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How
the Flue Pipe Speaks
18 November 2001 (last revised November 2005) Copyright © C E Pykett, 2001-2005 (Note: this
article should be easier to read if you allow it to occupy the full screen
width on your monitor) The organ flue pipe and similar wind instruments seem at first sight to be of the simplest structure imaginable. Yet the details of how it works have attracted the interest of musicians, organ builders and scientists for centuries, and it still continues to attract attention today. One reason is the persistence until quite recently of several fundamental misunderstandings, so this article presents a simplified review of the mechanisms involved in pipe speech, tone quality, voicing and scaling. The actual mechanisms are extremely complex, and in some cases not amenable to detailed analysis at all. The
basic mechanism: generator-resonator coupling It is necessary to understand the concept
of generator-resonator coupling in a flue pipe. The pitch of the pipe is largely
controlled by the air in the body of the pipe, the resonator. The energy required to set the air
in the resonator in motion is supplied by the wind issuing from the mouth,
the generator. Both the
generator and the resonator control the tone quality of the pipe, and they
are closely coupled together while the pipe is sounding. When the pallet of a pipe opens, wind
issues as a sheet of air from the slit-like flue at the base of the mouth
and moves towards the upper lip.
In doing so it also enters the pipe and pushes some of the
stationary air aside, and this initiates a pressure impulse which travels
up the column of the pipe at nearly the speed of sound (about 1100 feet
per second or 335 metres per second). The exact speed depends on factors
such as the amount of drag at the pipe walls. While the impulse is travelling in
the pipe the wind at the mouth also reaches the lip. Several things may happen at this
point, for example edge tones might be created (similar to wind whistling
through gaps in windows).
These contribute to the starting transient heard in some pipes.
When the impulse travelling up the pipe meets the air at the top it
increases in speed to exactly the speed of sound because it is now
unrestricted by the pipe walls. This results in the former region of
compressed air becoming stretched, and this causes a rarefaction or
partial vacuum to suddenly replace the region of positive pressure. Because “Nature abhors a vacuum”,
that region tries to restore pressure within itself by drawing in air from
around it, including air from just inside the pipe. So this causes a rarefaction
impulse to travel back down the pipe as the travelling region of rarefied
air continually tries to equalise the pressure to that of the
atmosphere. Therefore the
impulse moving back towards the mouth is a rarefaction rather than one of
positive pressure. Because the impulse is now a rarefaction
at the mouth, it tries to suck in the air sheet, so some of the air from
within the sheet moves into the region of rarefaction. This turns it back into a positive
pressure impulse again and this goes back up the pipe as before. The process will continue for a
number of up-and-down transits of the impulses within the body of the pipe
until it settles down to steady speech. During this settling-down phase
the characteristic starting transient of the pipe will be emitted.
Moreover the air sheet begins to oscillate steadily across the upper
lip: it receives assistance
by being sucked into the pipe each time it meets a returning region of
rarefaction, that is, once per cycle. And the reason that the process
continues is because of the energy of the air in the sheet, which
repeatedly flips the impulse back up the pipe. The time between each such
occurrence is that taken for a complete up-and-down transit of the
pipe. The length of this dual
transit equals one wavelength of the sounding pitch of the pipe and
therefore the length of an open pipe equals half a wavelength (neglecting
end corrections at the mouth and the top). It is emphasised that the statement just made, that open pipes are half a wavelength long, is only an approximation to reality. In fact, as any organ builder knows, the variation in length between pipes having the same musical pitch is considerable. This is because their scale, or cross-sectional area as a proportion of their length, is also important in determining pitch. Audsley, although neither an organ builder nor a scientist (he was an architect), could scarcely contain his venom when pointing this out in his usual verbose and opaque fashion 4. He quoted an example where two stopped pipes, one more than twice as long as the other, nevertheless gave the same note - one was short and fat and the other long and thin. The reason is the end correction which has to be applied, and this is greater for fat pipes than thin ones. An end correction occurs at the mouth as well as at the top (if open), which explains Audsley's result for stopped pipes. End corrections are also largely responsible for determining the timbre or tone quality of the pipe, as will be explained later. The misunderstanding also arises because of the tendency to regard an organ pipe merely as a gaseous version of a violin string, but this is seriously mistaken. A pipe is not a one-dimensional structure like a string but a three-dimensional resonating cavity, all of whose dimensions have to be considered if a satisfactory theory of its operation is to be developed.
What has been described is the currently
accepted jet-drive mechanism of the flue pipe. Until the 1970’s various other
theories were in vogue, including so-called air reeds and vortex/eddy
mechanisms. During steady
speech the air jet at the mouth is in the form of a flattish sheet
wriggling sinuously into and out of the pipe, flipping across the lip each
time (Figure 1). A wave-like
motion which begins at the flue and propagates upwards along the sheet is
responsible for these movements, which also push and pull on the air
outside the pipe thereby causing a sound wave to move from the mouth into
the surroundings. If the pipe
is open at the top, there will also be sound emission from this
region. Relatively little
energy is emitted from anywhere else. A stopped pipe works rather differently.
Consider the first impulse to travel up the pipe to be of positive
pressure, as before. But when
this meets the rigid stopper it is compressed even further, causing it to
rebound but in the form of a continuing pressure impulse rather than a
rarefaction. When the
descending pressure impulse reaches the mouth it pushes the sheet of air
further out of the pipe.
Because the mouth of the pipe looks rather like an open end as far
as the impulse is concerned, it is immediately transformed into a
rarefaction for the reasons described earlier. This then travels up and
then back down the pipe as a rarefaction, and when it returns to the mouth
it then sucks in the air sheet.
Therefore in this case, the air sheet is alternately pushed and
pulled at the mouth by the impulses travelling up and down, which take the
form of alternate high pressure regions and rarefactions. A complete cycle of events, that
is from one inwards suction of the air sheet to the next, involves two
complete up-and-down transits of the pipe rather than the one of the open
pipe. Therefore the length of
a closed pipe equals a quarter wavelength of the emitted sound,
consequently it sounds an octave lower than an open one. The fundamental physical
difference between an open pipe and a closed one is that two phase changes per up-and-down
transit of the travelling impulse take place in the open pipe (one at the
top and the other at the mouth).
In the closed pipe there is only one phase change, that at the
mouth. As already mentioned, note that the travelling compressions and rarefactions actually proceed a little beyond the physical confines of the pipe at the top if it is open, and always at the mouth. It is only in this way that they "realise" they have gone beyond the pipe into the open air. It's as if they say "hey - we've gone out of the pipe - let's get back inside". Seen in this light, it is not surprising that pipes have an end correction - they seem to be longer acoustically than they actually are. This phenomenon is extremely important, because not only does it mean that organ builders must know how long to make pipes if they are to emit the correct pitches, but the end corrections are responsible for shaping the harmonic structure of the tones and hence the timbre or tone colour of the pipe. This will now be explored in more detail. Tone
Quality or Timbre
When a pipe is emitting sound steadily
its subjective tone quality is determined principally by the proportions
of the fundamental frequency and its harmonics in the sound reaching the
ear. To avoid confusion we
need to canter through some basic definitions at this point, and we shall
use only the terms fundamental
and harmonics to describe the
various frequencies emitted by the pipe. Words such as partials and
overtones will not be used as they are defined differently and can cause
endless confusion. The
fundamental frequency (occasionally called the first harmonic) of a pipe
is the same as its musical pitch.
It equals the frequency of the air sheet as it flips in and out of
the mouth of the pipe. The
air sheet imparts motion to the internal air column as we have seen, but
only if this motion is absolutely sinusoidal in character will there be no
other harmonics, a situation that is never true for organ pipes (i.e. real
pipes never emit just a pure sine wave). Because the air in the pipe
vibrates in a non-sinusoidal manner there will always be some additional
harmonics present – mathematically this is stated by Fourier’s
theorem. Also according to
Fourier, for periodic waves (such as those emitted by a pipe during the
steady state) the harmonics are exact multiples of the fundamental
frequency, thus if the pipe is sounding middle C (262 Hertz or cycles per
second) the harmonics will lie at frequencies of 524, 786, 1048 Hz etc. The harmonic lying nearest to the
fundamental is called the second harmonic and it has the same frequency as
the octave above the fundamental.
The third harmonic, at three times the fundamental frequency, is at
the interval of a twelfth above the fundamental, and so on.
Figure 2. Representative Pipe Spectra It is instructive at this point to look
at the spectrum of some pipe
sounds; these are graphs showing the relative strengths of the harmonic
retinue. In Figure 2 are
presented some of my own measurements of flue pipe spectra. They show an open diapason, a
claribel flute and a viol d’orchestre from the large Rushworth and Dreaper
organ in Malvern Priory. Also
shown are spectra of one of the stopped diapasons from the 1858 Walker at
St Mary’s, Ponsbourne near Hatfield 1 and a quintaton pipe from a house
organ made by Brian Daniels of Crewkerne. Because of the rapid falling off
in strength of the harmonics, it was necessary to plot them using a
logarithmic scale in which 10 decibels equals a change in strength of
about 3.16, and 20 decibels a change by a factor of 10. Otherwise many of the harmonics
would not be visible. All
spectra represent the middle F sharp pipe of each stop. The first point to note concerns the
numbers and relative strengths of the harmonics. The two flutes (the claribel and
the stopped diapason) have the fewest, the diapason has more and the viol
more still. The quintaton has a third harmonic (that which sounds the
interval of a twelfth above the fundamental) at a relatively high
amplitude, together with its
second harmonic, the sixth in the series. This gives it a quinty voice which
is so prominent that the effect is almost as though a separate twelfth
rank was in use. In addition
there are somewhat more harmonics in total than for the diapason. The viol is the only pipe in which
the fundamental is not the strongest harmonic; in this case it is the
fourth. Such a strongly
developed harmonic structure gives string pipes their characteristic thin
and penetrating sounds. These
pronounced variations in the numbers of harmonics are partly due to the
different pipe diameters for reasons to be explained later. The second point concerns an interesting feature of the two flutes. In order to appreciate this we need to revert to the differences between open and closed pipes. As well as emitting a sound an octave lower than an open pipe, it is well known that a closed one also has a quite different hollow tone. This is because it cannot resonate at the even harmonics efficiently, leaving a preponderance of odd ones in the sound. The reason for this is as follows. Consider the first of the even harmonics, the second harmonic. This is a frequency corresponding to the octave above the fundamental, that is, at twice the frequency of the fundamental. There is certainly energy present at this frequency in the oscillatory air stream at the mouth, so let us see what happens to it. As before, consider the first impulse delivered to the air column to be a compression at the mouth. When this travels back down the pipe to reach the mouth again, we saw above that it is converted into a rarefaction. For the fundamental, it is this rarefaction which travels back up the pipe - the fundamental in a stopped pipe is characterised by alternate compressions and rarefactions. But because we are considering here an oscillation at twice this frequency, we have to consider the next impulse at the mouth to be compressive also. This largely cancels the rarefaction which the wave would actually require if it were to grow and be supported by the resonance of the stopped air column, hence the second harmonic is much weaker than the fundamental. Although it gets somewhat more difficult to visualise, the same reasoning applies to all of the even harmonics. The process has been explained in more detail by Charles Taylor 3. So it is clear why the stopped diapason
has even-numbered harmonics which are quite weak. Although it may not be immediately
obvious from the diagram, the even harmonics of the claribel flute, an
open pipe, are also significantly lower than would be expected. This attribute gives this stop too
an attractive hollow type of tone although one which is subjectively
different from the stopped diapason.
In order to understand how these harmonics can be reduced even in
open pipes it is necessary to discuss the subject of voicing. The Influence of Voicing Adjustments on Tone QualityIt is possible to adjust the lip position
relative to the flue so that the oscillating air jet produces stronger odd
harmonics, even in an open pipe.
Clearly, if strong odd harmonics are generated in the first place
then they will be strong in the final sound. The adjustment is done by moving
the lip of a pipe into or out of the plane of the pipe wall. It happens that the optimum lip
position so obtained generally interrupts the wind immediately it is
turned on, thereby making a flute pipe “quick” (i.e. it does not need long
to settle down to stable speech).
What is happening in this situation is that the wind sheet spends
the same amount of time inside the pipe as outside as it flips across the
lip, therefore the impulses imparted to the air column have a symmetric
nature. In other words the
“mark-space ratio” of the impulses is about 1:1. It is a similar situation to
the square waveform used in electronics, which contains only the odd
harmonics. Combining this
adjustment with a closed air column, as in the stopped diapason, will
further reduce the even harmonics as well as possibly making the voicing
adjustment less critical. The diagram shows that flutes also
require the higher harmonics, whether odd or even, to be considerably
reduced in strength otherwise they would not sound like flutes at
all. There are two ways to
achieve this. One is to use a
relatively wide pipe and the other is to use a high cut-up. Cut-up is the ratio of mouth
height to width, and a high mouth results in fewer harmonics being
generated by the air sheet.
This is because the sheet becomes more diffuse as it moves upwards
from the slit, and it therefore excites the resonator with less “sharp”
impulses of air at each cycle of oscillation. A lower cut-up is used for
diapasons (typically 1:4) and an even lower one for strings. Both of these stops also require
the upper lip to be pushed in further than for flutes so that a retinue of
both odd and even harmonics is produced by the oscillating wind sheet (the
excitation within the pipe thereby becomes more like short pulses rather
than like a square wave). The
position of the upper lip in these stops, particularly strings, generally
makes them slower of speech than flutes because the air sheet does not
fully engage with the lip until a greater time after the wind has been
turned on. The Influence of Pipe Width on Tone QualityThe influence of pipe width brings us to
the subject of scale, which is the ratio of the width of a pipe to its
length. But before moving
onto a discussion of scaling let us examine why the width of a pipe has
such an important effect on tone quality: most people are aware that
narrow pipes sound stringy and wide ones fluty, with diapasons lying
between. It is necessary to
understand the difference between the harmonics produced in forced excitation of a pipe and
its natural frequencies. A pipe which is sounding steadily
is one which is under forced excitation. It generates a fundamental
frequency equal to the musical pitch of the pipe plus an harmonic series
whose frequencies are exact
multiples of the fundamental as described earlier.
The natural frequencies of the pipe are quite different. They can be measured by exciting a pipe with a loudspeaker close to one end and picking up the sound with a microphone at the other (Figure 3). If the frequency of a sine wave applied to the loudspeaker is increased slowly from a low value, the microphone will receive a large signal when the pipe resonates at the fundamental pitch related to its length plus the end corrections. The response will then die away as the frequency continues to be increased until the next natural frequency is reached. This will be at somewhat more than twice that of the fundamental resonance. Subsequent natural frequencies will be detected at successively greater divergences from the exact frequency multiples of the harmonics in a forced excitation when the pipe is blown in the usual way.
The effect is illustrated in Figure 4
where two comb-like structures are drawn. The harmonic series of a pipe
speaking steadily (i.e. under forced excitation) forms a comb whose teeth
are regularly spaced, whereas the teeth on the natural-frequency comb lie
at successively greater separations such that the higher frequencies are
significantly sharpened in pitch.
It is the natural frequencies of the pipe which amplify by
resonance the corresponding forced harmonics of the oscillating air sheet,
to an extent depending on how well the teeth on the two combs approximate
to each other. The reason why the natural frequencies are not exact multiples of the fundamental is because of the end corrections of the pipe, which cause it to seem rather longer than its actual length to the travelling impulses. The end corrections at the mouth and top (if open) result partly from the impulses overshooting the pipe ends somewhat, and the mouth corrections dominate those at the open end. The total end correction (mouth plus end) is greatest for the lowest frequencies and it also increases for wider pipes. These two effects taken together mean that the natural frequencies for wide pipes do not coincide well with the harmonics of the forced excitation when the pipe is speaking. The teeth of the two combs lie progressively further apart in this case, as shown in Figure 4, although this diagram has been exaggerated for clarity. This means that such a pipe does not amplify the harmonics of the forced excitation very well beyond the fundamental and first few harmonics, thereby producing a frequency spectrum characteristic of a flute.
By contrast, for a narrow pipe, the teeth
of the two combs approximate better (Figure 5) and the pipe therefore
amplifies more of the harmonics in the forced excitation. The Effect of Pipe Material on Tone QualityThe material from which a pipe is made is
often said to have a pronounced effect on its tone. However most attempts to bring the
subject into the realm of objective measurement have shown that although
there are measurable effects, they are of lesser importance than the
issues already discussed relating to voicing and pipe width. Obviously a pipe has to be made of
material which is sufficiently durable to withstand many decades of
service and possible abuse.
If of metal, this means it has to be made of such a thickness that
pronounced resonances of the pipe walls will generally be inhibited, hence
the effect of the metal on tone quality is also likely to be relatively
small. Also this is one
reason why there is so little radiation of sound other than from the mouth
and the top – the substantial body of the pipe does not vibrate very
much. Perhaps
this controversial matter is best summed up by quoting from a scientific
paper on the subject which said that “it was particularly shocking to
hear a good diapason tone from a pipe with its cylinder made of wrapping
paper” 2 ! Nevertheless, there are distinct tonal
differences between certain wood and metal pipes but these arise for two
main reasons. The first is that the upper lip of wood pipes is of a
tapering wedge shape, whereas that of a metal one is thin and constant and
equal to the thickness of the metal. This affects some details of
the way the air jet interacts with the lip. The second reason again
derives from practical, constructional considerations which result in the
different cross-sectional shapes of the two sorts of pipe. Circular and rectangular cross
sections result in different natural resonance frequencies of the pipe,
and these have an important effect on tone quality as we have seen. For obvious reasons, it is usual
to make rectangular pipes of wood and circular ones of metal. These
factors may have led to the widely held belief that the tonal differences
are due to the materials themselves rather than the shapes of the pipes
and their mode of construction. Pipe
Scales
Because the tone quality or timbre of a
pipe is so dependent on its width, it is intuitive to assume that it
should remain constant if the ratio of width to length (i.e. the scale) is
also constant throughout a rank making up a complete stop. This is, however, not found to be
true subjectively. In such a
rank the sounds would be found too stringy or edgy in the treble for
reasons which relate to the spectral widths of the natural frequencies
of the pipe (technically
referred to as their Q-factors), and the subtle psycho-acoustic response
of the ear. In addition there would be a falling-off in power towards the
treble end that would be too rapid because of the rapidly decreasing
internal volume of the pipes, and hence their ability to generate power,
as the pipes become smaller. Therefore constant scaling is unacceptable
(remember that constant scale does not mean constant width; it means the
ratio of width to length
remains the same for all pipes). To remedy this, pipe diameters vary
somewhat more slowly across a rank than the lengths of the pipes
themselves, which halve at every octave. The result is that the pipes are
somewhat wider in the treble than they would be if constant scaling was
used, and this has the effect of progressively reducing the higher
harmonic content of the smaller pipes together with maintaining their
subjective loudness. In
the 19th century scaling laws for principal-type pipes emerged which
recommended that diameters should halve on the seventeenth note inclusive
(every 1 1/3 octaves). A variant with slightly wider
pipes but the same halving interval, sometimes called Normal Scaling,
found favour in the 1920’s. These are uniform scales, those in which the
variation of pipe widths follows a uniform logarithmic law just as do the
pipe lengths, but they are not always optimum. Therefore their adoption is
variable from builder to builder and from one instrument to another. This is quite natural
because the pipe scales used in a particular organ ought to be matched to
the acoustics of the building in which it is to speak, and to the other
stops of the main choruses.
In some of the most successful instruments almost none of the stops
are uniformly scaled, their scales having been designed as part of a
plenum for a given building.
The difference between non-uniform and
uniform scales can be better understood by referring to the scaling charts
used by organ builders. A
number of variants exist but one example is shown in Figure 6. The horizontal axis represents the
successive pipes making up a stop, with the numbers indicating the “C”
with which each octave begins.
The vertical axis represents how many semitones up or down a stop
deviates from Normal Scaling (NS), in which the pipe diameters halve on
the seventeenth note inclusive.
Thus Normal Scaling itself lies along the zero line. Any uniformly scaled stop will
appear as a straight line whose slope is related to the halving
interval. An open diapason stop of medium scale is
shown. Because the halving
interval varies in different parts of the compass the curve is not a
straight line, and the stop is therefore not uniformly scaled. Nevertheless it is made up from a
number of uniformly scaled segments whose departure from Normal Scaling
seldom exceeds more than about two notes either way. A four foot principal is also
shown, which broadly follows the pattern of the open diapason curve
although it is slightly narrower in scale throughout. This means that pipes of the same
pitch (e.g. middle C of the open diapason and tenor C of the principal)
are not of the same width.
The ranks are scaled individually to suit their place in the
principal chorus, and this shows how fundamentally flawed is the idea of
an extension organ if taken to extremes because it removes the ability to
design the ranks individually. The individual scaling applied to the
fifteenth is narrower still. By making measurements of the acoustic
response of a building using electronic equipment (e.g. by radiating noise
signals from a loudspeaker, picking them up with microphones and analysing
their frequency structures), it is possible to develop an individual
scaling strategy for an instrument.
If there are pronounced dips or humps in the frequency response of
the building, the scales of the main choruses can be designed to
compensate to some extent.
This is why the use of the same scales regardless of the building
is an unsatisfactory approach to organ design, though it often
characterises mass produced small organs. It is also an approach carried to
its most logical and unpleasant extreme by some of today’s electronic
copyists of pipe organ tone, whose lack of understanding of scaling is
revealed by their habit of selling identical instruments to every client
and by mixing sound samples taken from different pipe organs. A scaling chart can be used for
conservation or restoration purposes. If an old organ has missing or
badly damaged pipes, the dimensions of the good ones can be plotted on the
chart when some trends may become discernible. These can then be extrapolated to
fill in the gaps, thereby enabling the scales of the missing pipes to be
estimated. The slopes of any
straight-line segments will also indicate the halving intervals. In some cases the shapes of the
curves will even suggest the scaling “signatures” of particular
builders. And if there are
large disparities between the shapes of the curves for the different stops
making up the main choruses, this may indicate that some ranks may have
been interfered with or replaced.
For example, the common practice of changing a fifteenth rank into
a twelfth or vice versa may become obvious on a scaling chart. Thus the scaling chart for an
organ can be used to suggest ways in which such situations might be
remedied. Acknowledgements
Thanks are due to the church authorities at Malvern and Ponsbourne who allowed the acoustic measurements to be performed, and to Brian Daniels, organ builder, for permission to reproduce the spectrum of his quintaton stop. References
1. “Gleanings from the Cash Book: St Mary’s Hatfield: Church Expenses”, P Minchinton, Organists’ Review, May 1999, p 108 2.
“The Effect of Wall Materials on the Steady-State Acoustic Spectrum
of Flue Pipes”, C P Boner and R B Newman, J Ac Soc Am, July 1940, p
83 3. “Sounds of Music”, C Taylor, British Broadcasting Corporation 1976, ISBN 0 563 12228 5 4. "The Art of Organ-Building", G A
Audsley, New York 1905, chapter IX. (republished Dover 1965, ISBN 0 486
21314 5). Some
Further Reading
1. “End Corrections of Organ Pipes”, A T Jones, J Ac Soc Am, January 1941, p 387 2. “Acoustic Spectra of Organ Pipes”, C P Boner, J Ac Soc Am, July 1938, p 32 3. “The Physics of Organ Pipes”, N H Fletcher, Scientific American, January 1983, p 94 4. “Voicing Adjustments of Flue Organ Pipes”, A W Nolle, J Ac Soc Am, December 1979, p 1612 5. “St Barnabas Church Dulwich, the tonal concept”, W McVicker, The Organbuilder, Vol 16 1998, p 8
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