Wind instruments all rely upon the action of a pressure sensing valve which allows pressure into the system when a condition of high pressure exists inside the system. In reed instruments this is referred to as a "reed". However in brass instruments and flutes recorders and pipe organs a similar mechanism exists although without the physical presence of a reed. In the absence of a suitable short name for "a pressure sensing valve which allows pressure into the system when a condition of high pressure exists inside the system" the convention has arisen to refer to these other reeds as "lip reeds" in the case of brass instruments and "air reeds" or "jet reeds" in the case of flutes recorders and pipe organs. This sounds a bit strange but remember that the reference is not to the physcal existence of a reed but the functionality that a reed provides.
The discussion of the acoustics of wind instruments involves similar principles to those used in the discussion of stringed instruments. However, the nature of the sound source in wind instruments is rather different but the description of the sound modifiers in wind instruments has much in common with that relating to possible modes on a string. This section concentrates on the acoustics of organ pipes to illustrate the acoustics of sound production in wind instruments. Some of the acoustic mechanisms basic to other wind instruments are given later in the section.
Wind instruments can be split into those without and those with reeds. This is the basis on which they are presented in this section. For convenience, organ pipes are used as the model. Organ pipes can also be split into two types based on the sound source mechanism involved: 'flues' and 'reeds'. Figure 4.12 shows the main parts of flue and reed pipes. Each is constructed of a particular material, usually wood or a tin-lead alloy, and has a resonator of a set shape, size depending on the sound that pipe is designed to produce (e.g. Audsley, 1965; Sumner, 1975; Norman and Norman, 1980). The sources of sound in the flue and the reed pipe will be considered first, followed by the sound modification that occurs due to the resonator. 4.3.1 Sound source in organ flue pipes The source of sound in flue pipes is described in detail in Hall (1991) and his description is followed here. The important features of a flue sound source are a narrow slit (the flue) through which air flows, and a wedge-shaped obstacle placed in the airstream from the slit. Figure 4.13 shows the detail of this mechanism for a wooden organ flue pipe (the similarity with a metal organ flue pipe can be observed in Figure 4.12). A narrow slit exists between the lower lip and the languid, and this is known as the 'flue', and the wedge-shaped obstacle is the upper
Fig 4.12 The main parts of the flue (open metal and stopped wood) and reed organ pipes.
4.13 The main elements of the sound source in organ flue pipes based on a wooden flue pipe (left) and additional features found on some metal pipes (center and left)
lip which is positioned in the airstream from the flue. This obstacle is usually placed off-centre to the airflow.
Air enters the pipe from the organ bellows via the foot and a thin sheet of air emerges from the flue. If the upper lip were not present, the air emerging from the flue would be heard as noise. This indicates that the airstream is turbulent. A similar effect can be observed if you form the mouth shape for the 'ff' in far, in which the bottom lip is placed in contact with the upper teeth and produce the 'ff' sound. The airflow is turbulent, producing the acoustic noise which can be clearly heard. If the airstream flow rate is reduced, there is an air velocity below which turbulent flow ceases and acoustic noise is no longer heard. At this point the airflow has become smooth or 'laminar'. Turbulent airflow is the mechanism responsible for the non-pitched sounds in speech such as the 'sh' in shoe and the 'ss' in sea for which waveforms and spectra are shown in Figure 3.9.
When a wedge-like obstruction is placed in the airstream emerging from the flue a definite sound is heard known as an 'edgetone'. Hall suggests a method for demonstrating this by placing a thin card in front of the mouth and blowing on its edge. Researchers are not fully agreed on the mechanism which underlies the sound source in flues. The preferred explanation is illustrated in Figure 4.14, and it is described in relation to the sequence of snapshots in the figure as follows. Air flows to one side of the obstruction, causing a local increase in pressure on that side of it. This local pressure increase causes air in its local vicinity to be moved out of the way, and some finds its way in a circular motion into the pipe via the mouth. This has the effect of 'bending' the main stream of air increasingly, until it flips into the pipe. The process repeats itself, only this time the local pressure increase causes air to move in a circular motion out of the pipe via the mouth, gradually bending the main airstream until it flips outside the pipe again. The cycle then repeats providing a periodic sound source to the pipe itself. This process is sometimes referred to as a vibrating 'air reed' due to the regular flipping to and fro of the airstream.
Fig 4.14 Sequence of events to illustrate the sound source mechanism in a flue organ pipe
The f0 of the pulses generated by this air reed mechanism in the absence of a pipe resonator is directly proportional to the airflow velocity from the flue, and inversely proportional to the length of the cut-up:
f0 is proportional to (v1/Lcutup)
In other words, f0 can be raised by either increasing the airflow velocity or reducing the cut-up. As the airflow velocity is increased or the cut-up size is decreased, there comes a point where the to jumps up in value. This effect can be observed in the presence of a resonator with respect to increasing the airflow velocity by blowing with an increasing flow rate into a recorder (or if available, a flue organ pipe). It is often referred to as an 'overblown' mode.
The acoustic nature of the sound source in flues is set by the pipe voicer, whose job it is to determine the overall sound from individual pipes and to establish an even tone across complete ranks of pipes. The following comments on the voicer's art in relation to the sound source in flue pipes are summarised from Norman and Norman (1980), who give the main modifications made by the voicer in order of application as:
Adjusting the cut-up needs to be done accurately to achieve an even tone across a rank of pipes. This is achieved on metal pipes by using a sharp, short thick bladed knife. A high cut-up produces a louder and more 'hollow' sound, and a lower cut-up gives a softer and 'edgier' sound. The higher the cut-up, the greater the airflow required from the foot. However, the higher the air flow, the less prompt the speech of the pipe.
Nicking relates to a series of small nicks that are made in the approximating edges of the languid and the upper lip. This has the effect of reducing the high-frequency components in the sound source spectrum and giving the pipe a smoother, but slower, onset to its speech. More nicking is customarily applied to pipes which are heavily blown. A pipe which is not nicked has a characteristic consonantal attack to its sound, sometimes referred to as a 'chiff'. A current trend in organ voicing is the use of less or no nicking in order to take advantage of the onset chiff musically to give increased clarity to notes particularly in contrapuntal music (e.g. Hurford, 1994).
The height of the languid is fixed at manufacture for wooden pipes, but it can be altered for metal pipes. The languid controls, in part, the direction of the air flowing from the flue. If it is too high, the pipe will be slow to speak and may not speak at aU if the air misses the upper lip completely. If it is too low the pipe will speak too quickly, or speak in an uncontrolled manner. A pipe is adjusted to speak more rapidly if it is set to speak with a consonantal chiff by means of little or no nicking. Narrow scaled pipes (small diameter compared with the length), usually having a 'stringy' tone colour often have ears added (see Figure 4.12) which stabilise air reed oscillation, and some bass pipes also have a wooden roller or 'beard' placed between the ears to aid prompt pipe speech.
The sound modifier in a flue organ pipe is the main body of the pipe itself, or its 'resonator' (see Figure 4.12). Organ pipe resonators are made in a variety of shapes developed over a number of years to achieve subtleties of tone colour, but the most straightforward to consider are resonators whose dimensions do not vary along their length, or resonators of 'uniform cross-section'. Pipes made of metal are usually round in crosssection and those made of wood are generally square (some builders make triangular wooden pipes, partly to save on raw material). These shapes arise mainly from ease of construction with the material involved.
There are two basic types of organ flue pipes, those that are open and those that are stopped at the end farthest from the flue itself (see Figure 4.12). The flue end of the pipe is acoustically equivalent to an open end. Thus the open flue pipe is acoustically open at both ends, and the stopped flue pipe is acoustically open at one end and closed at the other. The air reed sound source mechanism in flue pipes as illustrated in Figure 4.14 launches a pulse of acoustic energy into the pipe When a compression (positive amplitude) pulse of sound pressure energy is launched into a pipe, for example at the instant in the air reed cycle illustrated in the lower-right snapshot in Figure 4.14, it travels down the pipe at the velocity of sound as a compression pulse.
Fig 4.15 The reflected pulses resulting from a compression (upper) and rarefaction (lower) pulse arriving at an open (left) and a stopped (right)end of a pipe of uniform cross section. Note: Time axes marked in equal arbitrary units
When the compression pulse reaches the far end of the pipe, it is reflected in one of the two ways described in the 'standing waves' section of Chapter 1, depending (Section 1.5.7) on whether the end is open or closed. At a closed end there is a pressure antinode and a compression pulse is reflected back down the pipe. At an open end there is a pressure node and a compression pulse is reflected back as a rarefaction pulse to maintain atmospheric pressure at the open end of the pipe. Similarly, a rarefaction pulse arriving at a closed end is reflected back as a rarefaction pulse, but as a compression pulse when reflected from an open end. All four conditions are illustrated in Figure 4.15.
When the action of the resonator on the air reed sound source in a flue organ pipe is considered (see Figure 4.14), it is found that the 10 of air reed vibration is entirely controlled by: (a) the length of the resonator, and (b) whether the pipe is open or stopped. This dependence of the It! of the air reed vibration can be appreciated by considering the arrival and departure of pulses at each end of the open and the stopped pipe.
Figure 4.16 shows a sequence of snapshots of pressure pulses generated by the air reed travelling down an open pipe of length L" (left) and a stopped pipe of length Lc (right) and how they drive the vibration of the air reed. (Air reed vibration is illustrated in a manner similar to that used in Figure 4.14.) The figure shows pulses moving from left to right in the upper third of each pipe, those moving from right to left in the centre third, and the summed pressure in the lower third. A time axis with
Fig 4.16 Pulses travelling in open (left) and stopped (right) pipes where they drive an air reed sound source. Note: Time axes marked in equal arbitrary units pulses travelling left to right are shown in the upper part of the pipe thoser going right to left are shown in the center and the sum is shown in the lower part.
arbitrary but equal units is marked in the figure to show equal time intervals. The pulses travel an equal distance in each frame of the figure since an acoustic pulse moves at a constant velocity. The flue end of the pipe acts as an open end in terms the manner in which pulses are reflected (see Figure 4.15). At every instant when a pulse arrives and is reflected from the flue end, the air reed is flipped from inside to outside when a compression pulse arrives and is reflected as a rarefaction pulse, and vice versa when a rarefaction pulse arrives. This can be observed in Figure 4.16.
For the open pipe, the sequence in the figure begins with a compression pulse being launched into the pipe, and another compression pulse just leaving the open end (the presence of this second pulse will be explained shortly). The next snapshot shows the instant when these two pulses reach the centre of the pipe, their summed pressure being a maximum at this point. The pulses effectively travel through each other and emerge with their original identities due to 'superposition' (see Chapter 1). In the third snapshot the compression pulse is being reflected from the open end of the pipe as a rarefaction pulse, and the air reed flips outside the pipe, generating a rarefaction pulse. (This may seem strange at first, but it is a necessary consequence of the event happening in the fifth snapshot.) The fourth snapshot shows two rarefactions at the centre giving a summed pressure which is a minimum at this instant of twice the rarefaction pulse amplitude. In the fifth snapshot, when the rarefaction pulse is reflected from the flue end as a compression pulse, the air reed is flipped from outside to inside the pipe. One cycle is complete
at this point since events in the fifth and fist snapshots are similar (A second cycle is illustrated to enable comparison with events in the stopped pipe.)
The fundamental period for the open pipe is the time taken to complete a complete cycle (i.e. the time between a compression pulse leaving the flue end of the pipe and the next compression pulse leaving the flue end of the pipe). In terms of Figure 4.l6 it is four time frames (snapshot one to snapshot five), being the time taken for the pulse to travel down to the other end and back again (see Figure 4.15), or twice the open pipe length.
T (open) = ((2L0)/c)
The f0 value for the open pipe is therefore (1/T (open)) = C/2L0
In the stopped pipe, the sequence in Figure 4.15 again begin, with a compression pulse being launched into the pipe, but there is no second pulse. Snapshot two shows the instant when the pulses reach the centre of the pipe, and the third snapshot the instant when the compression pulse is reflected from the stopped end as a compression pulse (see Figure 4.15) and the summed pressure is a maximum for the cycle of twice the amplitude of the compression pulse. The fourth snapshot shows the compression pulse at the centre and in the fifth, the compressior pulse is reflected fcom the flue end as a rarefaction pulse flipping the air reed from inside to outside the pipe. The sixth snapshot shows the rarefaction pulse halfway down the pipe and the seventh shows its reflection as a rarefaction pulse from the stopped end when the summed pressure there is the minimum for the cycle of twice the amplitude of the rarefaction pulse. The eighth snapshot shows the rarefaction pulse half-way back to the flue end, and by the ninth, one cycle is complete, since events in the ninth and first snapshot, are the same.
It is immediately clear that one cycle for the stopped pipe takes wice as long as one cycle for the open pipe if the pipe lengths are equal (ignoring a small end correction which has to be applied in practice), its fundamental period is therefore double that for the open pipe, and its f0 is therefore half that for the open pipe or an octave lower. This can be quantified by considering the time taken to complete a complete cycle is the time required for the pulse to travel to the other end of the pipe and back twice, or four times the stopped pipe length (see Figure 4.15):
T0(stopped) = (4Ls /c)
f0(stopped) = (1/T0(stopped))= (c/4Ls)
Example 4.2 If an open pipe and a stopped pipe are the same length, what is the relationship between their f0 values?
Let (Ls = Lu = L) and substitute into Equations 4.5 and 4.6:
f0(open) = (c/2L)
f0(stopped) = (c/4L)
f0stopped = 1/2(c/2L) = (f0open/2)
Therefore f0(stopped) is an octave lower than f0(open) (frequency ratio 1:2).
The natural modes of a pipe are constrained as described in the 'standing waves' section of Chapter 1. Equation 1.20 gives the frequencies of the modes of an open pipe and Equation 1.21 gives the frequencies of the modes of a stopped pipe. In both equations, the velocity is the velocity of sound (c).
The frequency of the first mode of the open pipe is given by Equation 1.20 when (n = 1):
fopen1 = (c/2L)
which is the same value obtained in Equation 4.5 by considering pulses in the open pipe. Using Equation 1.20, the frequencies of the other modes can be expressed in terms of its /., value as follows:
fopen(2) =(2c/2Lo)= 2 x fopen(1)
fopen(3) =(3c/2Lo)= 3 x fopen(1)
fopen(4) =(4c/2Lo)= 4 x fopen(1)
fopen(n) = n x fopen(1)
The modes of the open pipe are thus all hamonically related and all hamonics are present. The musical intervals between the modes can be read from Figure 3.3.
The fequency of the fundamental mode of the stopped pipe is given by equation 1.21 when (n=1)
fstopped(1) = (c/(4 x Ls))
This is the same value obtained in Equation 4.b by considering pulses in the stopped pipe. The frequencies of the other stopped pipe modes can he expressed in terms of its fstopped using Equation 1.21 as follows:
fstopped(2) = (3c/4Ls)= 3f stopped(1)
fstopped(3) = (5c/4Ls)= 5f stopped(1)
fstopped(4) = (7c/4Ls)= 7f stopped(1)
fstopped(n) = (2n-1)fstopped(1)
whece n = 1, 2, 3, 4, ...
Thus the modes of the stopped pipe are hamonically related, but only the odd-numbered harmonics are present. The musical intervals between the modes can be read from Figure 3.3. In open and stopped pipes the pipe's resonator acts as the sound modifier and the sound source is the air reed. The nature of the spectrum of the air reed source depends on the detailed shape of the pulses launched into the pipe, which in turn depends on the pipe's voicing summarised above. If a pipe is overblown, its f0 jumps to the next higher mode that the resonator can support, up one octave to the second harmonic for an open pipe and up an octave and a fifth to the third harmonic for the stopped pipe.
Fig 4.17 Waveforms and spectra for middle C (C4) played on a gedackt 8' stopped pipe and a principle 8' (open flue)
The length of the resonator controls the f0 of the air reed (see Figure 4.15) and the natural modes of the pipe are the frequencies that the pipe can support in its output. The amplitude relationship between the pipe modes is governed by the material from which the pipe is constructed and the diameter of the pipe with respect to its length. In particular, wide pipes tend to be weak in upper harmonics. Organ pipes are tuned by adjusting the length of their resonators. In open pipes this is usually done nowadays by means of a tuning slide fitted round the outside of the pipe at the open end, and for a stopped pipes by moving the stopper (see Figure 4.12).
A stopped organ pipe has a f0 value which is an octave below that of an open organ pipe (Example 4.2), and where space is limited in an organ, stopped pipes are often used in the bass register and played by the pedals. However, the trade-off is between the physical space saved and the acoustic result in that only the odd-numbered harmonics are supported. Figure 4.17 illustrates this with waveforms and spectra for middle C played on a gedackt 8' and a principal 8' (Section 5.4 describes organ stop footages: 8', 4' etc.). The gedackt stop has stopped wooden pipes, and the spectrum clearly shows the presence of odd harmonics only, in particular, the first, third and fifth. The principal stop consists of open metal pipes, and odd and even harmonics exist in its output spectrum. Although the pitch of these stops is equivalent, and they are therefore both labelled 8', the stopped gedackt pipe is half the length of the open principal pipe.
Other musical instruments which have an air reed sound source include the recorder and the flute. Useful additional material on woodwind flue instruments can be found in Benade (1976) and Fletcher and Rossing (1999). The air reed action is controlled by oscillatory changes in flow of air in and out of the flue (see Figure 4.16), often referred to as a flow-controlled valve, and therefore there must be a displacement antinode and a pressure node. Hence the flue end of the pipe is acting as an open end and woodwind flue instruments act acoustically as pipes open at both ends (see Figure 4.18),
Fig 4.18 The first four pressure and displacement modes of an open and a stopped pipe of uniform cross section. Note: the plots show maximim and minimum amplitudes of pressure and displacement.
Players are able to play a number of different notes on the same instrument by changing the effective acoustic length of the resonator. This can be achieved, for example, by means of the sliding piston associated with a swanee whistle or more commonly when particular notes are required by covering and uncovering holes in the pipe walls known as 'finger holes'. A hole in a pipe will act in an acoustically similar manner to an open pipe end (pressure node, displacement antinode). The extent to which it does this is determined by the diameter of the hole with respect to the pipe diameter. When these are large with respect to the pipe diameter, as in the flute, they equal the uncovered hole acts acoustically as if the pipe had an open end at that position. Smaller finger holes result acoustically in the effective open end being further (away from the flue end) down the pipe. This is an important factor in the practical design of instruments with long resonators since it can enable the finger holes to be placed within the physical reach of a player's hands. It does, however, have an importance consequence on the frequency relationship between the modes, and this is explored in detail below in connection with woodwind reed instruments. The other way to give a player control over finger holes which are out of reach, for example on a flute, is by providing each hole with a pad controlled by a key mechanism of rods and levers operated by the player's fingers to close or open the hole (depending whether the hole is normally open or closed by default).
In general, a row of finger holes is gradually uncovered to effectively shorten the acoustic length of the resonator as an ascending scale is played. Occasionally some cross-fingering is used in instruments with small holes or small pairs of holes such as the recorder as illustrated in Figure 4.19. Here, the pressure node is
Fig 4.19 Fingering chart for recorders in C (descants and tenors)
further away from the flue than the first uncovered hole itself such that the state of other holes beyond it will affect its position. The figure shows typical fingerings used to play a two octave C major scale on a descant or tenor recorder. Hole fingerings are available to enable notes to be played which cover a full chromatic scale across one octave. To playa second octave on woodwind flue instruments such as the recorder or flute, the flue is overblown. Since these instruments are acoustically open at both ends, the overblown flue jumps to the second mode which is one octave higher than the first (see Equation 4.8 and Figure 3.3). The finger holes can be reused to play the notes of the second octave.
Once an octave and a fifth above the bottom note has been reached, the flue can be overblown to the third mode (an octave and a fifth above the first mode) and the fingering can be started again to ascend higher. The fourth mode is available at the start of the third octave and so on. Overblowing is supported in instruments such as the recorder by opening a small 'register' or 'vent' hole which is positioned such that it is at the pressure antinode for unwanted modes and these modes will be suppressed. The register hole marked in Figure 4.19 is a small hole on the back of the instrument which is controlled by the thumb of the left hand which either covers it completely, half covers it by pressing the thumb nail end-on against it, or uncovers it completely. To suppress the first mode in this way without affecting the second, this hole should be drilled in a position where the undesired mode has a pressure maximum. When all the tone holes are covered, this would be exactly half-way down the resonator, a point where the first mode has a pressure maximum and is therefore reduced, but the second mode has a pressure node and is therefore unaffected (see Figure 4.18). Register holes can be placed at other positions to enable overblowing to different modes. In practice, register holes may be set in compromise positions because they have to support all the notes available in that register, for which the effective pipe length is altered by uncovering tone holes.
A flute has a playing range between B3 and 07 and the piccolo sounds one octave higher between B4 and 08 (see Figure 4.3). Flute and piccolo players can control the stability of the overblown modes by adjusting their lip position with respect to the embouchure hole as illustrated in Figure 4.20. The air reed mechanism can be compared with that of flue organ pipes illustrated in Figures 4.13 and 4.14 as well as the associated discus sion relating to organ pipe voicing. The flautist is able to adjust the distance between the flue outlet (the player's lips) and the
Fig 4.20 Illustration of lip to embouchure adjustments available to a flautist
edge of the mouthpiece, marked as the 'cut-up' in the Figure, a term borrowed from organ nomenclature (see Figure 4.13), by rolling the flute as indicated by the double-ended arrow. In addition, the airflow velocity can be varied as well as the fine detailed nature of the airstream dimensions by adjusting the shape, width and height of the opening between the lips. The flautist therefore has direct control over the stability of the overblown modes (Equation 4.4).
The basic components of an organ reed pipe is shown in Figure 4.12. The sound source results from the vibrations of the reed, which is slightly larger than the shallot opening, against the edges of the shallot. Very occasionally, organ reeds make use of 'free reeds' which are cut smaller than the shallot opening and they move in and out of the shallot without coming into contact with its edges. In its rest position as illustrated in Figure 4.12, there is a gap between the reed and shaifo'l, enabled by 'lhe slight curve in the reed itself. The vibrating length of the reed is governed by the position of the 'tuning wire', or 'tuning spring' which can be nudged up or down to make the vibrating length longer or shorter, accordingly lowering or raising the f0 of the reed vibration. The reed vibrates when the stop is selected and a key on the appropriate keyboard is pressed. This causes air to enter the boot and flow past the open reed via the shallot to the resonator. The gap between the reed and shallot is narrow and for air to flow, there must be a higher pressure in the boot than the shallot. The higher pressure in the boot than the shallot tends to close the reed fractionally, resulting in the gap between the reed and shallot being narrowed. When the gap is narrowed, the airflow rate is increased and the pressure difference which supports this increased airflow increases. The increase in pressure difference exerts a slightly greater closing force on the reed, and this series of events continues, accelerating the reed towards the shallot until it hits the edge of the shallot, closing the gap completely and rapidly.
The reed is springy and once the gap is closed and the flow has dropped to zero, the reed's restoring force causes the reed to spring back towards its equilibrium position, opening the gap. The reed overshoots its equilibrium position, stops and returns towards the shallot, in a manner similar to its vibration if it had been displaced from its equilibrium position and released by hand. Air flow is restored via the shallot and the cycle repeats.
In the absences of a resonator, the reed wound vibrate at its natural frequency. This is the frequency at which it would vibrate if it were plucked, or displaced from its equilibrium position and released by hand. If a plucked reed continues to vibrate for a long time, then it has a strong tendency to vibrate at a frequency within a narrow range, but if it vibrates for a short time, there is a wide range of frequencies over which it is able to vibrate. This effect is illustrated in Figure 4.21. This difference
Fig 4.21 Time (left) and frequency (right) responses of hard (upper) and soft(lower) reeds when plucked. Natural frequency (Fn) and natural period(Tn) are shown.
is exhibited depending on the material from which the reed is made and how it is supported. A reed which vibrates over a narrow frequency range is usually made from brass and supported rigidly, and is known as a 'hard' reed. A reed which vibrates over a wide range might be made from cane or plastic and held in a pliable support is known as a 'soft' reed. As shown
in the figure, the natural period (TN) is related to the natural frequency (FN) as:
Fn = 1/TN
A reed vibrating against a shallot shuts of the flow of air rapidly and totally, and the consequent acoustic pressure variations are the sound source provided to the resonator. The rapid shutting off of the airflow produces a rapid, instantaneous drop in acoustic pressure within the shallot (as air flowing fast into the shallot is suddenly cut off). A rapid amplitude change in a waveform indicates a relatively high proportion of high harmonics are present. The exact nature of the sound source spectrum depends on the particular reed, shallot and bellows pressure being considered. Free reeds which do not make contact with a shallot, as found, for example, in a harmonica or harmonium, do not produce as high a proportion of high harmonics since the airr10w is never completely shut off.
All reed pipes have resonators. The effect of a resonator has already been described and illustrated in Figure 4.16 in connection with air reeds. The same principles apply to reed pipes, but there is a major difference in that the shallot end of the resonator acts as a stopped end (as opposed to an open end as in the case of a flue). This is because during reed vibration, the pipe is either closed completely at the shallot end (when the reed is in contact with the shallot) or open with a very small aperture compared with the pipe diameter.
Organ reed pipes have hard reeds, which have a narrow natural frequency range (see Figure 4.21). Unlike the air reed, the presence of a resonator does not control the frequency of vibration of the hard reed. The sound-modifying effect of the resonator is based on the modes it supports (e.g. see Figure 4.18), bearing in mind the closed end at the shallot. Because the reed itself fixes the to of the pipe, the resonator does not need to reinforce the fundamental and fractional length resonators are sometimes used to support only the higher harmonics. Figure 4.22 shows waveforms and spectra for middle C (C4) played on a hautbois 8', or oboe 8', and a trompette 8', or trumpet 8'. Both spectra exhibit an overall peak around the sixth/seventh harmonic. For the trompette this peak is quite broad with the odd harmonics dominating the even ones up to the tenth harmonic, probably a feature of its resonator shape. The hautbois
Fig 4.22 Waveform and spectra for middle c (c4) played on a hautbois 8' and a trompette 8'
spectrum exhibits more dips in the spectrum than the trompette - these are all features which characterise the sounds of different instruments as being different.
Woodwind reed instruments make use of either a single or a double vibrating reed sound source which controls the flow of air from the player's lungs to the instrument. The action of a vibrating reed at the end of a pipe is controlled as a function of the relative air pressure on either side of it in tern-,s of when it opens and doses. It is therefore usually described as a pressllrecontrolled valve, and the reed end of the pipe acts as a stopped end (pressure antinode and displacement node-see Figure 4.18). Note that although the reed opens and closes such that airflow is not always zero, the reed opening is very much smaller than the pipe diameter elsewhere making a stopped end reasonable.
This is in direct contrast to the air reed in woodwind flue instruments such as the flute and recorder (see above), which is a flowcontrolled valve which provides a displacement antinode and a pressure node and hence the flue end of the pipe is acting as an open end (see Figure 4.18).
Soft reeds are employed in woodwind reed instruments which can vibrate over a wide frequency range (see Figure 4.21). The reeds in clarinets and saxophones are single reeds which can close against the edge of the mouthpiece as in organ reed pipes where they vibrate against their shallots. The oboe and bassoon on the other hand use double reeds, but the basic opening and closing action of the sound source mechanism is the same.
Woodwind reed instruments have resonators whose modal behaviour is crucial to the operation of the instrument and provide the sound modifier function.Woodwind instruments incorporate finger holes to enable chromatic scales to be played from the first mode to the second mode when the fingering can be used again as the reed excites the second mode. These mode changes continue up the chromatic scale to cover the full playing range of the instrument (see Figure 4.3). Clearly it is essential that the modes of the resonator retain their frequency ratios relative to each other as the tone holes are opened or else the instrument's tuning will be adversely affected as higher modes are reached. Benade (1976) summa rises this effect and indicates the resulting constraint as follows:
Preserving a constant frequency ratio between the vibrational modes as the holes are opened is essential in all woodwinds and provides a limitation on the types of air column (often referred to as the bore) that are musically useful.
The musically useful bores in this context are based on tubing that is either cylindrical as in the clarinet, or conical as in the oboe, cor Anglais, and members of the saxophone and bassoon families. The cylindrical resonator of a clarinet acts as a pipe that is stopped at the reed end (see above) that is open at the other. Odd-numbered modes only are supported by such a resonator (see Figure 4.18), and its f0 is an octave lower (see Example 4.2) than that of an instrument with a similar length pipe which is open at both ends, such as a flute (see Figure 4.3). The first overblown mode of a clarinet is therefore the third mode, an interval of an octave and a fifth (see Figure 3.3), and therefore unlike a flute or recorder, it has to have sufficient holes to enable at least 19 chromatic notes to be fingered within the first mode prior to transition to the second.
Conical resonators that are stopped at the reed end and open at the other support all modes in a harmonically related manner. Taylor (1976) gives a description of this effect as follows:
Suppose by some means we can start a compression from the narrow end; the pipe will behave just as our pipe open at both ends until the rarefaction has returned to the start. Now, because the pipe has shrunk to a very small bore, the speed of the wave slows down and no real reflection occurs. .., The result is that we need only consider one journey out and one back regardless of whether the pipe is open or closed at the narrow end. ... The conical pipe will behave something like an pipe open at both ends as far as its modes are concerned.
The conical resonator therefore supports all modes, and the overblown modes of instruments with conical resonators, such as the oboe, cor Anglais, bassoon and saxophone family, is therefore to the second mode, or up an octave. Sufficient holes are therefore required for at least 12 chromatic notes to be fingered to enable the player to arrive at the second mode from the first. The presence of a sequence of open tone holes in a pipe resonator of any shape is described by Benade (1976) as a tonehole lattice. The effective acoustical end-point of the pipe varies slightly as a function of frequency when there is a tone-hole lattice and therefore the effective pipe length is somewhat different for each mode. A pipe with an tone-hole lattice is acoustically shorter for low frequency standing wave modes compared with higher-frequency modes, and therefore the higherfrequency modes are increasingly lowered slightly in frequency (lengthening the wavelength lowers the frequency). Above a particular frequency, described by Benade (1976) the open-holes lattice cut-off frequency (given as around 350-500 Hz for quality bassoons, 1500 Hz for quality clarinets and between 1100 and 1500 Hz for quality oboes), sound waves are not reflected due to the presence of the lattice. Benade notes that this has a direct effect on the perceived timbre of woodwind instruments, correlating well with descriptions such as bright or dark given to instruments by players. It should also be noted that holes that are closed modify the acoustic properties of the pipe also, and this can be effectively modelled as a slight increase in pipe diameter at the position of the tone hole. The resulting acoustic change is considered below. In order to compensate for these slight variations in the frequencies of the modes produced by the presence of open and closed tone holes, alterations can be made to the shape of the pipe. These might include flaring the open end, adding a tapered
section, or small local voicing adjustments by enlarging or constricting the pipe which on a wooden instrument can be achieved by reaming out or adding wax respectively (e.g. Nederveen, 1969). The acoustic effect on individual pipe mode frequencies of either enlarging or constricting the size of the pipe depends directly on the mode's distribution of standing wave pressure nodes and antinodes (or displacement antinodes and nodes respectively). The main effect of a constriction in relation to pressure antinodes (displacement nodes) is as follows (Kent and Read, 1992:
A constriction at a pressure node (displacement antinode) has the effect of reducing the flow at the constriction since the local pressure difference across the constriction has not changed. Benade (1976) notes that this is equivalent to raising the local air density, and the discussion in Chapter 1 indicates that this will result in a lowering of the velocity of sound and therefore a lowereing in the mode frequency (see Equations 4.7 and 4.9). A constriction at a pressure antinode (displacement node), on the other hand, provides a local rise in acoustic pressure which produces a greater opposition to local air flow of the sound waves that combine to produce the standing wave modes. This is equivalent to raising the local springiness in the medium, which is shown in Chapter 1 to be equivalent for air of Young's modulus, which raises the velocity of sound (see Equation 1.5) and therefore raises the mode frequency (see Equations 4.7 and 4.91. By the same token, the effect of locally enlarging a pipe will be exactly opposite to that of constricting it.
Knowledge of the position of the pressure and displacement nodes and antinodes for the standing wave modes in a pipe therefore allows the effect on the mode frequencies of a local constriction or enlargement of a pipe to be predicted. Figure 4.23 shows the potential mode frequency variation for the first three modes of a cylindrical stopped pipe that could be caused by constriction or enlargement at any point along its length. (The equivalent diagram for a cylindrical pipe open at both ends could be readily produced with reference to Figures 4.18 and 4.23; this is left as an exeeeise for the interested reader.)
The upper part of Figme 4.23 (taken frm Figme 4.18) indicates the pressure and displacement node and antinode positions for the first three standing wave modes. The lower part of the Figure
Fig 4.23 The effect of locally constructing or enlarging a stopped pipe on the frequencies of its first three modes: a + indicates raised modal frequency - indicates lowered modal frequency and the magnitude of the change is indicated by the size of the +- signs/ The first three pressure and displacement nodes of a stopped pipe are shown for reference. N and A indicate node and antinode respectively.
exhibits plus and minus signs to indicate where that particular mode's frequency would be raised or lowered respectively by a local constriction or enlargement at that position in the pipe. The size of the signs indicate the sensitivity of the frequency variation based on how close the constriction is to the mode's pressure/displacement nodes and antinodes shown in the upper part of the Figure. For example, a constriction close to the closed end of a cylindrical pipe will raise the frequencies of all modes since there is a pressure antinode at a closed end, whereas an enlargement at that position would lower the frequencies of all modes. However, if a constriction or enlargement were made one third the way along a stopped cylindrical pipe from the closed end, the frequencies of the first and third modes would be raised somewhat, but that of the second would be lowered maximally. By creating local constrictions or enlargements, the skilled maker is able to set up a woodwind instrument to compensate for the presence of tone holes such that the modes remain close to being in integer frequency ratios over the playing range of the instrument.
Figure 4.24 shows waveforms and spectra for the note middle C played on a clarinet and a tenor saxophone. The saxophone spectrum contains all harmonics since its resonator is conical. The clarinet spectrum exhibits the odd harmonics dearly as its resonator is a cylindrical pipe dosed at one end (see Figure 4.18),
Fig 4.24 Waveforms and spectra for middle C (C4) played on a clarinet and a tenor saxophone.
but there is also energy clearly visible in some of the even harmonics. Although the resonator itself does not support the even modes, the spectrum of the sound source does contain all harmonics (the saxophone and the clarinet are both single reed instruments). Therefore some energy will be radiated by the clarinet at even harmonics.
Sundberg (1989) summarises this effect for the clarinet as follows:
This means that the even numbered modes are not welcome in the resonator ... A common misunderstanding is that these partials are all but missing in the spectrum. The truth is that the second partial may be about 40 dB below the fundamental, so it hardly contributes to the timbre. Higher up in the spectrum there differences between odd- and even..numbered neighbours are smaller. Further ... the differences can be found only for the instruments' lower tones.
This description is in accord with the spectrum in Figure 4.24, where the amplitude of the second harmonic is approximately 40 dB below that of the fundamental, and the odd/even differences become less with increased frequency.