite typical of the cause of many inquiries in musical acoustics. At the beginning of my acoustical studies, I was a reasonably good player of many woodwinds, good enough to feel a little confidence in my judgements concerning tone, tuning, and response and their dynamical implications (subject always to revision by comments from professional players and by data acquired in the laboratory). This familiarity with instruments made it easy to devise check-experiments, and to notice phenomena that might or might not be ready for closer study. However, formal consideration of any physiological parameters had to be set aside in favor of the more accessible physics problem. In this spirit and under the stimulus of Bouasse's writings, we in Cleveland pursued the idea of aligning the IAC resonances and carried the physics far enough to guide my production of musically interesting clarinets, flutes, and other instruments (designed to suit my own taste, after which they generally won the approval of distinguished players). The fact was not recognized by me for several years that I was exploiting the reed resonance phenomenon via embouchure tension changes in the course of my playing and while I was making physical adjustments to the instruments. In retrospect, I realize how fortunate it was that my chief musical advisers concerning the scientifically most accessible clarinet family were all of the old school, who were accustomed to using these resonances in their own playing, so that my unconscious efforts were not contradicted or confused by my advisers' comments. (This resource now turns out to be exploited by only a certain fraction of the younger clarinet players, in part because today's commercial instruments are made in such a way that this kind of exploitation is difficult if not well-nigh impossible.) Exploitation of the reed resonance phenomenon is normal practice for the other woodwinds. In any event, once the cooperation phenomena were well in hand, it was possible to pin down the physics of the reed resonance effects (Thompson 1979) and to relate these to musical technique. Meanwhile, my musical friends kept insisting that embouchure tension was by no means the only means of physiological control available to the player and, from the start, I had my own feeling that they were right. As my own ability as a player grew, I became more and more aware of the possibilities and also increasingly confident (as other things fell into place) that the time was approaching that a worthwhile study could be made. The intellectual parallel of the PWW and the reed resonance studies was recognized and used as an encouragement. Before this, a certain amount of conscious brashness was required to simply leave something out of consideration until it forced its way into visibility!

The moral of the story is that progress in musical acoustics is (best? only?) made by a special kind of two-legged creature: one of its necessary legs is scientific, while the other is musical, and it is important that both of them be strong and in good working order.

Supported by a grant from the National Science Foundation.


Backus, J. (1962). "Small vibration theory of the clarinet," J. Acoust. Soc. Am. 35, 301.
Benade, A.H. (1976). Fundamentals of Musical Acoustics.Oxford Univ. Press, New York.
Bouasse, H. (1920-30). Instruments à Vent. 2 Vols. Librarie Delagrave, Paris, I:115-116, 312-314, II:47.
Dawson, S.V. (1976). "Input impedance measurements of the respiratory system by impulse response," Tech. Prog. Rep. Harvard School of Public Health.
Fransson, F. (1965-75). A series of articles on the STL ionophone transducer published in the STL QPSR, Stockholm.
Fransson, F. and Jansson, E. (1975). "The STL ionophone transducer, properties and construction," J. Acoust. Soc. Am. 58, 910.
Ibisi, M.I. and Benade, A.H. (1982). "Impedance and impulse response measurements using low-cost components," J. Acoust. Soc. Am. 72, S63.
Krüger, W. (1979). "The impulse method of determining the characteristics of brass instrument response," Proc. of the Musical Acoust. Conf. (in German). Cesky Krumlov (Sept. 1979). Reprinted (in Czech) in Hudebni Nastroje 17, 52 (1980).
Oliver, B.M. (1964). "Time domain reflectometry," Hewlett-Packard J. 15/1. See also HP Application Note 75 "Selected articles on TDR applications.
Rosenberg, J. and Gordon, H. (1966). "The pulsed acoustic reflectometer," Physics Lab. Proj. Rep., Case Institute of Tech., Case Western Reserve Univ.
Thompson, S.C. (1978). "Reed resonance effects on woodwind nonlinear feedback oscillations," doctoral diss., Case Western Reserve Univ.
Thompson, S.C. (1979). "The effect of reed resonance on woodwind tone production," J. Acoust. Soc. Am. 66, 1299.


P. MILENKOVIC: Flute instructors also tell you to do things with your mouth. Do you have any ideas on how to attack this problem with the flutes?

A. BENADE: The effects of this kind are very, very small in flutes. I have said a few things about mouth interaction [A.H. Benade and J.W. French J. Acoust. Soc. Am. 37, 679, 1964], which are wrong. And John Coltman has said things [J. Acoust.. Soc. Am. 54, 417, 1973] which I think now aren't quite right. They're not wrong. These effects are very small because it's longitudinal. They matter, of course. Now I can spot how a flute player has got his mouth by listening to it. The musicians we talked with differ about a very small effect compared with what I'm talking about here. [Added in proof: Walther Krüger of the DDR has recently shown large effects on a recorder.]

M. ROTHENBERG: I can't help asking this question because it's been on my mind for so long. Some years ago I did some experiments with a harmonica. Have you ever worked with that instrument?

A. BENADE: By chance a letter asking essentially this same question came from Roy Childs of the American Harmonica Society at just about the time all this was beginning to occupy my mind in a serious way. The harmonica reed is nominally of the "free" type with its own preferred frequency of oscillation. Working out the theory along lines very similar to those outlined in this paper shows (and experiment confirms) that oscillation is only possible when there is some inertance in the airway channel, and that the frequency of oscillation can be moved around by changing the viscous resistance (real) part of the airway impedance.

M. ROTHENBERG: I once heard somebody playing on the harmonica in a music store, making all this beautiful music. And I knew how to play "Oh Susannah!" you know. So I went up closer; I expected to see a gigantic machine with all sorts of buttons, but he had a little marine-band harmonica. So I took a course actually, with a friend of mine who plays harmonica, and I found out that you could do that reed bending. Furthermore, my experiments in blowing the harmonica with compressed air, using an adjustable acoustic loading behind the mouthpiece, also convinced me that much of the beautiful music that a good player can obtain from a small, single key harmonica is due to acoustical loading of the reed performed by appropriately shaping the lips, vocal tract and larynx. However, even though I had lived in the vocal tract, so to speak, for about 15 years, and had, I thought, a great deal of insight on what tract manipulations could cause significant changes in acoustic loading at the lips, after about 10 hours of expert instruction I could only manage to perform the very easiest examples of "reed bending" (which I would define as a pitch shift toward the opposing, supposedly inactive reed).

A. BENADE: Well, here's one thing that distracts people like us. The real part of Z down here talks to the imaginary part of the reed and vice versa in many of these situations. And so in pulling the frequency, that you can do if you put a handkerchief over the windway or something, you can move it a long way. We have backwards reflexes here. Thank you for raising the question because I had intended to give a little bow in the direction of Roy Childs. An Irish harmonica player also visited, and supplied a little of the stimulus.