Acoustics of flute
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ContentsTable of figures ... 3 Introduction ... 5 Categories of Flute ... 4 Construction ... 5 Sound generation ... 6 Slit ... 6 Edge... 7 Air column ... 9 Over-blowing: ... 9 Rolling in: ... 10
Air cavity characteristics ... 10
Tone holes ... 11
Register holes ... 11
Cross fingering ... 12
Cut-off frequencies ... 13
The cork and head joint ... 13
Effect of head joint stopper ... 13
Acoustic impedance of the flute ... 15
Variation of impedance with pressure ... 16
Effect of lip-to-edge distance ... 17
Frequency response of the flute ... 18
Conclusion ... 19
Table of figures
Figure 1 Fipple flute ...5
Figure 2 Construction of a modern western flute ...6
Figure 3 Undulations produced by blowing air through a slit ...6
Figure 4 Effect of increasing speed of jet of air through a slit. Faster jet produces higher frequency ...7
Figure 5 Formation of swirl from an edge ...7
Figure 6 Pressure feedback from an edge to a slit ...8
Figure 7 Edgetone regimes of a slit-edge interaction based on variation of gap and velocity of air stream ...8
Figure 8 Frequency regimes as a result of slit-edge-air column interaction ...9
Figure 9 Over-blowing for jumping to a higher register ... 10
Figure 10 Rolling in to change the frequency ... 10
Figure 11 Effect of opening tone holes ... 11
Figure 12 Effect of opening Register holes ... 12
Figure 13 Cross fingering utilizing high pass filter ... 12
Figure 14 Embouchure and head joint with adjusting screw ... 13
Figure 15 Effect of varying of length of upstream air column ... 184
Figure 16 Apparatus for measurement of impedace of flute ... 185
Figure 17 Variation of acoustic impedance with pressure and phase ... 176
Figure 18 Variation of acoustic impedance with lip-to-edge distance ... 177
The flute is a musical instrument of the woodwind family. Unlike woodwind instruments with reeds, a flute is an aerophone or reedless wind instrument that produces its sound from the flow of air across an opening. According to the instrument classification of Hornbostel-Sachs, flutes are categorized as Edge-blown aerophones.
Aside from the voice, flutes are the earliest known musical instruments. A number of flutes dating to about 40,000 to 35,000 years ago have been found in the Swabian Alb region of Germany. These flutes demonstrate that a developed musical tradition existed from the earliest period of modern human presence in Europe.
A flute produces sound when a stream of air directed across a hole in the instrument creates a vibration of air at the hole. The air stream across this hole creates a Bernoulli. This excites the air contained in the usually cylindrical resonant cavity within the flute. This can be viewed similar to a Helmholtz resonator explained later. The pitch of the sound produced can be changed by opening and closing holes in the body of the instrument, thus changing the effective length of the resonator and its corresponding resonant frequency. Also by varying the air pressure, and hence air velocity, and changing lip-to-edge distance the pitch and registers of a note can be changed. A large flute, hence a larger air stream, or increased air stream velocity produces louder sound. A flute's volume can generally be increased by making its resonator and tone holes larger. The example being an organ pipe, which is several inches wide, sounding louder than a concert flute, which is only fraction of an inch wide.
CATEGORIES OF FLUTE
Basic three categories of flutes on the basis of mode of playing are:
1. Fippled flutes
These flutes, such have a duct that directs the air onto the edge. In fippled or ducted flutes, a precisely formed and placed windway will compress and channel the air to the labium ramp
edge across the open window. These are easy to play but take the degree of control away from the player.
2. End blown flutes
These are played from one end of the flute but should not be confused with the fippled flutes. These do not have any fipple like structure to guide the air in.
3. Side blown flutes
These are the most common flutes that we come across. These have a hole in the side of the flutes from where the wind is blown. The wind has to be correctly blown to produce sound. This type of flutes are particularly difficult for beginners but give a lot of control over the variations possible.
Flute is made in the form of an open cylinder about 26 inches in length with an inside diameter of about 3/4", open at one end. The embouchure (hole to blow the air in) is near one end (approximately 17 mm from one end) and constitutes a second open end, making the flute an open cylinder in harmonic content. The western flute is made up of silver and has a series of 16 openings in the tube wall, 11 of which can be closed directly by seven fingers and one by the left thumb. Its fundamental pitch is middle C (C4) and it has a range of about three octaves to C7. Indian variation of the flute is made up of bamboo and has usually 7 holes along the length of the
flute. The modern flute was developed by Theobald Boehm who experimented with it from 1832 to 1847.
Figure 2 Construction of a modern western flute
It is well known that flute is played by blowing air through its embouchure. But a beginner will not be able to play a flute effectively in first few tries due to several phenomena which make the flute work. Broadly, the air blown in through the embouchure resonates the air in the cylindrical cavity at the cavity’s resonance frequencies and hence produces sound. Covering and opening holes help change the pitch and the faster the air is blown inside, the louder the flute sounds. The flute produces sounds at the harmonics of an open cylindrical column.
Through literature we know that 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 jet continuously overcomes the amount of energy lost as viscous loss with the wall. Once the air in the flute is vibrating, some of the energy is radiated as sound out of the end and any open holes.
The generation of sound from flute includes some interesting phenomena related to slit, edge and air column. These are discussed in detail here.
When air is forced through a small opening or slit, the fast-moving airstream moves slower through air. It's edges meet resistance from that slower air and tend to peel back and form vortices. These vortices interact with the stream, exerting a sideways
influence on it and causing it to move in an undulating path. This undulation can actually produce a substantial amount of sound, as when someone whistles. An even greater amount of sound can be produced by coupling the slit to an edge. The undulations in the stream move along the stream at something less than one half the speed of the air in the stream. Then the faster the airstream speed, the faster the oscillations of the stream when it hits an edge. The role of slit is played by the slit made by the player’s lips.
Figure 4 Effect of increasing speed of jet of air through a slit. Faster jet produces higher frequency
For a slit and an edge, not coupled to an air column, the frequency of the air resonance in a slit-edge interaction tends to be about
𝑓 =0.2 𝑉𝑑 𝑗
where Vj is the speed of air stream and d is the distance between slit and edge.
The tone produced by the interaction of a slit and an edge is called an edgetone. The basic mechanism behind the production of the edgetone is
‘Air feedback’. When air is directed at an edge, it
does not divide smoothly, but tends to move to one side and form a swirl or vortex as shown in figure 5.
The resulting undulating flow may take the stream below the edge. The pressure created by this interaction with the edge feeds back to the area of the slit, tending to push the stream upward. The reverse happens when the stream moves to the top side of the edge and then the process repeats itself. As a result, a periodic flipping of the airstream from side to side produces edge tone, shown in figure 6. Efficiency of the sound increases if we couple slit, edge and an air column. The edgetone effect in such
an instrument serves to help initiate and sustain the tone, and can help make the transition to a higher harmonic of the air column. In case of air blown over an isolated edge a type of repetitive eddying called ‘vortex shedding’ takes place on alternate sides of the air jet, and a sound is produced if a sharp edge is used to separate the two sets of vortices. Vortex phenomena have only a secondary influence on flute-type sound production. The edgetone has regimes of frequencies as shown in figure 7. Region I of the edgetone behavior is the main operating mode for organ pipes and presumably also for the flute. The pitch can be made to go up by either increasing the airstream velocity, or by decreasing the distance from the slit to the edge (both applicable to playing the flute). Because of the feedback mechanism to the slit, the effective diameter of the edge does not figure directly in this pitch relationship.
Figure 6 Pressure feedback from an edge to a slit
The air that is vibrating at the embouchure doesn’t vibrate at any frequency. This vibration is forced by the air cavity, to which it is coupled to in the flute, to have only those frequencies which are the resonant frequencies of the air column. The position of the slit relative to the edge and the distance from the slit to the edge must be chosen so that the fundamental pitch is stable at the airstream velocities produced by the air cavity of the organ. If the airstream velocity is increased, the pipe can go sharp, or even jump to the next harmonic of the pipe. The jump has been shown in figure 8 as different frequency regimes.
These phenomena are utilized by the flute player through several actions, namely ‘Over-blowing’ and ‘Rolling-in’.
The lower notes of a flute are obtained by opening holes in the side of the instrument to shorten the air column, raising the fundamental frequency of the open air column. To achieve higher notes, players follow a process called "overblowing" the flute. The airstream velocity is increased and hence the frequency jumps to a higher regime (next higher register).
Figure 9 Over-blowing for jumping to a higher register
In "Rolling in" the player turns the flute inwards which serves the following purposes:
Varies the slit-edge distance
Covers up the embouchure hence decreases the size of opening to the atmosphere
Decrease the solid angle into which the sound waves can radiate
By rolling in and simultaneously increasing the airstream velocity, the air column can be made to pop cleanly from its fundamental to its second harmonic, raising the pitch by an octave
Figure 10 Rolling in to change the frequency
AIR CAVITY CHARACTERSTICS
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. 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.
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. 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.
Tone holes are basically the provided to change the pitch of sound. The number varies from flute to flute. On opening the tone holes, starting from the far end, the pressure node moves closer up the pipe. On the Boehm flute, each opened tone hole raises the pitch by a semitone. 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.
Figure 11 Effect of opening tone holes
High pass filter:
The tone holes act like short circuit only at low frequencies. At higher frequencies, the time available to the wave to push the small volume of air in the tone holes outside is very small. Hence these waves do not acknowledge these holes as short circuit; rather they pass through without ‘noticing’ them. Hence this is a kind of high pass filter effect. At very high frequencies waves can even neglect all of the 12 tone holes and the flute will behave like an open organ pipe only.
Holes can also serve as register holes. We can open certain specific holes and hence can directly jump one register. For example a hole opened midway through the flute 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 next higher register (C5), and will also play C6. 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. When the desired
wavelength is short (i.e. for high notes) one can open a register hole at a different fraction of the length. The jump to two and three register up the scale has been shown in figure 12.
Figure 12 Effect of opening Register holes
On the modern 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 the first register second register. Cross fingering basically utilizes the effect of high pass filter. The wave passes by an open hole and travels further as shown in figure 13. If it encounters another open hole after one closed hole the sound will be different from the sound that will be generated when all holes downstream are closed. Hence cross fingering can produce quarter notes and is used for playing ¼, ½, ¾ of a note.
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.
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.
THE CORK AND HEAD JOINT
Between the points 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. This 'upstream air' acts like a spring of a Helmholtz resonator. The air in the embouchure riser tube can be considered as a mass. Together they can resonate like a mass bouncing on a spring.
This has a resonance over a broad range of frequencies, but centered at about 5 kHz. At much lower frequencies over the playing range of the flute, it acts as impedance in parallel with the main part of the bore. This impedance’s magnitude decreases with frequency. 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 limit the upper range of the instrument.
Effect of head joint stopper
Figure 15 shows the results of an experiment in which the stopper was placed at various positions and the acoustic impedance was measured across frequencies. The flutes used had no
tone holes and these lengths gave a lowest note of C4, so in these curves the extrema are placed at 250 Hz over the whole frequency range.
Figure 15 Effect of varying the length of upstream air column (a) Stopper placed at 17.5 mm from the embouchure center (b) stopper placed level with the embouchure and (c) stopper placed at 70 mm from the embouchure center
With the stopper at the normal position of 17.5 mm from the centre of the embouchure hole (a) the global features of the impedance are similar to those of a modern flute. With the stopper positioned level with edge of the embouchure hole (b), the ‘shorting out’ of the extrema due to Helmholtz resonance is almost removed. With the stopper placed at 70 mm from embouchure center, the upstream air will act like a closed pipe itself, having its own natural frequencies. Hence this upstream air will cut down all the sound of the air cavity of flute at the natural frequencies of this closed pipe. Hence the graph at several places has zero amplitude of variation of impedance.
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. The acoustic impedance is the ratio of the sound pressure to the oscillating air flow. 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 behavior of the air jet, and so the flute will play only at a frequency very close to a resonance.
Figure 16 Apparatus for measurement of impedance of flute
An experiment was setup by John Coltman to study the variation of this impedance with blowing pressure. The setup includes a cylindrical flute head with its sliding tuning joint mounted on a short length of pipe representing the upper half of a flute resonating in a quarter-wave mode. At the position of the velocity node a massive piston, sealed with a thin rubber membrane, terminates the tube. The piston is driven at the desired frequency by a loudspeaker motor, and a pickup coil enclosed in a small magnetic yoke provides a measure of the piston velocity. Near the closed end is a low compliance microphone. Opposite this is a variable acoustic resistance consisting of a bundle of very small glass capillaries that could be closed off with a rubber pad. The microphone gives a measure of the oscillation pressure, and it could also be connected, as shown, to a null circuit to measure the ratio of acoustic pressure to acoustic current. Thus the real part of the impedance looking up the tube can be read from the potentiometer. This impedance can always be made real, either by shifting the frequency or changing the tube length at the
tuning slide. Adjustment of the acoustic resistor ensures that the tube is too lossy to go into self-oscillation when the jet of air is blown across the mouth hole. By comparing tube lengths and potentiometer readings for null output with and without the blowing jet the effective impedance of the jet as seen from the acoustic circuit at the plane of the mouth hole can be measured.
Variation of cavity impedance with pressure
The impedance ascribable to the jet is a smooth, well behaved function of blowing pressure over the entire range. Its magnitude decreases monotonically as the pressure is reduced, and its phase,
Figure 17 Variation of acoustic impedance with pressure and phase
which is determined by the arrival time of a disturbance on the jet, rotates clockwise over more than two complete cycles. Starting at the outer edge where the blowing pressure is about 0.56 inches of water, the impedance is real and negative. Such a condition would overcome the losses in the tube if the artificial resistance were removed, and would result in oscillation at large amplitude at the natural resonant frequency of the tube. At a higher blowing pressure the impedance has a capacitive component that will make the frequency sharp. At lower blowing pressure (0.3 inch), the inductive effect makes it go flat, and the real component is less negative, that is, it could not generate so much power. At a pressure of about 0.25 inch the phase crosses into the positive real domain the jet represents a loss mechanism and could not possibly sound the flute. Though the jet impedance varies smoothly with blowing pressure, spiraling around the origin while its magnitude decreases monotonically, it represents a possible sound-generating
mechanism only when it lies in the negative half plane. Only the outermost portion of the spiral, representing one half wave in this distance, is used in playing the flute.
Effect of lip-to-edge distance
Figure 18 Variation of acoustic impedance with lip-to-edge distance
The phase of the impedance is determined by travel time, that is determined not only by the jet velocity, but also by the lip-to-edge distance. Figure 18 shows two spirals where curve A shows a 7-mm lip-to-edge distance and curve B with 5 mm. The major effect of changing this spacing is to rotate the spiral in the plane. The two conditions are very different from the standpoint of the flutist. With a 7-mm spacing (curve A) he could achieve a large power input, and no frequency pulling, with a blowing pressure of 0.6 inch of water. With a 5-mm spacing (curve B) the same blowing pressure would produce less than half the power and would raise the pitch of the sound almost half a semitone. Proper lip spacing for the lowest frequency throws the curve for the next higher frequency mode almost entirely into the non-generating half-plane. For a shorter lip spacing the upper frequency shows favorable phases, while the impedance for the lower one lies in the lossy half-plane.
FREQUENCY RESPONSE OF THE FLUTE
The frequency response of a flute does not exactly follow the harmonics of an open organ pipe. Below about 2.5 kHz, the response of flute looks like that for a simple cylindrical pipe 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
Figure 19 Frequency response of a flute
However, above 2.5 or 3 kHz, the resonances become much weaker. This is because of the high pass filter mentioned above under cut off frequencies. Higher still, around 5 kHz, the resonances almost disappear completely, because they are shorted out by the Helmholtz resonator discussed above under the cork and the head joint. 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. The viscous loss increases with increase in frequency. The spacing of peaks or troughs in the graph at the low frequency end is about 600 Hz (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 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 high pass filter. Finally the curve 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 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.
Flute is an ancient musical instrument with a different sound than other instruments as it is played slightly away from the harmonics most of the time. Also the efficiency of a flute is only 2.4% which makes it difficult to be played loud. The flute’s operation looks simple but involves a lot of phenomena like air flowing through slit, on edges and coupling of slit, edge and air column. There are several tricks that a flutist can try to change the pitch and jump from one register to another. The control over pitch is one of the advantages in side blown flutes.
The structure of a flute consists of several holes, out of which some contribute just to change pitch by just one semitone, whereas register holes can make a jump of one complete register. High frequency waves generally produce a high pass filter effect at open holes and do not ‘notice’ them in the way. This is utilized in cross fingering which is used in playing half and quarter notes. The geometry of head joint plays an important role in deciding the frequency response of a flute and its cut-off frequency. A great amount of work has been done in the area of measuring of impedance of embouchure which shows allowable and non- allowable regimes of impedances i.e. where the flute will generate sound and where it cannot. \
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