Flute acoustics: an introduction
Flute acoustics: an introduction
How does a
flute work? This introduction gives first the simple explanations, then the subtleties. It requires no mathematics, nor any special acoustics
knowledge. Some more technical references
are listed near the end of this page.
&&&&&&To set the mood, listen
play some Debussy.
The flutist blows a rapid jet of air across the embouchure hole. The pressure
inside the player's mouth is above atmospheric (typically 1 kPa: enough
to support a 10 cm height difference in a water manometer). The work done
to accelerate the air in this jet is the source of power input to the
instrument. The player provides power continuously: in a useful analogy
with electricity, it is like DC electrical power. Sound, however, requires
an oscillating motion or air flow (like AC electricity). In the flute,
the air jet, in cooperation with the resonances in the air in the instrument,
produces an oscillating component of the flow. Once the air in the flute
is vibrating, some of the energy is radiated as sound out of the end and
any open holes. A much greater amount of energy is lost as a sort of friction
(viscous loss) with the wall. In a sustained note, this energy is replaced
by energy put in by the player. The column of air in the flute vibrates
much more easily at some frequencies than at others (i.e. it resonates
at certain frequencies). These resonances largely determine the playing
frequency and thus the pitch, and the player in effect chooses the desired
set of resonances by choosing a suitable combination of keys. In this
essay, we look at these effects one by one.
The air jet vibrates
The jet of air from the player's lips travels across the embouchure-hole
opening and strikes against the sharp further edge of the hole. If such
a jet is disturbed, then a wave-like displacement travels along it and
deflects it so that it may blow either into or out of the embouchure hole.
The speed of this displacement wave on the jet is just about half the
air-speed of the jet itself (which is typically in the range 20 to 60
metres per second, depending on the air pressure in the player's mouth).
The origin of the disturbance of the jet is the sound vibration in the
flute tube, which causes air to flow into and out of the embouchure hole.
If the jet speed is carefully matched to the frequency of the note being
played, then the jet will flow into and out of the embouchure hole at
its further edge in just the right phase to reinforce the sound and cause
the flute to produce a sustained note. To play a high note, the travel
time of waves on the jet must be reduced to match the higher frequency,
and this is done by increasing the blowing pressure (which increases the
jet speed) and moving the lips forward to shorten the distance along the
jet to the edge of the embouchure hole. These are the adjustments that
you gradually learn to make automatically when playing the flute. Flutists
are usually taught to reduce the lip aperture when playing high notes.
The figure at left shows a jet striking an edge and being alternately
deflected up and down. The sketch at right represents a cross section
of the flute at the embouchure.
The flute is an open pipe
The flute is open at both ends. It's obvious that it's open at the far
end. If you look closely at someone playing a flute, you'll see that,
although player's lower lip covers part of the embouchure hole, s/he leaves
a large part of the hole open to the atmosphere, as shown in the sketch
above. Let's begin by considering a pipe that is simpler than a flute.
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 head is cylindrical, and we shall replace the side
mounted embouchure hole with a hole at the end. In fact, this is more
than a flute. It's a crude approximation, but it preserves much of the
essential physics, and it's easier to discuss. (We shall introduce the
effects of finger holes and the embouchure geometry below, or you can
consult our
on this topic.)
The animation below is from , which gives a more detailed
explanation. It shows a shows a pulse of high pressure reflecting in a
pipe open to the air at both ends. Note that a complete cycle of vibration
is the time taken for the pulse to travel twice the length L of the flute
(once in each direction). The pulse travels at the speed of sound v, so
the cycle would repeat at a frequency of v/2L, as we shall see again below.
The natural vibrations of the air in the flute are due to resonances.
The reflecting pulse of air in the animation is an example of such a
resonance, the fundamental or lowest resonance of the flute. There is
more about resonances in the page on . What standing waves or resonances are possible in an open
cylindrical tube? We shall now answer this question in terms of sine
waves and harmonics.
The fact that the pipe is open to the air at the ends means that the
total pressure at the ends must be approximately atmospheric pressure
or, in other words, the acoustic pressure (the variation in pressure
due to sound waves) is zero. These points are called pressure nodes,
and they effectively lie past the end of the tube by a small distance
(about 0.6 times the radius, as shown: this distance is called the end
correction). Inside the tube, the pressure need not be atmospheric,
and indeed for the first resonance, the maximum variation in pressure
(the pressure anti-node) occurs at the middle. The standing wave is
sketched below. The bold line is the variation in pressure, and the
fine line represents the variation in the displacement of the air molecules.
The latter curve has anti-nodes at the ends: air molecules are free
to move in and out at the open ends. (Note that a node for pressure
and a node for air motion are not the same thing: indeed, pressure nodes
often coincide with motion anti-nodes and vice versa. See . The difference between closed and open pipes is
explained in , which compares them using
wave diagrams, air motion animations and frequency analysis, or some
The wave shown above is the longest standing wave that can satisfy
this condition of zero pressure at either end. In the figure below,
we see that it has a wavelength twice as long as the flute. The frequency
f equals the wave speed v divided by the wavelength l,
so this longest wave corresponds to the lowest note on the instrument:
C4 on a C foot instrument. (Flutists please note: this page uses the
not the names sometimes used by flutists.) You might want to measure
the length L of your flute, take the speed of sound as v = 350 metres
per second for sound in warm, moist air, and calculate the expected
frequency. Then check the answer in the . (You will find that the answer is only approximate, because
You can play C4 on the flute with this fingering, but you can also
play other notes by blowing harder, or by narrowing the lip aperture
(either gives a faster jet). These other notes correspond to the shorter
wavelength standing waves that are possible, subject to the condition
that the sound pressure be zero at both ends. The first several of these
are shown in the diagram below.
The series of notes with frequency fo, 2fo,
3fo etc is called the harmonic series, and notes with these
frequencies have the pitches shown below. With all the tone holes closed,
the first ten or so resonances of the flute are approximately in this
ratio, so you can play the first seven or eight of the series by closing
all the tone holes and blowing successively harder (or by narrowing
the lip aperture). Note the half sharp on the seventh harmonic - it
falls roughly midway between A6 and A#6. (You might be interested to
compare this with the , which has only the odd
harmonics present. There is also a more detailed discussion of the harmonic
series of .)
Eight 'harmonics' of the lowest note on a flute.
Each of the standing waves in the sketch above corresponds to a sine
wave. The sound of the flute is a little like a sine wave (a very pure
vibration) 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 C4 on
the flute contains some vibration at C4 (let's call its frequency fo),
some at C5 (2fo), some at G5 (3fo), some at C6
(4fo), etc. The 'recipe' of the sound in terms of its component
frequencies is called its spectrum. (See
for an explanation.) Looking at
(Open a ) you will see that, at pianissimo, the first
harmonic (fundamental) and the frequency of the note C4 dominates, and
that the higher harmonics become more important as the note is played
more loudly, and as the flute develops a richer tone and sounds less
and less like a sine wave. (For a detailed explanation, see .)
How the air jet and pipe work together
To sum up the preceding sections: the bore of the flute has several resonances,
which are approximately in the ratios of the harmonics, 1:2:3:4 etc, but
successively more approximate with increasing frequency--we'll see why
below under . The air
jet has its own natural frequency that depends on the speed and length
of the jet. To oversimplify somewhat, the flute normally plays at the
strongest bore resonance that is near the natural frequency of the jet.
(We shall see below how
used to weaken the lower resonance or resonances and thus make one of
the higher resonances the strongest.)
When the flute is playing, the jet is oscillating at one particular
frequency. But, especially if the vibration is large, as it is when
playing loudly, it generates harmonics (see ). For low notes, the first several harmonics
are supported by standing waves. However for high notes, the resonances
of the flute are no longer harmonic, so only a small number of harmonics---only
one in the third and fourth octave are supported by resonances of the
bore. Played loudly however, harmonics of the vibration are present
in the spectra, as you can see by looking at the spectra for .
Opening tone holes
If you open the tone holes, starting from the far end, you make the pressure
node move closer up the pipe - it's rather like making the pipe shorter.
On the Boehm flute, each opened tone hole raises the pitch by a semitone.
After you open 4 holes on a C foot flute, as shown below, you have the
fingering for ,
which is shown below. (Open a ).
For the moment, we can say 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 flute were 'sawn off' near the location of the tone hole.
We shall return to qualify these assumptions below when we discuss register
holes and cross fingerings. (For the technically minded, we could continue
the electrical analogy by saying that the open tone hole is actually
more like a low value inductance, and so it behaves more like a short
circuit at low frequencies than at high. We return to this point when
discussing
Register holes
Holes can also serve as register holes. For instance, if you play C4 and
then lift your left thumb, you are opening a hole halfway down the instrument.
This makes the fundamental and the odd harmonics impossible, but hardly
affects the even harmonics, which have a node there. So the flute 'jumps
up' to C5 (2f1), and will also play C6, G6 etc. Here the register
hole makes the played note (at least) one octave higher, because it is
halfway along the working length of the flute and so permits the second
harmonic of the fundamental C4. The example shown is not a standard fingering,
but a register hole at half the length is used for the standard fingerings
and others.
When the desired wavelength is short (i.e. for high notes) one can
open a register hole at a different fraction of the length. For example,
the fingering for
uses a register hole at approximately one third of the working length
for G4, and so facilitates the third harmonic of G4 (and thus produces
a note a twelfth higher than G4). The fingering for
also uses the working length for G4, but has a register hole about one
quarter of the way along, and so facilitates the fourth harmonic.
One of the alternative fingerings for
uses the working length for D#4 but has two register holes, at one quarter
and one half the wavelength. Notes in the third octaves of all flutes
rely heavily on using tone holes as register holes. Specific examples
are explained on the pages for these notes. (See
and choose a note above D#6.)
Acoustic impedance of the flute
The way in which the jet flows into and out of the flute depends upon
the acoustic impedance at the embouchure hole, which is why we measure
this quantity. The acoustic impedance is the ratio of the sound pressure
to the oscillating air flow. (See
for more detail.) If the impedance is low, air flows in
and out readily and a loud sound can be produced. In fact, the resonances,
which are the frequencies for which the acoustic impedance is very small,
are so important that they 'capture' the behaviour of the air jet, and
so the flute will play only at a frequency very close to a resonance.
We discuss the acoustic impedance below, under . There is further explanation on
and also a discussion
Cross fingering
On the modern or Boehm flute, successive semitones are played by opening
a tone hole dedicated to that purpose. There are twelve semitones in an
octave, so that one needs to open twelve keys in a chromatic scale before
going from say D4 in the first register to D5 in the second register.
(Because of the use of register holes for D5 and D#5, the fingerings do
not repeat exactly over an entire octave, but only between E4/5 and C#5/6.)
Twelve holes exceeds the number of fingers on standard players, particularly
when the right thumb is employed to support the instrument. Boehm's key
system employs clutches so that one finger can close more than one hole.
Flutes in the
periods had few keys. (See .) They had six open holes covered
by the three large fingers on each hand. Opening these holes in sequence
gave the 'natural' scale of the instrument, which was D major. Writing
X for closed hole and O for open, the fingering chart for such an instrument
is approximately:
Reproduction of a .
with E5 to B5 using the same fingerings as E4 to B5. (Much more detail
on fingerings and how they work is given on the pages for
On such instruments, cross fingerings are used to produce some of
the intervening notes. In a cross fingering, further holes are closed
downstream from the first open hole. For example, the fingering for
F4/5 : XXX XOX
Compare this with the fingering for ,
which is XXX XOO. For F#, the standing wave extends down the bore of the
instrument past the first open hole. The extended wave is stronger in
the baroque or classical instruments than in the modern flute because
the holes are smaller and so the acoustic impedance between the bore and
the outside field is smaller. (In the modern flute, the open hole acts
more like an acoustical short circuit, so the flute behaves almost as
though it were 'sawn off' at a point not far below the first open hole.
So the modern flute has relatively small end corrections.) Closing the
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. This has the effect of making the effective
length of the flute increase with increasing frequency. As a result,
the impedance minima at higher frequencies tend to become flatter than
strict harmonic ratios. One effect of this is that some cross fingerings
cannot be used in both first and second register: the cross fingering
used for G# in the first register will be too flat in the second register,
or may not even play a note near G# at all. (See for example the fingerings
on the baroque flute.)
A further effect of the disturbed harmonic ratios of the minima 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 jet at that frequency, and nor does it provide
impedance matching, so less of the high harmonics are present in the jet
and they are also less efficiently radiated as sound.) As a result, the
sound spectra for notes such as
on the baroque flute have weaker higher harmonics, and so these notes
are less loud and have darker or more mellow timbre than do the notes
on either side.
We noted previously that the 'natural' scale of these instruments
is D major: in D and in B minor they use no cross fingerings, so their
timbre is bright. In Eb major or C minor, their timbre is dark and they
play more quietly. These observations are also true of the baroque oboes,
baroque (and modern) recorders, and approximately true of the baroque
bassoon. I suspect that this has contributed to the different qualities
associated with different keys: keys with a couple of sharps are associated
with bright and relatively loud winds, whereas keys with a few flats
are associated with dark and quiet winds. For more information on the
acoustics of cross fingerings,
on the topic.
'Lipping' up and down
The design of a flute involves compromises and many notes require slight
pitch adjustment by the player. (See .) Players lower the pitch mainly by a combination of drawing
the chin back or pushing it forward, rolling the flute's embouchure hole
towards them or away and changing the jet geometry. These actions do several
things: (i) they increase the fraction of the embouchure hole that is
covered by the lower lip, thereby decreasing the size of the hole opening
to the atmosphere, (ii) they decrease the solid angle available into which
the sound wave can radiate (informally: they 'get in the way of' the radiation),
and (iii) they decrease the length and change the angle of the jet .
Effects (i) and (ii) increase the effective length of the flute and
so make the resonant frequencies lower and the note flatter. Rolling
the embouchure away and/or extending the lower jaw have the reverse
effects, and so raise the pitch. Technically, these actions work because
they change the radiation impedance at the embouchure: when a note is
'lipped down', the embouchure hole is "less open" (both the hole and
angle are smaller so there is more impedance to radiation from the bore
to the external field). The effects of the jet itself are more complicated.
We have measured these effects explicitly by installing our impedance
measuring equipment in a flute head and measuring the impedance at the
embouchure hole. (This is the impedance of the radiation
field, 'looking out' from inside the blowhole, which is partially blocked
by the lower lip. The flutist's lip and face also provide a baffle that
reduces the angle for radiation. These results are reported in a recent
conference paper - see our
site.) The interval that can be lipped depends on
the details of the impedance spectrum and on some properties of the
jet. It is easier to adjust the pitch of notes using a short length
of tube, whose impedance spectra have fewer and shallower harmonic minima
than do those of long tube fingerings. The analogous effects are much
bigger on the ,
and are described on that site.
The cork and the 'upstream space'
Between the point where the embouchure riser meets the main bore of the
flute and cork in the closed end of the instrument is a small volume of
air. The cork is normally positioned to be about 17 mm from the centre
of the embouchure hole (the exact value varies from player to player -
see ). Any very substantial variation seriously upsets the
internal tuning of the flute. So how does this work?
This 'upstream air' acts like a spring - when you compress it, the
pressure rises. The air in the embouchure riser tube can be considered
as a mass. Together they can resonate like a mass bouncing on a spring
(ie they form a ).This has a resonance over a broad range of frequencies,
but centred at about 5 kHz. At much lower frequencies, which is to say
over the playing range of the flute, it acts as an impedance in parallel
with the main part of the bore, but an impedance whose magnitude decreases
with frequency. The primary effect of this is good: with the cork correctly
placed, it compensates for the frequency dependent end effects at the
other end of the flute and so keeps the registers in tune with each
other. On the other hand, it does reduce the variation in impedance
with frequency when the frequency approaches the Helmholtz resonance,
and so is one of the effects that limits the upper range of the instrument.
If you push the cork in, as Charanga style players do, you can go further
up into the fourth octave, but at the expense of having an instrument
whose octaves are badly out of tune. If you want to know more about
this effect, download our
about it. To scale the highest reaches of the flute's range,
search for 'high playability' fingerings on the
and the report on
The important message for flutists, however, is this. Among the orchestral
winds, the flutes have the simplest method of adjusting their internal
intonation. If your octaves are narrow, try pushing the cork in a little.
If they are wide, pull it out. You will of course have to move the tuning
slide as well. See also .
Cut-off frequencies
When we first discussed , 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.
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 )
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.
The cut-off frequency for the Boehm flute is a little above 2 kHz.
For example, in the acoustic response curve for ,
you will see that the first four or five resonances become gradually
weaker with frequency--this is due to the increasing importance of energy
losses due to the 'friction' (viscous loss) between the air and the
wall. Above 2 kHz, however, the resonances are suddenly much weaker:
waves with these frequency propagate down the bore and are radiated
gradually from successive tone holes. The remaining weak standing waves
produce resonances with a different frequency spacing, as we shall see
in the next section. Before we move on, however, compare the A4 graph
with that for .
The latter is the lowest note on the flute, so there are no open tone
holes and therefore no cut-off frequency. Consequently, the resonances
fall gradually and uniformly with frequency over the whole range.
For the lowest note or two on the flute, there is no array of open
holes and so there is no cutoff frequency due to that effect. In principle,
if the higher harmonics were strong enough, one would expect this to
lead to a different timbre of these notes. One way to avoid this--a
way that is used for the oboe and clarinet--is to supply a bell that
radiates high frequencies but not low, and which has a cut-off frequency
comparable with that of the tone hole array. The flute has less radiated
power at high frequencies than do the oboe and clarinet, so the need
for a bell to 'homogenise' the timbre is rather less. However, a bell
would increase high frequency radiation, both for long and short tube
notes, and the pinschofon is the name of such an instrument.
(This technical paper gives measurements and analyses of
in baroque, classical and modern flutes.
There is also a more detailed explanation of cut-off frequencies and their effects .)
Frequency response of the flute
So now let's look at the acoustic impedance spectrum of the modern flute.
We'll choose the fingering used for C#5 and C#6, with nearly all tone
holes open. It is shown in the graph below. (This graph covers a wide
frequency range, but does not show much fine detail. For more detail,
Below about 2.5 kHz, this curve looks like that for a
with about half the length of the flute, because
the tone holes are open in the bottom half of the instrument. The first
three minima all support standing waves, and you can therefore play
the notes C#5, C#6 and G#6 with this fingering. However, above 2.5 or
3 kHz, the resonances become much weaker. This is because of the high
pass filter mentioned above under .
Higher still, around 5 kHz, the resonances almost disappear completely,
because they are shorted out by the Helmholtz resonator discussed above
under . Above this
frequency range, the Helmholtz resonator is no longer a short circuit,
so the resonances reappear, although they are weak because of the 'friction'
of the air with the walls (increased effect of viscothermal losses at
high frequencies).
Notice, however, an important difference. The spacing of peaks or
troughs in the graph at the low frequency end is about 600 Hz (roughly
the frequency of C#5, and corresponding to a standing wave in the half
of the flute with no tone holes). At high frequencies, the spacing of
peaks or troughs is about 260 Hz. This is the frequency of C4, and corresponds
to the standing wave over the whole length of the flute. At these high
frequencies, the wave in the bore of the flute propagates straight past
the open tone holes, not 'noticing' that they are there, because of
the inertia of the air discussed above under . A human player cannot blow air fast enough to excite
a fundamental frequency in this range. For a loud note in the normal
range, some high harmonics will fall in this range. However, these harmonics
have little need of the flute as a resonator, so the tuning of these
ultra high resonances has little practical importance.
Finally, notice the general shape of the curve, which has a broad
maximum at about 9 or 10 kHz. This is due to the relatively narrow embouchure
riser: the tube of air linking the main bore to the lip plate. The air
in this tube and a little bit outside at both ends (the end effects)
is itself a resonant tube, whose resonance occurs over a broad range
of frequencies because the tube's width is comparable with its length.
The solid line on the graph is the theoretical impedance of a truncated
cone having the geometry of the embouchure riser, including end effects.
Some more detail is given in our
about the end effects on the flute.
More detailed information
For more flute acoustics, return to the . Most of the individual note pages have descriptions
of the acoustical effects relevant to their particular fingerings. If
you are a flutist, you'll also want to check out the tools provided on
There is more detailed discussions of flute acoustics in several of
that concern the acoustics of flutes and other instruments.
The most recent paper concerns .
Several flute questions are addressed on our . These include multiphonics, undertones, end effects and the importance of the material from which
the flute is made.
For further reading, we recommend
'': the PhD thesis of Paul Dickens.
A technical reference: The Physics of Musical Instruments
by N.H. Fletcher and T.D. Rossing (New York: Springer-Verlag,
Other references, some less technical, are listed .
If you found this page already too technical, try .
For background on topics in acoustics (waves, frequencies,
resonances etc) see .
has much on the evolution and history of the flute.
Also in this series:
Research and scholarship possibilities.
See our page on
in music acoustics at UNSW.
including possible research projects
for music students.
phone 61-2- (UT + 10, +11 Oct-Mar)
birthday, theory of relativity!
As of June
2005, relativity is 100 years old. Our contribution is
It explains the key ideas in a short multimedia
presentation, which is supported by links to broader and deeper explanations.}