According to Galm Percussion instruments may be our oldest musical instruments with the exception of the human voice, but recently they have experienced a new surge in interest and popularity. Many novel percussion instruments have been developed recently and more are in the experimental stage. What is often termed "contemporary sound" makes extensive use of percussion instruments. Relatively little material has been published on the acoustics of percussion instruments. The term percussion means "struck" and strictly speaking percussion instruments are those in which sound is produced by striking. However, the percussion section in a modem orchestra employs many instruments that do not depend upon striking a blow. Indeed, the percussion section is expected to create any unusual sound effect that a composer has in mind. New instruments are constantly being invented and added to the percussionist's repertoire.
There are several ways that have been used to classify percussion instruments. Sometimes they are classified into four groups: idiophones (xylophone, marimba, chimes, cymbals, gongs, etc.); membranophoncs (drums); aerophones (whistles, sirens, etc.); and chordophones (piano, harpsichord). There may be differences of opinion as to whether aerophones and chordophones properly belong in the percussion family. Whistles and sirens are generally played by percussionists in the orchestra; the piano and harpsichord are not. At any rate, this book deals mainly with idiophones and membranophones.
Another way of classifying percussion instruments is by whether or not they convey a definite sense of pitch. Idiophones that convey a definite pitch include bells, chimes, xylophones, marimbas, gongs, and steelpans. Membranophones that convey a definite pitch include timpani, tabla, and mrdanga. Sometimes we described percussive sounds as having a "high" or "low" pitch even if they do not convey an identifiable pitch, but it would be more correct to describe this as high or low range or tessitura. Percussionists in a modem orchestra or band may have hundreds of instruments to play. Generally the timpanist plays only the timpani, but the other percussionists divide the remaining instruments depending upon (he demands of the music. Some works require as many as ten or more percussionists; Schoenberg's Gurrelieder, for example, calls for two timpanists and ten other percussionists. Percussion instruments generally use one or more of the following basic types of vibrators: strings, bars, membranes, plates, air columns, or air chambers. The first four are mechanical; the latter two are pneumatic. Two of them (the string and the air column) lend to produce harmonic overtones; bars, plates, and membranes, in general, do not. The inharmonic overtones of complex vibrators give percussion instruments their distinctive timbres.
Ficken (1976) has mentioned that most natural systems follow some type of rhythm: beating hearts, the motion of the planets, ocean waves, phases of the moon, the seasons, the list is long, It is only natural that primitive humans would begin sinking sticks or stones together rhythmically. Rhythm is one of the key ingredients in music, and percussion instruments often establish and maintain the rhythm in the performance of music. One of the best histories of percussion instruments is Percussion Instruments and Their History by renowned percussionist James Blades.]!] This book traces percussion instruments from their primitive origins to composers' use of modem percussion.
Blades points out that the earliest instruments were probably idiophones, instruments made of naturally sonorous material which can produce sound without the addition of stretched skin or column of air. These are of five types: shaken idiophones (rattles), stamped idiophones (pits, boards, hollow tubes); scraped idiophones (notched sticks or rasps); concussion idiophones (pairs of similar items such as sticks); and struck idiophones (one or more pieces of sonorous material struck with a stick or bone). Early in our lives we learned to play with rattles. It is interesting that rattles are among the earliest of percussion instruments. The gourd rattle, a seed pod in which the dried seeds remain, was widely used in primitive societies, especially in Africa. Rattles are still popular in orchestras and ensembles, especially for the performance of Latin American music.
Scraped instruments are found as far back as the early Stone Age. A stick could be drawn across a notched stone, bone, shell, or gourd to produce a raspy sound. The bone scraper has been closely associated with the hunt, erotic rituals, and funeral ceremonies. Scrapers were found among Indian tribes in North and South America, and also in Africa. The earliest drums were probably log drums of various types. Later, it was discovered that by stretching an animal skin across the cavity in the log. a louder sound could be made. Eventually the membrane drum came to be the most important percussion instrument. The earliest drums were probably struck with the hands, but the use of sticks as beaters was found to increase the loudncss of the sound. Later a second membrane was added. Today, there are thousands of different types of drums found throughout the world. Throughout the years, drums have been used for signaling, for sending messages, and for marshaling troops to battle as well as for performing music.
4.3. Percussion Ensembles
Although percussion instruments have most often been used in ensemble with string and wind instruments, a number of successful ensembles have relied on percussion instruments alone. Sometimes these ensembles use one type of instrument, such as steel bands and marimba orchestras. more often they employ a variety of percussion instruments, such as the Black Earth ensemble, whose members were once artists-in-residence at our university (Serra, 1986).
4.4 Drums with Definite Pitch
According to Van den Doel and Pai (1996) drums have played an important role in nearly all musical cultures They have been used to transmit messages, convey the time of day, send soldiers into battle, and warn of impending danger Drums are practically as old as the human race. The earliest drums were probably chunks of wood or stone placed over holes in the earth. Then it was discovered that more sound could be obtained from hollow tree trunks, the ancestors of our contemporary log drums. The most familiar type of drum consists of a membrane of animal skin or synthetic material stretched over some type of air enclosure.
4.5 Vibrations of Strings: A Little Bit of Physics
Richardson (1997) has mentioned that a guitar siring is probably the simplest of all musical vibrators. Yet its vibrations can be deceptively complex. When drawn to one side and released, the string vibrates in a rather complex way that can be described as a combination of normal modes of vibration. For example, if it is plucked at its centre, the nearly triangular shape it assumes, as it vibrates to and fro, can be thought of as being made up of simple modes having frequencies that correspond to the fundamental plus the odd-numbered harmonics, as listed in Figure 4.1:
Figure 4.1: Odd numbered vibration modes add up in appropriate amplitude and phase to the shape of a string plucked at its center
Source: Fletcher N H and Rossing T D (1991), The Physics of Musical instruments, Springer Verlag, New York
The above figure 4.1 illustrates how the modes associated with the odd-numbered harmonics, when each is present in the right proportion, can add up at one instant to give the initial shape of the string Modes 3. 7, 11, etc must be opposite in phase from modes 1.5. and 9 in order to give a maximum in the center. Also, the relative amplitudes are in the ratios I/n2, where n is the number of the mode. The force necessary to restore the string to its center (equilibrium) position when it is displaced comes from the force or tension applied to the string To tune the string to a higher frequency, the guitarist increases the tension by means of the tuning machines or pegs. Increasing the tension increases the frequencies of all the various modes of vibration but maintains their harmonic ratios
4.6 Vibrations of Membranes: Key to Understanding Drums
According to Yoo, Rossing and Larkin (2003) a membrane can be thought of as a two-dimensional string, in that its restoring force is due to tension applied from the edge. A membrane, like a string, can be tuned by changing the tension. Membranes, being two-dimensional, can vibrate in many modes that are not normally harmonic; that is. The frequencies of the higher modes are not simple integers times the fundamental frequency. In the four modes of a circular membrane the first mode (the fundamental), the entire membrane moves in the same direction, although the center moves through the greatest amplitude. In the other modes, there are one of more nodal circles or nodal diameters that act as boundaries or pivot lines. The parts of the membrane on either side of a nodal line move in opposite directions. A membrane, like a string, can be tuned by changing its tension. A major difference between vibrations in a membrane and in a siring, however, is that while the mode frequencies in a string are harmonics of the fundamental, in a two-dimensional membrane they are not. Another difference is that in a membrane, nodal lines (circles and diameters) replace nodal points along the string. In the first 12 modes of a membrane of nodal circles and nodal diameters the nodal circle always occurs at the edge of the membrane where it is supported. Above each diagram are given two numbers that designate the number of nodal diameters and circles, respectively. For example, the (21) mode has two nodal diameters and one nodal circle (at the edge).
In cither a string or a membrane, the modal frequencies vary as the square root of the tension. Thus to double the frequency, the tension would have to be quadrupled, which is quite impractical. A more practical example would be that to raise the frequency by 6 % (corresponding to a semitone on the musical scale) the tension would have to increase by about 12%. Actually the frequency of a membrane is determined by the ratio of tension to mass per unit area, so increasing the thickness of a drumhead by 12% lowers the mode frequencies by 6% just as increasing the tension by 12% raises it by the same amount. Increasing the radius by 12%, on the other hand, decreases the mode frequencies by a full 12%.
Rossing and Fletcher (1982) have described that the timpani or kettledrums are the most frequently used drums in the orchestra, one member of the percussion section usually devoting attention exclusively to them. During the last century, various mechanisms were developed for changing the tension to tune the drumheads rapidly. Most modern timpani have a pedal-operated tensioning mechanism in addition to six or eight tensioning screws around the rim of the kettle. The pedal typically allows the player to vary the tension over a range of 2:1 which corresponds to a tuning range of about a musical fourth. At one time all timpani heads were calfskin. But this material has gradually given way to Mylar (polyethylene terephthalate). Calfskin heads require a great deal of hand labour to prepare and great skill to tune properly. Some orchestral timpanists prefer them for concert work under conditions of controlled humidity, but use Mylar when touring. Mylar is insensitive to humidity and easier to tune, due (o its homogeneity. A thickness of 0.19 mm (0.0075 inch) is considered standard for Mylar timpani heads. Timpani kettles are roughly hemispherical; copper is the preferred material, although fibreglass and other materials are also used.
Borin, De Poli and Sarti (1992) have described that the modes of vibration of an ordinary membrane are not harmonic in frequency, a carefully tuned kettledrum sounds a strong fundamental plus two or more harmonic overtones. Lord Rayleigh recognized the principal note as coming from the (II) mode and identified overtones about a perfect fifth (3:2 frequency ratio), a major seventh (15:8 frequency ratio), and an octave (2:1 frequency ratio) above the principal tone. Timpanist H. W. Taylor identified a tenth (octave plus a third, 5:2 frequency ratio) by humming near the drumhead, a technique some timpanists use to fine-tune their instruments. How are the inharmonic modes of an ordinary membrane coaxed into a harmonic relationship in the timpani? Three effects contribute:
The membrane vibrates in a "sea of air." and the mass of this air lowers the frequency of the vibration at modes, especially those of low frequency;
The air enclosed by the kettle has resonances of its own that interact with the modes of the membrane that have similar shapes;
The stiffness of the membrane raises the frequencies of the higher overtones. Our studies show that the first effect (air loading) is mainly responsible for establishing die harmonic relationship of kettledrum modes; the other two effects only "fine tune" the frequencies but may have considerable effect on the decay rate of the sound (3).
2.4. Timpani Sound
According to Henrique and Antunes (2003) Vibration frequencies of a kettledrum, a drumhead with the kettle, and an "ideal" (unloaded) membrane are given in Table 2.1 along with ratios to the principal (11) mode. Note that the enclosed air in the kettle raises the frequencies of the (01), (02), and (03) modes which are circularly symmetrical. The (II). (21), (31). and (41) modes in the kettledrum have frequencies in the ratios I : 1.5 : 1.99:2.44. which is sufficiently close to the harmonic ratios 1 : 1.5 :2:2.5 to give the kettledrum a strong sense of pitch. To preferentially excite these modes, the timpanist generally strikes the head about one-fourth of the way from the edge to the center. Striking the head at its centre preferentially excites the rapidly-decaying (01) mode (along with the (02) and (03) modes, of course), producing a "thud."
Table 2.1. Vibration frequencies of a 65-cm (26-inch) kettledrum a drumhead without the kettle, and an ideal membrane (perfectly flexible and unloaded by air
Source: Smith J O (1992), Physical Modeling Using Digital Waveguides, Computer Music Journal, Vol. 16, No.4, p 74-91
Sound spectra obtained by striking a 65-cm (26-inch) diameter kettledrum in its normal place (about one-fourth of the way from the edge to the centre) and at the centre are shown in Figure 4.2:
Figure 4.2: Sound spectra from 65 cm kettledrum turned to E3
Source: Main I G (1993), Vibrations and Waves in Physics, 3rd edition, Cambridge University
Note that the (0, 1) mode appears much stronger when the drum is struck at the centre, as do the other symmetrical modes [(0,2) and (0,3)]. These modes die out rather quickly, however, so they do not produce a very clear sound. In fact, striking the drum at the centre produces quite a dull, thumping sound (Fontana and Rocchesso, 1996).
2.5. Interlude: Subjective Tones and Pitch of the Missing Fundamental
At this point, we would like to take time to describe a very interesting psycho acoustical effect mat has important applications in the perception of sound and music. When the ear is presented with a tone composed of exact harmonics, it is easy to predict what pitch will be heard. It is simply the lowest common factor in these frequencies, which is the fundamental. The ear identifies the pitch of the fundamental, even if the fundamental is very weak or missing altogether. For example, if the ear hears a tone having partials with frequencies of 600.800.1000, and 1200 Hz, the pitch will nearly always be identified as that of a 200-Hz tone, the "missing fundamental." This is an example of what is called virtual pilch, since the pitch doesn't correspond to any partial in the complex tone. The ability of the car to determine a virtual pitch makes it possible for the undersized loudspeaker of a portable radio to produce bass tones, and it also forms the basis for certain mixture stops on a pipe organ.
If a strong fundamental is not essential for perceiving the pitch of a musical tone, the question arises as to which harmonics are most important. Experiments have shown that for a complex tone with a fundamental frequency up to about 200 Hz the pitch is mainly determined by the fourth and fifth harmonics. As the fundamental frequency increases, the number of the dominant harmonics decreases, reaching the fundamental itself for -2500 Hz and above (Plomp, 1976). When the partials of the complex tone are not harmonic, however, the determination of virtual pitch is more subtle. The car can apparently pick out a series of nearly harmonic partials somewhere near the center of the audible range, and determine the pitch to be the largest near-common factor in the series. Several demonstrations of virtual pilch are presented on a Compact Disc by Houtsma, Rossing, and Wagcrumrs (1987),
Musical examples of virtual pitch from "near harmonics" in a complex tone are the sounds of bells and chimes. In each case, the pitch of the strike note is determined mainly by three partials that have frequencies almost in the ratio 2:3:4. In the case of the bell, there is usually another partial with a frequency near that of the strike note which reinforces it. In the case of the chime, however, there is none: The pitch is purely subjective.
2.6 The Kettle:
According to Van Duyne and Smith (1994) From Table 2.1, it is clear that the frequencies of the important (11), (21), (31) and (41) modes are not much influenced by the kettle. Without the kettle, these mode frequencies had ratios of 1.00:1.47:1,91-2.36, not quite as harmonic as the corresponding mode frequencies with the kettle but quite tolerable, liven without the kettle, an air-loaded timpani membrane conveys a fairly definite sense of pitch. The main role of the kettle is I hat of a baffle to acoustically separate the top and bottom sides of the membrane. This Increases the radiation efficiency which decreases the decay limes, especially in the modes of tower frequency. This is similar to the acoustical effect of an enclosure on a loudspeaker: without the enclosure radiation efficiency is diminished at the lower frequencies.
Vibration of Air Unclosed by the Kettle
One expects the air enclosed by the kettle to vibrate in a series of normal modes that can couple to membrane mode having maxima in the same relative positions, just as the enclosed air in a guitar body, for example, couples to the top and hack plates . in order to better understand how the kettle fine tunes the vibrational modes of the membrane, we did a series or experiments to determine die vibrational modes of the air enclosed by the kettle. In order to do this, we covered the kettle with a rigid cover having small holes through which a driving tube and probe microphone could be inserted. In the top line the normal modes of the air in a rigidly capped 65-cm (26-inch) diameter kettle. In every case, the kettle air modes are higher in frequency than the membrane modes (see Table 4.1) to which they couple. Also shown in Fig. 2.6 are the corresponding air modes in the kettle when the volume is reduced to one-half and one-quarter a f the original volume by partly filling the kettle with water. Note that some air modes are raised in frequency while some are lowered by reducing the kettle volume (Rossing, 2000).
Is Kettle Volume important in Determining Timpani Sound? Yes.
According to Cremer, Heckl and Ungar (1988) the effect of kettle volume and shape on timpani sound was the subject of a collaborative study carried out at Northern Illinois University and Purdue University. Experimental studies included measurements of sound spectra and decay limes with the kettle reduced in volume by adding water, as in the experiments on air modes described in the previous section. The experimental results agreed very well with theoretical calculations made using mathematical Green function . The results of these studies (summarized in Chapter IS of ref. (9) showed, not surprisingly, that modern timpani have kettles of just about the right volume to optimize the harmonicity of the principal partial S. A lot of dial and error and careful design has gone into modem timpani.
Is Kettle Shape Important in Determining Timpani Sound? No.
Careful studies indicate that the shape of the timpani kettle is quite unimportant in determining timpani sound, provided the volume is kept in the correct range. These results may surprise same timpanists. Since articles in the literature frequently refer to "shaping" the sound by using kettles with hemispherical, parabolic, or other shapes. A simple fact that helps explain why kettle shape is unimportant is that the sound wavelengths are so much larger than the kettle dimensions. (Al 140 Hz, for example, the wavelength is 2.5 meters, and even at 440 Hz, it is about 0,8m). Thus the mode frequencies of the enclosed air, which depend on kettle volume, are virtually unelected by the shape of the talk.
Another misunderstanding about kettles has to do with the vent hole generally placed at the bottom of the kettle to equalize air pressure inside and outside as the barometric pressure changes (and thus prevent the membrane from bulging up or dishing down). Al least one respected source reports that the vent hole has a strong influence on timpani sound through its damping of air vibrations in the kettle [11). Careful studies show that this is not the case, however. The vent hole is convenient for humidifier pads when using calfskin heads.
Sound Decay With and Without the Kettle
Ewins (2000) has mentioned that the decay times are quite long for the (M), (21), (31) and (41) modes but quite short for the (01) and (02) modes. Sound decay times for several modes with and without the kettle are shown in Table 2.2:
Table 4.2: Decay times for a 65 cm timpani membrane with and without a kettle
Source: Fontana F and Rocchesso D (1996), Physical Modeling of Membranes for Percussion Instruments, Submitted to Acustica
Following the practice generally employed in architectural acoustics to express reverberation time, tÃ¯ÂÂ³ is the time for the sound to decrease by 60 dB. Note dial limes of (he (01). (02), and (03) modes are very short with the kettle in place, Decay times for the (11) and (21) modes are shorter with the kettle dun without, although an exact comparison Is made mote difficult by the fact that different membrane tensions led to slightly different mode frequencies in the two experiments. On the basis or Table 2.2 and similar data it is possible to make the following observations:
(1) The harmonically tuned (1,1). (2.1), (3.1), and (4,1) modes decay much more slowly than other modes, especially the (0,1) (0,2) and (0,3) modes.
(2) The (0,1) mode, which acts as a monopole source, decays rapidly: the baffled (1,1) mode end the unbaffled (0,1) mode, which act as dipole source, decay less rapidly; the baffled (2.1) mode and the unruffled (1,1J mode decay still less rapidly.
2.7. Interlude : Sound Radiation
Campbell (1999) has mentioned that the, sound field radiated by musical instruments, like other sound sources, can often be approximated by considering radiation from simple sound sources. The simplest source ii a monopole source, which radiates equally in all directions. A balloon being inflated and deflated many times a second would be such a source, Almost any source that is much smaller than the wavelength of sound approximates a monopole The radiation efficiency of a monopole source increases as which indicates a lesser dependence on frequency than dipoles or higher order sources. A dipole source could be characterized by two balloons connected by a pipe, so that when one is inflated me pitch is deflated so that "here is no net displacement of air. In the plane that bisects the dipole. the sound Held will be zero due to cancellation from the two equally distant balloons. However, at all other locations the radiation from the nearer balloon is greater, and a net sound field results, as shown in Fig, 2,6 b. The radiation efficiency of a dipole source is very small at low frequency, but it increases as fââ‚¬â„¢ so becomes more of a factor at higher frequency.
Fig. 2.6. (a) A balloon alternately inflating and deflating acts as a monopolc source which radiates sound equally in all direction; (b) Two balloons connected by a pipe, so that one inflates when the other deflates, act as a dipole source which has maximum sound radiation in the axial direction (c) Two dipoles end to end form a liner quadrupole source; (d) Two parallel dipoles form a tesseral quadrupole source.
Source: Morse P M and Ingard K U (1968), Theoretical Acoustics, Princeton University Press, USA.
Smith (1992) has described that two dipoles can be combined in two different ways to form a quadrupole source, as shown in Fig. 2.6 c, d. A linear quadrupole. shown in Fig. 2.6 e consists of two dipoles end to end, so that the radiated field has a maximum along the axis, as does the field of a dipole, but now the plane of symmetry becomes a weaker maximum. When dipoles are placed side-by-side as in Fig. 2.6 d, a tessera quadrupole results, and the sound field has four maxima in the directions of the individual radiaiore. as shown, For either type of quadrupole, the radiation field is proportional to f* so they are effective radiators only at relatively high frequencies. A timpani membrane vibrating in its lowest 3 modes, with and without a kettle, roughly approximates monopole, dipole, and quadrupole sources, as shown in Fig. 2.7. The (0,1) mode, Tot example, approximates a monopole source when it vibrates over a kettle (2.7a) but approximates a dipole source (2.7b) when it vibrates without the kettle to act as a baffle. Thus it radiates more efficiently (and loses its vibrational energy more quickly) when the kettle attached, as we have al ready discussed.
Radiation from a violating timpani membrane: (a) (0,1) mode in a membrane baffled by the kettle approximates a monopole (b) (0,1) mode in an unbaffled membrane (kettle removed) approximates a dipole; (d) (1,1) mode without baffle (kettle) approximates a quadrupole; (e) (2,1) mode with kettle also approximates a quadrupole. Graphs showing the directionality of sound radiation from a 40-cm diameter kettledrum are shown. Note that the radiation from the (01) mode is the same in all directions in the plane of the membrane, while the (11) mode radiates more strongly in two directions, the (21) mode in & directions, and the (31) mode in 6 directions .
One lesson to be learned is that no two persons in the concert hall hear the same musical performance. Musical instruments radiate different sounds in different directions, and so each listener hears a different mix of sounds. Of course sound reflections from walls, ceiling, and other surfaces tend to mix the sounds and reduce the directional dependant? Nevertheless, music critics should realize that the listener a few meters away may be hearing quite a different concert. The directional patterns of musical instruments create quite a problem for recording engineers. These are especially acute if microphones are placed so close to instruments that the direct sound exceeds the reflected sound. Finally, the performer may have the "worst" location in the hall when it comes to hearing the sound from his/her own and nearby instruments.
Tabla and Mrdanga
Rossing et al (1992) has mentioned that foremost among the drums of India are the tabla of North India and the mrdanga (also known as mirdangam or mirdang) of South India shown. Both drums convey a strong sense of pitch due lo the harmonic liming of the partials in their sound spectra. Unlike timpani, which depend upon air loading of the drumhead to shift the inharmonic modes of vibration of an ordinary membrane into a nearly harmonic relationship, tabla and mrdanga load the membrane with a paste of starch, gum, iron oxide, charcoal or other materials. The small diameter of their drumheads make tuning by air loading alone impractical. The term tabla is sometimes used to describe a single drum hut more often it describes a pair of drums used in Hindustani (North Indian) classical music. The right-hand drum or daya (or dahina meaning right) is a tunable dram with p range of about an octave. The body is traditionally carved from a block of wood. The left-hand drum or bayan (also bamya, meaning left) is a larger drum, generally with a metal body.
Obata and Tesima (1935) have described that the most unusual feature of die daya is die three-piece design of the head. The main surfuce of the head (sur) is held in place by a collar (kinar) or goatskin whose edge is woven into a braid (gajra) which is held onto the body with straps, At the center of the drumhead (pudi) is a black disk, known as the syahi which is carefully shaped to accomplish the harmonic lulling of the partials. The head of the bay an al so has a syahi, though it is placed off ââ‚¬"center. When played, the palm of the left hand is placed near the center of the bayan, and the player can change the pitch of the drum by changing the pressure on the head. Tabla players make extensive use of vocal mnemonics for describing compositions. Ilach stroke is given a one-syllable name. Strokes on the Uayan are pmnuunccd toward ihc from of the mouth (na, ta, te, tin, mn), while strokes on the bnyan are pronounced toward the back (ge, ke). This allow fast rolls to be pronounced quickly,
The mrdanga (or mrdongam) is the main drum In the Carnatic tradition in South India. This instrument is j single piece of wood that is hollowed out and has playing heads on both sides so that it functions, in many respects, like the day on and bay on combined into one. The mridanga uses heavier hides than the tabla, and since they resonate over a common chamber, there is acoustical coupling between the two heads. The larger head of the mrdangam is generally loaded with a paste of wheat and water shortly before playing. The tabla often has a siring placed between die annular covering and the main skin, while the mridanga uses straw.