Tremulant Simulation in Digital Organs
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  Tremulant Simulation in Digital Organs  

 

by Colin Pykett

 

  "What went ye ... to see?  A reed shaken with the wind?"

(Holy Bible, St Matthew xi.7)

Written: 18 August 2009

Last revised: 28 September 2016

Copyright © C E Pykett 2009-2012

 

Abstract.  The problems posed by tremulant simulation in digital organs are discussed for synthesisers using the techniques of sampled sound, additive synthesis and physical modelling.  It is shown that the regular and smooth pulsations in the wind supply of an organ generated by a tremulant are transformed into complex and unpredictable frequency and amplitude modulations impressed on the pipe sounds largely because of air turbulence.  These effects become more pronounced the greater the modulation depth, and the effects differ from one beat of the tremulant to the next and for every pipe.  This feature endows a well-adjusted pipe organ tremulant with a richness comparable to that of the pipe sounds themselves.

 

Simulating all this at a detailed level is only possible for sound samplers which can capture the actual sounds of tremulated pipes, because turbulence and other random or chaotic effects are impossible to model realistically.  Therefore other types of synthesiser using any form of modelling, and this includes additive synthesis as well as physical modelling, can only approximate the effects of tremulants.  However even with samplers there are some practical difficulties which degrade the effectiveness with which the captured sounds can be reproduced.

 

The upshot is that shallow, gentle tremulants can be simulated quite well by any form of synthesis.  However the more complex effects of fast and deep tremulants can only be captured by sampling the sounds directly, notwithstanding the difficulties of reproducing them.  Moreover, because the effect of a tremulant differs for every pipe it is not surprising that simulating a theatre organ satisfactorily is especially difficult as far as its tremulants are concerned, regardless of the type of synthesis used.  Thus tremulants reveal the limitations of current synthesis techniques quite starkly, thereby providing yet another case study showing that digital organs can only approximate to real pipe sound.  In the last analysis only individual users can decide for themselves which of all the imperfect options offends them least.

 

Contents

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

 

Introduction

 

Pipe Organ Tremulants – how they work

 

            Retained-wind tremulants

            Spilled-wind tremulants

 

Tremulant Wind Pressure Variations

 

Effect of Tremulants on Pipe Speech

 

            Frequency variations

            Amplitude variations

            Timbral variations

 

Digital Simulation of Tremulants

 

            Sampled sound synthesisers

            Additive synthesis

            Physical modelling

 

Concluding Remarks

 

Notes and References

 

 

Introduction

 

When trying to simulate organs digitally one can get so immersed in the details of how organ pipes speak that the attributes of the various contrivances which modify or add expression to them can sometimes be overlooked.  Since medieval times these have ranged from the faintly ludicrous, such as “nightingale” stops whose pipes bubble their sounds through water, to today’s almost ubiquitous swell boxes and tremulants.  Although I have yet to implement a digital nightingale, I have studied swell boxes in some detail and posted the results on this website [1].  This article now looks at simulating tremulants (or tremolos). 

 

Not only are the characteristics of tremulants highly variable, but another striking feature is the spectrum of interest in the subject on the part of organ builders and players alike.  This ranges from indifference at one extreme to obsession at the other.  Not only that, but the tremulants in many organs are unusable because of the appalling racket they kick up or because they do not work at all, or because their effect is unsatisfactory in one way or another.  The latter problem posed a particular difficulty in writing this article because of its subjective element – some organists prefer a slow, gentle beat whereas others will have nothing but a strong and rapid throb, with the remainder lying somewhere in between.  These subjective elements are probably augmented because organists play either classical organs or theatre organs but seldom both.  Thus I have tried to cater for all tastes and preferences without inclining too strongly in any one direction, such as towards the “strong” and “gentle” tremulants of the classical organ or the theatre organ with its essential tremulated tibias.  Almost inevitably, therefore, the article will fail if only because one cannot please everybody all the time.  But at least I have tried!

 

We first look briefly at the technology of the pipe organ tremulant and its history over several centuries.  Then we move on to how a tremulant varies the sound of a pipe. Finally the problems of simulating these variations digitally are discussed with reference to the three simulation techniques of sampled sound, additive synthesis and physical modelling.

 

 

Pipe Organ Tremulants – how they work

 

Tremulants can be categorised in two ways – those which vary the wind supply to the pipes without losing air to the atmosphere, and those which dump wind externally before it gets to the pipes.  Following Dom Bédos in the 18th century [2], we shall call these retained-wind (tremblant à vent clos) and spilled-wind (tremblant à vent perdu) tremulants respectively.  In both cases the consequential periodic wind pressure variations result in various tremulant effects, and with a few exceptions the former type is more gentle than the latter.  We need to spend time looking at some of the mechanisms which have been devised over several centuries because some simulation techniques (particularly physical modelling) require them to be analysed and modelled mathematically.  However it is only possible to skim the surface of the subject here because of the vast number of methods which have been used by organ builders over this long timeframe.

 

Retained-wind tremulants

An increasingly popular retained-wind tremulant, at least in Britain where it seems to be one of today’s in-things, is the so-called Dom Bédos tremulant.  However this name is rather misleading because he described both types in his monumental treatise on organ building [2].  Illustrated in Figure 1, it is placed in series between the wind supply and the pipes and its working parts consist of a hinged valve with a spring-mounted lead weight on one face.  When not operative the valve is pushed out of the way by the drawstop mechanism, but when the tremulant stop is drawn the valve is allowed to bounce gently in the air stream passing through the box.  This imparts a mild undulating effect to the pipe sounds which is why it was called tremblant doux (gentle tremulant).  The pulsation rate and depth of the tremulant depend on the size of the weight and the springiness of its supporting wires, but also on the volume and speed of the wind passing through.  It has several advantages including noise-free operation and some musical “liveness” because of the dependence of the tremulant effect on how many stops are drawn and the number of notes keyed.  However it is tricky to set up, and the tremulant effect can sometimes vanish altogether if too many stops are drawn because the valve then gets blown permanently open.

 

 

 Figure 1.  Dom Bédos retained-wind tremulant (from [2] )

 

Other more recent retained-wind devices have included electric motors which shake the top board of the wind reservoir/pressure regulator via an unbalanced rotor.  When pressure control is done at the chests or soundboards using smaller Schwimmer regulators there is no separate reservoir so this method cannot be used, and in this case the Schwimmer regulator pan can be pushed and pulled electrically or pneumatically instead.  An advantage of these methods is that the speed and depth of the tremulant can be remotely controlled from the console.

 

Although most retained-wind tremulants are relatively gentle, an exception to the rule is the flutter tremulant sometimes used with fairground organs, particularly those not having an actual tremulant mechanism fitted.  In this case an extremely violent on-off tremulant effect is obtained by punching closely-spaced holes instead of continuous slots in the music books for the sustained notes.  Stephen Bicknell aptly remarked [3] that the effect is “so fast and deep that you expect the paint to flake off ”!

 

Spilled-wind tremulants

This type of tremulant allows regular pulses of air to escape into the atmosphere from the winding system at some point.  Generally the effect is more violent than that from a retained-wind tremulant and for this reason the species was called tremblant fort (strong tremulant) in the past.  Violence is also often done to the structure of the organ and the building itself, particularly those elements which resonate at the tremulant frequency such as furniture and windows.  Although these tremulants are the more usual type, at least in Britain with its wearisome preponderance of mediocre Victorian organs, they are unreliable and often very noisy.

 

 Figure 2.  Spilled-wind tremulant

 

A typical spilled-wind tremulant is sketched in Figure 2 though the drawing is not to scale as it is merely the intention to illustrate the principle.  The system is switched on when the lever arm magnet is energised, which is the situation depicted in the diagram.  This allows a large hinged pallet to admit wind to the middle section of the box, and it then passes through a second pallet into the large pneumatic motor at the top and out of the outlet port in its top board.  Because the outlet is smaller than the entry hole in the base the motor begins to inflate against a spring, thus partly closing the pallet and reducing the air flow.  At some point the flow reduces such that the spring can partially open the pallet again and the cycle therefore repeats in an oscillatory fashion, allowing periodic pulses of air to escape through the top of the box. 

 

Valves are usually fitted to the inlet and outlet ports; in the diagram the former is shown as a butterfly valve which can be adjusted from outside the wind trunk and the latter as a swinging vane.  Tremulant rate is determined partly by the lead weight attached to the top board of the oscillating motor and the strength of its pallet spring, but it is also affected by the valve settings.  The depth of the tremulant depends on the valve settings but also on its rate, so it is quite tricky to get the system working as desired because of these interactions.

 

Things are made even more difficult because of the interaction of the tremulant with the rest of the winding system.  To state the obvious, it will produce virtually no effect in an organ with very stable wind even if the tremulant mechanism itself oscillates properly.  It is necessary that the winding system itself is to some extent unstable by having a damped resonance at some frequency.  If present, the resonant frequency can be estimated by depressing and sharply releasing the top board of the main reservoir/pressure regulator when a flutter will be heard from the pipes.  The resonant frequency must first be tuned to the desired tremolo rate (difficult in itself), and the tremulant mechanism itself then tuned to it also.  When the tremulant is switched on it will then strongly excite the resonance in the main winding system and – hey presto – off it will go.  Adjusting the depth of the tremulant to taste is then necessary however, and this means patient backtracking over all the adjustments, usually several times.  Thus getting a tremulant to work properly demands an experienced organ technician, and the theory and practice of the processes involved have been well described for a Wurlitzer theatre organ by Dennis Hedberg [4].  Some photographs of tremulants of this type for Compton and Wurlitzer organs can be seen on the Internet at the website in reference [5].

 

Although the effect of this type of tremulant can be pretty strong, an even more pronounced one is obtained from the jazz flute or vibratone ranks on fairground and dance organs.  These usually consist of stopped pipes, each of which is fitted with a small tremulant valve covering a hole in the back face. When a “jazz tremolo” effect is called for, all the tremulant valves of the entire rank are opened and closed by a single motor at a frequency governed by the music book while the pipes are sounding.  Stephen Bicknell has entertained us with his pithy wit here also, describing it as “like a swanee whistle” [3].  To anyone who believes that tremulated Wurlitzer tibias are the ultimate in “giggle”, I recommend them to listen to these stops if they have not done so already [6].

 

 

Tremulant Wind Pressure Variations

 

All tremulants impose a periodic variation in wind pressure on the pipes, the important aspects being the amount of variation about the static value and the waveshapes of the excursions.  Although there are wide variations, a maximum peak to peak wind pressure deviation of some 75% about the static value is typical.  Examples are an excursion of  ± 30 mm wg about a static value of 80 mm for the fluework of a church organ, and ± 100 mm about a static value of 250 mm for that of a theatre organ.  Much more than this and at least some pipes might be thrown off speech.

 

As to pressure waveshape, three representative traces made from real time pressure measurements at the pipe valve of an organ are shown in Figure 3.  Figure 3(a) is smooth and practically sinusoidal and most often it characterises gentle Bédos-type tremulants. However it can also be produced by a spilled-wind tremulant when not oscillating too strongly, but when it is adjusted to give greater modulation depths the pressure waveform becomes distorted with Figures 3(b) and 3(c) being typical examples.  Although both of these can arguably sound more characterful than a smoother variation, that in Figure 3(c) should be avoided because the tremulated pipes can sound out of tune.  This is because, in this example, they spend significantly more time at the higher pressure than the low one and they would therefore sound sharp.  Therefore a tremulant should ideally be adjusted so that its waveform is roughly symmetrical about the horizontal axis as in Figures 3(a) or 3(b), but this can be difficult in the usual situation when the waveform cannot be observed during the adjustments.  In practice this often leads to highly unsymmetrical pressure waveforms being generated in pipe organs more by accident than design.  The issue is of course important for digital simulation in that there is little point simulating an unsatisfactory tremulant unless one particularly wants to do so.

 

 Figure 3.  Typical tremulant wind pressure waveforms

 

 

Effect of Tremulants on Pipe Speech

 

When the wind pressure varies the pitch and volume of the pipes also vary.  Flue pipes go sharper and get louder when the pressure rises, and flatter and quieter when it falls.  A third factor is a timbral (tone colour) modulation with some pipes in that the higher harmonics can be modulated in amplitude relative to the fundamental, though this effect is usually only noticeable with higher modulation depths (deeper tremulants).

 

Frequency variations

As an example of these variations, I measured the frequency versus wind pressure behaviour of a stopped pipe which spoke the F below middle C (though it was slightly off-tune during the experiment).  The results are shown in Figure 4.

 

Figure 4.  Fundamental frequency vs wind pressure for a stopped pipe

 

The variation of pitch with pressure is shown in blue and over much of its range it was close to linear as shown by the pink line.  This linear characteristic over a significant range of pressure (at least a factor of two in this case) is similar for virtually all flue pipes, and in a general sense it confirms the theoretical results of Fletcher [9] and other workers.  Below the linear region pipes cease to speak at all whereas above it their behaviour becomes unstable and less predictable, the pipe often flying suddenly to the octave (second harmonic) for an open pipe and to the twelfth (third harmonic) for a stopped one. 

 

Obviously, the pressure variations induced by a tremulant must be constrained to lie within the linear region of the curve if the pipes are not to be thrown off speech, and then the frequency of the fundamental (the first harmonic) is related to instantaneous pressure by a straight-line equation of the form:

 

f = kp + c

 

where f is frequency, p is pressure and k and c are constants.  For the pipe used for the experiment the values of k and c were obtained from measurements on the graph to give the empirical equation below,  f  being in Hz and p in mm wg:

 

f = 0.0829p + 170.2   (for p in the range 50 <  p < 110 mm wg)

 

This form of the equation is useful as an add-on to a wind system model in any type of digital simulation but particularly in additive synthesis and physical modelling.  However the numerical values of the constants will of course vary from pipe to pipe, but one can also dream up other values to try out various “invented” effects.

 

Because of the linear relationship between frequency and pressure, it might be supposed that the shape of the modulation waveform describing frequency deviations away from the static (untremulated) frequency will be the same in detail as that of the pressure waveform describing pressure deviations away from the static pressure.  If this was so then the typical pressure waveshapes shown earlier in Figure 3 will also apply to the corresponding frequency waveshapes.  In practice the pipe frequencies will be additionally perturbed by random or chaotic effects such as turbulence superimposed on the tremulant pressure waves, and these are discussed later in relation to amplitude modulations.

 

As well as the fundamental frequency we also need to look at the frequency deviations of the various harmonics making up the sound of a given pipe.

 

The frequency fn of the nth harmonic (n starts at 1 for the fundamental) is given by:

 

fn = nf = n(kp + c)

 

This shows that the constant of proportionality, k, relating frequency to pressure for the fundamental becomes nk for any harmonic above the fundamental because it is multiplied by the order of the harmonic n.  In other words, the higher the harmonic, the greater its tremulated frequency swing compared to the fundamental in order that the necessary relationships between the harmonics are preserved.  Thus if the fundamental is shifted by 5 Hz, say, at any instant during the tremolo cycle, then the third harmonic will be shifted by 15 Hz at the same instant and so on for all other harmonics.  This has implications for the depth of tremulant which needs to be used for different types of flue pipes in order for the effects to be subjectively satisfactory.  For instance, a theatre organ tibia has very limited harmonic development [10], and to evoke a perception of deep, throbbing vibrato in such a stop it is necessary to “swing” its fundamental frequency by a relatively large amount.  However if such a deep tremulant was applied to stops with many more harmonics such as strings, their upper harmonics (and amplitudes) would wobble around to such an extent that the effect would be ludicrous, assuming the pipes continued to speak at all.  The same would apply to delicate colour reeds such as the vox humana.  This is one reason why the tibias ideally need their own tremulant in a theatre organ and therefore why these instruments often have several tremulant tabs (another reason is the almost insatiable wind demand of a tibia rank).

 

Amplitude variations

So far, the foregoing discussion has related only to the frequency modulation of a pipe sound induced by a tremulant, but much the same analysis can be applied to its amplitude modulation as well.  An example of the amplitude envelope of a tremulated flute stop on the new (2009) Van den Heuvel organ in Danish Radio’s concert hall is shown in Figure 5.  This is a plot of the waveform of a single pipe over a duration of about 1300 msec as recorded by a microphone, but compressed in time so that only the envelope remains visible.

 

 Figure 5.  Amplitude modulation induced by a tremulant

 

For this particular pipe the effect of the tremulant was so strong that it almost ceased to speak at the amplitude minima, though some others in the same rank were less affected.  This was partly due to the use of divided soundboards in this organ in which the pressure in the treble is higher than in the bass.  The variations from pipe to pipe mean that individual tremulant characteristics have to be applied carefully to each simulated pipe in a digital organ if the overall effect is to be realistic.  Other important observations are the different envelope shapes from one tremulant cycle to the next, and the ragged appearance of the entire envelope.  At first sight these are surprising in view of the smooth and relatively uniform wind pulses usually generated by the tremulant itself, previously shown in Figure 3.

 

In fact both beat-to-beat variability and envelope raggedness are largely the result of turbulence set up in the wind system downstream of the tremulant and in the pipes themselves.  Even in a situation where the untremulated air flow remains laminar (non-turbulent) as far as the pipe mouth, turbulence can be triggered by the slightest perturbations anywhere in the winding system because its onset is highly sensitive to any variation in the aerodynamic conditions.  This is an example of the “butterfly effect” beloved of climate change pundits [11].  For digital organs the most important message to take away is that tremulant-induced turbulence usually exists and it is important, but it is impossible to model accurately using the rudimentary real time software and hardware of commercial music synthesisers.  Consequently this has obvious implications for the fidelity with which tremulants can be simulated and we shall return to this theme later.

 

Timbral variations

Often the amplitude of the fundamental (first harmonic) of a pipe is approximately linearly related to pressure, just as its frequency is, over the range of pressure due to the tremulant.  However this is not always true for the higher harmonics, whose amplitudes can vary non-linearly with pressure.  Even if they vary linearly, the higher harmonics may nevertheless vary more or less strongly than the fundamental with pressure. All these effects, if present, can result in a harmonic structure which varies over the tremulant cycle, thus there will be a corresponding timbral (tone colour) variation.  However whether a given timbral variation can be perceived as a real effect is another matter and it is therefore impossible to generalise further.  As well as listening to them, it is necessary to analyse in detail the tremulated waveforms of all the stops of interest to reach conclusions in this area.

 

 

Digital Simulation of Tremulants

 

In this section it is assumed that the reader is familiar with the basics of the three digital simulation techniques of sampled sound, additive synthesis and physical modelling as applied to pipe organ sounds.  Otherwise some background information can be obtained from two articles elsewhere on this site dealing respectively with the first two techniques [7] and the third [8].

 

Sampled sound synthesisers

To simulate a tremulant effectively in any type of synthesiser it is necessary to reproduce realistically the frequency and amplitude modulations it imposes on the pipe waveforms.  At first sight one might think that the simplest and most obvious way to do this for sound samplers is to have two sample sets available, one recorded with and the other without tremulant.  The main advantage is the significant one offered only by sampled sound synthesis as opposed to the other two methods, which is that the effect of the tremulant can be captured exactly on a pipe-by-pipe basis if desired, provided that the necessary number of samples is available and that the synthesiser can accommodate them all.  In this case the richness not only of the pipe sounds themselves but the tremulant as well can be captured.

 

However there are also several practical problems in this approach.  Firstly the considerable time, expense and effort involved in capturing the raw recordings and massaging them into a suitable form is doubled.  De-noising and looping tremulated waveforms can be particularly problematical.  Secondly the memory requirement is also approximately doubled, which can be a problem with a large sample set.  Thirdly there is an inherent issue of synchronisation when several notes are played on a tremulated stop – the tremulant modulation waveforms for each are unlikely to be in phase as they always will be in a pipe organ.  Fourthly the speed and depth of the simulated tremulant cannot be varied to taste – one is stuck with the tremulant enshrined once and for all within the simulation and one can be certain that not everybody will like it.

 

The synchronisation problem is perhaps the most important and it is difficult to solve, though not impossibly difficult if the control software is smart enough.  But if random tremulant phases across all notes currently keyed are unacceptable then other approaches have to be considered, though all of them apply an artificially derived tremulant envelope to the untremulated samples in various ways to be described presently.  Therefore they detract from the prime advantage of sampled sound which can otherwise reproduce more or less exactly the waveforms which were originally put into it.  The use of synthetic (modelled) modulation waveforms in a sound sampler takes them closer to synthesisers using additive synthesis or physical modelling, which some might see as a retrograde step.

 

All is not gloom and doom however.  The simplest modulation envelopes can sometimes produce very acceptable results, one example being the simulation of a Bédos-type tremulant.  Virtually all samplers have one or more LFO’s (low frequency oscillators) which produce sinusoids at the desired tremulant frequency, and they can usually be programmed so that the frequency and amplitudes of the samples are modulated simultaneously.  For slow, gentle tremulants this can give a delicious result that many pipe organ builders and organists would give their eye teeth for, whether they admit it or not.  Simulating the “live” behaviour of a Bédos tremulant more exactly would merely require the addition of a relatively simple mathematical model in which the speed and depth of the AM and FM depended on how many stops were drawn and the number of notes keyed.  The parameters of the model would of course be derived from measurements made on a real tremulant.

 

One can go a step further with more sophisticated samplers which allow the modulation waveforms to be derived from stored samples rather than being limited to sinusoids.  In this case the behaviour of virtually any tremulant can be simulated with the exception of timbral shifts during the tremulant cycle.  However even these can be approximated if sufficiently flexible filters are available in the synthesiser.  For example, a filter of suitable characteristics in series with the signal path will result in dynamic frequency shading being applied to the harmonic structure as the LFO modulates the frequency of the signal.  Another means of approximation is to derive the stored modulation waveforms from differences between the tremulated and untremulated samples.  These would have been derived offline beforehand according to some preferred algorithm.  But, as stated already, one must not forget that all these methods are only approximations to reality.

 

Additive synthesis

At some point in additive synthesis sound engines, waveforms have to be generated from tables of harmonics before they are sent to the loudspeakers.  This is because it is the waveforms that we actually listen to, not the lists of numbers.  For instance, in the original Bradford system the waveforms for a given stop are synthesised additively into a temporary memory when that stop is drawn.  They are then read out according to which notes are subsequently keyed.  Therefore at this point artificially derived tremulant modulation envelopes can be imposed on the waveforms in much the same way as for sampled sound synthesisers, at least in principle, therefore much of the discussion just given will apply here.  But, also as before, this is only an approximation to the real thing - it is impossible in additive synthesis to do better by exactly copying a complex tremulated waveform from a pipe in the way that a sampler can. 

 

More sophisticated additive synths allow arbitrary modulation envelopes in amplitude and frequency to be applied independently to each partial of the sound throughout its duration, and in this case any desired tremulant characteristic is sometimes claimed to be possible.  However in practice the limitations of additive synthesis become most apparent with time-varying waveforms of which tremulated waveforms are an example.  The limitations are of two kinds – one is a practical one due to finite computing power which restricts the amount of real time number crunching available.  The other is a limitation in theory because it is impossible to accurately model other than the simplest modulations of a waveform using only a finite number of harmonics or partials, and even then the result can only be approximate.  Only unmodulated periodic waveforms, those which repeat identically from one cycle to the next, can be synthesised accurately using a finite Fourier series of harmonics.  These are called deterministic functions because their exact values at any time can be calculated mathematically, but pipe waveforms are of course much more complicated.

 

To do the additive synthesis job properly requires the ability to simulate stochastic  waveforms as well.  Stochastic waves in this context are deterministic ones which have been perturbed by random or chaotic (in the sense of chaos theory) processes such as noise or turbulence, and synthesis then entails the use of Fourier integrals rather than Fourier series.  Such a synthesiser has yet to be made because it goes well beyond what today’s technology can offer.  Fourier integrals invoke an infinite number of sine waves spaced infinitesimally in frequency, rather than the limited number of widely spaced partials used in current additive synthesis engines.  This is why single-shot rapidly changing waveforms which do not periodically repeat, such as attack and release transients, are not good candidates for the approximations to  additive synthesis currently implemented, and the same constraints apply to some extent with all real-world stochastic waveforms including tremulated ones.  For the same reason wind noise, which consists of an infinite number of frequency components, cannot be synthesised additively either.

 

These limitations become more severe the more complex the tremulant envelope.  Additive synthesis will usually cope reasonably well with gentle Bédos-type tremulants which only require shallow and near-sinusoidal modulation envelopes.  But it can only approximate to the effect of fast and deep tremulants such as those applied to theatre organ tibias, especially if dynamic timbral variations are involved.  Thus whether the approximation is satisfactory will largely depend on the fussiness of the user.  A glance back at the ragged modulation envelope of Figure 5 which shows a real tremulant in operation is a good test for any synthesiser.  This could be reproduced more or less exactly by a sound sampler, but it would pose insurmountable difficulties for additive synthesis because turbulence and other random or chaotic effects are  impossible to model.

 

Physical modelling

Before diving into the physical modelling of tremulants let us backtrack for a moment.  Sampled sound synthesisers care nothing about acoustical physics and mathematics because all they do is to accept a captured pipe waveform, warts and all, and reproduce it on demand, warts and all.  The original sound waveform is everything to them, their alpha and omega, the thing they operate on.  Additive synthesis likewise starts out with some captured pipe waveforms but they are first processed mathematically offline to determine their harmonic structure.  The harmonic strengths are then put into the additive synthesis engine which uses essentially the same maths in reverse to reproduce the sounds – but only approximately.  Some of the important “warts and all” which we all know and love in the sounds of organ pipes cannot be reproduced by additive synthesis for the reasons outlined earlier.

 

Physical modelling goes a big step further in the same direction.  It no longer cares anything at all about sounds as such in terms of their waveforms or harmonics, only about the physics and mathematics of organ pipe theory.  A cloth-eared physicist who couldn’t care less about the organ and who couldn’t loop a waveform for toffee could nevertheless model the acoustic processes going on in an organ pipe, if only because it has already been described in the open literature by many authors (see, for example, [12]).  S/he could then program a computer to emit that pipe waveform on demand when the model runs – but again only approximately.  So sound samplers and physical modelling synths are at opposite poles, with additive synthesis lying somewhere in between.

 

This does not mean that physical modelling is no good, as it has enabled sound synthesis for orchestral and other instruments to make enormous strides over the 15 years or so since it was introduced.  However its applicability to the organ is limited for the reasons discussed in detail in [8].  It is therefore not surprising that some of the theoretical problems of simulating tremulants already outlined for additive synthesis will surface again in physical modelling.  These problems centre on the impossibility of modelling stochastic  processes realistically (those with an important random or chaotic element), such as turbulence, in a cheap and cheerful bit of kit sitting inside an organ console which has to work in real time.  Those who disagree might reflect that even with the roomfuls of supercomputers used by meteorologists which grind away for hours on end, the reliability of weather forecasts leaves a lot to be desired.  If we can’t do weather forecasting accurately then we can’t do the wind modelling accurately which is so necessary for simulating tremulants.

 

In spite of this it is nevertheless necessary to incorporate a wind system model of some sort in a physical modelling synth, part of which must include models of the tremulant or tremulants offered by the system.  This is because the physical models of the pipes themselves cannot run until their wind supply, including the variations on it imposed by the tremulant, can be represented mathematically within a computer program.  This is no different than saying that an organ pipe can’t speak until you admit wind to it.  However the models can only be approximate, involving various heuristics and short cuts so that many copies of them (one for each simulated pipe speaking at a given instant) will be able to run simultaneously in real time on relatively small computers.  As with additive synthesis, physical modelling can give a reasonable account of itself with simple tremulants of the Bédos type, including the way they react to the number of stops drawn and the number of notes played.  But when one looks at the problem of simulating more complex tremulants with amplitude envelopes such as those depicted in Figure 5, it is a different matter.  The need to model turbulence accurately defeats physical modelling, so only approximations to real world tremulants can be incorporated in physical modelling synthesisers.

 

 

Concluding Remarks

 

A pipe organ tremulant generates regular and fairly smooth pulsations in the wind supply of an organ whose waveshapes do not differ much from one pulse to the next.  However the way the pulses affect the pipe sounds is a different story because the continual pressure changes can trigger turbulence downstream of the tremulant or at the pipes themselves.  The result is that the pressure changes are transformed into complex and unpredictable frequency and amplitude modulations impressed on the pipe sounds.  These effects become more pronounced the greater the modulation depth, and the effects differ from one beat of the tremulant to the next and for every pipe.  This feature endows a well-adjusted pipe organ tremulant with a richness comparable to that of the pipe sounds themselves, a fact particularly well known and important to those whose first love is the theatre organ.

 

Simulating all this at a detailed level is only possible for sound samplers which can capture the actual sounds of tremulated pipes, because turbulence and other random or chaotic effects are impossible to model realistically.  Therefore other types of synthesiser using any form of modelling, and this includes additive synthesis as well as physical modelling, can only approximate the effects of tremulants.  However even with samplers there are some practical difficulties which degrade the effectiveness with which the captured sounds can be reproduced.

 

The upshot is that shallow, gentle tremulants can be simulated quite well by any form of synthesis.  However the more complex effects of fast and deep tremulants can only be captured by sampling the sounds directly, notwithstanding the difficulties of reproducing them.  Moreover, because the effect of a tremulant differs for every pipe it is not surprising that simulating a theatre organ satisfactorily is especially difficult as far as its tremulants are concerned, regardless of the type of synthesis used.

 

Thus tremulants reveal the limitations of current synthesis techniques quite starkly,  thereby providing yet another case study showing that digital organs can only approximate to real pipe sound.  In the last analysis only individual users can decide for themselves which of all the imperfect options offends them least. 

 

 

Notes and References

 

1.  “Swell Control in Electronic Organs”, C E Pykett, 2006.  Currently on this website (read). 

 

2.  “L’Art du facteur d’orgues” (The Art of Organ Building), Dom François Bédos de Celles, Paris, 1766.

 

3.  “Fairground Organs”, Stephen Bicknell.

     See: http://www.stephenbicknell.org/3.6.11.php (accessed 24 December 2009).

 

4.  “The Physics of Tremolo”, Dennis Hedberg, Theatre Organ, Nov/Dec 1987.  (Theatre Organ is the journal of the       American Theatre Organ Society).

 

5.  See:  http://www.toff.org.uk/CHAMBERS/tremulant.html (accessed 24 December 2009; broken 28 September 2016).

 

6.  For those able to visit it, a tremulated jazz flute can be seen and heard on one of the dance organs in the St Albans Organ Museum in the UK.

     See: http://www.stalbansorgantheatre.org.uk/site/frame.html (accessed 24 December 2009).

 

7.  “Voicing Electronic Organs”,  C E Pykett, 2003.  Currently on this website (read).

 

8.  “Physical Modelling in Digital Organs”, C E Pykett, 2009.  Currently on this website (read).

 

9.  “Sound Production by Organ Flue Pipes”, N H Fletcher, J Acoust Soc Am, (60), 1976.

 

10.  “The Tonal Structure of Organ Flutes”, C E Pykett, 2003.  Currently on this website (read).

 

11.  The term “butterfly effect” was coined by US meteorologist Edward Lorenz in the 1950’s to illustrate that an infinitesimal change in initial conditions in aerodynamic systems can produce wildly different end results.  His butterfly flapping its wings on one side of the world has been used ever since by the media and assorted pundits to “explain” why hurricanes arise on the other.

 

12.  “The Physics of Musical Instruments”, N H Fletcher and T D Rossing, Springer, 1998.