In this chapter a simple model is developed which allows the acoustics of all musical instruments to be discussed and, it is hoped, readily understood. The model is used to explain the acoustics of stringed, wind and percussion instruments as well as the singing voice. Any acoustic instrument has two main components:
For the purposes of our simple model, the sound source is known as the 'input' and the sound modifiers are known as the 'system'. The result of the input passing through the system is known as the 'output'. Figure 4.1 shows the complete input! system/ output model. This model provides a framework within which the acoustics of musical instruments can be usefully discussed, reviewed and understood. Notice that the 'output' relates to the actual output
Fiug 4.1 input/system/output model for describing the acoustics of musical instruments
Fig 4.2 The input/system/output model applied to an instrumnent being played in a room
from the instrument which is not that which the listener hears since it is modified by the acoustics of the environment in which the instrument is being played. The input/system/output model can be extended to include the acoustic effects of the environment as follows.
If we are modelling the effect of an instrument being played in a room, then the output we require is the sound heard by the listener and not the output from the instrument itself. The environment itself acts as a sound modifier and therefore it too acts as a 'system' in terms of the input/ system/ output model. The input to the model of the environment is the output from the instrument being played. Thus the complete practical input/system/output model for an instrument being played in a room is shown in Figure 4.2. Here, the output from the instrument is equal to the input to the room.
In order to make use of the model in practice, acoustic details are required for the 'input' and 'system' boxes to enable the output(s) to be determined. The effects of the room are described in Chapter 6. In this chapter, the 'input' and 'system' characteristics for stringed, wind and percussion instruments as well as the singing voice are discussed. Such details can be calculated theoretically from first principles, or measured experimentally, in which case they must be carried out in an environment which either has no effect on the acoustic recording, or has a known effect which can be accounted for mathematically. An environment which has no acoustic effect is one where there are no reflections of sound-ideally this is known as 'free space'. In practice, free space is achieved in a laboratory in an anechoic ('no echo') room in which all sound reaching the walls, floor and ceiling is totally absorbed by large wedges of sound-absorbing material. However, anechoic rooms are rare, and a useful practical approximation to free space for experimental purposes is outside on grass during a windless day, with the experiment being conducted at a reasonable height above the ground.
This chapter considers the acoustics of stringed, wind, and percussion instruments. In each case, the sound source and the sound modifiers are discussed. These discussions are not intended to be exhaustive since the subject is a large one. Rather they focus on one or two instruments by way of examples as to how their acoustics can be described using the sound source and sound modifier model outlined above. References are included to other textbooks in which additional information can be found for those wishing to explore a particular area more fully.
Finally the singing voice is considered. It is often the case that budding music technologists are able to make good approximations with their voices to sounds they wish to synthesise electronically or acoustically, and a basic understanding of the acoustics of the human voice can facilitate this. As a starting point for the consideration of the acoustics of musical instruments the playing fundamental frequency ranges of a number of orchestral instruments as well as the organ, piano and singers are illustrated in Figure 4.3. A nine octave keyboard is provided for reference on which middle C is marked with a black square.
Fig 4.3 Playing fundamental frequency ranges of selected acoustic instruments and singers
The string family of musical instruments includes the violin, viola, violoncello and double bass and all their predecessors, as well as keyboard instruments which make use of strings, such as the piano, harpsichord, clavichord and spinet. In each case, the acoustic output from the instrument can be considered in terms of an input sound source and sound modifiers as illustrated in Figure 4.1. A more detailed discussion on stringed instruments can be found in Hutchins (1975a,b), Benade (1976), Rossing (1989), Hall (1991) and Fletcher and Rossing (1999). The playing fundamental frequency ([0) ranges of the orchestral stringed instruments are shown in Figure 4.3.
All stringed instruments consist of one or more strings stretched between two points, and the f0 produced by the string is dependent on its mass per unit length, length and tension. For any practical musical instrument, the mass per unit length of an individual string is constant and changes are made to the tension and/ or the length to enable different notes to be played. Figure 4.4 shows a string fixed at one end, supported on two singlepoint contact bridges and passed over a pulley with a variable mass hanging on the other end. The variable mass enables the string tension to be altered and the length of the string can be altered by moving the right-hand bridge. In a practical musical instrument, the tension of each string is usually altered by means of a peg with which the string can be wound or winched in to tune the string, and the position of one of the points of support is varied to enable different notes to be played except in instruments such as stringed keyboard instruments where individual strings are provided to play each note.
The string is set into vibration to provide the sound source to the instrument. A vibrating string on its own is extremely quiet because little energy is imparted to the surrounding air due to the small size of a string with respect to the air particle movement it can initiate. All practical stringed instruments have a body which is set in motion by the vibrations of the string(s) of the instrument, giving a large area from which vibration can be imparted to the surrounding air. The body of the instrument is the sound modifier. It imparts its own mechanical properties
Fig 4.5 input/system/output model for a stringed instrument
onto the acoustic input provided by the vibrating string (see Figure 4.5).
There are three main methods by which energy is provided to a stringed instrument. The strings are either 'plucked', 'bowed' or 'struck'. Instruments which are usually plucked or bowed include those in the violin family, instruments which are generally plucked only include the guitar, lute, and harpsichord, and the piano is an instrument whose strings are struck.
A vibrating string fixed at both ends, for example by being stretched across two bridge-like support as illustrated in Figure 4.4, has a unique set of standing waves (see Chapter 1). Any observed instantaneous shape adopted by the string can be analysed (and synthesised) as a combination of some or all of
Fig 4.6 The first ten possible modes of vibration of a string of length L fixed at both ends.
these standing wave modes. The first ten modes of a string fixed at both ends are shown in Figure 4.6. In each case the mode is illustrated in terms of the extreme positions of the string between which it oscillates. Every mode of a string fixed at both ends is constrained not to move, or it cannot be 'displaced', at the ends themselves, and these points are known as 'displacement nodes'. Points of maximum movement are known as 'displacement antinodes'. It can be seen in Figure 4.6 that the first mode has two displacement nodes (at the ends of the string) and one displacement antinode (in the centre). The sixth mode has seven displacement nodes and six displacement antinodes. In general, a particular mode (11) of a string fixed at both ends has (II + 1) displacement nodes and (II) displacement antinodes. The frequencies of the standing wave modes are related to the length of the string and the velocity of a transverse wave in a string (see Equation 1.7) by Equation 1.20.
When a string is plucked it is pulled a small distance away from its rest position and released. The nature of the sound source it provides to the body of the instrument depends in part on the position on the string at which is plucked. This is directly related to the component modes a string can adopt. For example, if the string is plucked at the centre, as indicated by the central dashed vertical line in Figure 4.6, modes which have a node at the centre of the string (the 2nd, 4th, 6th, 8th, 10th, etc., or the even modes) are not excited, and those with an antinode at the centre (the 1st, 3rd, 5th, 7th, 9th, etc., or the odd modes) are maximally excited. If the string is plucked at a quarter of its length from either end (as indicated by the other dashed vertical lines in the Figure), modes with a node at the plucking point (the 4th, 8th etc.) are not excited and other modes are excited to a greater or lesser degree. In general, the modes that are not excited for a plucking point a distance (d) from the closest end of a string fixed at both ends, are those with a node at the plucking position. They are given by:
Modes not excited = m(L/d)
Thus if the plucking point is a third of the way along the string, the modes not excited are the 3rd, 6th, 9th, 12th, 15th, etc. For a component mode not to be excited at all, it should be noted that the plucking distance has to be exactly an integer fraction of the length of the string in order that it exactly coincide, with nodes of that component.
This gives the sound input to the body of a stringed instrument when it is plucked. The frequencies (fn) of the component modes of a string supported at both ends can be related to the length, tension (T) and mass per unit length (mu) of the string by substituting Equation 1.7 for the transverse wave velocity in Equation 1.20 to give:
fn = (n/2L) x sqrt(T /mu)
The frequency of the lowest mode is given by Equation 4.2 when (n=1),
f1 = (1/2L) x sqrt(T /mu)
This is the f0 of the string which is also known as the first harmonic ("e Table 3.]). Thus the first mode (f1) in Equation 4.2 is the f0 of string vibration. Equation 4.2 shows that the frequencies of the higher modes are harmonically related to f0.
When a stringed instrument is struck such as in a piano, the same relationship exist, between the point at which the strike occurs and the modes that will be missing in the sound source. There is however, an additional effect that is particularly marked in the piano to consider. Piano strings are very hard and they are under enormous tension compared with the string on plucked instruments. When a piano string is stuck, it behaves partly like a bar because it is not completely flexible since it has some stiffness. This results in a slight raising in frequency of all the component modes with the effect being greater for the higher modes, resulting in the modes no longe being exact integer multiples of the fundamental mode. This effect, known as 'inharmonicity', varies as the square of the component mode (n^2), or harmonic number, and as the fourth power of the string radius (R^4). Thus for a particular string, the third mode is shifted nine times as much as the first, or fundamental, mode, and a doubling in string radius increases inharmonicity by a factor of sixteen (24). The effect would therefore be considerably greater for bass strings if they were simply made thicker to give them greater mass, and in many stringed instruments, including pianos, guitars and violins, the bass strings are wrapped with wire to increase their mass without increasing their stiffness. (A detailed discussion of the acoustics of pianos is given in Benade, 1976; Askenfelt, 1990; Fletcher and Rossing, 1999.)
The notes of a piano are usually tuned to equal temperament (see Chapter 3) and octaves are then tuned by minimising the beats between pairs of notes an octave apart. When tuning two notes an octave apart, the components which give rise to the strongest sensation of beats are the first harmonic of the upper note and the second harmonic of the lower note. These are tuned in unison to minimise the beats between the notes. This results in the fo of the lower note being slightly lower than half the /... of the higher note due to the inharmonicity between the first and second components of the lower note.
Example 4.1 If the f0 of a piano note is 400 Hz and inharmonicity results in the second component being stretched to 801 Hz, how many cents sharp will the note an octave above be if it is tuned for no beats between it and the octave below?
Tuning for no beats will results in the f0 of the upper note being 801 Hz, slightly greater than an exact octave above 400 Hz which would be 800 Hz. The frequency ratio (801 /800) can be converted to cents using equation A2.4 in Appendix 2:
Number of cents = 3986.3137 log10(801/800) = 2.16 cents
Inharmonicity on a piano increases as the strings become shorter and therefore the octave stretching effect increases with note pitch. The stretching effect is usually related to middle C and it becomes greater the further away the note of interest is in pitch. Figure 4.7 illustrates the effect in terms of the average deviation from equal-tempered tuning across the keyboard of a small piano. Thus high and low notes on the piano are tuned sharp and flat respectively to what they would have been if all octaves were tuned pure with a frequency ratio of 2:1. From the Figure it can be seen that this stretching effect amounts to approximately 35 cents sharp at C8 and 35 cents flat at Cl with respect to middle C.
Fig 4.7 Approximate form of the average deviations from equal temperament due to inharmonicity in a small piano. Middle C marked with a dot
The sound source that results from bowing a string is periodic and a continuous note can be produced while the bow travels in one direction. A bow supports many strands of hair, traditionally horsehair. Hair tends to grip in one direction but not in the other. This can be demonstrated with your own hair. Support the end of one hair firmly with one hand, and then grip the hair in the middle with the thumb and index finger of the other hand and slide that hand up and down the hair. You should feel the way the hair grips in one direction but slides easily in the other.
The hairs of a bow are laid out such that approximately half grip when the bow is moved in one direction and half when it is moved in the other. Rosin is applied to the hairs of a bow to increase its gripping ability. As the bow is moved across a string in either direction, the string is gripped and moved away from its rest position until the string releases itself, moving past its rest position until the bow hairs grip it again to repeat the cycle. One complete cycle of the motion of the string immediately under a bow moving in one direction is illustrated in the graph on the right hand side of figure 4.8 (when the bow moves moves in the other direction the pattern is reversed.) The string moves at a constant velocity when it is gripped by the bow hairs and then returns rapidly through its rest position until it is gripped by the bow hairs again. If the minute detail of the motion of the bowed string is observed closely, for example by means of stroboscopic illumination, it is seen to consist of two straight-line segments joining at a point which moves at a constant velocity around the dotted track as shown in the snapshot sequence in Figure 4.8. The time taken to complete one cycle, or the fundamental period (T0) is the time taken for the point joining the two line segments to travel twice the length of the string (2L):
T0 = (2L)/v
Fig 4.8 One complete cycle of vibration of a bowed string and graph of string displacement at the bowing point as a function of time.
Substituting equation 1.7 for the transverse wave velocity gives:
T0 = 2L sqrt(mu/T)
The f0 of vibration of the bowed string is therefore:
f0 = (1/T0) = (1/2L) x sqrt(T/mu)
Comparison with Equation 4.2 when (n=1) shows that this is the frequency of the first component mode of the string. Thus the f0 for a bowed string is the frequency of the first natural mode of the string, and bowing is therefore an efficient way to excite the vibrational modes of a string
The sound source from a bowed string is that of the waveform of string motion which excites the bridge of the instrument. Each of the snapshots in Figure 4.8 correspond to equal time instants on the graph of string displacement at the bowing point in the Figure, from which the resulting force acting on the bridge of the instrument can be inferred to be of a similar shape to that at the bowing point. In its ideal form, this is a sawtooth waveform (see Figure 4.9). The spectrum of an ideal sawtooth wavefom1 contains all harmonics and their amplitudes decrease with ascending frequency as (1/ n), where n is the harmonic number. The spectrum of an ideal sawtooth waveform is plotted in Figure 4.9 and the amplitudes are shown relative to the amplitude of the AI component.
The sound source provided by a plucked or bowed string is coupled to the sound modifiers of the instrument via a bridge. The vibrational properties of all elements of the body of the instrument play a part in determining the sound modification that takes place. In the case of the violin family, the components which contribute most significantly are the top plate (the plate under the strings which the bridge stands on and which has the f holes in it), the back plate (the back of the instrument), and the air contained within the main body of the instrument. The remainder of the instrument contributes to a very much lesser extent to the sound-modification process, and there is still lively debate in some quarters about the importance or otherwise of the glues, varnish, choice of wood and wood treatment used by highly regarded violin makers of the past. Two acoustic resonances dominate the sound modification due to the body of instruments in the violin family at low frequencies: the resonance of the air contained within the body of the instrument or the 'air resonance', and the main resonance of the top plate or 'top resonance'. Hall (1991) summarises the important resonance features of a typical violin as follows:
Apart from the air resonance which is defined by the internal dimensions of the instrument and the shape and size of the f holes the detailed nature of the response of these instmments is related to the main vibrational modes of the top and back plates. As these plate, are being shaped by detailed carving, the maker will hold each plate at particular point and tap it to hear how the so called 'tap tones' are developing to guide the shaping process. This ability is a vital part of the art of the experienced instrument maker in setting up what will become the resonant properties of the complete instrument when it is assembled.
The acoustic output from the instrument is the result of the sound input being modified by the acoustic properties of the instrument itself. Figure 4.10 (from Hall, 1991) shows the input spectrum for a bowed G3 (f0 =196 Hz) with a typical response curve for a violin, and the resulting output spectrum. Note that the frequency scales are logarithmic, therefore the harmonics in the input and output spectra bunch together at high frequencies. The output spectrum is derived by multiplying the amplitude of each component of the input spectrum by the response of the body of the instrument at that frequency. In the figure, this multiplication becomes an addition since the amplitudes are expressed logarithmically as dB values and adding logarithms of numbers is mathematically equivalent to multiplying the numbers themselves.
There are basic differences between the members of the orchestral string family (violin, viola, cello and double bass) differ from each other acoustically in that the size of the body of each instrument becomes smaIler relative to the f0 values of the open strings (e.g. Hutchins, 1978). The air and tap resonances approximately coincide as follows: for the violin with f0 of the D4 (2nd string) and A4 (3rd string) strings respectively, for the viola with f0 values approximately midway between the G3 and D4 (2nd and 3rd strings) and D4 and A4 (3rd and 4th srings) strings respectively, for the cello with f0 of the G2 string and approximately midway between the D3 and A3 (3rd and 4th strings) respectively and for the double bass with f0 of the D2 (3rd string) and G2 (4th string) strings respectively. Thus there is more acoustic support for the lower notes of the violin than for those of the viola or the double bass, and the varying distribution of these two resonances between the instruments of the string family is part of the acoustic reason why each member of the family has its own characteristic sound.
Figure 4.11 shows waveforms and spectra for notes played on two plucked instruments, C3 on a lute and F3 on a guitar. The decay of the note can be seen on the waveforms, and in each case the note lasts just over a second. The pluck position can be estimated from the spectra by looking for those harmonics which are reduced in amplitude that are integer multiples of each other (see Equation 4.2). The lute spectrum suggests a pluck point at approximately one sixth of the string length due to the clear amplitude dips in the 6th and 12th harmonics, but there are also clear dips at the 15th and 20th harmonics. An important point
Fig 4.11 Waveforms and spectra for C3 played on a lute and and F3 played on a six string guitar
to note is that this is the spectrum of the output from the instrument, and therefore it includes the effects of the sound modifiers (e.g. air and plate resonances), so harmonic amplitudes are affected by the sound modifiers as well as the sound source. Also, the 15th and 20th harmonics are nearly 40 dB lower than the low harmonics in amplitude and therefore background noise will have a greater effect on their amplitudes. The guitar spectrum also suggests particularly clearly a pluck point at approximately one sixth of the string length, given the dips in the amplitudes of the 6th, 12th and 18th harmonics.
Sound from stringed instruments does not radiate out in all directions to an equal extent and this can make a considerable difference if, for example, one is critically listening to or making recordings of members of the family. The acoustic output from any stringed instrument will contain frequency components across a wide range, whether it is plucked, struck or bowed. In general, low frequencies are radiated in essentially all directions, with the pattern of radiation becoming more directionally focused as frequency increases from the mid to high range. In the case of the violin, low frequencies in this context are those up to approximately 500 Hz, and high frequencies, which tend to radiate outwards from the top plate, are those above approximately 1000 Hz. The position of the listener's ear or a recording microphone is therefore an important factor in terms of the overall perceived sound of the instrument.
You Need to Know
Sound source from a plucked string
Sound source from a struck string
Sound source from a bowed string
Sound modifiers in stringed instruments