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As we play more loudly, we increase the pressure (which moves the operating point to the right) and we also increase the range of pressure. This means that the (larger) section of the curve we use is no longer approximately linear. This produces an asymmetric oscillation, whose spectrum has more higher harmonics. (Centre diagram.)
When we blow even harder, the valve closes for part of the part of the cycle when the pressure in the mouthpiece is low due to the standing wave inside the instrument. So the flow is zero for part of the cycle. The resultant waveform is 'clipped' on one side (diagram at right), and contains even more high harmonics. As well as making the timbre brighter, adding more harmonics makes the sound louder as well, because the higher harmonics fall in the frequency range where our hearing is most senstitive (See What is a decibel? for details). Conversely, playing in the linear range of this playing curve gives few high harmonics, so the minimum playing level of the clarinet can be very quiet indeed, a feature often used by composers.
While talking about decibels, we should mention that spectra, including those on our clarinet site are usually shown on a decibel scale. This means that one notices easily on the spectrum a harmonic that is say 20 dB weaker than the fundamental, even though it has 10 times less pressure and 100 times less power. What is important is that your ear notices it too, because of the frequency dependence referred to above. However, it is much more difficult to notice the presence of harmonics if you look at the waveform.
The behaviour of closed and open pipes are explained in Open vs closed
pipes (Flutes vs clarinets), which gives more explanation of this animation.
For the purposes of this simple introduction to clarinet acoustics, we shall
now make some serious approximations. First, we shall pretend that it is a
simple cylindrical pipe---in other words we shall assume that all holes are
closed (down to a certain point, at least), that the bore is cylindrical, and
that the mouthpiece end is completely closed. This is a crude approximation, but
it preserves much of the essential physics, and it is easier to discuss. (We
look at many of the complications in turn below and when explaining the real
results.) The natural vibrations of the air in the clarinet, the ones that cause it to
play notes, are due to standing waves. (If you need an introduction to this
important concept, see standing waves.) What
are the standing waves that are possible in such a tube?
The fact that the clarinet is open to the air at the far end means that the
total pressure at that end of the pipe must be approximately atmospheric
pressure. In other words, the acoustic pressure (the variation in pressure due
to sound waves) is zero. The mouthpiece end, on the other hand, can have a
maximum variation in pressure. Now the distance between a zero and a maximum on
a sine wave is one quarter of a wavelength. So, the longest standing wave that
can satisfy these conditions is one that has a wavelength four times the length
of the instrument, as shown at the top of the next figure: it has zero pressure
at the open end (called a pressure node). Inside the tube, the pressure
need not be atmospheric, and indeed the maximum variation in pressure (the
pressure anti-node) occurs in the mouthpiece. The standing wave is sketched
below. The bold line is the variation in pressure, and the fine line represents
the displacement or the amplitude of the vibration of the air molecules. The
displacement curve has an anti-node at the bell: air molecules are free to move
in and out at the bell but, in the approximation where the mouthpiece is closed,
there is little acoustic flow in the mouthpiece. (There is of course DC flow,
but that does not affect the standing waves directly.)
The frequency equals the wave speed divided by the wavelength, so this
longest wave corresponds to the lowest note on the instrument: D3 on a Bb
clarinet. (See standard
note names, and remember that clarinets are transposing instruments, so that
D3 is written as E3 for the Bb clarinet. Hereafter we refer only to the written
pitch.) You might want to measure the length of your instrument, take v = 350
m/s for sound in warm, moist air, and calculate the expected frequency. Then
check the answer in the note table. (You will find
that the answer is only approximate, because of end corrections.)
You can play (written) E3 with this fingering, but you can also play other
notes by overblowing--by changing your embouchure and changing the blowing
pressure. These other notes correspond to the shorter wavelength standing waves
that are possible, subject to the condition that the sound pressure be zero at
the bell and a maximum in the mouthpiece. The first three of these (solid lines)
are shown in the diagram below.
These three notes are approximately members of the harmonic series. Notes
with these frequencies have the (written) pitches shown below. The complete
harmonic series has the frequencies fo, 2fo,
3fo, 4fo, 5fo etc. The clarinet, under these
conditions, plays (approximately) the odd members of the series only. The
missing even harmonics and their waves are shown in dashed lines and in
parentheses in the diagrams. (You might like to compare this with the analogous
diagram for the flute,
which has all harmonics present.) There is also a more detailed discussion of
the harmonic series of open and closed
pipes.
Harmonics of the lowest note on a
clarinet. Recording
of notes played using only the fingering for the lowest note. The notes in the diagram are the harmonics of the fundamental. The notes in
the sound file are those played by overblowing, without using register keys
(which is one reason why they don't sound pretty and don't start cleanly). You
will observe that the played notes are successively flatter than the harmonic
frequencies. The third is slightly flatter than B4, the fifth is about a
semitone flat, the seventh more flat again. This is due to effects of the reed
(itself a deformable element approximately in parallel with the bore) and
effects of the bell (longer wavelengths penetrate less into the bell before
being reflected.)
When the clarinet is playing, the reed is vibrating at one particular
frequency. But, especially if the vibration is large, as it is when playing
loudly, it generates harmonics (see What is a sound
spectrum?). The reed vibration tends to have both odd and even harmonics.
However, in the low registers at least, only the odd harmonics set up, and are
in turn reinforced by, standing waves. Consequently, the sound spectrum in the
low registers has strong first and third harmonics, but weak second and fourth.
Observe that this note, which uses only a little more than half of the length
of the clarinet, is still lower than the lowest note on a flute. (For a Bb
clarinet, written C4 = sounding Bb3, the lowest note on a flute is B3 or C4.)
This is one of the big advantages of a closed pipe: you get low notes with a
shorter pipe.
For the moment, we can say the an open tone hole is almost like a 'short
circuit' to the outside air, so the first open tone hole acts approximately as
though the clarinet were 'sawn off' near the location of the tone hole. This
approximation is crude, and in practice the wave extends somewhat beyond the
first open tone hole: an end effect.
(For the technically minded, we could continue the electrical analogy by
saying that the air in the open tone hole has inertia and is therefore actually
more like a low value inductance. The impedance of an inductor in electricity,
or an inertance in acoustics, is proportional to frequency. So the tone hole
behaves more like a short circuit at low frequencies than at high. This leads to
the possibility of cross
fingering, which we have studied in more detail in classical and baroque
flutes.)
The frequency dependence of this end effect means that the low note played
with a particular fingering has a larger end effect than does the corresponding
note in the next register. If the clarinet really were a perfect cylinder, then
the registers would be out of tune: the intervals would be too wide. This effect
is removed by variations or perturbations of the cylindrical shape, including
the shape of the mouthpiece, an enlarging of the upper region of the bore, and a
gradual flare in the bottom half of the instrument, leading to the bell.
We mention in passing that the saxophone has two dedicated register holes for
the second register and an automated mechanism that allows only one key to
operate the hole appropriate to each end of the range. Multiple register keys
have been tried on the clarinet, but have never become popular.
The speaker key is the register key used for the second register. In higher
registers, other register holes are used: the altissimo register uses the hole
that is normally closed by the index finger of the left hand. This hole is
designed primarily as a tone hole, so it is bigger than it need or should be for
an ideal register hole. This defect is not so important because it is used only
for high frequencies, where its inertance is large. However some players partly
cover this hole (half-holing) when using it as a register hole.
When we play E3, the reed vibrates at the frequency of E3 (about 147 Hz for
a Bb clarinet). In a steady vibration, only harmonics (odd or even) of this
frequency are possible, and they are exactly harmonic. (See How harmonic are
harmonics?.) The odd harmonics are supported by the resonances of the
bore, and so the resultant sound spectrum is rich in odd harmonics but has
weaker even harmonics, at least at low frequencies. (See How the
reed and pipe work together.)
When we play B4, the reed
vibrates at the frequency for this note (about 440 Hz for a Bb clarinet),
which is three times the frequency of E3. Again, in steady vibration only
harmonics (odd or even) of this frequency are possible. The fifth resonance of
E3 (approximately G#5) is still present, as the impedance spectrum for B4 shows, but
there is nothing to put energy into a vibration at that frequency.
Observe also that we are now in the clarino register (about which see the
general comments on the page for B4). The
resonances that one might have expected to support the 3rd and 5th harmonics
of B4---ie the 9th and 15th harmonics of E3---are not close enough in
frequency to be of any help. However, the acoustic
response of the clarinet is strong enough to help all of the harmonics of
the reed to some extent, and the resultant sound spectrum has no strong
differences between even and odd harmonics. We return to discuss register holes in more detail below,
after we have discused the frequency response.
An open tone hole connects the bore to the air outside, whose acoustic
pressure is approximately zero. But the connection is not a 'short circuit': the
air in and near the tone hole has mass and requires a force to be moved. So the
pressure inside the bore under a tone hole is not at zero acoustic pressure, and
so the standing wave in the instrument extends a little way past the first open
tone hole. (There's more about this effect under Cut-off
frequencies.) Closing a downstream hole extends the standing wave even
further and so increases the effective length of the instrument for that
fingering, which makes the resonant frequencies lower and the pitch flatter.
The effect of cross fingerings is frequency dependent. The extent of the
standing wave beyond an open hole increases with the frequency, especially for
small holes, because it takes more force to move the air in the tone hole at
high frequencies. This has the effect of making the effective length of the bore
increase with increasing frequency. As a result, the resonances at higher
frequencies tend to become flatter than strict harmonic ratios. Because of this,
often one cannot use the same cross fingerings in two different registers.
A further effect of the disturbed harmonic ratios of the maxima in impedance
is that the harmonics that sound when a low note is played will not 'receive
much help' from resonances in the instrument. (Technically, the bore does not
provide feedback for the reed at that frequency, and nor does it provide
impedance matching, so less of the high harmonics are present in the reed motion
and they are also less efficiently radiated as sound. See Frequency
response and acoustic impedance. To be technical, there is also less of the
mode locking that occurs due to the non-linear vibration of the reed.) As a
result, cross fingerings in general are less loud and have darker or more mellow
timbre than do the notes on either side. You will also see that the impedance
spectrum is more complicated for cross fingerings than for simple fingerings,
especially in the region around 1.3 to 2 kHz.
We have studied cross fingerings more extensively on flutes than on
clarinets, by comparing baroque, classical and modern instruments. (There is of
course no baroque clarinet.) See cross
fingering on flutes or download a
scientific paper about crossfingering.
So high frequency waves are impeded by the air in the tone hole: it doesn't
'look so open' to them as it does to the waves of low frequency. Low frequency
waves are reflected at the first open tone hole, higher frequency waves travel
further (which can allow cross
fingering) and sufficiently high frequency waves travel down the tube past
the open holes. Thus an array of open tone holes acts as a high pass filter:
some thing that lets high frequencies pass but rejects low frequencies. (See filter
examples.) This is one of the things that limits the ability to play high
notes on the clarinet. The stiffness of the reed is another: a clarinet will
only play notes with frequencies lower than the natural frequency of the reed.
The player can alter the cut-off frequency, and this is one of the effects
used to achieve the spectacular glissando in the opening bars of Gershwin's
Rhapsody in Blue. The player begins by gradually sliding the fingers off
the tone holes, with the principal effect of changing smoothly the end effect at
the open tone hole, and so the pitch. This provides most of the glissando up to
the thorat register. From this register up, the player changes the position and
force of the lower lip on the reed, thereby changing its natural frequency and
also uses the resonances in the vocal tract. Normally, the resonances of the
instrument are so strong (have such high acoustic impedance--see below) compared
with those of the vocal tract that the latter make only modest changes to the
pitch. However, the player can reduce the strength of the instrument resonances
by lowering the cut-off frequency below that of the note being played. To do
this, the fingers are kept very near to the tone holes, partially covering them,
so that the tone holes are effectively very small. In this state, the resonances
of the vocal tract can be stronger than those of the instrument, so the note
played tends to follow that of the tract resonances, which the player increases
smoothly--with some considerable help from the change in the natural frequency
of the reed as the player's bite changes simultaneously.
This figure shows in black the calculated impedance spectrum for a simple
cylinder (the impedance is given in decibels:
20 log10(Z/Pa.s.m-3). A suitable reed attached to the
input of this tube would play near the frequencies of the peaks, which are in
the ratios of the odd harmonics 1:3:5:7 etc (see open and closed
pipes for more explanation, and/or compare with the comparable
curves for a saxophone). The curve in red is for the same cylinder, with a
simple bell at the end. Note that the bell makes the pipe longer, so each peak
and trough has been moved to lower frequencies, as expected. Note however the
change in the overal shape: all of the resonances are now weaker (extrema are
smaller). This is because the bell helps the sound waves in the bore to radiate
out into the air. (Incidentally, the presence of a large, effective bell is what
makes brass instruments loud: try playing a trombone with the tuning slide taken
off.) More sound radiated means less sound reflected, so the standing waves are
weaker.
This effect is less noticeable at low frequencies: the first maximum and
minimum are not weakened very much, because the bell is much smaller than the
wavelengths of the low frequency waves, and so is not very effective at
radiating these waves. (Incidentally, this frequency dependent effect of the
bell is what makes brass instruments brassy: the bell-less trombone has a sound
that is darker, as well as softer.)
The effect of the bell is that of a high pass filter: it allows high
frequencies to radiate out of the instrument, rather than reflecting back up the
bore to help establish the standing wave. In this, it acts much like the cut-off
frequency effect of the tone holes. In fact, one purpose of the clarinet's
bell chiefly is primarily to provide a high pass filter for the lowest few
notes, so that they have a cut-off frequency and so behave more similarly to the
notes produced with several tone holes open.
We could state this another way: for all of the notes on a clarinet except
the lowest few in the chalumeau and clarino registers, the array of open tone
holes acts as a high pass filter. For the few notes mentioned, however, few or
no tone holes are open, so the reflection condition is different, and so a
purely cylindrical clarinet would have a noticeably different timbre for these
few notes. So the bell performs a similar function for these notes. At
frequencies well above the cut-off, the bell has other functions, including
directional radiation of high frequencies.
So that is one reason why the clarinet departs from its approximately
cylindrical shape for the last few tone holes. It is also non-cylindrical for a
few centimetres at the other end. Which brings us to the effect of the
mouthpiece.
Here, the red curve is the same as the one we saw above: that for a cylinder
with a bell. Now the black curve is for a cylinder plus a bell plus a conical
constriction at the opposite end, which has approximately the same effect as a
mouthpiece: it reduces the cross sectional area of the bore gradually as it
approaches the reed. This has several effects.
First, it raises the impedance overall. This is because it functions like a
horn or an impedance matching transformer, connecting a small area (where the
same flow would require more pressure and therefore high impedance) to a large
area.
Second, it does this more effectively at high frequency for maxima (all of
which are shifted by 8-10 dB) than for minima (where the high frequency minima
are much less deep than are the low frequency minima). This however is not
important to the clarinet, because it operates at maxima, but is an important
consideration in the design of flutes.
Third, it makes the peaks and troughs asymmetric. At high frequencies, the
minima move to lower frequencies while the maxima occur at higher frequencies.
Of course the bore is not exactly cylindrical: there are local variations in
radius, particularly near the barrel. The effects of these are to make subtle
differences to the relative tuning of the registers.
On this figure, the single dots are the experimentally measured impedance
spectrum for E3, with a value
of the compliance corresponding to a hard reed. The continuous line (actually
the experimental points joined together) shows the spectrum for a soft reed. At
low frequencies, there is not much difference, but you can already see a slight
difference in frequency: the hard reed plays sharper, all else equal. As you go
to higher frequencies, you see that the soft reed gives lower peaks. Lower peaks
are harder to play, so the hard reed makes it easier to play high notes.
(Unfortunately, a hard reed also makes it easier to play squeaks.)
To understand more about the detailed shape of these impedance curves, see
the discussion of the experimental results for E3.
At low frequencies, however, the effect is greater, and this register hole
substantially reduces the height of the second maximum. It also raises its
frequency, and takes it out of the harmonic series with other peaks. These
effects, especially the former, make the second peak harder to play, and so
(provided you use the appropriate embouchure), the instrument will play the
third maximum, which is C#6.
We have not mentioned the first maximum. It has already been weakened and
shifted by the speaker key, which is open here. However, its maximum is not all
that weak and a careless embouchure and low blowing pressure could find you
dropping down to a muffled low note near that frequency. Further, the fourth
peak is pretty high, too. With a hard reed and blowing hard, there's a danger of
jumping up to this note, too. Which is why the altissimo register is hard to
play. (There is more discussion of the altissimo register on the experimental page
for C#6).
The clarinet is a 'closed' pipe
The clarinet is open at the far end or
bell. But it is (almost) closed at the other end. For a sound wave, the tiny
aperture between reed and mouthpiece---a much smaller cross section than the
bore of the instrument---is enough to cause a reflection almost like that from a
completely closed end. The rest of the clarinet is approximately cylindrical. Of
course there are a few irregularities in the bore (discussed later), and there's
the bell, but if you take the bell off it doesn't make a huge difference to the
sound of most notes. (In fact if you take the lower joint off as well as the
bell, you still get a reasonable clarinet sound for the available ranges.)
How the reed and pipe work together
To sum up the preceding sections:
the bore of the clarinet has several resonances, which are approximately in the
ratios of the odd harmonics, 1:3:5, but successively more approximate with
increasing frequency--we'll see why below under frequency
response. The reed has its own resonance--which is approximately what you
hear when you produce a squeak. One good way to produce a squeak is to put your
teeth on the reed. In normal playing, with your lower lip touching the reed, you
damp (ie reduce the strength of) the reed's resonances considerably. This allows
the resonances of the bore to 'take control'. To oversimplify somewhat, the
clarinet normally plays at the strongest bore resonance whose frequency is lower
than that of the reed. (We shall see below how register
holes are used to weaken the lower resonance or resonances and thus make one
of the higher resonances the strongest.)
Spectrum and registers of the clarinet
In the section on harmonics
above, each of the standing waves in the sketch above corresponds to a sine
wave. The sound of the clarinet is a little like a sine wave when played softly,
but successively less like it as it is played louder. To make a repeated or
periodic wave that is not a simple sine wave, one can add sine waves from the
harmonic series. So E3 on the clarinet
contains some vibration at E3 (fo), some at B4 (3fo), some
at G#5 (5fo) etc. The 'recipe' of the sound in terms of its component
frequencies is called its sound spectrum.
The predominant presence of odd harmonics in the lowest or chalumeau
register gives this register its characteristic 'hollow' timbre. (See the
discussion of general features of the chalumeau register on the page for the
note E3.) From
about E4 up to
A#4, the
even harmonics become more important. This range overlaps approximately with
what clarinettists call the throat register. The notes in this range have
only two bore resonances that coincide well with harmonics, and so pitch of
notes in this range is easier to 'bend' than that of notes in the chalumeau
register. Once the speaker key is used, the systematic difference between odd
and even harmonics almost disappears, and the timbre becomes bright and clear.
(See the discussion of general features of the clarino register on the
page for the note B4.) This
difference in timbre is one awkwardness associated with the 'break' between A#4
and B4; the other is the fingering difficulty 'to cross the break'---moving
several fingers and a thumb simultaneously. The register that uses only the
speaker key as a register hole (see below) is called the clarino
register. The altissimo register uses the hole for the left index
finger as a register hole as well. (See the discussion of general features of
the altissimo register on the page for the note C#6.)
Opening tone holes
If you open the tone holes, starting from the far
end, you make the pressure node move up the pipe, closer to the mouthpiece---it
is very much like making the pipe shorter. Starting near the bell, each opened
tone hole raises the pitch by a semitone, which requires a pipe that is about 6%
shorter. After you open all of the right hand finger holes, as shown below, you
have the fingering for C4, which is shown
below.
Register holes
Holes can also serve as register holes. For instance, if
you play Bb3
(call this frequency fo) and then open the register key (or speaker
key), you are opening a hole one third of the way down the (closed part of the)
instrument. (See the middle diagram below.) This hole disrupts the fundamental,
but has little effect on the higher harmonics, so the clarinet 'jumps up' to F5
(3fo). One can imagine a clarinet that had a separate register hole
for each note, but that would be a lot of keys. In fact, only one register hole
is used for the second register from B4 to C6. Looking at the diagram below, we
see that it its position is a compromise: for B4 it is well below the pressure
node, and for C6 well above. This is not a big problem in practice. The register
hole is small, so it is not really a 'short circuit', except at low frequencies.
So it does not too much affect the third and higher harmonics. It does however
disrupt the fundamental, and that is its purpose: to stop the instrument
dropping down to its bottom register.
Let us open an aside to answer a question that has been asked a few times:
"If the speaker key destroys the first resonance, but none of the others, why
is it that the sound spectrum in the clarino register has only harmonics 3, 6,
9 etc? What happened to the fifth harmonic?" One can verify the basis of this
question by looking at the spectra for E3 and B4.
Cross fingering
On the modern clarinet, successive semitones are usually
played by opening a tone
hole dedicated to that purpose. Being a closed,
cylindrical pipe the clarinet overblows a twelfth, and so one would need
eighteen tone holes to cover the chalumeau and throat registers before repeating
fingerings using the speaker key as a register
hole. Because players don't have this many fingers, the clarinet requires
keys and clutch mechanisms, so that one finger can close or open two or more
holes. Cross fingering is also used. For instance, one of the fingerings for the
note B3 is a
simple fingering: the right hand index finger closes its hole, and the right
ring finger opens tone hole with a key (upper figure below). The other is a
cross fingering: all three fingers on the left hand, plus the middle finger on
the right close their holes. Why do these both give the same note?
Other effects of the reed
As well as controlling the flow of air, the
reed has a passive role in clarinet acoustics. When the pressure inside the
mouthpiece rises, the reed is pushed outwards. Conversely, suction draws the
reed in towards the bore. Thus the reed increases and decreases the mouthpiece
volume with high or low pressure. (Techncially, we say it is a mechanical
compliance in parallel with the bore.) Indeed, it behaves a bit like an
extra volume of air, which could also be compressed and expanded by changing
pressure in the mouthpiece. It has the effect of lowering the frequency of each
resonance a little. However, soft reeds move more than hard reeds, so soft reeds
lower the frequency more than do hard reeds. Further, this effect is greater on
high notes than on low, so soft reeds make intervals narrower and hard reeds
make them wider. This is useful to know if you have intonation problems. (See
also tuning.)
Cut-off frequencies
When we first discussed tone
holes, we said that, because a tone hole opens the bore up to the outside
air, it shortened the effective length of the tube. For low frequencies, this is
true: the wave is reflected at or near this point because the hole provides a
low impedance 'short circuit' to the outside air. For high frequencies, however,
it is more complicated. The air in and near the tone hole has mass. For a sound
wave to pass through the tone hole it has to accelerate this mass, and the
required acceleration (all else equal) increases as the square of the frequency:
for a high frequency wave there is little time in half a cycle to get it moving.
Frequency response and acoustic impedance of the clarinet
The way in
which the reed opens and closes to control the air flow into the instrument
depends upon the acoustic impedance at the position of the reed, which is why we
measure this quantity. The acoustic impedance is simply the ratio of the sound
pressure at the measurement point divided by the acoustic volume flow (which is
just the area multiplied by the particle velocity). If the impedance is high,
the pressure variation is large and so it can control the reed. In fact, the
resonances, which are the frequencies for which the acoustic impedance is high,
are so important that they 'control' the vibration of the reed, and the
instrument will play only at a frequency close to a resonance. (There is further
explanation on What is
acoustic impedance and why is it important?). The section below shows how
the major features of the clarinet's shape give rise to its acoustic impedance
spectrum, and thus to how it operates.
The effect of reed hardness
In the preceding section we have ignored the
compliance
of the reed, discussed above. This acts in parallel with the bore, and its
impedance decreases at high frequency, so its effect is to reduce the rise in
impedance with frequency: softer reeds give lower overall impedance at high
frequency. Further, the very high resonances are weaker and occur at lower
frequency when you use a soft reed.
More about register holes
Now that we know about impedance spectra, we
can better understand the effect of register holes, which we met above.
In the graph below, the single dots are the experimentally measured impedance
spectrum for E5, which plays at
the second maximum of the curve. The continuous line is that for C#6, which
plays at the third maximum. The only difference in fingering is that the hole
for the left index finger is opened, and here acts a register hole. Notice that
its effect is small at high frequencies. (As we saw above,
the inertia of air in the hole effectively 'seals' the hole, so that high
frequency waves pass by as though it were closed).
More to comeWe shall continue to add to this site as we find time and as our research results are published. In particular, we shall soon add a section on vocal tract effects, which is the PhD project of Claudia Fritz (see the article below). One FAQ is 'Why don't we provide a Virtual Clarinet service like the Virtual Flute?' The answer is that we are working on this and hope to have one this year.The question about the importance of the material from which the instrument is made is one of those addressed on our FAQ in Music Acoustics. |
Acoustical Society of America's Award for Technical Writing for Acoustics Professionals. |
We have recently published two conference papers concerning the influence of the player's vocal tract on the pitch and timbre of wind instruments:
For further reading, we recommend
Also in this series:
Research and scholarship possibilities.
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AcknowledgmentOur research work on clarinets is supported by the Australian Research Council and by Yamaha Music Australia.If your right thumb is sore from supporting clarinets, see this link. |
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