[ Continuing the topic which began in 000403 MMD, "Free Reeds
 [ and Beating Reeds", by Prof. Johan.  -- Robbie

The reason you use reed pipes at all is that they give louder sound and
have higher and stronger harmonics as compared to flue pipes.  But they
are more difficult to fabricate and handle.  In particular, with a
change in temperature they are detuned by a smaller amount than flue
pipes -- the pipe frequency is partly controlled by the temperature
indifferent reed, not only by the resonator column.

The inward beating reed (or sometimes the shallot, as with clarinets)
is curved such that the passage is open at rest.  Increasing blowing
pressure and/or mechanical force (clarinetist's lips, organ tuning
crutch) shortens the free length of the reed and frequency rises.  The
difficult thing in voicing a reed is to perfect this curvature and to
have the reed vibrate 'cleanly'.

The frequency is quite a lot higher than the natural one for the reed
(as determined only from its mass and elasticity) because it does not
oscillate in a sinusoidal fashion like a free reed.  It rather goes
in a sequence of half-periods -- when trying to enter the 'negative'
half-period it bangs into the shallot and immediately bounces back to
start another 'positive' half-period.

The flow through the reed resembles a full-wave rectified sine wave.
Because of its rather sharp minima it is very rich in harmonics.  It
is this bouncing against the shallot (or the other reed in double reed
instruments) that makes the beating reeds behave very unlike the free
reeds and makes them much more difficult to analyze.

When you have the proper bend in the reed and there is no resonator
connected it may be able to work over a big frequency range just by
varying the blowing pressure.  An important partial mechanism in
maintaining the oscillation is the 'Bernoulli force' -- in the closing
phase, when the airflow passes between reed and shallot, it gets a high
linear speed creating a local underpressure that increases closing
speed.  When the reed opens again this force is smaller because the air
has not yet acquired such speed because of its inertia.

When you hook on the resonator, the acoustic pressure in this will give
an additional force on the reed.  But this force is oscillating with
the natural frequency of the resonator, not the one of the reed.  When
the force is big enough it tends to synchronize the reed oscillation
to the resonator frequency.  So when you blow with increasing pressure
the frequency will no more rise evenly; in a graph of frequency vs.
pressure it will have plateaus at the resonance frequencies.

Mostly you want tight control of frequency so you want to maximize
the feedback force: the resonator should determine the frequency rather
than the reed and the pressure.  The magnitude of the feedback force
depends upon the reed area exposed toward the resonator and the
resonator acoustical impedance at the point of connection.  High
impedance means high pressure in relation to the flow.

A reed is a periodic open/shut valve that delivers pulses of high
pressure P (the blowing pressure) and little flow Q into the resonator.
It is a _high_ impedance device, impedance Z = P/Q.  As a contrary,
a flue mechanism converts the blowing pressure into a jet of high
velocity but a very low pressure, almost zero, same magnitude as the
audible sound pressure.  The flue generator is a _low_ impedance
device.

To have any good resonator action you must connect the generator at
a point where there is a reasonable impedance match to the resonator.
A high impedance source (reed) connects to where there is a pressure
maximum and flow minimum, essentially a _closed_ end of the resonator.
When you measure the resonator impedance here as function of frequency
then this corresponds to an impedance _maximum_.  Example instruments
are all the brasses, the clarinet (cylindrical resonator), bassoon and
oboe and saxophone (conical resonator), and the speech vocal tract.

To be efficient a low impedance source should be connected at a
pressure minimum and flow maximum, that is, to an _open_ end of the
resonator.  When you measure the resonator impedance as function of
frequency then this corresponds to an impedance _minimum_.  Examples
are the flute, recorder, ocarina, organ flue pipe.  The pipe cutup is
an open end.

The common pipe resonator is a tube of uniform cross section.  Its
resonances are determined by its length L, the speed of sound c, and
the 'boundary conditions' -- it may be open both ends or only one end.
(Closed both ends would not give any external sound; instead, this
case leads toward the different discipline of room acoustics).

When open _both_ ends the fundamental resonance frequency f1 = c/2L and
the higher resonances come at 2, 3, 4, 5, .. etc. times this frequency.
This is an example often illustrated in physics textbooks as sinusoidal
patterns of standing waves; at the resonances there is always an
integer number of half-periods over the length of the tube.  At all
those frequencies there are sharp impedance minima at both ends.
Always low pressure (because of the openness), but at resonance there
are flow maxima.  This suits the flue generator, but never the high
impedance reed generator.

When open at only _one_ end there is always a flow node at the closed
end.  The possible resonance standing wave patterns -- zero flow at the
closed end, maximum flow at the open -- now come out with an odd number
of quarter periods along the tube.  The fundamental resonance frequency
f1 = c/4L and the higher resonances come at 3, 5, 7, .. etc. times this
frequency.  Putting a flue generator at the open end you get the
classical stoppered organ pipe which has its dominant harmonics at
those frequencies [the odd harmonic series].  Instead, looking at the
closed end, at those same frequencies you have impedance maxima -- high
pressures and zero flow because of the closure.  This is the proper
place to connect a high impedance reed generator.

Responding to Robert Linnstaedt's question in MMD 2000.04.06-03 there
is another screw to the resonator problem.  Instead of cylindrical you
can make the resonator conical like in the oboe, bassoon, and
saxophone.  Then the waves inside the resonator are no more the simple
plane waves, they are spherical with their center at the tip of the
cone.  Close analysis calls for the solution to the spherical wave
equation which is too complicated to discuss here.  Let us leave with
the correct comments by John Nolte in MMD 2000.04.11-07 that the
resonances come at 2, 3, 4, .. etc. times the fundamental and that the
tube has to be some 1.5 times longer than the cylindrical one with the
same fundamental resonance.

The clarinet and the (straight soprano) saxophone are exceptionally
similar except for one crucial thing: the bore that is cylindrical and
conical respectively.  This conicity then also necessitates the holes
and pads in the saxophone to be progressively bigger toward its low
end.  The unlike distributions of the higher resonances explain their
difference in timbre.

Another most important consequence appears when you overblow the
conical instruments: the sound goes to the octave (fundamental*2),
whereas the cylindrical clarinet goes to the octave and a fifth
(fundamental*3).  When overblowing the player increases blowing and
lip pressures to raise the reed frequency, and he normally also opens
a special valve [the register key] placed near a pressure node for
the overblown note.  This has little consequence at the overblown
frequency, but it destroys the Q value for the fundamental resonance.

It is important that the reed really does have a high impedance in
order not to destroy the proper action of the closed end resonator.
When the reed is mounted the closed end is only approximately so.  The
consequence is a somewhat impaired resonance Q and the need to shorten
the resonator slightly.

The most sophisticated and difficult to analyze resonators are those of
the brass instruments.  With those the fundamental resonance is never
used because it is grossly out of tune with the higher resonances that
instead are accurately in the relations 2, 3, 4, 5, etc.

It is a great feat in instrument evolution to find out the proper shape
of the bell flare in order to reach this accurate intonation.  Because
the player overblows to any of them, all must be in tune.  It is weird
to know that there are infinitely many shapes of horns that can fulfill
this criterion, still each such shape must be made with high precision.

Good examples are the trumpet, the cornet and the fluegelhorn -- three
instruments which to a naive observer might appear identical looking at
working principle, resonator length, and tonal range.  But they have
distinctly different flare shapes and this settles their differences in
timbre, the various higher resonances get differing sets of Q values.

Some band organs feature trumpet-like flared resonators.  This is
attractive to look at, but perhaps not always necessary from a
technical point because with each pipe you never give more than one
note.  It is then not so important that the higher resonances are
precisely tuned so the more unpretentious straight conical horn
speaking at its fundamental is often found adequate, and at the same
time this saves space.

The 'inverted' layout using the boot as a resonator mentioned MMD
2000.04.08-05 by Bill Chapman, and commented on by John Nolte in MMD
2000.04.13-08, is a very interesting variant I have never seen.  It
appears to me the boot resonator is made to lock the reed fundamental
at the correct pitch for the note.  The small top resonator then works
quite independently of pitch just to shape the spectrum envelope of all
the higher harmonics.

This has some similarity to speech production where the pitch is
primarily controlled by mass and tension in the vocal cords (mechanical
rather than acoustical resonator).  The vocal tract 'top resonator' has
its fundamental resonance in the range 200 to 800 Hz, generally a lot
higher than the vocal cord pitch and with no direct connection to it.
The Vox Humana designation is just to the point.

It would be interesting to know what is the internal height of the
boot, and the pitch of one of those pipes.  Then it should be easy
enough to characterize the type of boot resonance.  It is also of
special interest to know the size of the foot hole.  This is probably
important for the Q value of the boot resonance and should give some
decoupling from a rather unpredictable influence from the windchest and
playing valve.

Here I come to think of my trumpet instructor who long ago taught me
to tune my mouth cavity with my tongue -- the same kind of thing you
do to control the note when whistling.  I believe all wind instrument
players do this.

Once I also saw a research video of the vocal cords taken with a fiber
endoscope through the nose of a flutist and a trumpet blower.  They did
things with their vocal cords to control the upstream airflow (an
adjustable foothole!), something the players had probably learned by
practice, but were completely unaware of.

Johan Liljencrants