The physics of free reeds
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  The physics of free reeds  

 

Colin Pykett

 

Posted: 24 January 2023
Revised: 24 January 2023
Copyright C E Pykett 2023


Abstract.  Without using mathematics, this article describes the physics of the venerable 'free' (as opposed to 'beating') reed used for millennia in many forms of wind instrument. It shows that the reed is an acoustic negative resistance oscillator, oscillation being maintained by an impulse at mid-cycle arising from forces due to static blowing pressure, the Bernoulli effect and the inertia both of the vibrating tongue and air flow. The reed tongue then moves freely each side of the impulsing region to oscillate virtually sinusoidally with a high Q-factor. Thus oscillation is practically isochronous, frequency being virtually independent of amplitude (loudness) and hence blowing pressure. Periodic air pulses are emitted from the time-varying windway of the tongue as it oscillates, the pulse shapes defining the harmonic content of the waveform and thus the timbre of the sound. Pulse shapes vary with blowing pressure.

 

The free reed can couple strongly to adjacent resonant cavities. Therefore, notwithstanding its frequency stability, its pitch can be 'bent' by harmonica players because of the cavities formed by the player's vocal tract and their cupped hands. In the case of organ pipes using free reeds it is shown that the 'boot' acts as a Helmholtz resonator with the reed assembly forming the tuned port. If the boot is too small it can result in 'pulling' when tuning the pipe, explaining why the boot of free reed organ pipes has to be made unusually large.


Specific free reed waveforms recorded for this article show three distinct regions per cycle. Two are probably related to the air pulses arising from the windway of the tongue as it oscillates about its mid-point, but the third is associated with a high frequency oscillatory burst at about 12.5 kHz. The bursts are tentatively associated with the tongue when it receives the impulse which maintains its oscillation. It is suggested that the impulse might transiently excite higher-order modes of tongue vibration at these high frequencies. These results differ markedly from those reported elsewhere where the waveforms are sometimes said to resemble square waves, and that only two regions per cycle are visible rather than the three seen here.  Frequency spectra corresponding to these waveforms are also presented. They confirm the large number of harmonics associated with free reeds which suggest that considerable acoustic power exists well beyond the limit of human hearing. Both odd and even harmonics were strongly represented in the sounds of these reeds.

 

Reed tuning and voicing matters are discussed, including some differences between beating and free reeds. The shallot aperture of a beating reed is important in that its shape, and hence the acoustic spectrum generated by the reed, is varied widely by pipe organ builders. Therefore the lack of anything comparable for a free reed considerably limits its voicing flexibility. However the sensitivity of free reeds to adjacent resonators such as organ pipes, mentioned above, is discussed by relating it to the appearance of pipe organ stops in the nineteenth century using free reeds with an apparently attractive range of tone colours.

 

 

Contents

(click on the headings below to access the desired section)


Introduction

 

Free and beating reeds

 

How the free reed speaks:

 

    Reed oscillation mechanism

    Q-factor, frequency stability and pitch bending

    Frequency and pitch

    Acoustic waveforms

    Acoustic spectra

    Tuning and voicing

 

Free reeds in the pipe organ

 

Summary and conclusions

 

Notes and references

 

 

Introduction

 

Free reeds are musical sound-producing devices which have been used for at least three millennia, such as in the ancient varieties of small mouth-blown instruments of Eastern origin which still exist in diverse forms. However it took Western culture far longer to catch up, so they did not make much of an appearance on that stage until about two centuries ago in instruments such as the concertina, accordion, harmonica and the reed organs of the nineteenth century [1]. But these instruments then swept rapidly across the globe, their influence powered by the several European-culture-dumping empires which then held sway. Another significant, if short-lived, application of free reed technology was its brief appearance in various reed stops of the Victorian-era pipe organ. Today a paradoxical consequence of this ebb and flow of attention is that effort devoted to understanding how free reeds emit sound has never been higher, even though they are hardly ever found now in pipe organs, and even though surviving examples of the long-defunct reed organ struggle to maintain the narrowest of niches in an ever-narrowing organ world. Consequently, when the free reed revolution was at its most fervent in Victorian times, it is now evident with hindsight that nobody really understood how to make those instruments sound to best advantage. However the recent resurgence of curiosity about the acoustical physics of free reeds is not antiquarian or backwards-looking. Rather, it is driven largely by the rising popularity of various types of 'squeeze box' such as the concertina and accordion, and especially by the enduring esteem for the harmonica where considerable effort is now devoted to understanding and expanding its subtle expressive capabilities such as pitch-bending. (Incidentally, this recent work can deliver a mental jolt to those brought up to believe that the tuning stability of free reeds is so high that they could not go out of tune sufficiently to deliver a noticeable pitch-bend at all). Here, we take account of these instruments, while also looking further back to include reed organs such as the harmonium and the American organ and those rare pipe organ stops which also used them. Their common factor is the free reed itself, whose physics is the subject of the article and for which one's respect increases the more one delves into them.

 

 

Free and beating reeds

 

Almost all reed stops in pipe organs use beating reeds rather than free reeds. These are discussed in two articles elsewhere on this website so the material will not be repeated here in detail ([2], [3]). They are so named because the oscillating reed tongue covers and uncovers an aperture smaller than itself cut into the flattened wall of a narrow tube (the shallot). However the initially-curved tongue does not actually beat against the shallot in the sense of striking it; instead it rolls out by way of a bending process towards flatness when it meets the shallot and then takes up its curvature again during the course of its oscillation cycle.

 

A free reed also has a vibrating tongue fixed over an aperture, but this is cut into a frame whose thickness is greater than that of the tongue. A typical free reed taken from a keyboard reed organ, a harmonium, is shown in Figure 1 which shows the brass tongue riveted to a frame of the same material. The main difference between a beating and a free reed is that the aperture in the reed frame of the latter is slightly larger (rather than smaller) than the tongue, thus the tongue is able to swing freely through the aperture during each oscillation cycle. 

 

 

Figure 1. Harmonium reed


Besides the keyed accordion, reed organs played from a keyboard were of two main types. Although the nomenclature can vary in different countries, in Britain the term 'harmonium' usually denotes an instrument which uses wind blown into the reed at positive pressure, whereas the 'American organ' uses suction [4]. However, in all cases the reed is 'blown-closed', meaning that the initial motion of the reed tongue when wind is first applied is towards the aperture cut into the reed frame seen in the figure. If wind is applied in the reverse direction the reed does not sound because the tongue is merely blown further beyond the frame. This feature is exploited in instruments such as the harmonica with its 'blow' and 'draw' reeds which the performer can play separately by applying pressure or suction.

 

Another major difference between instruments using free and beating reeds is that the sound of the naked reed is never heard with a beating reed in a pipe organ. Its sound is invariably directed into the hollow shallot and thence into a resonating tube in the form of an organ pipe. If that were not so it would sound little different to the unmusical squawk of a clarinet reed assembly yet to be inserted into the instrument. But the opposite is true of most free reed instruments, where one hears essentially only the sound of the reed itself unmodified by a resonator. Although that sound will be affected to some small extent by the physical structure surrounding it, such structure (e.g. the reed cell honeycomb of the American organ) was usually intended more to support and control the reedwork rather than to modify its tone (but see reference [5]). This is also largely true of instruments such as the concertina and accordion, and definitely so of the harmonica, where the player's cupped hands and vocal tract provide important and subtle means of modifying the timbre (and pitch) of the otherwise naked reeds. It was only in situations where free reeds were occasionally combined with tubular resonators in the pipe organ that one's ears were acoustically isolated from the sound of the reeds as far as that instrument was concerned. In passing it should be noted that some of those free reed pipe organ stops were considered to have great beauty, and they are discussed later on.

 

Despite their differences, both beating and free reeds produce sound by periodically interrupting the air flowing through the assembly containing them. They are analogue air valves which apply a continuously varying throttling effect to the air passing through, and it is the energy in the resulting repetitive puffs of air which produces the sounds we hear. Very little acoustic energy arises from the motion of the reed itself. This can be understood by referring to a tuning fork, whose tines move in the same way as a free reed (the physics, that of a vibrating cantilever, is the same) but without any air flow to sustain their motion indefinitely and to generate acoustic power. Its reticent sound confirms these remarks. Consequently the acoustic waveshape resulting from the puffs of air passing through its windway is responsible for the timbre of the sound emitted by both free and beating reeds, pulses with rapid rise times resulting in greater harmonic development than those with a more leisurely onset.

 

The term 'windway' applied to free reeds deserves some expansion at this point, since it does not refer to the rectangular opening in the reed frame through which the tongue swings. Instead it means the total area of the aperture measured around the periphery of the tongue through which the wind passes while the reed is sounding, which is quite a different thing. Since the dimensions of this aperture obviously change continuously during each cycle of oscillation as the tongue moves, it is this effect which gives shape to the air impulses which we hear as sound. These matters will be discussed for the free reed in more detail later on, but here it is noted that the physical shape of either type of reed tongue will also influence the windway, hence the acoustic waveshape, and thus the tone colour produced. For beating reeds the curvature applied to the stationary tongue influences the emitted sound considerably. This is also true of the free reed, for which various kinks and twists can also be formed deliberately in the tongue as in Figure 1, an anathema never permitted for the beating reeds of the pipe organ. Another key factor for beating reeds concerns the shape of the aperture cut into the shallot as described in reference [2] (see especially Figure 3 therein), whereas there is no equivalent feature for a free reed because these have no shallot and thus no shallot aperture. This means that voicing options for free reeds are significantly restricted compared to those for beating reeds. This has led to the disappointment sometimes expressed by those unimpressed by the sound of the reed organ along the lines of 'the stops all sound much the same and like a mouth organ'. However this opinion reflects only part of the story, because the wide range of beautiful sounds which arise from the beating reed stops in the pipe organ is largely because they do indeed have pipes. Without these even the most carefully voiced beating reed, regardless of how its tongue and shallot are treated, would sound little better than a buzzing insect. So, in the absence of shallots and resonating chambers, voicing options for free reed organs were hobbled from the start because all we can ever hear from them is more or less limited to the sounds of their naked reeds. Conversely, those free reeds which did occasionally speak into resonating chambers such as organ pipes were capable of some beautiful effects in their heyday, as will be discussed later.

 

 

How the free reed speaks

 

Reed oscillation mechanism

 

 

The tongue of a free reed either lies flat across the frame aperture while it is not speaking, or it might have been given some gentle curvature, or various kinks might have been formed in it (as in Figure 1). However, in all these cases there will usually be a windway of some sort between the stationary tongue and the frame through which air will initially pass when the wind is first applied by the performer. Even if the windway at this position is negligible before wind is applied, it will develop once the tongue begins to oscillate. From the unassailable fact that a free reed makes musical sounds, it is obvious that the tongue must move towards the frame when wind at the proper pressure is admitted, and it will continue to do so until the increasing force due to its elasticity or springiness causes it to move back again, thus causing the cycle to repeat indefinitely. However, this is about the limit of the usual explanation one gets of free reed behaviour, so in this article we need to go quite a lot further to understand the situation in more detail.

 

First let us observe that the tongue will only oscillate over a restricted range of wind pressure - if the pressure is too low the force exerted by the air on the tongue will be insufficient to push it close enough to the frame and consequently nothing will happen. If too high, the pressure will not allow the tongue to move back fully through the frame, so again either nothing will happen, or the reed might speak with an unsatisfactory tone. In the correct pressure regime between these two extremes the reed, in effect, acts as a negative resistance oscillator of the type encountered in electronics. Negative resistance in a circuit occurs when an increase in voltage results in a decrease in current, the reverse of the usual situation described by Ohm's Law, and it compensates for the energy lost as heat in the normal (positive) resistive parts of the circuit. Negative resistance can only arise when an active device such as an amplifier is present to provide the compensating energy, and under these conditions the circuit can be made to oscillate indefinitely instead of the oscillation dying away owing to energy losses.

 

The free reed (and the beating reed) is an acoustic version of this type of oscillator, where air under positive or negative pressure (suction) provides the energy necessary to overcome losses elsewhere in the reed oscillatory system. We can see how an acoustic negative resistance arises by considering what happens as the air pressure (analogous to voltage in the electronic case) is increased slowly from a low value without causing the tongue to oscillate. Initially the air flow rate (analogous to current) through the existing windway will merely increase with the pressure as we might expect, but nothing else will happen. However, at a certain value of pressure the tongue will begin to move towards the frame, the windway available to the air will be made smaller, and therefore the flow rate will start to decrease. This decrease in flow with an increase of pressure is a negative resistance situation which encourages the instability necessary for continuous oscillation. Thus the free reed is a negative resistance oscillator.

 

Once the tongue has moved fully into the closely-fitting frame the windway is at its minimum. Little air can escape around the periphery of the tongue and so it continues to move, pushed both by the static wind pressure and (when oscillating) additionally impelled by its own inertia. Frequently it will then exit the frame on the other side of the aperture whereupon the windway suddenly increases again, allowing the air flow rate to increase. Simultaneously, the force on the tongue suddenly reduces as the air is able to escape into the atmosphere. So here again we have a second occurrence of negative resistance, but this time in the opposite direction - flow rate has now increased for a decrease in pressure on the tongue.

 

Another rather esoteric factor can also come into play during oscillation which needs to be mentioned for completeness. We can ask: what makes the tongue move towards its frame in the first place? This might sound a trivial question but in fact the answer is quite subtle. Consider the case where wind is first applied. Intuitively we can see that the drag experienced by the tongue due to this initial impulse will tend to bend it towards the frame as the air grabs it. As soon as this happens the windway for the air flowing into the reed frame will be reduced as noted above. This reduces the flow rate but it also increases the speed at which the air passes through the reduced windway. The situation is no different to a garden hose in which one reduces the size of the orifice to get a faster water jet. So far so good, but now for the subtle bit. The increase in air speed results in a reduction of air pressure behind the tongue, that is, on its surface which faces the frame aperture. The reason why this happens is because of the Bernoulli effect [6]. But the pressure on the opposite surface of the tongue facing the incoming air will remain the same. Therefore there is now a net pressure difference which causes the tongue to move further towards the frame. Moreover, the Bernoulli pressure difference will continually increase as the tongue moves because the air rushing through the ever-decreasing windway moves ever-faster. Therefore the Bernoulli pressure effect gets continually stronger, and it forces the tongue ever closer and ever faster towards the frame. You might think that this motion will continue until the tongue completely fills the aperture, and at this point it will reverse its direction of motion by virtue of its elasticity because the Bernoulli rarefaction effect practically vanishes owing to the virtual cessation of air flow. However, because a reed tongue moves with significant speed in mid-cycle while speaking, its inertia (plus the static wind pressure) often carries it right through the frame aperture, resulting in the direction reversal occurring on the far side of the frame. In some cases there might also be an additional impulse due to the air flow having been abruptly halted when the windway almost vanishes, resulting in the inertia of the air itself helping to push the tongue further into the frame.

 

So there are several factors causing the tongue to move into, and often beyond, the frame. These are the static wind pressure, the inertia of the moving tongue, the Bernoulli effect, and the inertia of the air as the windway suddenly closes against it. Together they generate the force impulse necessary to maintain the oscillation of the tongue. The way the impulse fleetingly develops and decays over time as the tongue moves through the frame is of importance to those developing a physical model of the free reed. This endeavour is further complicated by the number of parameters which have to be taken into account such as blowing pressure, the oscillation mode(s) of the tongue and how the reed unit is constructed and from which materials.

 

On its recovery stroke the tongue will generally overshoot its original position, again because of inertial effects, before reversing its direction of travel once more owing to its springiness. At this point the cycle then repeats as before when the air begins to act on it again. The reed tongue is therefore a valve which admits a periodic (repetitive at a constant frequency) series of air impulses through its continually varying windway. Each impulse has the same characteristic shape in terms of the variation of air pressure with time, and this waveshape defines the harmonic content of the waveform emitted by the reed assembly into the atmosphere.

 

It is worth comparing the free reed oscillation mechanism to that of a child on a swing. In both cases maximum speed occurs mid-cycle - for the swing this is at its lowest point and for the reed tongue it happens inside the frame aperture. There is a difference in that the swing is normally maintained in oscillation by someone pushing it at an extremity of its cycle, whereas the tongue is pushed hardest by the wind and its own inertia when it is inside the frame. However this is of little consequence since it would be possible, if less convenient, for the swing to be impulsed at its lowest point. The swing is a simple harmonic oscillator which oscillates almost sinusoidally like a clock pendulum (which, incidentally, keeps time best when it is impulsed at its lowest point), and the oscillation period of both swing and pendulum are scarcely affected by how strongly they are pushed within reason (their amplitude). The free reed is similar in that its tongue also oscillates almost sinusoidally, and its frequency hardly varies with variations in wind pressure which affect its loudness. These types of oscillator are therefore referred to as isochronous in that they remain stable in frequency with changes in amplitude. The reason is that the free reed, like the swing and a pendulum, only receives a brief maintaining impulse as it passes through the frame. Over the remainder of each oscillation cycle it is largely free to oscillate at its natural frequency. This is not the case for a beating reed, whose tongue is impelled continuously by a Bernoulli force towards the shallot. Unlike the brief impulse received by the free reed, this force is never absent so it dominates the tongue's motion at all points of its oscillation cycle. It is like pushing a swing where you never let go. The instantaneous speed of a beating reed tongue is therefore controlled strongly by the Bernoulli force, which in turn depends on wind pressure, rather than by the elastic properties and other parameters of the tongue as for the free reed. Therefore its frequency is sensitive to wind pressure and thus loudness variations, and it is not isochronous.

 

 

Q-factor, frequency stability and pitch bending

 

The 'Q' or 'quality' factor of an oscillator is a number which reveals how sharply it resonates. A high-Q oscillator will only resonate at, or very close to, its design frequency whereas one with lower Q can oscillate over a wider frequency range. Not surprisingly, it is desirable that a free reed should have a high Q so that it will not be persuaded to go too much out of tune by the myriad factors which afflict musical instruments, their players and their environment. Bypassing a lot of detail, this means that the reed tongue should not be strongly damped, in other words there should not be processes which cause it to lose excessive energy while oscillating. Taking a ridiculous example, the tongue would be highly damped by viscosity if we tried to make it oscillate in oil rather than air. We can relate Q to damping because it can be defined mathematically in those terms if desired, and thus related to the energy lost in oscillation. Fortunately there is little which impedes the motion of the oscillating free reed tongue other than minor factors such as friction with the air, and therefore it does indeed have a high Q. The mere fact that its frequency is so stable, as discussed above, confirms that it is a high-Q oscillator (more so than the beating reed).

 

This is different to the case of analogous orchestral instruments such as the clarinet. Here the reed is more strongly damped because of the player's lips clamped around it, and they absorb energy from the vibrating reed. Therefore the clarinet reed has a lower Q and consequently it can be persuaded more easily to vary in frequency while playing a single note. In the case of the clarinet this does not matter because the player's ears provide a continuous and unconscious feedback mechanism which enable the pitch of the note to remain sufficiently stable for musical purposes. At the same time the amount of pitch shift available to the woodwind player by subtle changes to the way the instrument is blown is advantageous, enabling desirable effects such as vibrato (a periodic frequency variation) to be added to the performance. This is exploited universally by oboists in particular.

 

Another implication of its high Q is that the free reed comes onto speech relatively slowly. The number of cycles required for any oscillator to reach its stable working amplitude is yet another way to define its Q-factor. This might be fine if it happens to fit the music being played in a subjective sense, but there are also situations where it is disadvantageous. To counteract this, some reed organs had a 'percussion' stop which actually hit the reeds with small hammers, though this was intended more to kick the otherwise leisurely-onset reeds into prompt action rather than to generate an ersatz xylophonic tinkle.

 

From the foregoing emphasis on the high frequency stability of free reeds, it might seem surprising that there are situations where the pitch of a reed can nevertheless be 'bent' significantly by (for example) a harmonica player. This takes us onto the subject of frequency and pitch in more detail where pitch bending will be explained.

 

 

Frequency and pitch

 

The free reed tongue oscillates at or close to its fundamental natural frequency, the frequency you would get if you gently 'pinged' it. There are also higher frequency modes of oscillation, though these are inharmonic (not harmonically related to the fundamental). The frequency of the second mode, for example, is around six times that of the fundamental. It is often said that the reed tongue's motion is little affected by these remote upper modes, so that it moves with a pure sinusoidal motion in and out of the reed frame. However, some results presented later suggest that the reed might exhibit significant higher-order modes of oscillation which are easily visible in its acoustic waveform. But regardless of how the tongue itself actually moves, the puffs of air it emits from its windway are certainly far from sinusoidal and it is these which we hear and which produce the harmonics in the sound. Deriving an equation for the natural frequencies of a reed tongue involves some pretty heavy maths and physics and this will not be covered here. In fact it is probably true to say that the physics of the vibrating reed was not really sorted until well into the twentieth century, implying that making reeds speak with the desired frequency was for long a largely empirical art.

 

Free reed frequency depends on the length of the tongue, its cross-sectional area and the material it is made of. For a tongue with the usual rectangular cross section, frequency increases as the square root of area and inversely as the square of length (beating reeds are different). Tongue width affects to some extent the volume of the resulting sound because a wider reed provides a larger windway, giving acoustic waveforms of similar shape but with a higher amplitude. However the loudness increase is not as great as might be imagined, because increasing the smallest contributor (tongue width) to the total windway does not increase the windway very much. So to significantly increase the windway to make a louder reed it would be more sensible to increase tongue length, and then adjust the design frequency by changing its cross section. Looking at some of the bass reeds in Victorian reed organs, or others intended to be loud, makes me wonder whether this was understood in those days [7]. 

 

We have seen that wind pressure affects the speaking pitch of a free reed hardly at all, so its tuning is independent of pressure for practical purposes. This is why devices such as the 'expression' stop can be used in a harmonium - this bypasses the air reservoir and allows the pressure applied to the reeds to be instantaneously controlled by the player's feet on the treadles. Thus the volume of harmonium reeds can be varied without affecting the tuning. Yet, as mentioned earlier, harmonica players can nevertheless bend the pitch of their instrument by a considerable amount, so how do they do it if the pitch of free reeds is supposed to be invariant? The answer is that the harmonica reed is strongly coupled in an acoustical sense to the resonant volume of the player's oral cavity, and the characteristics of this can of course be varied. Free reeds are markedly more sensitive to acoustic coupling than beating reeds, and we shall return to this attribute later in connection with free reed stops in the pipe organ where they were coupled to tubular resonators (i.e. organ pipes). It is reed-resonator coupling which allows the player to bend the pitch of a free reed.

 

 

Acoustic waveforms

 

The acoustic waveforms emitted by free reeds are highly variable from one reed to another even for reeds within the same set and close to each other on a keyboard instrument, though this behaviour is not unusual in musical instruments including the pipe organ. Cottingham reported an accordion reed waveform which looked closely similar to a square wave, which he explained by relating it to the puffs of air emitted by a siren. His article was available on the internet at the time of writing (see the link at reference [8]). Each puff was said to correspond to the tongue opening its windway on one or other side of the frame as it oscillated. The frequency of the reed was not quoted, but estimating it from the waveform plot (Fig. 2a in [8]) shows that it was near to 87 Hz, suggesting that the note in question was bottom F on a reed rank sounding at 8 foot pitch. The frequency spectrum of the waveform was not presented, though I would expect it to have shown significant suppression of the even-numbered harmonics compared with the odds because in a perfect square wave there are no even-numbered harmonics at all.

 

However, that example is nothing like one which I recorded myself, shown below in Figure 2.

 

 

Figure 2. A free reed acoustic waveform (treble C 8' pitch; 516.8 Hz)

 

This harmonica reed spoke at the much higher pitch of treble C (an octave above middle C) and its frequency was measured at 516.8 Hz using a wave editor (WaveLab), slightly flat to today's usual standard pitch where middle A is tuned to 440 Hz. As in Cottingham's experiment, the microphone was placed close to the reed. The waveform here bears no resemblance to the quasi-square wave he observed, being far more complicated. Broadly speaking, each cycle consists of three sections. A large peak is followed by a secondary one of lower amplitude, followed by a third which includes a distinct high frequency oscillatory region. In fact there is evidence of high frequency oscillation, though with lower amplitudes, superimposed across the entire waveform. One can reasonably relate the first major peak to the rise and fall of wind passing through the windway as the tongue moves towards and then away from its extremity of motion relative to one side of the reed frame. The smaller peak which follows might then result from the tongue having passed through the frame, opening up its windway on the other side. That this asymmetry (a large peak followed by a smaller one) exists is understandable, since the tongue of this reed was riveted to one side of the frame as is usual. This means that the windways on either side of the frame would also be expected to show some geometrical asymmetry, which would be reflected in the generated sound waveform. So this waveform (apart from its high frequency oscillatory component which we will come to in a moment) is relatable to Cottingham's air-puffs hypothesis, of which two should exist per cycle of oscillation, one on each side of the reed frame provided that the tongue was oscillating sufficiently strongly to emerge on both sides. Moreover, the first two peaks per cycle here match qualitatively the shape of his theoretical plot of air flow rate calculated for yet another reed (Figure 2b of his paper [8]).

 

The high frequency oscillations were something of a mystery, however. They cannot be parasitic bursts of oscillation due to air turbulence as these would look completely different in each cycle. Turbulence is a stochastic process whereas what we see here is definitely deterministic in origin. So I analysed the sound of another reed to see whether the results would be reproducible, and its waveform is shown in Figure 3.

 

 

Figure 3. Another free reed acoustic waveform (treble E 8' pitch; 646.6 Hz)

 

This reed from the same harmonica spoke treble E, a major third above the previous example. Its measured frequency of 646.6 Hz was again slightly flat to standard A440 pitch. The waveform again demonstrates the existence of three oscillatory regions per cycle, two of them plausibly corresponding as before to the windway of the tongue appearing successively on either side of the reed frame with each generating an acoustic pressure pulse. The third region is even more strongly oscillatory than before, but again with traces elsewhere on the waveform. The frequency of the high frequency oscillation in this case was estimated in WaveLab to be about 12.5 kHz, much higher than the fundamental reed frequency.

 

Although it remains to be confirmed, it is possible that the high frequency oscillations seen in the waveforms of both these reeds arose from one or more higher-order flexural (bending) or torsional (twisting) modes of the reed tongue. Such modes are inharmonic to and well separated from the fundamental frequency of the tongue. However, confirmation of this hypothesis will require additional work so it cannot be taken much further at this point. But it is also of interest to enquire of the whereabouts of the tongue when the most prominent oscillation bursts took place. If it was within the reed frame where the impulsive maintaining force due to wind pressure and other factors acting on it was strongest (see the previous reed oscillation mechanism section), it is tempting to suggest that this transient 'kick' did indeed result in higher-order mode(s) being excited. Such impulses must exist at this point otherwise the oscillation would not be maintained. Regardless of how such uncertainties might be resolved, the results here leave no doubt that three distinct temporal regimes of oscillation per cycle need to be included in a realistic physical model of free reed oscillation rather than the two sometimes assumed. Moreover it is credible to associate two of the regimes with the tongue emerging either side of the reed frame, together with the third related to its motion as it moves within the frame.

 

But it is worth bearing in mind that the subjective, aural, effect of the oscillatory behaviour is likely to be barely detectable, if at all. People with presbyacusis will be deaf to such high frequencies of relatively low amplitudes, and others with better hearing also might not perceive them when they listen to the sounds at normal listening distances much greater than the reed-to-microphone separation used in these experiments. In any event, now that we are speaking of frequencies it is appropriate that we turn the discussion away from waveforms and towards the acoustic spectra, and this follows.

 

 

Acoustic spectra

 

The frequency spectrum of the harmonica reed waveform in Figure 3 is shown in Figure 4.

 

 

Figure 4. Free reed acoustic spectrum (treble E 8' pitch; 646.6 Hz)


Note the presence of significant acoustic power nearly up to the Nyquist limit on the frequency axis (this equals half the sampling frequency of 44.1 kHz). As is necessary in digital sampling, the analogue microphone signal would have been first low-pass filtered in the digital audio recorder to prevent the appearance of aliased frequencies, so the spectrum display terminates abruptly and artificially after the 29th harmonic as one would expect. However, from the trend of the spectrum up to that point there is every reason to suppose that considerable power will still exist beyond this frequency even though it would be inaudible to humans.

 

This is unusual for musical instruments. Pipe organ spectra, for example, seldom go anywhere near such high frequencies because they employ resonators (i.e. the pipes) whose resonant modes fall off relatively quickly with frequency. This remains true even for organ pipes generally considered to have many harmonics such as those of reed and string stops. However, here we are hearing the unmodified sound of a naked reed, and the extended spectrum might assist in understanding the opinions of those who dislike the sounds of free reed musical instruments. Even for single notes there is much dissonance between the powerful upper harmonics in such sounds, and when chords are played on free reed keyboard instruments the capability of the ear to make sense of the musical harmony is also degraded for the same reason.

 

The interesting high frequency component of the sound at about 12.5 kHz, discussed above, fell closest to the 19th harmonic of the fundamental (646.6 Hz) without being coincident with it. Although the spectrum shows that this harmonic is not the strongest peak in the immediate region, it is nevertheless associated with a generally rising power characteristic which then falls off noticeably afterwards. So the spectrum confirms the presence of significant power near this frequency, but whose origin is currently speculative as the discussion above noted.

 

For completeness the spectrum of the other reed analysed, speaking treble C, is shown in Figure 5.

 

 

Figure 5. Free reed acoustic spectrum (treble C 8' pitch; 516.8 Hz)

 

Obviously the two spectra are not identical because the corresponding waveforms were also different in detail. However some general features are common to both, such as the extended range of harmonics up to (and probably beyond) the Nyquist limit of the recordings, and the existence of both odd and even harmonics. The frequencies at which the power begins to fall off are also similar in both cases.

 

Since all harmonics, both odd and even, exist in the spectra of both reeds, this confirms what we saw previously from the waveforms themselves (Figures 2 and 3) that they were not generating anything like the quasi-square wave reported by Cottingham [8]. However he was using a reed from a different instrument (an accordion rather than an harmonica), speaking at a much lower frequency, and blown at a pressure which might have differed considerably from that used here. So direct comparison of the two data sets cannot sensibly be taken further.


Tuning and voicing

 

Reeds are tuned by removing metal from the tongue. Removing it from the tip raises the pitch, and scraping it from nearer the riveted end lowers it. Some larger reeds are loaded with weights (often lumps of solder) at the tip, and these can be adjusted fairly readily.

 

Voicing to adjust tone quality is done in various ways. One technique is to adjust the windway by slightly bending the tongue into the frame. This reduces the high frequency content of the sound. The converse operation, bending it slightly away from the frame, increases the high frequency content. Other operations can be carried out by forming loops in the tongue of the type seen in Figure 1, or twisting a corner of the tip out of line with the rest of the tongue. (The latter technique might also encourage the formation of the torsional modes of high frequency oscillation mentioned above). In all cases the windway is being modified which will result in changes to the shape of the corresponding air impulses emitted by the reed. Normally one would not decide to retune or revoice more than the few reeds which clearly need attention, because apart from anything else they are obviously susceptible to damage during the process, and obtaining replacement reeds for an old instrument can be next to impossible.

 

Even with skill and experience the range of tone colour adjustment is nevertheless rather limited. Put bluntly, a disadvantage of the naked free reed is its harsh and otherwise unattractive sound! Although this will doubtless offend some readers, others are likely to agree. The holy grail of free reed voicing has always been twofold - to reduce the grossly excessive number of harmonics going well beyond the range of audibility, and to shape the frequency spectrum in more detail and more comprehensively than has yet been achieved, at least for the reeded keyboard instruments. These aims motivated the better reed organ makers such as Mustel during the nineteenth century, who saw that harmonium tone could be improved by making the reeds speak into resonating chambers of various designs rather than directly into the atmosphere. Even so, marketing even the most sophisticated and expensive reed instruments with stops labelled 'flute' was an example more of intent than achievement. It could never have been anything else. The problem of getting more attractive tone colours from free reeds also underlies the development of extraordinary musical skills such as those of harmonica players who exploit two types of resonating chamber - their vocal cavity and their hands, each type working on different sides of the reed.

 

This conveniently brings us onto the next section of the article which considers yet another type of resonating chamber, namely those formed by organ pipes.

 

 

Free reeds in the pipe organ

 

Almost all reed stops in the pipe organ have used beating reeds from their inception hundreds of years ago. However a small minority of free reed stops appeared in the early nineteenth century, mainly in German and French organs. Some of these ranks found their way to Britain, and in the United States some native builders also made them. In the famous organ built for Worcester cathedral in England by Robert Hope-Jones in 1896 there were two Cor Anglais stops, one of which used beating reeds and the other free reeds. Hope-Jones employed some celebrated voicers, one of whom (W C Jones, no relation ) was particularly skilled at reed voicing.

 

In a few cases, such as the so-called Physharmonica stop from Germany, harmonium-type techniques were used in that the reeds emitted their sounds directly into the atmosphere via a small chamber which contained the pallets and other mechanism. Although these were stops in a pipe organ, no pipes were involved. It is likely that the chamber was designed solely for mundane mechanical purposes rather than to modify the reed tones via resonance, although it would probably have exerted some influence for better or worse on the sound. The stop was said to have 'imparted the effect of a powerful harmonium' according to Audsley [9], who was also led to 'gravely question its value in a properly-appointed instrument'.

 

Of more interest here were the free reed stops which employed tubular resonators similar in function, if not always in shape, to those of beating reed pipes of nominally similar tone quality. The general construction of the pipes followed that of their beating reed cousins in that the reed assembly was affixed to a 'block' supported in a 'boot' inserted in the soundboard of the organ through which it drew its wind when the note was keyed. The free reed unit spoke into the resonating tube which sat on top of the block. The reed assembly was more or less identical with those used in reed organs apart from some minor details related to its mounting, save for the significant difference that a sliding tuning wire accessible from outside the pipe allowed the reed to be tuned in the same way as in a beating reed pipe. 

 

 

Figure 6. Victorian-era Cor Anglais free reed organ pipe by Laukhuff

(after Audsley [9])

 

Figure 6 is one of the hundreds of exquisite line drawings in Audsley's treatise [9] (he was an architect) which illustrate the points just made. It shows a Cor Anglais free reed pipe made by the German firm of August Laukhuff speaking middle C at 8' pitch, implying that the resonator would have been about two feet long, and Audsley opined that 'the free reed is the most suitable for the production of the smooth dreamy voice of the true instrument'. Other apparently successful free reed tonalities included Roosevelt's Contra Saxophone at 16' pitch (a 'fine stop' according to Audsley), his Euphone or Euphonium at 16' ('fine and extremely valuable') and also at 8' ('yields a refined and pleasing tone'). However Audsley dismissed the free reed Oboe, Clarinet and Fagotto as 'absolutely worthless' compared to their beating reed relatives.

 

It is difficult to go beyond Audsley's opinions to discover whence they arose and how widely they might have been shared, though he was well travelled and well connected and he can probably be credited with an educated and keen ear for these matters. However opinions are just that - opinions - regardless of their source without some basis in fact to substantiate them, and Audsley was nothing if not opinionated. Reminding ourselves of what has been said previously, we noted that the free reed lacks the shallot aperture of the beating reed which can be shaped in many subtle ways to help derive the plethora of beautiful reed tones we associate with the pipe organ ([2], [3]). So even when driving the tubular resonators of the pipe organ there is evidence that the free reed in its brief heyday had difficulty competing with the beating reed across a broad enough front. Nevertheless, there is also evidence that suitable resonance chambers (organ pipes) were able to at least filter out the less attractive aspects of naked free reed tone, and at best the same evidence persuades us that the limited results were probably worth the development effort that organ builders had put into achieving them. However it is obvious with hindsight that the general opinion of the pipe organ world was that the free reed was not really up to the mark, given its short-lived days of glory.

 

But the bottom line is this. The Victorian-era reed organ appeared at a time when there was a clamour for much smaller and cheaper imitations of the pipe organ which could be played in the same way. People flocked to church, they liked organs and their music and they wanted to play it in their homes. So reed organs were marketed in vast numbers and they satisfied a demand of the times which is plain to see. Few would have insisted that they sounded indistinguishable from the pipe organ because they plainly did not, but they were nevertheless genuinely musical instruments in their own right which attracted the interest of some great composers. Although their tonal qualities could have been much improved by using a proper resonance chamber for each reed, this would have swept away the key advantages of small size and low cost, so there can be no surprise that it was not attempted other than on the limited experimental scale outlined above. A major factor which contributed to their final demise was the appearance of the electronic organ accompanied by an expanding popular musical literature in the 1930s, novelties which were greeted with an enthusiasm similar to that accorded to reed organs of a century earlier. However with their disappearance went their musicality, a vacuum which the electronic organ has still not really filled in the judgment of many today. Who might be the later counterparts of Franck, Vierne and Karg-Elert who have written specifically for the electronic organ?

 

An interesting acoustic detail arising from the free reed period of organ pipe design concerns the boot or foot of the pipe. Several sources relate that the pipes would only work properly when the boot was exceptionally large, much larger than that usually used for beating reed pipework. In Figure 6, for example, the boot is about half the total length of the pipe. Although literature on the matter from that period is vague, it seems that there were major tuning and intonation issues. Audsley [9] said that 'the boots of free reed pipes, while they vary considerably in size in the works of different makers, are invariably much larger than those required and commonly used for striking-reed pipes. Large boots containing considerable volumes of air have been found by practical experience to be conducive, if not absolutely necessary, to the prompt and satisfactory speech of free reeds, but the reason for this has not been settled beyond question'. My view is that the problem hinged on the acoustic coupling of free reeds to resonant cavities. The boot is such a cavity which will resonate at a frequency depending on its volume, and if that frequency is too close to the desired frequency of the reed then it will try to 'pull' the reed into tune with itself while the tuner is attempting to tune it. Raising the resonant frequency of the boot by making it smaller would not solve the problem because it would recur whenever the boot resonated near a harmonic of the reed. Thus the only solution seemed to be that of making the boot so large that it resonated at a frequency so low that it no longer interfered with the fundamental frequency of the reed.

But this explanation begs two questions. Why is acoustic coupling to a resonator a problem for free reeds, but why is it not for beating reeds? Answering the first question requires that we consider the boot in more detail as a Helmholtz resonator. This consists of two parts - a cavity plus a port open to the atmosphere. The system is tuned to a particular frequency by adjusting the dimensions of both cavity and port in the same way that a bass reflex enclosure must be tuned to match a particular loudspeaker unit. However, for the organ pipe the boot is the enclosure and the tuned port is the tuned free reed assembly itself. Earlier we discussed how the reed oscillated (see reed oscillation mechanism) by assuming a wind supply at a stable pressure. It was shown how the force acting on the reed tongue reached a maximum value during each cycle as it passed through the frame because at this point the wind pressure acted on the whole surface area of the reed. When the tongue emerged either side of the frame the wind was able to flow around it, thereby reducing the aerodynamic force on the tongue. However this happy situation becomes perturbed if the wind pressure is not stable. Since the reed is now immersed in the intense acoustic field of a resonant cavity - indeed, it forms part of it - its wind is now anything but. The air chugging in and out of the resonator can only pass through the reed assembly. So the outcome is that there will be a range of frequency close to the reed's normal speaking frequency where the air flowing in and out of the resonator takes control of the tongue rather than the tongue controlling its own motion. In practice that range is typically around 10% of the natural frequency of the reed in round figures, or about a semitone either side of it. The situation repeats for higher cavity resonances close to each harmonic of the normal reed speaking frequency, which will arise if the cavity is made progressively smaller. Consequently, in practice the cavity can only be made much bigger to take its resonant frequency well below that of the reed, as the Victorians found empirically when making the boots for their free reed organ pipes. And for higher cavity resonances (smaller boots) at frequencies between any pair of harmonics, without getting too close to either, it would not have been surprising if the pipes refused to speak at all. Also, going back again to the harmonica, this is why skilled players can achieve pitch bends by exploiting the coupling of the reed to their vocal cavity (whose dimensions can be varied) in the same way as in organ pipes.

 

As to the second question posed above, why are the cavity resonances of the boot apparently not a problem for beating reeds? Because they work in a different way in that the reed tongue is impelled towards the shallot largely by Bernoulli forces [2]. Although these are also involved as a secondary factor for the free reed as we saw earlier, they are not the dominant source of the impulses which maintain tongue oscillation in the free reed case. But the Bernoulli force is exerted on a beating reed tongue continuously rather than impulsively as it moves towards the shallot, in fact it gets ever larger as the tongue gets ever closer. Although cavity resonances are still set up within the boot as with the free reed, they have less effect on what is a more powerful and energy-intensive oscillation mechanism. Their high power tone generator is also why beating reed organ pipes can be made so loud when desired.

 

 

Summary and conclusions

 

This survey of the physics of the free reed has covered the following ground:

 

1. The free reed is a negative resistance oscillator, oscillation being maintained by an impulse centred near mid-cycle when the tongue is within the reed frame where its windway is at a minimum. The impulse arises from four main effects, namely the forces due to wind pressure, the Bernoulli effect, the inertia of the tongue and a transient force due to air deceleration as the windway reduces rapidly. However not all of these will necessarily apply to all reeds at all blowing pressures, and their relative magnitudes will also vary.

 

2. Since the reed tongue moves freely as a vibrating cantilever beyond the impulsing region, it oscillates sinusoidally for practical purposes and has a high Q-factor. Reed frequency is practically independent of amplitude (loudness) and hence blowing pressure over a useful pressure range. Like a pendulum, it is therefore isochronous.

 

3. Beyond the impulsing region where the windway is at a minimum, it then expands to emit an air pulse as the reed tongue exits the frame on one or both sides, depending on the amplitude of oscillation. The two pulse shapes, usually different in detail, define the harmonic content of the waveform and hence the timbre or tone quality of the sound. The pulse shapes vary with blowing pressure.

 

4. Notwithstanding its isochronicity, the pitch of a free reed can nevertheless be 'bent' by (for example) harmonica players because of its sensitivity to acoustic coupling to neighbouring resonant cavities. In the case of the harmonica these are the vocal tract of the player and their cupped hands. In the case of organ pipes using free reeds it was shown that the 'boot' acts as a Helmholtz resonator with the reed assembly forming the tuned port. If the boot is too small it can resonate at or near to the reed's speaking frequency or its harmonics, creating 'pulling' difficulties when trying to tune the reed. The phenomenon was explained in detail, showing why the boot of free reed organ pipes has to be made unusually large to take its resonant frequency below the fundamental frequency of the reed.

 

5. Recordings were made of the waveforms of two free reeds, showing the presence of three distinct regions per cycle in both cases. Two of these could be related plausibly to the puffs of air arising from the windway of the reed tongue as it emerged from one side or other of the frame. However the third was associated with a high frequency oscillatory burst at about 12.5 kHz. These bursts were clearly deterministic, repeating in the same form in each cycle, rather than the stochastic character which might otherwise have suggested air turbulence. The bursts were tentatively associated with the tongue while within the reed frame, where it receives the impulse which maintains its oscillation. It was suggested that the impulse might have excited higher-order torsional (twisting) or flexural (bending) modes of tongue vibration at these high frequencies. These results differed markedly from those reported elsewhere where the waveforms are sometimes said to resemble square waves, and moreover that only two distinct regions per cycle can be identified.

 

6. Frequency spectra corresponding to the waveforms were also presented. Both spectra confirmed the large number of harmonics associated with free reeds which, in these two cases, gave every reason to suppose that considerable acoustic power will still exist beyond the Nyquist limit of the data at 22.05 kHz. Both odd and even harmonics were strongly represented, confirming that the waveform of neither reed analysed was similar to a square wave.

 

7. Reed tuning and voicing matters were discussed, embracing some differences between beating and free reeds. In particular the shallot aperture of a beating reed was shown to be important in a voicing context in that its shape, and hence the acoustic spectrum of the reed, can be varied within wide limits. The lack of anything comparable for a free reed was shown to considerably limit its voicing flexibility. However the sensitivity of free reeds to neighbouring resonant cavities, mentioned above, was related to the appearance of organ pipes in the nineteenth century using free reeds with an apparently attractive range of tone colours.

 

 

Notes and references

 

1. In terms of physics, the tuning fork is essentially a stripped-down version of a free reed which merely oscillates as a vibrating cantilever at its own natural resonant frequency. It appeared c. 1711, predating by about a century the later explosive growth of wind-blown Western free reed instruments.

 

2. 'How the reed pipe speaks', an article on this website, C E Pykett 2009

 

3. 'The tonal structure of organ reeds', an article on this website, C E Pykett 2011

 

4. This nomenclature denoting the two types of reed organ in Britain became well established in the nineteenth century and into the twentieth. For example, a popular instructional series for reed organ playing appeared in a handsome set of volumes entitled 'The Musical Educator' (ed. J Greig, Caxton, London c. 1910) under the title 'The Harmonium and American Organ' by J C Grieve. The set with its reed organ series ran through several editions to survive beyond World War II, when it became 'The New Musical Educator' edited by Harvey Grace.

 

The tone of suction instruments was deemed to be less strident than that of positive pressure ones (because 'the sound gets drawn into the organ'), therefore they were marketed as better suited to the more intimate domestic environment.

 

5. The reed cell assembly in American organs often incorporated felted shutters or shades which reduced the volume of certain reed ranks as well as subduing their tone colour. The associated stop names would then include adjectives such as 'dolce', making this practice a rather extreme form of borrowing in the sense of the same reed set being used for two stops of the same pitch on the same keyboard. Since the rank and its 'dolce' variant were not independent they could not be sensibly used together when registering a piece.

Some better quality harmoniums in the nineteenth century, such as those by Mustel, used resonating chambers into which the reeds spoke.

 

6. The Bernoulli effect is one form of the principle of conservation of energy, which states that energy can neither be created nor destroyed and therefore that the total energy of a system remains constant. When the speed of a fluid increases as the aperture through which it is passing gets smaller, its kinetic (motional) energy increases. The energy conservation principle therefore demands that its potential energy, the energy available to do work, must reduce to compensate. In the case of an air stream this means its pressure must reduce as its speed increases.

 

7. A comparable situation concerning wind delivery through a free reed windway is that of an organ pallet valve. Like a free reed, a pallet is long and thin, it covers a rectangular aperture, it is fixed at one end, and therefore it has a windway of the same general shape. In this case Victorian organ builders seemed unaware that increasing pallet width did little to increase wind delivery to the pipes. All it did was to massively increase the force required to open the pallet against its 'pluck', thereby making some large organs virtually unplayable. The wide free reeds one sometimes sees in reed organs from the same era suggests that a similar misapprehension regarding wind delivery and the loudness of reeds might have been at work there.

 

8. 'Acoustics of free reed instruments', J Cottingham, Physics Today, p. 44, March 2011, American Institute of Physics

 

https://physicstoday.scitation.org/doi/pdf/10.1063/1.3563819 (accessed 18 January 2023)

 

9. 'The Art of Organ Building', G A Audsley, New York 1905. Reprinted in facsimile by Dover, New York 1965.