The Physics of Organ Actions
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The Physics of Organ Actions

by Colin Pykett

Posted: January 2003

Last revised: 9 July 2016

© Copyright C E Pykett 2003-2016

 Abstract This is a review paper which draws together work published in the public domain on the design and performance of mechanical and electric actions for pipe organs.  The subject is approached by considering the fundamental physical principles which govern the performance of such actions.  In the case of mechanical actions the subject of repetition rate is discussed in some detail in view of the paucity of the literature on this aspect. Other matters include pluck and hence pallet design.  Among many other aspects, the apparently widely-held view that the key always dominates the inertia of an action because it is the most massive component is shown to be flawed.  This is most eloquently demonstrated by examining the behaviour of a suspended tracker action in which the keys are usually long and massive.  In the case of electric actions another widely-held view, that direct electric actions are invariably slower than electro-pneumatic ones, is also shown to be unsupported by experimental data.

 

CONTENTS

(click on the titles in this list to access the desired section of the paper)

FOREWORD

PART 1 - MECHANICAL ACTIONS

DESIGN REQUIREMENTS

Fore-touch Weight

After-touch Weight

Repetition Rate

Release Time

MATHEMATICAL MODELLING

BACKFALL ACTIONS

Maximum allowable tracker length

The effect of key mass

The effect of pallet & backfall masses

The effect of tracker mass

The effect of changes in repetition rate

SUMMARY - Backfall Actions

SUSPENDED ACTIONS

Maximum allowable tracker length

The effect of key mass

The effect of changes in repetition rate

SUMMARY - Suspended Actions

PART 2 - ELECTRIC ACTIONS

REPETITION RATE IN ELECTRIC ACTIONS

Parameters of example actions

CONCLUSIONS - Direct electric actions

CONCLUSIONS - Electro-pneumatic actions

QUANTITATIVE OUTCOMES

PERCEIVED SLUGGISHNESS of direct electric actions

REFERENCES

 

 

Foreword

"One feels like a boy who has been long strumming on the silent keyboard of a deserted organ into the chest of which an unseen power begins to blow a vivifying breath.  Astonished, he now finds that the touch of a finger elicits a responsive note, and he hesitates, half-delighted, half-affrighted, lest he be deafened by the chords which it seems he can now summon almost at his will"

So said Sir Oliver Lodge in a lecture to the Royal Institution in the 1890's when he had been speaking about the astonishing tide of technical and scientific progress that was engulfing the nation at the close of the Victorian age.  What is interesting today is that he found the organ to be the apposite metaphor to emphasise his message, interesting in that it could be mentioned in the same breath as that progress.  Also it is interesting to reflect on how far things have moved since then.  We are still engulfed by change, if not progress, but it is doubtful whether the organ would be used now to underscore a similar lecture on computers or space travel, say.

Yet despite its splendours the Victorian organ concealed some murky secrets.  It is interesting to speculate how many organ builders knew how to design an efficient pallet valve for example.  Interesting, because such knowledge was widespread elsewhere in steam, hydraulic and gas engine technology. But organ builders had been released at a stroke from the necessity to pursue such tiresome details by the advent of pneumatic and electric actions.  It has been left to us to start again when we want to make an organ with a mechanical action today, and the history of organ building since the 1950's or so is therefore not without its failures and blind alleys.  Because we have not yet fully emerged from these, this article was written.

Thus an understanding of how pipe organ actions work in terms of the physical principles involved clearly benefits an organ builder.  Just as important, it also assists his clients, who are thereby enabled to write more rigorous specifications for the work to be done and how it is to be tested on completion.  This article draws together work done by some other authors and myself  in recent years on both mechanical and electric actions in the form of a detailed technical review.

 

PART 1 - Mechanical Actions

A wholly mechanical linkage between key and pallet, without electric or pneumatic assisters, is regarded as ideal by many players, advisers and builders, though there are also many who disagree.  (The use of assisted mechanical actions has been considered elsewhere.  See for example [1], [8] ).  This article will deliberately avoid the controversies in this area by merely pointing out that there are some cases where a purely mechanical action can work well, and others where it is doomed to fail before it leaves the drawing board.  By addressing the physical design principles the two cases will become easier to identify at an early stage of design.  Armed with such foreknowledge, responsibility for the outcome for good or ill will then lie squarely with those who insist that a mechanical action be used.

We shall disregard subjective matters such as querying rhetorically whether a pianist would be content with a non-mechanical action for his or her instrument, whereas many organists seem not to mind. Objectively, however, there can be no doubt that a purely  mechanical action does give the organist control over some elements of pipe speech which is degraded or impossible with most other types [2].  The paper referred to illustrated how the starting and termination transients of pipe speech can be modulated by the rate at which the pallet opens and closes.  This is no mere academic matter in view of the perceptual importance which the ear assigns to such artefacts in the sounds.  (The musical importance attributed to such artefacts clearly varies from one individual to another, however, and it is a factor in the debates which still continue).

 

Design Requirements for a Mechanical Action

There are many factors which could be used to specify the requirements for a mechanical action, and broadly speaking they can be divided into two groups.  One group is involved mainly with the fundamental physics of the action, and the other mainly with issues of engineering.  If we do not get the physics right to start with, the action will never work as desired regardless of how good the engineering might be.  On the other hand, provided the design is sound and realistic in a physical sense it will often be possible to get close to the design requirements through the use of impeccable engineering.  Thus in the first group we might wish to specify the maximum allowable pluck that the player will experience at a given key, whereas in the second we might include factors such as the maximum amount of lost motion which will be tolerated  at the same key before the pallet begins to open.  Partly to reduce the complexity of this paper we shall not consider factors in the second group, nor shall we include all possible factors in the first.  The design factors to be considered will be limited to:

1.      Fore-touch weight measured at the key.  This is the initial force required at the key to induce the pallet to open.  It includes pluck plus the total force exerted by all springs in the action.  It does not include frictional force as this is an "engineering" parameter.  It needs to be made clear in the specification for an organ whether fore-touch may be allowed to vary across the compass, and if so, how.

2.      After-touch weight measured at the key.  This force is that required to just keep a key fully depressed.  It will usually result from the total force exerted by all the springs in the action.  Again, the variation of after-touch across the compass needs to be addressed in the design specification if necessary.

3.      Release time measured at the key.  This is the time taken for the key to return to its rest position, and thus the time taken for the pallet to close, when the finger has been lifted rapidly.  It is an important parameter governing the repetition rate of the action.  In this paper we shall infer release time from a specification of repetition rate because the latter is easier to measure, and players have a better understanding of it.  As with the former factors, repetition rate and thus release time  must usually be specified as a function of position in the compass. 

Attack time is excluded even though it also is an important contributor to repetition rate, because it can be varied within wide limits by the player in a mechanical action.  In other words, attack time is not purely a function of the action itself.  (This is not true of most electric actions, where the attack time is determined by the action rather than the player and therefore it has to be included as an additional factor in the design process).

These three factors will be considered in turn.

Fore-touch Weight

The initial force required to cause a key to move downwards is a matter of first importance to an organist.  It is certainly the most discussed, sometimes to the detriment of other factors.  The difference between fore-touch and after-touch is largely due to pluck, that force acting on the pallet as a consequence of the wind pressure trying to keep it closed.  In an organ not in wind, therefore, the fore-touch and after-touch weights are about the same.  Most players seem to like a certain amount of pluck, to the extent it was lauded in charge-pneumatic actions (rare today) whose touch boxes also resulted in pluck.  When the Westminster Cathedral organ was rebuilt its touch boxes were retained so that its attractive playing experience was not lost.  A modicum of pluck certainly assists a clean playing technique, and it is often simulated in electric actions and even electronic organs.  So how much is enough?  John Norman has suggested a value of 60 gm, about 2 ounces, at the key excluding the spring force [9].  (At some risk of pedantry, we need to bear in mind the difference between mass and force or weight.  Mass expresses a quantity of matter, which only becomes a weight or a force in the earth's gravitational field.  Touch weight is a force, not a mass.  There is a multiplier of 981 between the two units if mass is expressed in gm - quite a difference.  Bridges have fallen down when this has been forgotten!  In this article the difference is vital when we start to explore mathematical models of organ actions shortly).

Too much pluck, however, is a decided hindrance and in extreme cases it causes fatigue and can even damage the muscles.  The plethora of quotations from sorely-tried players of yesteryear such as Dr Camidge (re York Minster) and Mendelssohn (re Birmingham Town Hall) are so well known they will not be repeated here.  This bad press from the nineteenth century was a material factor in hastening the arrival of novel and exciting pneumatic and electric actions, with a corresponding lack of effort devoted to what was really wrong with the mechanical actions of the day.  This does not mean that nothing at all was done to mitigate their shortcomings, but it tended to be along the lines of "how can we compensate for pluck using yet more complication" rather than "how can we design an action so it has less pluck to start with".  The former approach, which can be termed unintelligent engineering, resulted in a huge variety of devices and patents, some of which could not possibly work even in theory.  The latter, that of starting from first principles and guided by physics, is still not fully mature today despite the rediscovered penchant for trackers.  Therein lies a justification for this article..

The most important matter in designing for minimum pluck is to design the pallets or valves properly.  They should be no larger than necessary to admit sufficient air to the pipes when all stops are drawn, and they must be the correct shape.  Valves which have the same aperture as far as air admission is concerned (the windway) can vary markedly in the amount of pluck they impart to the touch.  Some intriguing results, which will be presented without mathematical proof here, demonstrate this:

Example 1 .  Consider a conventional long, thin pallet hinged at one end.  The ratio of windway to pluck is given by Q = 2d/wp approximately, where d is the distance the pallet descends; w is the width of the aperture it covers; and p is wind pressure.  An efficient valve clearly must  have as high a value of Q as possible, thus we can call Q the quality factor of the pallet.  

Note the curious fact that the length of the aperture does not arise here, therefore Q is dominated by aperture width rather than length.  It follows that, in the limit where a pallet aperture is infinitely thin (w tends to zero), pluck vanishes but the windway remains!  Obviously there are limits beyond which simple theory such as this breaks down, but it nevertheless demonstrates principles which are well known in aerodynamics.  To summarise, the conventional centuries-old pallet can make an efficient valve if designed properly, and that's the rub.

Example 2 .  Now consider another type of widely-used organ valve, the disc or cone valve.  It can be shown that Q = 4d/wp in this case, which on the face of it suggests that a disc is twice as efficient as a pallet (w is the diameter of the aperture covered by the disc).  However, for the windway of a disc valve to be the same as that of a pallet, we find that its diameter needs to be much greater than the width of a pallet.  It may also need to descend further.  Neither of these assists in the layout or engineering of a soundboard, so we find that the pallet scores again.

As far as I am aware, no analyses of this nature were done by Victorian organ builders, despite the fact they only involve simple algebra and use principles well known in other areas of contemporary engineering (in particular steam engines).  If they were done, they do not seem to have been published, nor do they seem to have found their way into organ building at large.  This statement does not imply contempt for the organ builders of that era, who were as innovative and competent as any other engineer-artists of their time.  My guess is they were simply swept along by the hydraulic, pneumatic and electric novelties being applied in other fields, and saw these as the way ahead for organ building rather than trying to improve a mechanical technology which would have seemed dated.  For that reason it probably would not have sold either, no matter how good it was.

Today times have changed and we strive to make some enormous organs with mechanical actions, some of which still fail just as they did 150 or more years ago.  There is still relatively little in the public domain to guide such endeavours in the key area of valve design, as well as other issues.  A paper by John Norman written in 1978 [6] is one of the few exceptions.  For this reason I undertook a detailed study of pallet design from first principles, and the results are available elsewhere on this website [7].  Using guidelines such as these it should be possible to design actions in which the fore-touch weight is acceptable, or (and this is equally important) to determine at an early stage of design that it will not be possible.  A further study reviewed some methods for pluck compensation in the event the latter conclusion is reached [8].  These were also the subject of  Alan Woolley's study already mentioned [1].

One of the simplest yet most important outcomes of this work is that pallets must not be wider than they need to be.  There is a danger this will happen if they are simply made to cover the gaps between the soundboard bars, particularly in the bass.  The bar spacings are determined by factors other than wind delivery, such as the geometries of pipe plantation.  An over-wide pallet will only deliver marginally more wind while magnifying pluck unnecessarily.  It is curious that this did not seem to be picked up by most (if not all) Victorian organ builders, though perhaps one reason was that it did not much matter once they had transferred to the use of pneumatic or electric actions.  A way to overcome the problem is to design the pallets and the soundboard as separate exercises.  Then the pallets can be made to close on a "pallet table" attached to the underside of the bars if necessary.  The correct aperture widths can be cut in this table.  Although not an original suggestion, it is obvious that some organs are still built today without the benefit of this approach.

After-touch Weight

Little which is not obvious needs to be said about this parameter.  One or more springs are required to close the pallet, and their forces are transferred to the key.  In the absence of the springs the action may have a net unbalance which causes the key to rise or fall.  Such forces will add to or subtract from the spring force measured at the key.  If there is a mechanical advantage different to unity between key and pallet, i.e. if the key dip and pallet descent at the pull-down are different, then the pallet spring force will need to be multiplied by this factor to arrive at its value at the key.

The importance of after-touch weight as far as this article is concerned is that it is the force which causes the action to accelerate when the finger leaves the key.  It therefore controls the release time, and hence the repetition time, of the action.

A typical value for after-touch at the key might be about 75 gm.

Repetition Rate

This is by far the most complex of the three parameters under discussion, and probably the least understood.  First we need to consider what might seem to be a statement of the blindingly obvious: repetition rate is not the same as the rate at which music can be played.  Unfortunately I have found that many people seem to confuse the two, even some who are professional musicians, and eventually the discussions get hopelessly confused until the difference is resolved.  So here goes.  Repetition rate is defined here as the maximum rate at which a single key can be manipulated while still allowing the valve(s) it controls to open and close fully.  We shall specify it in terms of the corresponding number of key strokes per second.  The speed at which music can be played is quite different, because most of the time the successive notes in music are different rather than being the same.  Often, the rate in terms of notes per second at which a good player can play a scale, say, is significantly faster than the rate at which he/she can play a single note repeatedly.  Taking an extreme example, a glissando may well execute impeccably whereas trying to persuade the action of a single note to reiterate at the same rate will be impossible (even assuming the player's technique is up to it).

So what repetition rate do we need?  Even when the uncertainty just mentioned has been resolved, considerable controversy remains.  This might be because repetition rate seldom arises when discussing actions, at least in a precise and quantitative manner, whereas other parameters such as pluck are regularly debated and therefore better understood.  Not all organ builders seem prepared to specify the repetition performance of their actions, and at the same time not all musicians seem to have a good idea of what their repertoire demands from an action.  This stands in sorry contrast to the piano, where extreme ingenuity has been devoted almost from the first to improving the repetition of the action based on the demands of composers and performers, particularly as the 19th century advanced.  It is therefore all the more extraordinary that attempts merely to specify a repetition rate for the organ, let alone attempts to achieve the specified figure, get bogged down almost before they begin.  As soon as a figure for repetition rate is mentioned it usually attracts instant disagreement, thereby preventing the discussion advancing.

I attempted to derive a figure for a reasonable minimum repetition rate in another article currently on this website [3].  (This article considered electric actions, although the arguments for repetition were based only on musical considerations.  Therefore they can also be used here where we are discussing mechanical actions).  The required repetition rate, a minimum one remember, turned out to be about 8 repetitions per second.  Because the issue is so important the relevant parts of the other article are reproduced here for convenience and completeness:

" ...  we need to establish what response speed is needed in an action. In other words what time delay is acceptable between pressing a key and having the pallet open fully? The converse delay between key release and closure of the pallet is of equal importance. We can only look to music to supply these answers, because there would be little point trying to make an action work more quickly than it needs to in rendering the repertory of the organ. On the other hand it is futile attempting to justify the shortcomings of an action that cannot respond quickly enough.

This is not the place to enter into a detailed musicological discussion, but nor is it necessary. Merely by taking one example we can show how to extend the logic to others, and at the same time arrive at useful results. Consider the well known Toccata from Widor’s Fifth Organ Symphony. Most players will know the articulative difficulties posed by the detached semiquaver chord-pairs which recur throughout this piece, but not all will have mastered the muscular flexibility and dexterity needed to realise them as separate sound entities. Even fewer, perhaps, will realise that the difficulties are imposed by the instrument more often than might be supposed. For if the action is not responsive enough then even the lightest and most rapid touch will not suffice.

The 100 crotchets per minute metronome marking on the Hamelle edition of 1901 is pedestrian compared to the 120 or more at which many modern players take this piece. At this speed (crotchet = 120) there are 8 semiquavers per second. Each semiquaver must occupy less than 125 milliseconds (1/8th of a second) if it is to be separated from its neighbour ...... This is actually quite a tall order for an action, as one can see by trying to play 8 repetitions per second of the same note in the middle of the keyboard of a good piano (set the metronome to 120 and try to get four repetitions per beat out of the instrument). It is even more of a test for the organ which has none of the ingenuity found in the piano action for improving repetition. Therefore we shall take a repetition capability of at least 8 per second as our design requirement .... "

Even this modest attempt to come up with a minimum repetition rate seeded controversy! [4].  This is of course healthy, but it also illustrated the point made earlier concerning the difficulty of moving forward.  However, as part of the ensuing dialogue at that time, a number of eminent musicians were kind enough to write privately offering their views.  A consensus view seems to be that a repetition rate of about 10 per second (but not less) is considered sufficient, a figure which is not far away from mine derived so simply above.

It is vital, absolutely essential, for an organ builder not to agree to a repetition figure higher than needed, and this may not be generally appreciated.  We shall see later that a number of fundamental relationships describe the motion of an action, but for the present a useful design law can be stated in the following terms :

The release time of an action is approximately proportional to the square root of the masses in the action.

The importance of this can hardly be over-estimated because it means, for example,  that to halve the release time it is necessary to reduce at least some of the action masses by a factor of four.  Such a reduction would be scarcely practicable in most cases, even at the design stage and certainly not once an organ had been built.  Therefore an organ builder who decides to make his actions repeat at a rate of 20 per second, say, when the evidence from musicians quoted above suggests a reasonable figure would be only half this value is making a heap of trouble for himself.  But the story does not stop here, because a repetition rate reasonable in the middle and upper reaches of the keyboard will not usually be appropriate in the bass.  This is because the larger pipes speak more slowly.

As an extreme but far from unusual example of this, the lowest F of the 16 foot pedal Violone of the beautiful Rushworth & Dreaper organ at Malvern Priory took just over a second to come onto stable speech when I measured it in 1979.  This is not a criticism of the organ but an inescapable consequence of the way flue pipes work [5].  Obviously one would scarcely struggle to make the action for a pedal stop of this type repeat at 10 or even 8 per second - it would be completely pointless.  However manual doubles and many unison stops also speak slowly in the bass for the same reason.  In general, therefore, we need to be realistic and allow for some decrease in the specification for repetition rate lower in the compass, based partly on the speaking characteristics of the stops concerned. But even so, the action cannot be allowed to become too sluggish towards the bass otherwise the upperwork in this region of the keyboard will be unable to play its part (literally) in a sensible manner.

For the purposes of this article we shall take a minimum repetition rate of 10 per second as our design aim in the middle and upper regions of the manual compass, based on the foregoing.  Lower down it is best to examine the art of the possible before setting a rate which might turn out to be impractical.  This will be done later.  Apart from the speaking characteristics of the pipes, repetition in the bass is complicated by the fact that the pallets are larger and therefore heavier than higher in the keyboard, and moreover there may be more than one pallet per note.  Also tracker and roller runs in the bass might be longer and involve more convoluted paths.  In turn this might mean using more squares or other methods of enabling trackers to navigate corners.  All these factors point to an action mass for the bass notes which is usually larger than for the treble ones, and the law enunciated above means that the bass notes cannot repeat as fast in these circumstances.  In view of this it seems reasonable to use an average value for repetition across the keyboard as a whole as 8 per second, and this figure will be used in what follows in order that we can get some idea of how this will affect the design of an action.

Release Time

For any type of action, repetition rate (measured in notes - depressions of the same key - per second) is the factor which is immediately comprehended by a player.  However, for a mechanical action (not an electric one), we noted earlier that it is dominated by the release time.  This is because the reciprocal of repetition rate (seconds per note) is the sum of attack time and release time, and attack time is as much a function of the player as the action - the player can vary the force exerted on the keys within wide limits and hence the speed with which they descend.  Release time, on the other hand, is limited by the action alone in those cases where the player removes his finger from the key faster than the key itself can rise.  So how can we get from a specification of repetition rate to one for release time?

The simplest approach is to adopt an average value for attack time, and subtract this from the reciprocal of the desired repetition rate.  In terms of a simple equation,

trel = (1/Rrep ) - tatt         (1)

where trel is release time in seconds; Rrep is repetition rate in notes per second; and tatt is attack time in seconds.

Taking some practical values, the release time for a well made mechanical action in an organ of about 25 stops (no couplers in use) is typically about 50 msec. Attack time can vary widely of course but it rather depends on the fore-touch weight, of which pluck is a significant element.  Therefore the action has to wait until the finger muscles have developed sufficient force to overcome fore-touch in response to a command from the brain, at which point the key descends more or less rapidly once the pluck has been broken.  Under normal playing conditions we might reasonably adopt a value for tatt   equal to trel   in the absence of better data.  Putting these values into equation 1 gives a value for repetition rate for our small organ of about 10 notes per second.  This meets our design requirement for a rate of 10 per second in the middle of the keyboard and exceeds our average rate for the whole keyboard of 8 per second, which confirms that for small organs a mechanical action is well up to the job.

However, for the purpose of designing an action we have to use equation 1 in a different way.  By inserting values for average  repetition rate (8 per second), and an attack time equal to the release time, we find that a maximum allowable release time of 62.5 msec is required, regardless of how large or small the organ might be. We now have to design an action that will achieve this figure, and this means a mathematical model of the action is required.

Mathematical Modelling

Developing a mathematical model of an action begins simply but gets more complicated later on.  Therefore we shall only go so far from first principles in this article, after which we shall just have to accept some results as given.  It also relates mainly to release time and therefore repetition rate, because it requires some careful analysis to associate these parameters with the elements of an action.  The other design parameters, fore-touch and after-touch weights, scarcely demand a mathematical model as such provided pallet design has been undertaken carefully.  Once this has been done it is a simple matter to relate pluck and spring tensions to the playing forces at the key.

The starting point in modelling is to consider two equations from school days - the first is Newton's first law of motion:

v  = u  +  at            (2)

where v is the velocity of a body at a time t after it has been subjected to a constant acceleration a.  u is the initial velocity at t = 0.

Integrating this equation with respect to time and ignoring the arbitrary constant of integration gives:

s = ut + 0.5 at2        (3)

where s is the distance travelled in time t.

We can apply equation 3 to a key on the keyboard.  If s be the key dip, i.e. the distance the key rises when it is released, then t becomes the release time we have been speaking of above.  Since the key accelerates from rest, u =0.  Therefore equation 3 can be re-arranged to give:

a = 2s/trel2        (4)

where a is now the acceleration of the key as it rises.

The second equation to be dusted off from school is Newton's second law of motion, stating that force equals mass times acceleration.  Applied to an organ action, acceleration (a) is the same as that used above, viz. the acceleration of the key as it rises after release.  Force (F) is the restoring force of the springs in the action which causes the key to accelerate, measured at the key [10].  Mass (M) in this context is more tricky, but for now we shall just call it mass.  Thus:

F = Ma         (5)

Eliminating a in equations 4 and 5 then gives release time from the following relation:

trel2 = 2sM/F        (6)

This equation is the basis of the law discussed earlier which stated that the release time of an action is proportional to the square root of the masses in the action.  But there is a complication.  We now need to note that the mass M is not simply the total masses in the action added together; it is a mathematical function involving those masses.  Moreover, the exact form of the mathematical function is different for each and every type of action.  Therefore beyond this point the analysis gets too complex to be examined in detail here.  Not only is there a different form of mathematical model for each type of action, but the mathematics itself gets more involved as well.  As well as masses, forces and accelerations we have to consider moments of inertia, torques and angular accelerations for all components which are hinged or pivoted.  These include keys, backfalls, pallets, rollers, etc.  For present purposes it will be sufficient to note that the factor M in the above equations can be termed equivalent dynamic mass (EDM) to reflect the complexity.  EDM has the dimension of mass but it varies depending on how an action is constructed.  Nevertheless we can proceed by examining different types of action and simply taking the corresponding form of its EDM as a given.

We need to specify an action in minute detail before we can begin to examine the principles which govern its operation.  There are often several variants of mechanical action even within the same organ, because the geometrical layout of an organ dictates how far the pipes are from the keys and therefore how complicated the action has to be in terms of the length of the trackers, whether rollers should be used or whether splayed backfalls would suffice, how many corners have to be navigated by the action, whether a suspended form of action should be used, etc.  In fact this brings us immediately to an important conclusion - it is impossible to develop a single set of design rules for mechanical  actions in view of their diversity.  At a more detailed level it is therefore also impossible to develop a single mathematical model of an action which could be presented here.  This might disappoint some readers who might have been expecting to see such a model, which would enable them to predict parameters such as release time.  Because each organ is unique, the action to each key on each organ is also potentially unique.  Therefore any attempt to model it mathematically requires that the model be synthesised from a library of sub-models, each one describing a single element of the action such as a tracker, a square, a key, a roller, a pallet, etc.  The sub-models will be called up as required to describe a particular action, and even some of the sub-models can be quite complicated.  For example, the sub-model for a key needs to contain details of the dimensions of the key, the density of its constituent material(s), where the tracker or sticker which links the key to the rest of the action is situated, the key dip, the required fore- and after- weights of touch (if these are not specified elsewhere), the desired release time (again if not defined elsewhere) and exactly how the key is pivoted. Thus before the complete model can be run, it is necessary to provide it with a large quantity of data which together precisely define the action under consideration.  We shall therefore consider two types of action - a backfall action in which the key is balanced and pivoted near to its centre, and a suspended action in which the key is pivoted at the rear.  We shall only examine the repetition performance of these actions in detail, because other parameters of interest (in particular pluck) have been discussed elsewhere in the references mentioned already. 

Backfall Actions

The action to be considered first is a typical "backfall  action" as in Figure 1.

For simplicity it is assumed the mechanical advantage between key and pallet is unity, that is, they move by the same amount. This will often be the case in practice, or nearly so.  In cases where it is not, we would have to consider where in the action the leverage changed.  The change could be introduced at the key, the backfall, the roller or the pallet, or any combination of these.  Although it is straightforward to model these changes, it is less straightforward to draw general conclusions about the performance of the action in these circumstances.  Therefore we shall ignore the matter to simplify the discussion which follows. 

For a backfall action we find the Equivalent Dynamic Mass (EDM) to be made up of three components:

1. Those whose contribution to EDM is negligible. These include rollers, even if made of heavy material such as old-fashioned iron tubing. As a more relevant  example, a modern roller assembly made of aluminium tubing 8 mm in diameter with a 6 mm bore, a length of 1 metre and having roller arms 50mm long has an EDM of about 0.15 gm.  This compares with its actual mass of  about 32 gm.  The large difference demonstrates the importance of undertaking a careful analysis.  In simple terms, the difference reflects the everyday experience that it is much easier to rotate something which is balanced and supported on  low-friction bearings (e.g. a car wheel) than to lift the same object bodily off the ground.

Also the masses of stickers, provided they are short and lightweight, can likewise be ignored.  Otherwise their masses need to be added to those of the trackers (see step 3 below).

2. Those components which are pivoted or hinged such as backfalls, keys and pallets. These contribute to EDM as a larger fraction, typically one third, of their actual masses although the exact divisor depends on the way each component is pivoted.

3. Those whose actual masses are included in EDM.  These are usually the trackers.  The masses of all the trackers are simply summed when computing EDM.

Using equation 6 and inserting the proper value for EDM we can now examine the repetition behaviour of various actions.  The approach adopted will be to compute first of all the maximum length of tracker runs that will support our requirement for at least 8 repetitions per second, and then run the model repeatedly with some variations in the parameters (a parametric study).

The first example is based on typical Victorian organ building practice.  Many of these instruments were built with superb craftsmanship but to an uninspired, even clumsy, mechanical design.  Thus many of them lasted a long time and are still with us, but they continue to frustrate many an organist.

Some parameters typical of such instruments are:

After-touch weight. 75 gm measured at the key.

Key dip. 10 mm.

Keys. Balanced with the pin in the centre.  60 cm long.  Stock: 12 mm by 25 mm cross-section, unloaded. Pine construction.  Mass 90 gm.  

Stickers. These will be ignored.

Backfalls. 45 cm long of hardwood (density 0.7 gm/cc).  Mass 75 gm.  Centrally pivoted.

Rollers. Although of iron tube, the EDM per metre length is only 0.43 gm, thus negligible unless the rollers are very long.  Therefore these will be ignored.

Trackers. Hardwood (0.7 gm/cc).  Cross-section 10 mm by 3 mm.  Hence mass/metre = 21 gm/m.

Pallets. 325 mm long of pine (density 0.5 gm/cc).  Mass 50 gm. Descent 10 mm at the pull-down.

Some performance parameters of this action, and variants of it, will now be discussed.  

Maximum allowable tracker length .  Using values for a repetition rate of 8 notes/sec (trel = 62.5 msec); s = 10 mm; and F = 75 gm in equation 6, together with the EDM values computed via a mathematical model from the data above, we find that the maximum total tracker mass which can be tolerated is about 72 gm.  As the trackers have a mass of 21 gm/m, this means the total allowable tracker length is 3.4 m if we are to achieve a repetition of 8 notes per second.  This length does not give much leeway in designing any but the smallest instruments. Tracker lengths beyond this figure will result in a much-degraded repetition capability owing to the increased inertia of the action. It is obvious that the action specified above is a clumsy one in terms of some of the materials employed and their dimensions, although these were taken from actual Victorian organs.   Therefore it is instructive to undertake a sensitivity analysis by varying some of the data and running the model repeatedly, to see how the outcome changes. 

The effect of key mass .  Because the keys in this and many other actions are by far the heaviest components, this seems to have led to a view that the keys dominate the design and performance of an action.  Unfortunately it is difficult to do much about the problem in reality without using quite different construction techniques for the keys, but by using mathematical modelling we can assign any mass we like to them and observe the effect.  Let us assume that we can reduce the key mass by a factor of 3, to 30 gm.

The outcome is that the maximum allowable tracker length while still retaining an 8 note/sec repetition capability increases to about 4.4 m, an increase of one metre.  This is useful but hardly startling.  One might have expected that the trackers could increase in mass by the amount that the keys had reduced (60 gm).  Had this been the case they could have been extended in length by nearly 3 metres, i.e. an increase of nearly 90%.  However in view of the practical difficulty of reducing key mass, this result is rather academic anyway.  Nevertheless it does illustrate that the widely held view that key mass is a dominant factor is misguided, at least for backfall actions.

The effect of pallet and backfall masses .  Because the keys, backfalls and pallets all contribute to EDM in a similar way as a fraction of their actual masses, it follows that a similar result will be obtained if we vary any of them.  It is easier to vary the mass of a pallet, for example, than that of a key and this explains why much ingenuity has been applied in recent years to making the pallets as light as possible.  In the limit let assume we have made a pallet with zero mass, thus we can delete its mass of 50 gm as used in the model thus far.  The result is that the maximum allowable tracker length increases to 4.2 m, less than a metre more than the original value when the pallet weighed a (reasonable) 50 gm.  Similar results apply when varying the backfall mass.  Therefore the conclusion is that none of the hinged or pivoted components in this particular action have a strong effect on repetition behaviour.

The effect of tracker mass .  In modern organ building practice it would be unwise for the trackers to be made of dense material, or if they are, efforts should be made to reduce their cross-section as far as possible.  Having said this, rectangular trackers of fibreglass with a cross-sectional area about 2/3 of that specified above are widely used.   2 mm diameter phosphor-bronze wire is even heavier for the same length than the Victorian hardwood trackers mentioned above, and aluminium of the same diameter approaches 9 gm/metre.  Such material and dimensions might need to be used with caution when the runs are long.  The following might assist in understanding why.

Because of the figures quoted above some recent instruments use low-density wood trackers (density = 0.5 gm/cc), which are in the form of thin ribbons with a cross-section around 1.5 mm by 8 mm and therefore a mass of about 6 gm/m.  This is less than 30% of the tracker mass per unit length used in the discussion so far.  In this circumstance the model predicts that the maximum allowable tracker length using this material would increase to 12 metres for the same repetition capability of 8 notes per second.  Thus the allowable tracker length has increased by about 3.5 times for a reduction in tracker mass per unit length of the same amount.  This is an important result.  Firstly, of course, we now have much more flexibility to design an organ layout.  Secondly, for all other factors remaining the same, we see that any variation in tracker mass reflects directly into changes in their length rather than some fraction, as was the case with the other components of the action.  This is because it is the actual masses of the trackers which form part of EDM as noted above.

There might be some practical difficulties when using thin trackers however.  Unless they run vertically and are under constant tension at all times, there could be some risk of excessive lost motion at the keys leading to a "flappy" feel to the action.  This danger would likely increase if the runs were horizontal with the consequential difficulties in supporting them.

The effect of changes in repetition rate .  It is most important of all to consider the implications of varying the repetition rate, and therefore release time, demanded of the action.  So far we have used a value of 8 notes/sec, this being an average figure based on a value of 10 notes/sec in the middle/upper parts of the keyboard and some lower values in the bass.  Being more specific, let us now demand a minimum repetition rate of 10 notes/sec anywhere in the keyboard.  The components of the action will remain the same as for the Victorian organ above, except we shall now be using the thin trackers specified in the previous paragraph.

The result is that, by coincidence, the maximum allowable tracker length reduces from 12 m back to its earlier value of 3.4 m, even though we are using thin trackers!  Therefore there has been a reduction in maximum allowable tracker length of nearly 71% for an increase in repetition rate of only 25%.  This is obviously a most important result, and it confirms the statement made earlier that organ builders should not lightly accept a specification for repetition rate higher than is musically reasonable.

Summary - backfall actions

A summary of the main points made above is:

1.  The equivalent dynamic mass (EDM) of a backfall action is dominated by the total tracker masses.  The masses of all other components contribute less to the EDM even if their actual masses are greater than those of the trackers.  Rollers are almost always negligible in terms of their effect on action inertia.

2.  The repetition performance of the action is not usually dominated by the key, even though this might be the most massive component in the action.

3.  Thus repetition performance can most easily be improved by reducing the total tracker mass.

4.  There is a strong inverse relation between tracker mass and repetition rate.  Typically, a decrease of about 70% in tracker mass will increase repetition rate by only about 25%.  Conversely, a modest improvement in repetition rate will require a reduction in tracker mass by an amount which might well be impractical.  

5.  Therefore the repetition rate specified for an organ with this type of action must be reasonable.  Rates over 10 notes per second may not be achievable.  In the bass, with the longer tracker runs, the achievable rates may be less than this figure.

Suspended Actions

We turn now to a second type of action, the suspended form, so called because the key is suspended on the end of a tracker (Figure 2).  A feature which dominates this arrangement is the rather long back-hung key, for reasons which become obvious if the geometry is considered.  There are two issues:  firstly the distance (a - b) in the diagram cannot be less than about 200 mm, otherwise the amount by which the keys project beyond the upstand of the keyboard will be inadequate.  Often (a - b) will be significantly greater than this figure.  Secondly the leverage, more specifically the mechanical advantage, of the action cannot be too large otherwise the pallet will not open adequately for a reasonable value of key dip.  A reasonable maximum for key dip is 10 mm and the pallets will need to descend typically by at least 7 mm.  Taken together these figures result in a value for a, the key length, of about 675 mm.  As stated already, this will often be exceeded in real actions.

The basic behaviour of the action is still described by equation 6.  However, because of the long key and the necessity for a mechanical advantage always greater than unity, the characteristics of this action are different in detail to those for the backfall type.  These can best be illustrated by discussing the elements making up the equivalent dynamic mass (EDM), M in equation 6.  An important factor in the mathematical model for this action is the ratio b/a.  Typically b/a will vary between about 0.7 and some higher value for practical reasons, but of course it can never quite reach unity regardless of how long the key is made.  As with the backfall action, the static mass of the pallet is reduced in the formula for EDM because it is pivoted.  However an important difference for the suspended action is that the EDM for the key is now magnified by the reciprocal leverage ratio a/b, which is always greater than unity.  This would be expected intuitively; if we consider the pallet spring which has to accelerate the key when it is released, its job gets harder for smaller values of b.  Thus the relative values of the divisors which convert actual masses into EDM values are indicated qualitatively below, where it is assumed the tracker is connected to the pallet near to the end opposite its hinge (i.e. no significant leverage is introduced at the pallet itself):

1.  The divisor for pallet mass is largest.

2.  The divisor for key mass is intermediate in value.

3. The divisor for tracker mass is unity, i.e. the EDM for the tracker is its actual mass.

This does not mean necessarily that the influence of pallet mass is always of least importance and that of the tracker always greatest.  It is necessary to insert the actual masses of the components and divide them by the appropriate number before such a ranking can be derived, and this can vary for different actions.  Therefore, as for the backfall action, we shall take an example of an action and perform some parametric studies.  The action has the following parameters:

After-touch weight. 75 gm measured at the key.

Key dip. 10 mm.

Keys.  Length = a = 677 mm;  b = 474 mm.  Therefore b/a = 0.7.  Stock: 12 mm by 25 mm cross-section, unloaded. Pine construction. Hence mass = 102 gm.

Trackers. Instead of specifying a particular type of tracker, we shall use the model to evaluate the maximum allowable tracker mass for a range of other parameters.

Pallets. 325 mm long of pine (density 0.5 gm/cc).  Mass 50 gm. Descent 7 mm at the pull-down.

Maximum allowable tracker length .  For a repetition rate of 8 notes/sec the maximum allowable tracker mass is 228 gm.  Even with the heavy trackers (21 gm/m) considered earlier in the Victorian organ example (see under Backfall Actions above) this would enable trackers up to nearly 11 metres long to be used.  

The effect of key mass . The key dimensions above are rather small compared to those in many a suspended action.  Let us see what happens when the key length is doubled to 1354 mm, thereby doubling the key mass to about 203 gm.  

With a repetition rate of 8 notes/sec the maximum allowable tracker mass reduces to 180 gm, which is a reduction of about 21% compared to the previous case.  However this still gives considerable design flexibility.

The effect of changes in repetition rate .  With the key as specified originally (i.e. 677 mm long) and increasing the repetition rate demand to 10 notes/sec, the maximum tracker mass decreases to 122 gm.    Therefore it would still be possible to use trackers nearly 6m long with the same mass per unit length, which would still allow for considerable design flexibility.  We also see that the maximum allowable tracker mass decreases by about 54% for an increase of 25% in the repetition rate.

With the double-length key (1354 mm long) and the higher repetition rate of 10 notes/sec the maximum tracker mass reduces again to about 74 gm.  At this point it would only be possible to use trackers 3.5m long if they weighed 21 gm/m.  However much longer runs would still be possible by using lighter trackers.

Summary - suspended actions

1.  It is more difficult to draw general conclusions about the relative importance of the masses of the action components for a suspended action than for the backfall action considered earlier.  This is because of the inevitable introduction of a leverage ratio at the key which results in more cases having to be examined in a parametric study.

2.  The EDM for tracker mass is still the same as its actual mass, as for the backfall action.

3.  The EDM's for the key and pallet are reduced compared to their actual masses because they are pivoted, again as for the backfall action.  However, an additional  multiplier (>1) then has to be applied to the EDM for the key to take account of the leverage.

4.  By studying some examples, it is found that the repetition performance of the action is less dependent on tracker mass than for the backfall action e.g. the maximum allowable tracker mass decreases by 54% for an increase in repetition rate of 25%.  For the backfall action the mass decrease was 70%.

5.  Doubling the length of the key, and hence doubling its mass, reduces the maximum allowable tracker mass by 21%

6.  On the whole the results of the study show that a suspended action ought to be capable of a good performance.  For example, with a repetition rate of 10 notes/sec and a key length of 1354 mm the maximum allowable tracker mass is about 74 gm.  With lightweight trackers this will still allow considerable run lengths.

7.  The main reason for the conclusion in 6 is that the trackers are "decoupled" from the key motion by virtue of the leverage ratio b/a.  This means they exert less influence on factors such as repetition rate than in the case of a backfall action with unity leverage ratio.  This remains true even when the keys in the suspended action are relatively long and massive, much more so than in the backfall action.

8.  In view of their good performance, the widespread use of suspended actions in pre-Revolutionary French organs must have encouraged contemporary composers to write in the customary florid style of that era, with much ornamentation.  The engineering simplicity of the action also implies less opportunity for unfortunate attributes such as lost motion, which lends further weight to this hypothesis.

 

PART 2 - Electric Actions

The two types of electric action considered here are direct electric and electro-pneumatic actions.  The former type uses a more or less powerful electromagnet to open the valve directly, whereas the latter uses a smaller magnet which admits air to pneumatic motors, which in turn do the work of opening the valve.  Proportional electric actions in which the movement of the valve is related to the instantaneous position or velocity of the key movement are not considered.

The parameters specifying the performance of electric actions are fundamentally different to those for mechanical ones.  Fore-touch and after-touch weights are irrelevant because they are purely attributes of the keyboard and have no influence on the performance of the action.  Repetition rate is probably the most important parameter and it is the only one discussed here.

Repetition Rate in electric actions

As with mechanical actions, repetition rate is affected both by the attack time and the release time, therefore equation 1 earlier in this article still holds for electric actions.  However there is a fundamental difference in this case in that both these times are irredeemably determined by the action itself; neither can be varied by the player.  Otherwise there are some similarities and in particular the same law enunciated earlier applies, that the masses in the action affect repetition rate via a similar square law to that of equation 6.  For direct electric actions there is also an electrical inertia associated with the growth and decay of stored energy in the magnet.  Therefore it would be unwise for an organ builder to agree to a specification for repetition rate higher than necessary, just as for mechanical actions.  A figure of 8 - 10 repetitions per second (of the same note, and depending on position in the compass) is a reasonable minimum design aim for electric actions as well as for mechanical ones.

It is unnecessary to enter into a detailed discussion of the physics involved in electric actions as this was done in another research paper on this website [3].  Particular attention was devoted to an examination of the widely-held belief that direct actions are slower that electro-pneumatic ones, and this was not found to be the case.  As this is a review article it will be appropriate to reproduce the other outcomes of the earlier study as well.

For both direct electric and electro-pneumatic actions the same parameters were used, as in the following table:

Pallet mass

112 gm

Power motor mass (in e-p case)

112 gm

Pallet aperture

300 x 18 mm

Pallet descent

7 mm

Pallet spring force at pull-down

107 gm

Power motor/heavy duty magnet force at pull-down

320 gm

Note the deliberate use of a large and heavy pallet.  It was made of hardwood, and a reduction in mass of at least 50% could be achieved by using a lighter material.  The rationale behind the choice of pallet was to explore the extremes of the performance envelope of the actions.  In particular, if the actions worked satisfactorily with this pallet then they could be expected to perform better with a lighter one.

For direct electric actions the conclusions were:

 

Measurements were made on a representative direct electric action with a large and heavy pallet at various wind pressures. They showed that the attack time was independent of pressure below the pull-in limit of the magnet, and equal to 60 msec.

 

The release time was 47 msec and also pressure independent.

 

 Release time in the action used for the measurements was due to mechanical inertia (about 80%) and current decay in the coil due to the spark suppression diode (about 20%).

 

 Dynamic repetition frequency measurements showed that the action could support repetition up to about 10 Hz, compared to a prediction from static measurements of 9.4 Hz. With a smaller pallet this figure would increase.

 

 The dynamic measurements showed that the performance of the action can be degraded by other factors such as an inadequate wind supply which can execute uncontrolled pressure excursions.

 

And for electro-pneumatic actions the conclusions were:

 

 Predictions of the response times of a representative 3 – stage electro-pneumatic action were derived from a combination of experimental measurements and theoretical modelling. The pallet was relatively large and heavy as in the direct electric case, and other parameters were also the same.

 

 The attack time was predicted to be about 46 msec, slightly shorter than the direct electric action.

 

 The release time was predicted to be about 55 msec, slightly longer than the direct electric action.

 

 The maximum repetition frequency the action could support derived from these values was about 10 Hz, the same as for the direct electric action.

 

The main quantitative outcomes of the work are summarised in tabular form below:

 

 

Parameter

Direct electric action

Electro-pneumatic action

Attack time (ms)

60

46

Release time (ms)

47

55

Max repetition rate (Hz)

10

10

Because a direct electric action can be made which comfortably meets the required repetition performance there seems little foundation for the belief that direct electric actions are always intrinsically "slow". Moreover an electro-pneumatic action is not likely to do any better. We should note again that the pallet used in these experiments was relatively large and heavy and representative of one which might be used for the bass notes of an organ. Higher in the compass the pallets would be smaller and lighter and so they could be expected to respond even more rapidly.

The perceived sluggishness of direct actions might therefore have arisen from other factors such as:

1. They have a performance envelope set by the maximum pluck that the magnet can tolerate. Inadequate actions may therefore be working too close to the limit in which there would indeed be perceptible slowness as well as other unreliabilities

2. The attack time of 60 msec for the direct action used here might be perceptible if compared with that for a small mechanical action instrument for example, although it is only 30% longer than that predicted for a typical electro-pneumatic action. Even in a mechanical action the key needs a certain time to fall and in any case the pipes, at least the smaller ones, will begin speech before the pallet has opened fully for any type of action.  On balance therefore it is unlikely that the figures here could be said to constitute an unreasonable attack time for any action. In any case, with the smaller pallets used higher in the compass the attack time would be smaller because of the lower masses and hence lower inertia.  It is possible this reduction might be more pronounced for an electro-pneumatic action than a direct electric one, provided the former was meticulously designed and optimised.  However we have to ask whether the subjective difference, even if there was one, would really amount to much in practice.

3. The release time of 47 msec might be objected to similarly, though the arguments in the previous paragraph also apply here in the sense of getting the figure into perspective. But it is 15% faster than for the electro-pneumatic case. For direct electric action the release time could in principle be reduced even further by some 20% by removing the spark suppression diode, but it has been demonstrated that this is unwise unless an equivalent circuit exists elsewhere. Some manufacturers of magnet driver equipment (solid state transmissions) recommend removal of the diodes, but one hopes they will have incorporated over-voltage suppression in their product. The stored energy in these large magnets has to be dissipated safely somehow at key release despite any other claims which are made.

4. Actions which are said to be slow might be suffering from performance degradation due to inadequate wind pressure regulation, a phenomenon observed during these experiments. The action itself might be perfectly satisfactory. (Such effects could also degrade the response of electro-pneumatic actions).

It was then enquired whether direct electric actions should be used in conjunction with mechanical actions, or whether electro-pneumatic actions should be preferred. If there are two consoles, one mechanical and the other direct electric, the work showed the latter could provide adequate repetition performance provided the mechanical inertia of the action attached to the magnets was not excessive. It would be unwise to place the magnets at a distance from the chests so that they had to overcome the inertia of the tracker work as well as the pallets.

Using heavy duty magnets to assist the coupling or to open auxiliary pallets could introduce noticeable delays between the mechanical and electric elements of the action, depending partly on the responsiveness of the mechanical action itself. However on the basis of the results obtained  in the study it is debatable whether these would be any worse than with electro-pneumatic helpers.

Finally it was asked which is the better action – direct electric or electro-pneumatic. The results obtained were inconclusive in this regard only because the two types appeared to perform similarly. But one thing they showed unequivocally was that direct electric actions can work at least as fast as the demands likely to be placed on them by the performer, just as can electro-pneumatic actions. The paper concluded that since direct actions are cheaper it is not surprising their cost-effectiveness often makes them the system of choice.

References

1.  "Actions and Reactions", A Woolley, JIMIT, September 2001

2.  "Touch Sensitivity and Transient Effects in Mechanical Action Organs", C E Pykett, Organists' Review, November 1996, p. 285, currently on this website (read).

3.  "Response Speed of Electric Actions", C E Pykett, 2002, currently on this website.  (read)

4.  "Tracker or Electric?", A Moyes, Organists' Review, November 2002, p. 373

5.  "How the Flue Pipe Speaks", C E Pykett, 2001, currently on this website.  (read)

6. "The Design of Organ Soundboard Pallet Valves", J Norman, ISOB Journal, vol. 4(1), p. 44, Autumn 1978.

7. "Calculating Pallet Size", C E Pykett, 2001, currently on this website.  (read)

8. "Touch Relief in Mechanical Actions", C E Pykett, 2001, currently on this website.  (read)

9. "Towards a uniform tracker key touch", John Norman, The Organbuilder, p. 6, volume 17, November 1999

10.  Some amplification of this statement might be helpful.  The restoring force while the key is down is a net figure equal to the difference between the force(s) exerted by the spring(s) and that resulting from any imbalance in the weights of the components in the action.  In the analysis here both contributions are measured at the key, and their algebraic sum (the net restoring force) will always be non-zero and positive in the upwards direction otherwise the key would not rise when released from the finger.  It is possible, indeed likely, that the restoring force will vary somewhat with key position rather than remaining constant as assumed here.  However the magnitude of the variation in relation to its value at key-down is considered small enough to be ignored in view of the relatively small key movement (generally less than a centimetre).  Pallet springs, for example, are pre-tensioned so that they continue to exert significant force even when the pallet is closed.  The pre-tension in this position will commonly correspond to an equivalent spring movement significantly greater than the additional movement due to pallet descent.  Therefore it is not unreasonable to ignore the increase in spring tension during pallet descent.  Similar considerations apply to the possible variation in unbalanced action weights, if present, over the key movement.  It is worth recalling that actions in which the restoring force increases noticeably as the key descends feel spongy and unpleasant to the player, therefore measures are taken in any half-decent keyboard instrument to minimise this effect, or even to replace it with a characteristic in which the force either remains constant or slightly decreases with key descent.

When an organ is in wind an additional factor comes into play.  When the pallet approaches its rest position as the key rises, it begins to experience an additional force when the decreasing windway results in an air pressure drop across the pallet.  I have observed this effect during the experiments carried out as part of the investigations leading to this article.  In simple parlance the wind 'grabs' the pallet increasingly strongly as it closes.  This additional force assumes its maximum value when the pallet is fully closed, and at this point it then equals the force due to pallet pluck discussed in the article.  The fact that the force increases as the pallet closes counteracts the decreasing spring force, thereby providing an additional argument in favour of neglecting the variation of spring force with pallet and key position.

In an exact analysis all these factors could be parameterised in principle and incorporated in a model of the action.  However the point is whether the additional complication would be worthwhile, particularly as the factors are subject to a range of imponderables which would be difficult to quantify successfully.