How the Reed Pipe Speaks

by Colin Pykett

Date: 17 January 2009

Last revised: 16 April 2012

Copyright © C E Pykett

“Schnarrwerk, ist unterweilen Narrwerck”

(Reed-work is often the work of fools)

Andreas Werckmeister, 1698

Abstract.  The important sound producing mechanisms involved in the organ reed pipe are discussed in detail but without recourse to mathematics.  The breadth and depth of the treatment are thought to be unique if only because it seems to be the first time that this quantity of material has been gathered together in one place.  Examples of waveforms and frequency spectra of real reed pipes are included and their details explained in terms of the physical processes described in the article.  The variable quality of organ reed work is remarked on, and it is considered likely that further research could improve the situation and reduce costs, as it has for some other instruments.  However it is concluded that the prospect is remote that this will occur in view of the continuing decline of interest in the organ, at least in Britain.

Contents

(click on the headings to access the desired section)

Introduction

 

Reed Pipe Construction

 

The Mechanism of the Reed Pipe

 

The Reed

Reed oscillation mechanism

Frequency of a vibrating reed

Q-factor of the reed

Acoustic output impedance of the reed and shallot

The Shallot

 

The Resonator

The need for impedance matching

Radiation efficiency

Conical resonators 

                        Length of a conical resonator

                        Pipe scales and degree of flare of a conical resonator

                        Tuning reeds with conical resonators

Cylindrical resonators

Reed Pipe Waveforms and Frequency Spectra

The waveform generated by the reed

Open shallots

Closed shallots

Emitted waveforms and spectra

Pipes with conical resonators

Pipes with cylindrical resonators

Concluding Remarks

 

Notes and References

Introduction

While flue stops are essential, an organ does not need reeds to be an organ and a number of great organ builders often excluded them,  Gottfried Silbermann being one.  The famous pun quoted above from his near-contemporary Werckmeister is rather taken out of context though, for he went on to say that when the reeds of an organ turn out well it is a matter for rejoicing (perhaps born of relief).  However his point evokes echoes today because reeds are more often either good or bad rather than merely indifferent.  For over a century writers on the organ have remarked on the absence of widely accepted rules governing how to make and voice reed pipes successfully, and even in the current millennium articles have continued to appear sporadically in the specialist literature describing some detail or other of reed pipe construction or their acoustics.  Therefore it is probably fair to say that there is still an inadequate understanding of how they actually work, and in turn this might be related to the fact that what knowledge does exist seems never to have been written down in a single easily accessible place, at least in the English language.

Some years ago when writing an earlier article on this website [1] which described how flue pipes work, I was confronted with an analogous situation though it was not as extreme.  Therefore this article considers reed pipes in a similar manner, that is without mathematics while presenting a physical picture of the most important sound producing mechanisms involved.  As with flue pipes, this has not been easy.  In both cases this is partly because some of the mechanisms have traditionally only been cast in terms of advanced mathematics.  Yet even if one can understand it, the mathematics sometimes does little to assist the construction of an intuitive or pictorial view of the situation, and I find such a perspective sterile and incomplete [2].  Another reason for the difficulties experienced in compiling this article is the unsatisfactory state of the literature on the subject in other respects.  As well as the aspects just touched on, some of the acoustic properties of reed pipes are widely misunderstood [3] and even experts have made statements which are factually incorrect [4].  Also some standard works on musical acoustics overlook the reed pipe, one example being that by Benade [5] who compounded the felony by ignoring the organ completely.

Given these difficulties it is perhaps not surprising that many authors, including some organ builders, have sidestepped other than the simplest and most obvious aspects of how reed pipes work.  Instead they have concentrated entirely on their history and morphology - how they are made and of which materials, their shapes and sizes, how they are voiced and how these things have changed through the ages.  Such data are of course useful, but the approach does not fill the gaps nor right the wrongs mentioned above.  Some surveys of this type are those of Norman and Norman [7], Audsley [8] and Bonavia-Hunt [15], and while all have various shortcomings in isolation, they present a reasonable picture at a qualitative level when taken together.  However I have felt for a long time that one deserves something rather better than that offered by the standard diet served up in the organ literature, especially as one has to pay good money for the privilege of reading it.  Therefore, for all these reasons, this article tries to go further and by covering most of the ground at one go it might illuminate some hitherto murky corners in a way which is fairly easy to assimilate provided you have the time and the inclination to wade through it.    To keep its scope within reasonable bounds it only addresses the physics rather than going into excessive detail of matters such as how reed pipes are made and voiced, because these are covered elsewhere such as in the references mentioned above. Nevertheless it is a long article, so thank you for reading it but do take it gently!

Reed Pipe Construction

We shall only consider in detail here the type of pipe which uses a beating or striking reed, so called because the reed tongue vibrates against a rigid structure called a shallot.  In the past other types of reed stops in pipe organs have used harmonium-style free reeds which vibrate freely within an aperture, these being coupled to a resonating tube in the usual way.  Or the so-called valvular reed in a diaphone pipe has been used.

Neither of these are found today except in a few organs where they still exist as legacy stops, and even in their heyday about a century ago they were rare.  In some ways this is a pity because they endowed the romantic organ with an authentic range of tones which are virtually lost to modern players.  In Britain Robert Hope-Jones helped to promote both of these now-obsolete types of stop before he emigrated to America, and in his Worcester cathedral organ of 1896 there were actually two Cor Anglais registers, one of which used conventional beating reeds and the other free reeds.  One cannot dismiss such ranks out of hand because Hope-Jones employed some of the finest reed voicers of his day, and there remains to us a number of accounts extolling the beauty of his reed work from those in a position to judge. 

As to valvular reeds, Hope-Jones was the first to use diaphones in organs and, again, it is facile to describe them as mere foghorns just to make a cheap point.  John Compton continued to use them for decades afterwards, and Nicholson’s have retained one of his in their recent rebuild of the organ at Christchurch Priory.  Having played this example, I found its quiet grip and definition very useful on the pedals, and it illustrates that a diaphone does not have to be loud and deafening.  In fact, one of the advantages of the diaphone is the range of dynamics and different tones it can provide.   (For those who want to try the loud and deafening ones though, there are still a few around in Hope-Jones’s surviving church organs, such as those at Pilton [9] and Llanrhaeadr!).

Perhaps the digital organ fraternity might reflect on whether it would be useful to the organ community as a whole to re-create these sounds of yesteryear in their instruments.  But here we must move on by returning to the main theme of beating reeds.

A typical reed pipe, which emits a trumpet type of tone in this case, is illustrated in Figure 1.  The reed assembly is enclosed in what is called the boot at the bottom of the diagram, and a conical resonating tube sits on top.  However the resonators can be of many different shapes, and historically some of these have verged on the bizarre (as have the corresponding sounds).  A representative illustrated survey appeared in [7].  In this article we shall restrict ourselves to the two main types of resonator, conical and cylindrical, and these will be discussed in more detail later.  The resonator shown in the diagram has an open top, so it is sometimes turned sideways by mitreing the pipe to prevent dust and dirt falling into the reed.  Mitred tubes are also commonly used to enable their overall height to be reduced by coiling them up, as with brass orchestral instruments.  Such convoluted tubes are a tour-de-force of the pipe maker’s craft, and they exhibit breathtakingly accurate metalwork and superbly neat soldering.

A more detailed view of the boot and the reed assembly within is at Figure 2 below.  The reed tongue itself is usually of hard brass, although a wide range of different materials has been tried including aluminium, bakelite and even wood.  It vibrates against the shallot, a tubular structure also usually made of brass which has one side cut away to form an aperture   The size and shape of the aperture and of the shallot itself vary depending on the type of tone desired.  The reed is wedged against the shallot using a hardwood wedge, and its effective vibrating length is adjusted by the tuning spring.  This can be knocked up and down from outside the boot by the tuner using a “reed knife”.  The block is typically cast from lead or pipe metal, and it sits inside the boot which in turn sits in the wind hole in the soundboard of the organ.  The resonator tube rests inside a short socket inserted into the block hole.

 Figure 2.  The reed assembly (after Audsley [8])

The Mechanism of the Reed Pipe

As with the flue pipe, reed pipes consist of two parts – a generator and a resonator.  The reed is the generator of the acoustic energy, and the tube into which it feeds the sound is the resonator.  However there is an important difference between the two cases.  In the flue pipe the generator and resonator are intimately coupled to the extent that the pipe will not work, indeed it cannot exist in any real sense, if there is no resonating tube sitting above the mouth.  The flue pipe is a transit time oscillator whose frequency is determined by the time taken for air impulses to travel up and down within the resonator, which also modifies the emitted sound at the same time.  The reed pipe, on the other hand, can be persuaded to emit some sort of sound even if the tube is completely removed.  This is also true of its orchestral counterparts such as the oboe, whose players habitually “crow” their reeds before inserting them into the instrument.

We can therefore consider the reed and the resonator in isolation to derive many of the features which make a reed pipe sound as it does, although there is still some degree of interaction which cannot be ignored in a more detailed treatment.  Notwithstanding the latter aspect, the former suggests a welcome simplification, but unfortunately it is immediately countered when we come to realise the complexity of the behaviour both of the reed and the resonator.  As with most musical instruments, it is next to impossible to develop a complete analytical theory which predicts accurately all the important characteristics of reed pipes.  Approximate equations are the best one can come up with, and some of them can be useful when designing pipes.  Nevertheless the corpus of experience and empiricism built up over centuries is still as necessary today as it always was to the organ builder if he is to include successful reed stops in his organs.  However this does not mean that there is no room for further research, and this thread is extended in the concluding remarks to this article.

The Reed

Reed oscillation mechanism

From the unassailable fact that a reed pipe works, it is obvious that the reed tongue must move towards the shallot when wind at the proper pressure is admitted to the pipe, and it will continue to do so until the shallot aperture is more or less completely covered.  The reed will then commence to move back again as a consequence of its elasticity or springiness, and the cycle thus repeats indefinitely.  However, this is about the limit of the explanation of reed behaviour in most descriptions of reed pipes, 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 reed will only oscillate over a restricted range of wind pressures – if the pressure is too low the force exerted by the air on the reed is insufficient to push it close enough to the shallot and consequently nothing will happen.  If too high, the pressure will not allow the reed to spring back once it has covered the aperture and again nothing will happen.

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 cancels out the energy dissipated 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 additional energy to compensate for that which is dissipated elsewhere, and under these conditions the circuit can be made to oscillate indefinitely instead of the oscillation dying away due to energy losses.

The reed is an acoustic version of this type of oscillator where the air under pressure from the organ blower provides the necessary energy source to overcome losses elsewhere in the reed oscillatory system.  We can see how the oscillator works by considering what happens as the air pressure (analogous to voltage in the electronic case) is increased from a low value.  Initially the air flow rate (analogous to current) into the shallot aperture will merely increase with the pressure as we might expect, but nothing else will happen.  However, at a certain value of pressure the reed will begin to move towards the shallot, the windway available to the air will be made smaller, and therefore the volume 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 reed pipe is a negative resistance oscillator whereas the flue pipe is a transit time one.

What makes the reed move towards the shallot in the first place?  This might sound a trivial question but in fact the answer is quite subtle.  Neglecting turbulence and local eddies and vortices, the bulk air flow within the boot is in the direction from the boot hole towards the shallot aperture.  Intuitively we can see that the drag experienced by the reed within this flow will therefore tend to bend it towards the shallot as the air grabs the reed tongue.  As soon as this happens the effective exit aperture for the air, formed between the shallot and the reed, will be reduced.  This reduces the flow rate but it also increases the speed at which the air passes through the reducing aperture.  The situation is no different to the familiar garden hose in which one reduces the size of the orifice to get a smaller but faster water jet.

So far so good, but now for the subtle bit.  For the reed, the increase in air speed results in a reduction of air pressure behind the reed, that is on its surface which faces the shallot aperture.  The reason why this happens is because of the Bernoulli effect [12].  But the pressure on the opposite surface of the reed facing the incoming air will remain constant.  Therefore there is now a net pressure difference which causes the reed to move further towards the shallot.  Moreover, the pressure difference will continually increase as the reed moves because the air rushing through the ever-decreasing aperture moves ever-faster.  Therefore the Bernoulli pressure effect gets continually stronger, and it forces the reed ever closer and ever faster towards the shallot.  This motion will continue until the reed has either completely or nearly snapped shut over the shallot aperture, and at this point it will “bounce” by virtue of its elasticity because the Bernoulli rarefaction effect then vanishes due to cessation of the air flow – unless of course the air pressure is so great that the reed cannot spring back again.

On its recovery stroke the reed will generally overshoot its original position slightly because of its inertia before reversing its direction of travel once more.  At this point the cycle then repeats as before when the air begins to act on it again.  The reed is therefore a valve which admits a periodic (repetitive at a constant frequency) series of air impulses to the shallot and thence to the resonator.  Each impulse has the same characteristic shape in terms of the variation of air pressure with time, and this wave shape defines the harmonic content of the waveform applied to the resonator.

Although it is called a beating reed, the tongue does not bodily strike the shallot as it moves.  Rather, it progressively rolls and unrolls across the shallot aperture as a consequence of the curve given to it by the voicer.  The type of curve materially affects the tone of the pipe in terms of its harmonic content, because it affects the shape of the pressure waveform applied to the resonator.  Achieving the correct curve demands a high degree of skill and experience on the part of the voicer, and Bonavia-Hunt [15] has given as good a description of this process as any.

Frequency of a vibrating reed

The frequency of the vibrating reed depends mainly on the wind pressure, the length of the reed, its thickness and the material it is made of.  The width of the reed affects mainly the amplitude of the resulting sound rather than the frequency, because a wider reed allows more air to enter the correspondingly wider shallot, and hence the resonator, than does a narrow one.  Although a wide reed obviously needs more force to be exerted on it than a narrow one, all other factors being equal, this happens automatically because there is a larger surface area for the wind pressure to push against.  Thus the width of the reed does not influence its frequency nearly as strongly as its length and thickness.

Simple theory applied to the case of a free reed, one which vibrates freely rather than striking a shallot, predicts that the frequency should increase linearly with the thickness of the reed and inversely as the square of its length (see, for example, [10]).  Thus a thicker reed vibrates faster than a thinner one (both of the same length), and a longer reed vibrates slower than a shorter one (both of the same thickness) as we would expect intuitively.  As remarked above, the reed width does not appear at all in the equation, but nor does wind pressure.  The latter explains why the tuning of a free reed is independent of pressure and thus 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 and tone quality of harmonium reeds can be varied without affecting the tuning.

The free reed vibrates sinusoidally or nearly so, thus the harmonics which are present in the sounds of a harmonium cannot arise from the motion of the reed itself.  Rather, they are the result of the variation of the windway through the reed assembly during its oscillation cycle.  The windway does not vary sinusoidally, thus the reed generates relatively sharp air pulses containing many harmonics even though it moves sinusoidally itself.

However a beating reed is not a free reed, and it is more difficult to derive an equation which adequately predicts its vibrational frequency.  This is partly because the effective length of the reed varies during its oscillation cycle because of the way it rolls and unrolls across the shallot aperture.  Also the shallot gets in the way of the reed as it moves, which means the reed experiences forces which do not vary smoothly during its oscillation cycle.  These rather violent and jerky forces introduce what are called nonlinearities into the mathematics, which make it difficult to solve the resulting equations.  One upshot is that the reed cannot vibrate sinusoidally as stated in reference [6].  Another upshot is that it is easier to do experiments from which an empirical (trial and error) equation can be derived, rather than to try and derive such an equation purely by theory.  The experimental work described in reference [11] showed that the vibrational frequency is proportional to reed thickness as for the case of the free reed.  However it varies inversely with length rather than as the inverse square of length as for the free reed.  Frequency is also proportional to wind pressure for the beating reed, quite unlike the free reed.  Note that all these results are approximate rather than exact, but they are close enough to be useful for design purposes.

Q-factor of the reed

Do not be put off by this technical term, which can be explained fairly easily in terms of familiar concepts rather than mathematics.  However it is desirable to be aware of it and what it means because differences in Q illuminate one of the differences between organ reed pipes and their orchestral counterparts, even though they might look superficially similar.

The Q-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 reed pipe 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 organ pipes and their environment.  This means that the reed should not be strongly damped, in other words there should not be processes which cause it to lose excessive energy.  We can say this because an alternative definition of Q relates to the damping, and thus the energy lost in oscillation.  Fortunately there is little which impedes the motion of the oscillating organ reed other than minor factors such as friction with the air inside the boot, therefore it does indeed have a high Q and is consequently stable enough in frequency to be used for musical purposes over extended periods without re-tuning – several months or so.

This is different to the case of analogous orchestral instruments such as the clarinet.  Here the reed assembly is superficially similar in construction to the organ reed, yet it is more strongly damped because of the player’s lips clamped around it.  The lips absorb energy from the vibrating reed.  Therefore the clarinet reed has a lower Q and consequently it can be persuaded easily to vary considerably in frequency while playing a single note.  In the case of the clarinet this does not matter because the player’s ears constitute 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.

Acoustic output impedance of the reed and shallot

This is another technical term, this time governing the efficient transfer of acoustic energy from the reed to the resonator.

For flue pipes the cross-sectional area of the pipe at the mouth is the same as that of the resonator above it.  Therefore, both in absolute terms and relative to the resonator, the acoustic channel between the oscillator and resonator is large and unobstructed for flue pipes.  This is not the case for reeds, where even the largest reed-plus-shallot assemblies are relatively small in cross-section compared to the ultimate size of a flared resonator tube.  Therefore the narrow shallot cannot pass nearly as much air into the resonator as can the mouth of a flue pipe emitting a note of the same pitch, and therefore we can say that the emitted air comes from a source of high acoustic resistance.

To see what this means it is useful to consider an electrical analogy.  In electric circuits the value of resistance is calculated using Ohm’s Law, which says that resistance equals voltage divided by the current.  When dealing with the alternating currents and voltages involved in oscillators, we use the more general term impedance instead of resistance for reasons which need not concern us here.  In acoustics the same applies – we define acoustic impedance as the ratio of pressure (equivalent to voltage) and air flow rate (equivalent to current).  Note that the term “air flow” as used here relates only to the to and fro vibrational motion of the air molecules involved in transmitting sound waves.  It does not relate to the unidirectional bulk flow of the air which passes up the resonator from the organ blower.  The two are quite different.  Taking an analogy from electronics once more, the vibrational motion is analogous to the alternating (AC) component of a current and the bulk flow to the direct (DC) component.  Both are frequently present in a given conductor at the same time.

With a narrow orifice such as that of the shallot at the point where it feeds energy into the resonator tube, we have just noted the restriction to the amount of air which it can transfer.  With a wider orifice such as that of the flue pipe just above the mouth, which equals the area of the pipe itself, there is no such constriction.  Therefore the acoustic output impedance of the reed and shallot assembly is much higher than that of the mouth a flue pipe for equal pressures in the two cases.

We shall not take the discussion further at this point but shall need to return to it later when discussing the action of the resonator.  For the present we just need to note that the output impedance of the reed plus its shallot assembly is high.  Sometimes one meets statements such as “a reed is a pressure driver”, and this is just another way of saying the same thing.

The Shallot 

The shallot is the structure against which the reed tongue vibrates, and it has been mentioned frequently without saying much so far about what it actually consists of.  At this point only a brief description of the various types of shallot will be given without dwelling in detail on why they give different types of tone.  This will be covered later on.

Figure 3.  Various types of shallot (after Bonavia-Hunt [15])

Several types of shallot are shown in Figure 3.  Each of them consists of a brass tube with a cut-away or flattened face which may be partly occluded, thereby resulting in an aperture of one shape or another.  The tube itself might be cylindrical or tapered as shown.  The shallot marked ‘A’ is the oldest and simplest type and it probably persisted in France for longer than elsewhere on account of the range of characteristic and beautiful tones which Cavaillé-Coll managed to coax from it in the nineteenth century.  However William Hill in Britain also continued to use it for his chorus reeds for some while after his illustrious contemporary Willis had embraced the other types to be described in a moment (though it was George and Vincent Willis who did much of the painstaking innovative development work on novel shallots rather than "Father" Henry himself ).  It is cylindrical and commonly referred to as an “open” shallot because the aperture runs the whole length of the tube.

The aperture in type ‘B’ does not reach the base of the tube and it also tapers to a point half way up.  For the latter reason this type and similar ones are termed “closed” shallots because their flat faces are closed off by the plate in which the aperture is cut.  According to Bonavia-Hunt [15] it was introduced by Vincent Willis who applied it to smooth-sounding reeds giving tromba and horn types of tone.  Two points are worthy of remark concerning this type of shallot.  The first is that any tapered aperture will usually result in the generation of fewer harmonics than the open type (‘A’) for reasons which will be given in a later section.  Because the higher order harmonics are progressively attenuated, the resulting tones are smoother rather than strident and harsh.  The second point is that the filled-in region below the aperture coincides with the region where the vibrating reed itself sometimes does not fully cover the aperture in the other types of shallot.  This means the waveform generated by a type ‘B’ shallot might be better defined because the air supply is completely cut off once every vibration cycle. 

Type ‘C’ is similar except that the aperture starts at the base of the tube and, although tapered, it is not as long as that of type ‘B’.  The wider tube and shorter aperture will tend to further emphasise the fundamental and lower order harmonics at the expense of the higher ones.

Type ‘D’ is a typical “trumpet” shallot.  It gives rounded tones with more harmonics that those of ‘B’ and ‘C’, but without the splashy raucousness which people sometimes complain of in the open type ‘A’.

Type ‘E’  is used for quieter reeds because the diameter of the tube is narrower, thus less acoustic power is emitted.  Apart from this, an extensive retinue of higher harmonics characterises the tone as it does with type 'D' shallots.  They are used for quasi-orchestral solo reeds such as the Orchestral Oboe.  

Type ‘F’ is similar to type ‘A’ in that it has a full length aperture.  However the taper controls the production of the extreme high harmonics to some extent.  It is used for loud and arresting “fanfare trumpet” types of tone.

The Resonator

The main functions of the resonating tube which sits on top of the shallot and receives the acoustic energy from it will now be described.  Unlike a flue pipe which has a separate mouth, it can never be completely closed because there must always be an escape path of significant size for the air passing into it from the organ blower, and there must be an aperture through which the sound itself can escape.  This factor is partly responsible for the shapes of the tubes and it limits the number of possibilities which can be envisaged in practice.

The need for impedance matching

We have seen that the reed assembly on its own has a high acoustic output impedance, and in practice this means it cannot emit much vibrating air into the environment because of the narrowness of the shallot.  In turn, this means it cannot by itself make much sound in the auditorium.  One of the main functions of the resonator is to improve this state of affairs by matching the high impedance of the reed to the low impedance of the surrounding air.  Why do the surroundings have a low impedance?  We have seen already that acoustic impedance is the ratio of vibrational air pressure to air flow rate, and the vibrational pressure emitted by the reed would immediately dissipate to a very small value if it emitted its sounds directly into the vastness of the environment.  Such small pressure values are the only ones which can exist in the environment, and they imply that the impedance of the surroundings is therefore low.

Radiation efficiency

Although the resonator improves the transfer of acoustic energy from the reed into the surroundings, it does not do so independently of frequency.  The open top of the resonator is subject to much the same rules of acoustics as a loudspeaker, and it is well known that loudspeakers have to be physically large when they are called upon to handle low audio frequencies (woofers and sub-woofers) whereas they need only be small for the high ones (tweeters).  A loudspeaker designer uses the term “radiation resistance” to denote the equivalent electrical load which a loudspeaker throws onto an amplifier, and this means that the amplifier has to do work to move the speaker cone against the air for it to radiate sound.  This work is expressed as the electrical energy dissipated in the radiation resistance.  Moreover, the radiation resistance varies with frequency for a given loudspeaker.  However an organ pipe is not driven electrically, and therefore it is better to use the more general term “radiation efficiency” in this article to avoid confusion.  Nevertheless, the issues involved are much the same in the two cases as will now be explained.

Let us imagine what happens to a sound pressure impulse travelling up the resonator towards the open aperture at the top.  When it reaches the top of the tube it first compresses a column of the surrounding air in front of itself.  Immediately, the local pressure enhancement within the column begins to dissipate by air movement at the speed of sound from inside to outside the column, leading to outward propagation of a wavelike disturbance beyond the column itself into the auditorium.  Rather like ripples on a pond, it is this which causes the sensation of sound which we hear at a distance from the pipe. Intuitively we may see that the wider the column (i.e. the larger the resonator cross-section at the top) then the longer the pressure equalisation process will take, because a high pressure area in the middle of a fat column has further to move before it dissipates within the atmosphere at large than if it was in the middle of a thin column. This dissipation time governs the time for which outward sound propagation from the resonator will occur in response to an emerging sound pressure impulse, and hence the amplitude of the disturbance at a given distance from the pipe. Thinking further, it is possible to deduce that for maximally efficient radiation of a particular frequency, the cross-section of the pipe should be some appreciable fraction of a wavelength. If it is not then pressure dissipation, at the speed of sound, will take place in a shorter a time for the frequency being radiated and the disturbance will die out quicker. This is a fundamental size requirement for all structures, whether acoustic or electromagnetic, which have to launch a disturbance into the environment efficiently. Therefore it also applies to loudspeakers as we have already noted, and to television aerials (antennas) whose rod elements have to be around half a wavelength long.

Fortunately it is unnecessary for the aperture of a resonator to be as large as half a wavelength in diameter or even in circumference, otherwise it would be of ludicrously impractical dimensions for pipes at the lowest pitches.  However it remains a fact that all resonator tubes emit sound more efficiently at the higher harmonics than at the lower ones, just as with loudspeakers, therefore for powerful reeds it is certainly a case of the larger the better for the fundamental frequency to be radiated efficiently.  This is why double or even quadruple length tubes need to be used for these loud stops, simply because the tubes flare outwards to larger sizes.  At the other extreme, the thin tone of many imitative reeds is partly because of the narrowness of their cylindrical resonator tubes, which do not flare at all.   The third harmonic of a clarinet pipe, for example, is commonly much stronger than the fundamental because the radiation efficiency of the tube is higher for the higher harmonics.

Conical resonators

A conical resonator or tube placed on top of the shallot makes an effective impedance matching device, and in fact it acts as an acoustic transformer which transforms the high output impedance of the reed into the low input impedance of the surrounding air.  Without the resonator the reed would make nothing more than a thin squawk.

At the narrow end of the cone where it stands on top of the shallot, we have already noted that the vibrational air pressure is quite high as it comes out in periodic puffs from the reed.  At the same time the total air flow is relatively small because it is concentrated within a region of small cross-sectional area.  At the risk of becoming repetitious, this means the acoustic impedance at this point is high.

The situation for both of these quantities – pressure and flow - changes smoothly and progressively as the sound puffs travel up the resonator tube.  At the flared top of the tube the area is now much greater than before, thus the total amount of air involved across a cross-sectional slice (measured, say, in terms of the number of molecules) is also much greater.  Therefore the total flow due to the vibrating molecules is also greater.  At the same time the pressure, formerly quite high, has been reduced because pressure depends inversely on area – it is always measured in terms of force per unit area.  This means that flow has increased whereas pressure has decreased, therefore the acoustic impedance has been transformed from a high to a low value as required.

The resonator also performs another function – as its name implies, it resonates at certain frequencies.  We need to know why it resonates at all, and at which frequencies.  Consider first a pressure impulse from the reed injected into the small end of the resonator from the shallot.  It will travel up the resonator tube, but when it reaches the top some of the energy will get reflected back down the tube again.  This is because there remains an imperfect impedance match between the resonator and the surrounding air, even though a partial match exists by virtue of the transformer action of the resonator. However the reflected impulse experiences a phase change, which means that it now becomes an impulse of negative pressure – a rarefaction or partial vacuum.  There is also an end-correction which needs to be applied to the length of the tube to arrive at its effective acoustic length.  All these things occur at the top of open flue pipes in just the same way, and the reasons for them are described at length in the article on this website dealing with flue pipes [1] so they will not be repeated here.

As the reflected impulse travels back down the tube its pressure and flow are transformed by the action of the narrowing tube in the opposite way to that just described for the upwards transit, and at the narrow end there is little or no further reflection.  This is because the energy arriving at the narrow end, and thus at the shallot, has pretty much the required high acoustic impedance for it to be swallowed up completely by the shallot.  Therefore there is a damped-out resonance effect due to the double up-and-down transit which would, if left to itself, only persist for one or two complete cycles.  The exact number of cycles will depend on the exactness of the impedance match between the narrow end of the resonator and the shallot, but there will never be less than one.

The time taken for the double transit obviously depends on the length of the resonator tube and the speed of sound inside it.  If the double transit time is adjusted by choosing the tube length correctly, i.e. to be the same as the time between the periodic impulses emitted by the vibrating reed, then a resonance will build up at the fundamental frequency of the reed.  This is because a reflected impulse of negative pressure travelling back down the tube will reach the shallot at the same time as the reed is emitting the next pulse of positive pressure into the shallot.  As with an open flue pipe, the required tube length equals one half of the wavelength of the emitted sound approximately.

The resonance mechanism described occurs for all the harmonics in the sound generated by the reed, just as it does for open flue pipes.  Therefore we have two sets of propagating waves in the pipe at each harmonic frequency, one set being the impulses generated by the reed and emerging from the shallot and the other being the reflected impulses travelling in the opposite direction.  Consequently the well known standing wave patterns of nodes and antinodes at all the harmonic frequencies which are often drawn for open flue pipes exist for conical reed resonators also.

As with the flue pipe, the natural resonance frequencies of the resonator tube are anharmonic, that is, they are not quite exact integer multiples of the fundamental frequency. This arises because of the end correction of the tube.  On the other hand the harmonics of the vibrating reed are exact.  The way the two sets of frequencies interact has a major bearing on the timbre or tone quality of reed pipe in the same way as for a flue pipe.  This is an important phenomenon which is fully explained in [1].

Observe that the important resonance effect could not occur in the first place if the tube was a perfect transformer, that is, if it provided a perfect impedance match to the surroundings at the top.  If it did, no energy would be reflected back down the tube, no standing waves would be generated and therefore no resonances would be set up at the harmonics of the vibrating reed.  It is therefore essential that the tube should remain an imperfect transformer.  This will always be the case in practice, because we have already seen from the previous section on radiation resistance that a perfect impedance match into the surroundings would imply a tube of impractically large dimensions.

Length of a conical resonator

In the previous paragraph I said that the tube length should equal half a wavelength at the fundamental frequency of the note being emitted.  As with flue pipes, this is only approximate because it depends on the end correction of the tube, which in turn depends on factors such as the degree of flare.  However it does mean that the longest pipe of an 8 foot trumpet stop, say, will indeed be about 8 feet long.

But there are other issues relating to the length of reed pipes which need to be touched on. Sometimes double length tubes are used in loud high pressure stops such as the Harmonic Trumpet, mainly in the upper half of the rank.  This technique was introduced in the 19th century to assist in maintaining the power of the stop in the treble.  Even quadruple length pipes are used on occasion, such as for powerful Tuba stops.  The reasoning was discussed earlier in terms of radiation efficiency - the longer the flared resonator for a given note, the wider it becomes at the top, and this improves still further the impedance match from the tube into the surroundings.  This is the same thing as saying that the radiation efficiency is enhanced by a wider tube.  In turn this improves the energy transfer into the auditorium and therefore makes the pipe sound louder.  However such pipes require a lot of soundboard space so they cannot always be used, and they are of course more expensive to make in the first place.

The double length tube still augments all the harmonics in the sound from the reed just as a normal length tube does, because its natural resonant frequencies are spaced only half as far apart as those of a resonator of normal length.  Therefore there remains a natural frequency of the tube to correspond with each harmonic generated by the reed.  However, unlike those flue pipes to which the harmonic principle is applied with double length resonators, such as the Harmonic Flute, it is unnecessary with harmonic reeds to suppress the unwanted harmonics of the tube by boring a hole at an antinodal point.  This is because the resonator of a reed pipe is not actually necessary for the reed to vibrate, therefore it does not control its frequency to anything like the same extent.

Half length tubes can also be used, such as in the bottom octave of a 16 foot reed of low to medium power.  Stops so treated include the Contra Fagotto (virtually always), Double Trumpet (sometimes) and even a pedal Trombone where great power is not required, such as in a smallish church.  The rebuilt organ at Bradford Abbas in Dorset for which I was the consultant was fitted with such a reed [13].  The half length principle is also applied frequently to the lowest pipes of 32 foot reeds.  However the retinue of natural frequencies in half length tubes is incomplete compared to the harmonics of the reed itself, and in particular there is no resonance corresponding to the fundamental frequency of the reed.  This does not prevent the reed vibrating at the fundamental frequency and transferring it to the resonator, but the fundamental of a half length reed pipe will clearly be weaker because it is not amplified by the tube.  Also the impedance match at the top of the tube and its radiation efficiency will be poor at the fundamental frequency because the cross-section is smaller.

Interestingly, a fortuitous psycho-acoustic property of the human auditory system compensates for this to some extent – in a sound with many harmonics (this is important) but with a weak fundamental, the ear will in effect amplify the fundamental.  This is a well established effect and I have proved to my own satisfaction that the ear will even re-insert a fundamental which is completely absent.  Using a digital synthesiser I removed the fundamental in the spectrum of the sound of a 16 foot Trombone stop whose sounds contained seventy harmonics or so, yet its absence made not the slightest subjective difference to how it sounded!  There is a widely held belief that the same principle applies regardless of the number of harmonics, which is why quinted 32 foot “resultant bass” stops are so often found.  However when there are only a few harmonics available in the quiet 16 foot flue stops which are so treated, the effect is unsatisfactory because the ear and brain do not re-insert the missing fundamental as commonly assumed [14].

Pipe scales and degree of flare of a conical resonator

The area of the top of a conical resonator depends on the degree of flare applied to the tube, and in organ building terminology it is analogous to the scale (width to length ratio) of a flue pipe.  There are well defined rules complemented by a vast body of experience dealing with flue pipe scaling, and a summary is included in reference [1].  This is not so for reeds where, typically, the subject is not discussed at all in the literature, or dismissed in terms of  “no rules can possibly be laid down” [15].

In fact the situation is not so dire as it might at first appear.  We have seen already that an important property of the resonator is the impedance match between the open top of the tube and the surrounding air.  A wider tube offers a better match, it therefore transfers more energy into the environment and consequently the pipe will be louder.  This is important for reeds, where a historical problem for centuries was their tendency to be too loud in the bass but insignificant in the treble.  In the nineteenth century this problem was addressed vigorously by builders such as Henry Willis.  He obtained a more satisfactory variation in power across the compass by grading the diameters of the resonators carefully and quite gradually, typically halving them every 30 - 40 notes or so.  This is a much slower variation than the more critical scaling used for principal pipes where a diameter-halving interval every 16 or 17 notes is common.  This example, and there are many others, shows that reed pipe scales can vary widely – they are not as critical as those for flue pipes.

Varying the pipe scale alone was not enough to increase the power of the smaller pipes however, and Cavaillé-Coll found it necessary to use higher pressures in the treble only.  Still further measures were introduced, including double length pipes in the treble, half length pipes in the bass (both discussed already) and weighted reed tongues for the lowest notes.  The stops which eventually resulted from this treatment, at least in Britain, were powerful when necessary, yet rounded and smooth in tone and evenly regulated from note to note.  The enormous amount of work which went into the development of this type of reed tone for the romantic organ was reflected in the additional repertoire of skills demanded of the hard-pressed reed voicer, and it is perhaps no surprise that their caricature as the secretive and irascible prima donnas of the organ building world arose from this period.  Here and in America such reeds were universally liked at the time and until well into the 20th century, but in some quarters there has since been a move back towards the freer tone and greater individuality of the older style of reed stop.

Tuning reeds with conical resonators

The relative independence of the reed and the resonator has been emphasised above, but they cannot be considered entirely in isolation.  This becomes apparent when one tries to tune a reed pipe. 

Because the impedance match between the shallot and the narrow end of the resonator is good but not perfect, except perhaps in a few serendipitous circumstances, the vibrating reed usually “knows” when it is driving a resonator.  Otherwise the pitch of a vibrating reed would not vary when the resonator was tuned, whereas in practice it usually does.  Tuning a resonator simply means varying its length, and this is commonly done by adjusting a tongue of metal near the top of the tube. The impedance match between reed and resonator is not merely the result of the transformer action of an aperiodic tube with no frequency response of its own.  In practice we have a more complicated type of transformer which resonates at many frequencies as discussed already.  The input impedance of the transformer as seen by the shallot varies with frequency, and the impedance versus frequency curve has maxima at the resonance frequencies of the tube.  Appreciable acoustic energy will only be coupled to the tube at the impedance maxima as described previously.

Therefore it is not surprising that the tube will generally “pull” the frequency of the reed either sharp or flat, because the frequency values of the impedance maxima will vary slightly as the effective length of the resonator is adjusted.  Both types of behaviour have been reported and are apparently used in organ building, however both Fletcher and Rossing [6] and Norman and Norman [7] say that the reed itself should first be tuned at the spring slightly below the desired pitch.  The pipe is then tuned by sharpening the resonance of the tube itself, which is done by opening the tuning slot at the top.  Nevertheless the reverse technique can also be adopted because the pipe will work either way.  Although the two approaches are said to lead to noticeable differences in the final tone quality, it is dubious whether hard and fast rules can be laid down.

Cylindrical resonators

The cylindrical resonator is the second main class of reed pipe tube and it has existed in many forms for several centuries.  In the following discussion issues such as tuning will not be covered because much of the material just given for conical resonators also applies to cylindrical ones.

The clarinet pipe illustrated in Figure 4 is representative of the type of cylindrical resonator frequently made today.  The cylindrical tube works in two entirely different ways to the conical one.  Firstly it does not provide, by design and intention, nearly as good an impedance match between the shallot and the resonator.  Its radiation efficiency is also worse, particularly at the fundamental frequency, therefore its impedance match to the surroundings is also poorer.  Secondly it is only approximately half as long as a conical resonator of the same pitch.  Taken together, these differences mean that the resonator emphasises mainly the odd harmonics rather than all of them.  Let us see why.

The immediately identifiable sound of the orchestral clarinet is because some of its even-numbered harmonics are of lower amplitudes than those of other instruments, this feature endowing it with its characteristic hollow and woody sound.  Pipe organ stops of the same name, although not identical to their orchestral counterparts, must nevertheless have a similar harmonic structure if they are to sound acceptable.  This also applies to other hollow-sounding stops such as the Corno di Bassetto and Vox Humana.  The prime function of the resonator in all these cases is to reduce the even harmonics, and the acoustic transformer action so important to the conical resonator is virtually absent for the cylindrical one.  In fact the one is achieved at the expense of the other. 

As can be seen from Figure 4, the cylindrical tube sits in a short conical socket on top of the shallot.  This is for mechanical reasons as much as anything else because the diameter of the cylinder for most types of stop needs to be somewhat greater than that of the shallot if the pipe is to emit sound at a reasonable volume.  Although the short conical section does exert some acoustic effect (an impedance transforming action as before), it is negligible compared to that which occurs in the much longer conical tubes used for trumpet type stops.  Therefore there is an appreciable acoustic impedance mismatch between the base of the cylinder and the shallot.  This means the cylinder is mismatched at both ends, because the sudden discontinuity between the narrow tube and the surrounding atmosphere at the top also represents a sudden change in impedance.

The situation is comparable to the stopped flue pipe which exhibits similar abrupt changes in acoustic impedance, both at the stopped end and at the mouth where it opens suddenly into the surroundings.  Therefore it is not surprising that the cylindrical resonator of a reed pipe works in a similar manner to the stopped flue pipe.  In an acoustical sense the cylindrical resonator can be thought of as an upside down stopped flue pipe, the open top of the reed resonator being equivalent to the mouth of the flue pipe and its base being equivalent to the stopper.  The latter might be difficult to comprehend at first, so let us consider a reflected sound wave moving downwards inside the cylinder.  When it gets to the bottom it is confronted with a short section of conical tubing which rapidly gets narrower.  The length of the conical tube is so short that it can perform little acoustic transformer action, so the situation is not much different to suddenly reducing the diameter of the cylindrical tube without any conical section at all.  This represents a sudden impedance mismatch so the descending sound wave, or most of it, just bounces back up the tube.  Only a small proportion of the energy will get through to the shallot, though the amount which does get through has some bearing on the proper functioning of the pipe. 

Another important point is that there is no phase change on the reflection at the bottom of the pipe – if the descending wave consists of a region of positive pressure it will be further compressed when it meets the impedance mismatch, and this in fact is the cause of the reflection.  Rather like bouncing a rubber ball on the ground, it is the momentary extra compression at the point of impact which causes the rebound.  Thus when the reflected wave starts to move back up the tube, it will still consist of a region of compression as it did when moving down.  This is the meaning of the term “no phase change”.  The same will occur if the descending wave is a region of rarefaction though it is perhaps a little more difficult to visualise.  At the top of the pipe where the sound waves meet the surrounding air there is a phase change – a compression is converted into a rarefaction and vice versa.  The reason for this was explained in the article on flue pipes [1].

In order for the resonator to emphasise the odd harmonics rather than the even numbered ones, the pipe is made about half the length of a conical one for the same pitch.  This is another similarity to the stopped flue pipe.  Therefore, to summarise, both cylindrical reed and stopped flue pipes are characterised by (a) impedance mismatches at both ends, (b) a phase change at the open end only, and (c) being of half length.  The reason why such a pipe suppresses the even numbered harmonics was explained in detail in the article on flue pipes [1] and it will not be repeated here.  Note that it is a common misconception that a reed pipe has to be cylindrical for this to happen, whereas in fact the only requirement is that its cross-section must not vary.  Therefore a square or rectangular resonator would work just as well, as it does with the stopped flue pipe.  Such a pipe would merely be more difficult to make and interface to the shallot which is why cylinders are used more frequently for reeds.  The important point is that it must not flare, otherwise the impedance discontinuity at the shallot end which is so essential to its working would not be pronounced enough.

Reed Pipe Waveforms and Frequency Spectra

Some remarks have been made already about the differences in tone quality between reed pipes with conical and cylindrical resonators, and these depend mainly on the relative proportions of the odd-numbered harmonics to the even-numbered ones.  The influence of the shallot was also mentioned, particularly whether it was open (with an aperture occupying the full length of the shallot) or closed (the aperture occupying only a fraction of the length and moreover having a characteristic shape).  These features will now be discussed in more detail insofar as they affect the actual acoustic waveforms emitted by pipes of different types.

The waveform generated by the reed

It is unprofitable to devote too much time discussing details of the sound pressure waveform within the interior of the shallot as generated by the vibrating reed.  This is because we only hear the waveform when it has been modified by the resonator and emitted from the top of the pipe, and by that time it has changed considerably.  However some broad considerations can be helpful to understand the harmonic structure of the generated wave and how the type of shallot influences it.

Open shallots

We shall consider two types of shallot – type ‘A’ (open) and type ‘B’ (closed) as illustrated earlier in Figure 3.  Taking the open type first, the reed tongue covers and uncovers the shallot periodically as it vibrates, and although sometimes the reed does not fully close off the base of the shallot on its forward stroke, we shall ignore this for the time being.

Figure 5.  Waveforms and spectra for two idealised pressure waveforms generated by the reed for an “open” shallot

A: square wave, B: triangular wave

At the simplest possible level one could envisage the reed, which is effectively an oscillating air valve, shutting off the air supply and then opening it again, in both cases virtually instantaneously.  This would give rise to an impulsive air pressure variation, the edges of the pulse being sharp and vertical on a time-pressure graph,  or nearly so.  If we additionally assume that the durations for which the aperture is closed and open are equal, then the impulse will have the form of a square wave as shown at Figure 5A.  The corresponding spectrum is also sketched, showing that only the odd-numbered harmonics are present which is always true for a square wave.  The amplitudes of the odd harmonics decrease at a rate of 6 dB per octave (that is, for every doubling of frequency the harmonic amplitude halves).

We can say with certainty that this type of waveform cannot exist for stops having a conical resonator because no even-numbered harmonics would be generated.  Stops requiring a trumpet type of tone would sound ludicrous in the absence of the even harmonics, like an extremely loud and raucous clarinet, yet we know from experience that they do not.  A square wave contains only the odd harmonics, and although a conical resonator resonates at all frequencies, it could not emit harmonics which were not generated in the first place.  Therefore we can discount the square wave on the basis of experience with real pipes.  Nevertheless, if it could be generated, such a waveform would be useful for pipes with cylindrical resonators for tones where only the odd harmonics are desired.

However there are several reasons why a square wave could not be generated by the reed.  One is that the air pressure impulse which is transmitted to the interior of the shallot depends strongly on the type of curve given to the reed tongue by the voicer, therefore the aperture will be closed in a more or less gradual manner rather than instantaneously.  Let us assume that the air pressure within the shallot decreases linearly with time as a result of the curved tongue gradually rolling over it.  If we further assume that the tongue rebounds in the same way, then we get the triangular (serrasoidal) wave depicted in Figure 5B.  Unfortunately this also would be an unlikely candidate because the triangular wave likewise contains no even-numbered harmonics.  The spectrum shows that in this case the odd harmonics decrease at 12 dB per octave (for every doubling of frequency the amplitude is reduced by four), which is twice as rapid as for the square wave.  Nevertheless although the waveform itself is unrepresentative, we have arrived at an important point.  The spectrum illustrates that, having assumed a curved tongue and thus a slower pressure variation inside the shallot, the amplitudes of the higher harmonics have been strongly modified.  For instance, the amplitude of the 9th harmonic for the triangular wave is nearly 20 dB lower (a factor of ten) than for the square wave.  This is a large amount and it illustrates the critical influence over the higher harmonics which is exercised by tongue curvature, a fact well known in practice.

Figure 6. Waveforms and spectra for two more idealised pressure waveforms generated by the reed

 A: sawtooth wave for an “open” shallot, B: “smooth” wave for a “closed” shallot

We can proceed further by noting that there is little justification for the assumption that the tongue will rebound from the shallot over exactly the same time period as that taken for it to close the aperture.  Therefore in general the two halves of the triangular waveform will be asymmetrical, and for simplicity we might assume the tongue could rebound much quicker than when it is closing the aperture on its forward stroke.  (We could equally well assume that it rebounds much more slowly because it would make no difference to the broad conclusions in what follows.  It is only the assymetry of the waveform that matters).  The asymmetry introduced into the triangular wave would make it more like a sawtooth as depicted in Figure 6A.  Such a waveform possesses all harmonics both odd and even and, although it remains idealised, a sawtooth wave is the closest approach to reality out of the three possibilities considered so far.  It is no more unrealistic than assuming, as is commonly done, that the stick-slip action of a rosined violin bow excites the string in a sawtooth fashion.  The harmonic slope of the sawtooth spectrum is, like the square wave, -6 dB per octave and therefore it has an extended retinue of many high-order harmonics which decrease relatively slowly in amplitude.  In this qualitative sense the sawtooth spectrum is a reasonable representation of what is going on in a pipe with an open shallot.  Because of their large numbers of high frequency harmonics such pipes are well known for their free, splashy and somewhat raucous tone to an extent which makes them unpleasant to some ears.  This characteristic harmonic structure is generated by the periodic air pressure variations within the shallot broadly in the manner described.

Closed shallots

So far we have only considered the open type of shallot.  In cases where it is closed by a specially shaped aperture the generated waveform will vary yet again.  Often the aperture will not extend to the top of the shallot, which is sensible because the tuning wire will often prevent the top part of the reed vibrating in any case.  Also some apertures do not extend to the base of the shallot either, which again is sensible because the reed cannot always be assumed to cover the shallot completely.  This type of aperture, illustrated in Figure 3B, is the one we shall now consider.

As the reed begins to roll over the aperture on its forward stroke we can reasonably postulate that the pressure inside the shallot will not change much at first because the tip of the aperture is narrow.  However it will reduce progressively more quickly as the reed shuts off the wider regions of the aperture towards its base.  In addition it is unlikely that the rebound of the tongue will be as sudden as assumed previously in the sawtooth case, and a somewhat slower rebound is more realistic for any reed and shallot combination.  This might result in a pressure waveform inside the shallot of the general form illustrated in Figure 6B.  It must be emphasised again that this picture is idealised and it has been included merely to illustrate the remarks just made.  Such a waveform will possess fewer high-order harmonics than the sawtooth on account of its “curviness” because, in crude terms, it has some of the smooth attributes of a sine wave as well as those of a sharp sawtooth.  This will reduce the amplitudes of the higher harmonics in the same way that those of the triangular wave were lower than of a square wave.  Without refining the assumptions more closely there is not much point trying to derive an exact spectrum in this case, and only an approximate one has been sketched in Figure 6B to make the point.  Such an air pressure spectrum, generated within a closed shallot, would help to explain why the corresponding reed pipe sounds smoother and less aggressive than one whose shallot is open because the amplitudes of the higher harmonics are lower.

Emitted waveforms and spectra

Only two examples of real waveforms and frequency spectra will be discussed, one each for conical and cylindrical pipes.  A more complete discussion of the many different types of reed stops encountered in organs must await a separate article.

It must be understood that it is the frequency spectrum, not the particular time waveform, which determines unambiguously what we hear.  The spectrum, of course, shows how the amplitudes of the harmonics vary, and it is unambiguous in the sense that if the harmonic amplitudes change then so will the nature of the sound we perceive.  However it is possible for an identical spectrum to be produced for any number of different waveforms.  This means that the time waveform is highly ambiguous because the several waveforms having the same spectrum will sound exactly the same.  The reason is that waveform shape depends on the phase as well as the amplitude of each harmonic, whereas phase information is thrown away when the frequency spectrum is computed.  We can throw it away because the ear is insensitive to phase.  (As an aside, and to be rigorous, we should really call the frequency spectrum the power spectrum of the waveform, but having raised this for correctness we can proceed to forget it again as far as this article is concerned).

Another necessary reminder is that the waveforms and spectra we are about to see might bear little resemblance to those just discussed, which related to the waveforms and spectra inside the shallot rather than those emitted from the resonator into the auditorium.  Apart from the simplistic nature of the previous analysis, differences will occur owing to the way the slightly anharmonic natural resonant frequencies of the resonator match the exact harmonic frequencies of the waveform generated by the reed.  This important phenomenon has already been mentioned and it is discussed in detail for the flue pipe in [1].  The frequency-dependent radiation efficiency of the resonator will also affect the shape of the emitted sound spectrum as described previously in this article.

Pipes with conical resonators

An actual waveform of a trumpet type of pipe is shown at Figure 7.  This is the middle F# pipe of the Swell 8 foot Trumpet on the beautiful four manual organ at Malvern Priory, built by Rushworth and Dreaper in the 1920’s.  The recording was made not long after the late Ralph Downes had declared the instrument so fine that it should never be touched tonally, a recommendation made before the major interventions of more recent years.

Figure 7. Trumpet 8’ waveform (Malvern Priory, F#3, Swell organ)

The pitch of the pipe was 370 Hz, therefore each cycle on the graph occupies 2.7 milliseconds as shown by the red lines.  For the reasons mentioned a moment ago there is little point trying to extract much from a time waveform, but it is worth noting that the repetition period of the waveform stands out.  This means that if we compute the corresponding spectrum there should be a well defined peak at the fundamental or first harmonic.

Figure 8.  Trumpet 8’ spectrum (Malvern Priory, F#3, Swell organ)

Looking at the spectrum in Figure 8 this is indeed what we see.  This is a plot of the amplitude of each harmonic in terms of sound pressure level (SPL).  The SPL values are not absolute but relative to some arbitrary datum value, and they are expressed in decibels.  The most important point to bear in mind about decibels is that they convey information about ratios rather than absolute numbers, with a change of +6 dB meaning a ratio of two.  Thus a change of –6dB means a factor of ½.

Although the fundamental is of high amplitude, it is not the dominant harmonic – the third is the strongest, though not by a significant amount compared to the major variations elsewhere in the spectrum.  This spectrum is typical of those for reeds on a romantically conceived organ which could be called “smooth” rather than “splashy”, having harmonics which fall off gradually in strength until about the 7th, beyond which they then decay more rapidly.

Pipes with cylindrical resonators

The waveform of an 8 foot clarinet type of pipe is shown at Figure 9.  In fact this stop was called Corno di Bassetto and it also was recorded at Malvern Priory.  Also as before the pipe was the middle F# note on the keyboard.

Figure 9. Corno di Bassetto 8’ waveform (Malvern Priory, F#3, Solo organ)

The time axis of this diagram is the same as that for the Trumpet, yet at first sight the frequency might look much higher than it ought to be for a pipe of the same pitch.  However by putting in the red lines delineating each cycle we can understand better what is happening.  Within each cycle there appears to be no peak relating to a strong fundamental as there was with the Trumpet, and instead there seem to be three strong peaks per cycle.  From such a waveform one would expect the fundamental to be weaker than for the Trumpet, with a stronger 3rd harmonic at three times the frequency.

Figure 10.  Corno di Bassetto 8’ spectrum (Malvern Priory, F#3, Solo organ)

The corresponding spectrum is at Figure 10, and this is exactly what we see.  The third harmonic dominates the spectrum, being stronger than the fundamental by about 12 dB or a factor of four in amplitude.  This is a big difference.  The 5th and 7th harmonics are also very pronounced.  Note that all these are odd-numbered harmonics, and such a spectrum is understandable in terms of the mechanism outlined earlier which favours the odd harmonics. The only even-numbered one which has a comparable amplitude is the 2nd, and although the even ones are generated by the reed as with the Trumpet, most of them are not strongly enhanced by the cylindrical resonator.  The reason for the anomalously strong 2nd harmonic is because the resonator is of half length, thus the fundamental of a tube of this size coincides with the 2nd harmonic of the emitted sound.  Although a cylindrical tube should not resonate at this frequency by virtue of the way it is intended to work, the powerful retinue of harmonics generated by the reed manages to excite it nevertheless [16].

Because the diameter of the resonator is so narrow at the top it was relatively inefficient at radiating the lowest frequency (the fundamental) into the church, which partly accounts for its low amplitude.  This also was explained earlier.

Concluding Remarks

A fairly complete account of the principles underlying sound production in reed pipes has been given, including some novel material,  together with pointers into the literature whence more detail can be found.  Hopefully this will have proved interesting to those who wish to expand their knowledge, if only because the breadth and detail of material included here has not so far appeared elsewhere in one place.  However it does not mean that there is no scope for further research, thus it is legitimate to reflect on whether the possession of greater understanding might assist organ builders to produce better reeds, stops which have gained a notorious reputation for their variable quality over several centuries.  Only they can answer this question by relating it to their own circumstances, and it is probably better to recast it in a more general form here.  Thus one might ask whether physics can assist musical instrument makers in general to produce better instruments, or to have greater confidence that they will meet the necessary standards.

There is no doubt that the answer to this is affirmative, though with some qualifications.  On the one hand, research still continues in an effort to understand how the great violin makers achieved their results during the 17th and 18th centuries.  On the other, improvements were effected quite rapidly in the orchestral clarinet and some other instruments in the latter half of the 20th century once the physics of their operation had become sufficiently well understood, and these are considerably more complicated than organ reed pipes.  For example, the effects of the player's lips on the clarinet reed introduce extra dimensions of uncertainty which the reed pipe does not have, and the subtle acoustic mechanism of the tone holes likewise does not exist for the organ reed.  Because these problems were largely solved many years ago it cannot easily be argued that the simpler physics of the reed pipe would not likewise yield to research, and it would be easy to envisage suitable experiments.  Experiments would be necessary because theory alone would not suffice, and doing experiments properly costs money.  Nevertheless the investment could be worthwhile because as well as reducing the probability of unsatisfactory reed work, such research might also result in cost reductions which would of course be attractive to customers as well as to organ builders.

However there is a major difference between the situation for the organ and most other instruments.  Unfortunately the organ appears to be edging towards near-extinction in Britain, an insidious situation which is examined in detail elsewhere on this website [17].  Once installed at enormous expense, an organ today has little intrinsic value beyond the profit a scrap dealer could turn on the pipes, and we are seeing this increasingly frequently.  To pose the question in an extreme if rhetorical form, who would pay anything to rehouse the new organ in Worcester cathedral if it were to appear next week on eBay, as many fine organs do?  Hammond tone wheel organs frequently attract higher prices than any pipe organ on that forum.  Yet most other traditional instruments are still much in demand by musicians and educational institutions, with the result that more research is done today on making cheap plastic recorders even cheaper than on any aspect of the organ.

The question posed above as to the value of further research thus reduces more to one of market forces rather than physics alone.  As just remarked, there is a reasonable probability that further research could improve organ reed work, or at least make the lot of the organ builder less hit and miss when he is trying to manufacture it.  However science is not the beginning and end of the matter, and the unpalatable fact is that nobody is likely to fund much research today for a product which so few people apparently want.  Although reed pipes are as fascinating as anything else about the organ, the community which finds them so is probably dwindling along with interest in the instrument itself.  Unlike other artefacts with their roots in yesteryear such as steam engines and antiques, the level of interest expressed both by musicians and the British public at large in the organ has reached a dangerous low.  I wish I could see an answer to the problem because I take no pleasure in pointing it out so starkly. 

Notes and References

1. “How the Flue Pipe Speaks”, C E Pykett, 2001.  Currently on this website (read).

2.  An example of how mathematics does not always illuminate what is happening concerns the way that the different types of resonator tube used on reed pipes actually work.  A mathematician or physicist might be satisfied having solved an equation for the normal modes of the resonator, which involves Bessel functions.  Then s/he would have to determine the acoustic input impedance of the resonator for each mode and select only those which present the highest acoustic impedances to the reed. 

It is true that the solutions of these equations, if you can follow them, duly predict that a conical tube will reinforce all the harmonics generated by the reed whereas a cylindrical one only reinforces the odd-numbered ones.  However I have yet to see a satisfactory theoretical treatment which includes the end effect at the top of the resonator.  This is necessary if the anharmonicity of the natural resonances of the tube is to be modelled, and hence how the tube will modulate the spectrum of the harmonics emitted by the reed which are exactly spaced in frequency.

Therefore a lot more effort is necessary to explain to one’s own satisfaction, and thus to a non-mathematician, how all this happens in terms of a picture of the sound waves travelling inside the tube, and how the reed generates them in the first place.  This is one example where I have tried to present such a picture in this article.

3. A common misunderstanding is that organ reed pipes, such as those used for clarinet and oboe stops, are assumed to work in the same way as their orchestral counterparts.  In fact there are several major differences, one of which is the Q-factor of the oscillating reed as explained elsewhere in this article.

4.  Fletcher and Rossing wrote that the organ reed “moves in a nearly sinusoidal manner” (see 6 below), whereas in fact a beating reed does not as explained in this article.

5.  “Fundamentals of Musical Acoustics”, A H Benade, Dover, New York, 1990.

At first sight this is a laudable attempt to present a survey of the major aspects of acoustics as it affects virtually every musical instrument except the organ.  Written by an acknowledged expert in acoustics but for musicians rather than physicists, it is obviously of little value for the subject matter of this article.

6.  “The Physics of Musical Instruments”, N H Fletcher and T D Rossing, Springer-Verlag, New York, 1999.

Not for the faint-hearted in mathematics, nor for those with less than a deep pocket.  Refuge is sought mainly in equations rather than in explanations, some of which are incorrect in any case.  The disingenuousness of the authors, who regard the fruits of their labours as suitable for those who are “not frightened by a little mathematics”, is unhelpful.  Much of the mathematics is in fact of university level, and far from being “a little” there is an awful lot of it.  Therefore I feel bound to point out that this expensive book cannot really be recommended for non-specialist readers who just wish to expand their knowledge somewhat, though it is without question useful to the physicist.

7.  “The Organ Today”, H and  H J Norman, Barrie and Rockliff, London, 1966.

An interesting and readable account of the main issues involved in organ building by two members of the former well-known organ building dynasty, though it reflects a style which is rather complacent, dated and chauvinistic.  It is descriptive and qualitative rather than numerical and quantitative, and this is not a criticism.  However one comes across some facile statements from time to time, which is surprising considering the background of the authors.  The contents are also exasperatingly muddled, typified by the paragraph on the relative merits of tracker action which concludes the section on reed voicing!  The neo-gothic drawings are curious and some are obscure.

8.  “The Art of Organ–Building”, G A Audsley, New York, 1905.

A detailed though excessively pedantic survey of organ building in two volumes, written by an architect whose unawareness of his limitations is continually amusing.  Perhaps the best description of the work is that it is good in parts, and one has to exercise judgement in extracting material from it.  As would be expected of an architect though, most of the diagrams and drawings are exquisite.

9.  “The Hope-Jones Organ in Pilton Parish Church”, C E Pykett, Organists’ Review, November 1993.  Currently also on this website (read).

A digital re-creation of this organ including some sound clips is in the article currently on this website entitled “Re-creating Vanished Organs”, C E Pykett, 2005 (read).

10.  “Organ Reed Pipe Tongue Dimensions”, J Liljencrants,

 http://www.mmdigest.com/Tech/reedPipeDimensions.html  (accessed 23 December 2009).

This useful and interesting article derives a simple expression for the frequency of a free reed and then goes on to discuss measurements on a beating reed, which gives different results. 

11.  “Vibration characteristics of pipe organ reed tongues and the effect of the shallot, resonator, and reed curvature”, G R Plitnik, Journal of the Acoustical Society of America, 107(6), June 2000, pp 3460-73.

12.  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 it pressure must reduce as its speed increases. 

13.  “The Organ at Bradford Abbas Parish Church, Dorset”, C E Pykett, 2004.  Currently on this website (read).

14.  It is also a common misconception that the beats which arise at 32 foot pitch from two quinted pipes from a 16 foot flue rank are exactly the same as desired 32 foot fundamental tone.  This is quite wrong because there is no sound energy in a beat.  What is happening is that the beats are so slow at these low frequencies that the ear is merely able to follow their time pattern, but this is quite different to actually hearing the missing frequency.

15.  “The Modern British Organ”, N A Bonavia-Hunt, Weekes, London, 1947.

One of the many tomes on organ building written by a dilettante with quite another day job (in his case a parson).  Therefore, although it is useful and readable, one nevertheless has to be careful not to accept everything it says at face value.  Too much material in this book describes the work of hobbyists who, far from being long-forgotten, were never known in the first place.  When reading Bonavia-Hunt I cannot avoid reflecting how remarkable it was, and is, that so little time can apparently be spent by some members of the clergy curing souls in relation to that spent fiddling about with organs.

16. One reason why the resonator of the Corno di Bassetto pipe reinforced the second harmonic at Malvern Priory, and to some extent the other even-numbered ones, is that the base of a cylindrical tube does not behave exactly like the stopper in a stopped flue pipe as assumed in the analysis given earlier in the article.  In practice it allows a small amount of standing wave energy to leak away into the shallot rather than being fully reflected back up the tube.  This endows the tube with some vestigial characteristics similar to those of a conical resonator, which is acoustically open at the base where it meets the shallot.  For this reason the conical resonator reinforces all harmonics both odd and even as explained previously.

17.  "The End of the Pipe Organ?", C E Pykett, 2005.  Currently on this website (read).