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Acoust. Sci. & Tech. 25, 6 (2004) INVITED REVIEW

Acoustics of percussion instruments: An update

Thomas D. Rossing, Junehee Yoo and Andrew Morrison Physics Department, Northern Illinois University, DeKalb, IL 60115, USA

Abstract: Methods for studying the modes of vibration and sound radiation from percussion instruments are reviewed. Recent studies on the acoustics of , , , tamtams, lithophones, , and are described. Vibrational modes and sound radiation from the , a new steel are presented.

Keywords: Percussion, Marimbas, Cymbals, Gongs, Tamtams, Lithophones, Steelpans, bells, HANG PACS number: 43.75.Kk, 43.40.Dx [DOI: 10.1250/ast.25.406]

frequency (eigenfrequency). It should be possible to excite 1. INTRODUCTION a normal mode of vibration at any point in a structure that Percussion instruments are an important part of every is not a node and to observe motion at any other point that musical culture. Although there has been less research on is not a node. It is a characteristic only of the structure the acoustics of percussion instruments, as compared to itself, independent of the way it is excited or observed. In wind or string instruments, quite a number of scientists practice, it is difficult to avoid small distortions of the continue to study these instruments. normal modes due to interaction with the exciter, the In 2001 we published a review paper on the acoustics sensor, and especially the supports. of percussion instruments in this journal [1]. It is the Normal modes shapes are unique for a structure, purpose of this paper to continue the discussion of whereas the deflection of a structure at a particular percussion instruments, and to summarize some research frequency, called its operating deflection shape (ODS), done in recent years. may result from the excitation of more than one normal mode [2]. Normal mode testing has traditionally been done 2. METHODS FOR STUDYING using sinusoidal excitation, either mechanical or acoustical. PERCUSSION INSTRUMENTS Detection of motion may be accomplished by attaching Recent studies of the acoustics of percussion instru- small accelerometers, although optical and acoustical ments have included: 1) theoretical studies of modes of methods are less obtrusive. Modal testing with impact vibration; 2) experimental studies of modes of vibration; 3) excitation, which became popular in the 1970s, offers a sound radiation studies; 4) physical modeling; 5) studies of fast, convenient way to determine the normal modes of a nonlinear behavior. structure. In this technique, an accelerometer is generally attached to one point on the structure, and a hammer with a 2.1. Finite Element and Boundary Element Methods load cell is used to impact the structure at carefully For all but the simplest vibrator shapes, it is difficult to determined positions. Estimates of modal parameters are calculate vibrational modes analytically. Fortunately, there obtained by applying some type of curve fitting program. are powerful numerical methods that can be carried out Experimentally, all modal testing is done by measuring quite nicely by use of digital computers. These generally operating deflection shapes and then interpreting them in a are described as finite element methods or boundary specific manner to define mode shapes [2]. Strictly speak- element methods. ing, some type of curve-fitting program should be used to determine the normal modes from the observed ODSs, 2.2. Experimental Studies of Modes of Vibration even when an instrument is excited at a single frequency. When a percussion instrument is excited by striking (or In practice, however, if the mode overlap is small, the bowing or plucking), it vibrates in a rather complicated single-frequency ODSs provide a pretty good approxima- way. The motion can be conveniently described in terms of tion to the normal modes. normal modes of vibration. A normal mode of vibration 2.2.1. Scanning with a microphone or an accelerometer represents the motion of a linear system at a normal Probably the simplest method for determining ODSs

406 T. D. ROSSING et al.: ACOUSTICS OF PERCUSSION INSTRUMENTS

(and hence normal modes) is to excite the structure at 2.2.3. Experimental modal testing single frequency with either a sinusoidal force or a Modal testing may be done with sinusoidal, random, sinusoidal sound field, and to scan the structure with an pseudorandom, or impulsive excitation. In the case of accelerometer or else to scan the near-field sound with a sinusoidal excitation, the force may be applied at a single small microphone [3]. With practice, it is possible to point or at several locations. The response may be determine mode shapes rather accurately by this method. measured mechanically (with accelerometers or velocity 2.2.2. Holographic interferometry sensors), optically, or indirectly by observing the radiated Holographic interferometry offers by far the best spatial sound field. resolution of operating deflection shapes (and hence of In modal testing with impact excitation, an acceler- normal modes). Whereas experimental modal testing and ometer is typically attached to the instrument at some key various procedures for mechanical, acoustical, or optical point, and the instrument is tapped at a number of points on scanning may look at the motion at hundreds (or even a grid with a hammer having a force transducer (load cell). thousands) of points, optical holography looks at almost an Each force and acceleration waveform is Fourier trans- unlimited number of points. formed and a transfer function Hij is calculated. Several Recording holograms on photographic plates or film (as different algorithms may be used to extract the mode in most of the holographic interferograms in [1]) tends to shape and modal parameters from the measured transfer be rather time consuming since each mode of vibration functions [5]. must be recorded and viewed separately. TV holography, on the other hand, is a fast, convenient way to record ODSs 2.3. Radiated Sound Field and to determine the normal modes [4]. The best way to describe sound radiation from complex An optical system for TV holography is shown in Fig. 1. sources such as percussion instruments is by mapping the A beam splitter BS divides the laser light to produce a sound intensity field. Sound intensity is the rate at which reference beam and an object beam. The reference beam sound energy flows outward from various points on the reaches the CCD camera via an optical fiber, while the instrument. The sound intensity field represents the object beam is reflected by mirror PM so that it illuminates direction and the magnitude of the sound intensity at every the object to be studied. Reflected light from the object point in the space around the source. reaches the CCD camera, where it interferes with the A single microphone measures the sound pressure at a reference beam to produce the holographic image. The point, but not the direction of the sound energy flow. In speckle-averaging mechanism SAM alters the illumination order to determine the sound intensity it is necessary to angle in small steps in order to reduce laser speckle noise in compare the signals from two identical microphones the interferograms. spaced a small distance apart. The resulting pressure Generally holographic interferograms show only var- gradient can be used to determine sound intensity. When iations in amplitude. It is possible, however, to recover this is done at a large number of locations, a map of the phase information by modulating the phase of the reference sound intensity field results [6]. beam by moving mirror PM at the driving frequency. This is a useful technique for observing motion of very small 2.4. Physical Modeling amplitude or resolving normal modes of vibration which Synthesizing sounds by physical modeling has attracted are very close in frequency. a great deal of interest in recent years. The basic notion of physical modeling is to write equations that describe how particular sets of physical objects vibrate and then to solve those equations in order to synthesize the resulting sound. Percussion instruments have proven particularly difficult to model completely enough to be able to synthesize their sounds entirely based on a physical model. Physical modeling is complicated by their nonlinear behavior and by the strong role which transients play in their sound. 3. MARIMBAS In most of the World, the term denotes a deep-toned instrument with tuned bars and resonator tubes that evolved from the early Latin American instrument. 1 The marimba typically includes 3 to 4 2 octaves of tuned Fig. 1 Optical system for TV holography. bars of rosewood or synthetic material with a deep arch cut

407 Acoust. Sci. & Tech. 25, 6 (2004) in the underside to tune the . The first , not be excited to any great extent. On the other hand, if the which is radiated by the second bending mode, is normally bars are struck away from the center, deliberately or not, tuned to the 4th harmonic of the fundamental in the first 3 the torsional modes could contribute to the . 1 to 3 2 octaves, after which the interval decreases [1]. Applying finite element methods to marimba and Some companies now make large 5-octave bars showed that a small curvature in the bars marimbas which cover the range C2 to C7. In two such has very little effect on the relative frequencies of the instruments, the second bending mode is accurately tuned vibrational modes [8]. Henrique and Antunes have used 1 to the 4th harmonic in the first 3 2 octaves. The third finite element methods both to optimize the shape of bending mode is tuned to the 10th harmonic in the first 2 marimba and xylophone bars and to model the sound. They octaves, after which the interval decreases. The fourth employ a physical modeling approach that addresses the mode varies from the 20th harmonic in the lowest bars to spatial aspects of the problem and is suitable for both non- about the 6th harmonic in the highest bars [7]. dispersive systems and dispersive systems [9]. Relative frequencies of the first 4 bending modes in a The sound field radiated by a simulated marimba bar Malletech marimba are shown in Fig. 2(a), while those of has been calculated by assuming the vibrating bar to be several torsional modes in the same marimba are in equivalent to a linear array of oscillating spheres. This Fig. 2(b). The first torsional mode frequency ranges from sound pressure excites a monodimensional lossy tube of about 1.9 times the nominal frequency (largest bars) to finite length terminated by a radiation impedance at its about 1.2 times the nominal frequency (smallest bars). In open end, which represents the tubular resonator. The normal playing, the bars are struck near their centers, amount of frequency decrease as the resonator is moved where the torsional modes have nodes, and thus they will closer to the bar is calculated [10]. 4. CYMBALS, GONGS, AND PLATES The strong ‘‘aftersound’’ that gives the its characteristic shimmer is known to involve nonlinear processes [1]. There is considerable evidence that the vibrations exhibit chaotic behavior. A mathematical analy- sis of cymbal vibrations using nonlinear signal processing methods reveals that there are between 3 and 7 active degrees of freedom and that physical modeling will require a like number of equations [11]. One procedure is to calculate Lyapunov exponents from experimental time series, so that the complete spectrum of exponents can be obtained. The chaotic regime can be quantified in terms of the largest Lyapunov exponent [12]. (a) Gongs of many different sizes and shapes are popular in both Eastern and Western music. They are usually cast of with a deep rim and a protruding dome. Tamtams are similar to gongs and are often confused with them. The main differences between the two are that tamtams do not have the dome of the , their rim is not as deep, and the metal is thinner. Tamtams generally sound a less definite pitch than do gongs. In fact, the sound of a tamtam may be described as somewhere between the sounds of a gong and a cymbal. The sound of a large tamtam develops slowly, changing from a sound of low pitch at strike to a collection of high- frequency vibrations, which are described as shimmer. These high-frequency modes fail to develop if the tamtam is not hit hard enough, indicating that the conversion of (b) energy takes place through a nonlinear process [13]. Among the many gongs in Chinese music are a pair of Fig. 2 Relative frequencies of vibrational modes in a Malletech 5-octave marimba: (a) bending modes; (b) gongs used in , shown in Fig. 3. torsional modes. These gongs show a pronounced nonlinear behavior. The

408 T. D. ROSSING et al.: ACOUSTICS OF PERCUSSION INSTRUMENTS

Fig. 5 Holographic interferogram showing vibrational modes of a jade stone.

Fig. 3 A pair of gongs used in Chinese opera. frequency close to one mode, lead to a bifurcation with the appearance of lower frequencies corresponding to other modes. Varying the excitation frequency at constant force yielded subharmonics which were not observed at constant excitation frequency. This is quite similar to the nonlinear behavior of cymbals [1,11]. 5. LITHOPHONES Lithophones are stones that vibrate and produce sound. The ancient Chinese were fond of stone chimes, many of which have been found in ancient Chinese tombs. A typical stone chime was shaped to have arms of different lengths joined at an obtuse angle. The stones were generally struck on their longer arm with a wooden . Sometimes the stones were richly ornamented. A lithophone of 32 stone chimes found in the tomb of the Marquis Yi (which also contained a magnificent set of 65 bells) are scaled in size, although their dimensions do not appear to follow a strict scaling law [16]. In later times, the Chinese made stone chimes of jade. Holographic interferograms showing some of the modes of vibration of a small jade chime are shown in Fig. 5. Korean chime stones, called pyeon-gyoung, were originally brought from to Korea in the 12th century. A set of 16 stone chimes from the Chosun Dynasty is shown in Fig. 6. Unlike the Chinese stone chimes, these stones all have the same size but differ from each other Fig. 4 Holographic interferograms showing several only in thickness. The fundamental frequency is essentially vibrational modes of the larger gong in Fig. 3. proportional to the thickness, just as in a rectangular bar such as a marimba bar. The second mode in each stone is pitch of the larger gong glides downward as much as three approximately 1.5 times the fundamental, while the third semitones after striking, whereas that of the smaller gong mode is about 2.3 times the nominal frequency. The fourth glides upward by about two semitones [14]. Several mode is about 3 times the nominal frequency up to the 12th vibrational modes of the larger gong are shown in Fig. 4. stone, after which the ratio drops to about 2.7 [17]. Some of the modes are confined pretty much to the flat Holographic interferograms of several modes of vibration inner portion of the gong, some to the sloping shoulders, in a pyeon-gyoung stone tuned to D6# are shown in Fig. 7. and some involve considerable motion in both parts. When Relative frequencies of the modes in the pyeon-gyoung are the gong is hit near the center, the central modes (178, 362, shown in Fig. 8. 504, 546 Hz) clearly dominate the sound. When the gong is 6. STEEL INSTRUMENTS hit lightly on the shoulder, the lowest mode at 118 Hz is heard. 6.1. Sound Radiation from Caribbean Steelpans The vibrations of a large tamtam were studied by The Caribbean continues to be an object of Chaigne, et al. [15]. They found that the nonlinear acoustical study, both in its home country of Trinidad and phenomena have the character of quadratic nonlinearity. Tobago and in the United States. Modern steel bands Forced excitation at sufficiently large amplitude at a include a variety of instruments, such as tenor, double

409 Acoust. Sci. & Tech. 25, 6 (2004)

14

12

10

8

6 Ratio to the note frequency 4

2

0 pg1 pg2 pg3 pg4 pg5 pg6 pg7 pg8 pg9 pg10 pg11 pg12 pg13 pg14 pg15 pg16 Notes

Fig. 8 Relative frequencies of the pyeon-gyoung stone in Fig. 7.

Fig. 6 Set of 16 pyeon-gyoung (stone chimes) from the Chosun Dynasty in Korea.

Fig. 9 Map of active and reactive sound intensity for the F#3 note of a tenor pan excited at its fundamental frequency.

An effective aid to understanding sound radiation is to map the sound intensity field around the instrument. Since sound intensity is the product of sound pressure (a scalar quantity) and the acoustic fluid velocity (a vector), a two- microphone system is used. The acoustic fluid velocity can be readily calculated from the difference in sound pressure at the two accurately spaced microphones. Both the active intensity and the reactive intensity can be obtained at the desired points in the sound field. The active intensity represents the outward flow of energy, while the reactive intensity represents energy that is stored in the sound field Fig. 7 Holographic interferograms showing vibrational near the instrument. While the active intensity is the most modes of a Korean pyeon-gyoung stone (D6#). significant field in a concert hall, both active and reactive intensity fields have to be considered in recording a second, double tenor, , cello, quadrophonic, and . steelpan. Our earlier review paper [1] included holographic inter- Figure 9 shows a map of active and reactive sound ferograms of several instruments showing how individual intensity in a plane that bisects a double second steelpan notes vibrate, how the entire instrument vibrates, and how when a single note (F]3) is excited at its fundamental the skirts of the instruments vibrate. Another piece of the frequency [18]. puzzle, so to speak, is to understand how the vibrating components radiate sound.

410 T. D. ROSSING et al.: ACOUSTICS OF PERCUSSION INSTRUMENTS

Fig. 10 The HANG, a hand-played steel instrument.

6.2. The HANG The HANG is a new steel percussion instrument, consisting of two spherical shells of steel, suitable for playin with the hands. Seven to nine notes are harmonically tuned around a central deep note which is formed by the Helmholtz (cavity) resonance of the instrument body. The HANG shown in Fig. 10 has nine notes which can be tuned Fig. 11 Holographic interferograms showing vibration- in any tonal systems between A3 and G5, including 30 al modes in a HANG (G3 note area). tonal systems suggested by the tuners. The central note is usually tuned a fifth or fourth below the lowest note of the scale. Although it is a new instrument, thousands of them have been shipped all over the world by PanArt, its creaters. Holographic interferograms in Fig. 11 show the first five vibrational modes in the G3 note area of the HANG. The second and third modes are tuned to the second and (a) (b) (c) third harmonics of the fundamental mode, respectively [19]. Fig. 12 Active sound intensity field above the HANG Figure 12 shows the active sound intensity in a plane when the E4 note area is excited at: (a) its fundamental frequency; (b) its second harmonic frequency; (c) the 8 cm above the E4 note. The arrowheads show the direction frequency of its third mode. of the sound intensity at each point in the plane, while the gray scale shows the sound pressure level. Note the sound level is greatest at the fundamental frequency, and the the clapper surface is ground or filed to restore its curved sound intensity is strongly upward from the note, while at shape at the impact site. the frequency of the second and third modes, considerable A theoretical treatment using Hertzian impact theory sound is radiated laterally. agrees rather well with measurements on the change in the sound of several bells [20] The effects of re- 7. EASTERN AND WESTERN BELLS voicing that appear to make an improvement on the sound It is well known that the sound of bells changes with of the appear to be: (1) the impact is lengthened so that time due to the development of elliptical flat areas on both the turnover frequency is reduced, giving a more ‘‘mellow’’ bell and clapper in the impact region. This change is most sound; and (2) the impact time becomes more dependent rapid during the first year or so of use, and then becomes upon impact velocity, so that ‘‘soft’’ notes have relatively more stable as the inside surface of the bell becomes more little harmonic development while ‘‘loud’’ notes sound resistant to plastic deformation through work hardening. A brighter, as in most musical instruments. re-voicing operation is periodically undertaken in which Acoustical studies on 46 bells in the historical carillon

411 Acoust. Sci. & Tech. 25, 6 (2004) in Perpignan, before and after taking them down, vibrations,’’ Appl. Opt., 35, 3791–3798 (1996). sanding their oxide layer, and rehanging them show that the [5] D. J. Ewins, Modal Testing: Theory and Practice, 2nd ed. (Research Studies Press, Baldock, 2000). bells rang 15% longer, on average, after restoration. This is [6] F. J. Fahy, Sound Intensity (Elsevier Science Publ., Amster- due to removal of the oxide layer and also to tightening the dam, 1989). bells on their support [21]. [7] J. Yoo, T. D. Rossing and B. Larkin, ‘‘Vibrational modes of With the aid of finite element analysis, a series of new five-octave concert marimbas,’’ Proc. Stockholm Music Acous- tics Conf. (SMAC03), Stockholm pp. 355–357 (2003). bells has been designed that contains up to several partials [8] A. Chaigne, M. Bertagnolio and C. Besnainou, ‘‘Tuning of in the harmonic series beginning at the fundamental xylophone bars: Influence of curvature and inhomogeneities,’’ frequency. This was achieved by choosing geometries in Proc. ISMA 2001, Perugia, pp. 531–534 (2001). which as many circumferential bending modes as possible [9] L. L. Henrique and J. Antunes, ‘‘Optimal design and physical occurred at frequencies below any mode with an axial ring modelling of instruments,’’ Acta Acustica/ Acustica, 89, 948–963 (2003). node [22]. Several different profiles have resulted. A [10] V. Doutaut, A. Chaigne and G. Bedrane, ‘‘Time-domain conical bell has considerable less than a European simulations of the sound pressure radiated by mallet percussion church bell or carillon bell sounding the same pitch, and instruments,’’ Proc. ISMA, Dourdan, pp. 519–524 (1995). thus it has considerably less loudness. Concave bells have [11] C. Touze´, A. Chaigne, T. Rossing and S. Schedin, ‘‘Analysis of cymbal vibration and sound using nonlinear signal processing greater mass and surface area than conical bells, and thus methods,’’ Proc. ISMA 98, Leavenworth, pp. 377–382 (1998). sound louder. Other variants have even greater mass and [12] C. Touze´ and A. Chaigne, ‘‘Lyapunov exponents from are expected to be appropriate for church and carillon experimental time series: Application to cymbal vibrations,’’ towers [23]. Acustica/Acta Acustica, 86, 557–567 (2000). [13] T. D. Rossing and N. H. Fletcher, ‘‘Acoustics of a Tamtam,’’ Large temple bells have been cast in Korea for more Bull. Australian Acoust. Soc., 10(1), 21 (1982). than 1200 years. The largest bell is the magnificent King [14] T. D. Rossing and N. H. Fletcher, ‘‘Nonlinear vibrations in Songdok or Emilie bell, cast in 771 with a mass of nearly plates and gongs,’’ J. Acoust. Soc. Am., 73, 345–351 (1983). 20,000 kg. The acoustics of this bell were described at the [15] A. Chaigne, C. Touze´ and O. Thomas, ‘‘Nonlinear axisym- 133rd meeting of the Acoustical Society of America [24]. metric vibrations of gongs,’’ Proc. ISMA 2001, Perugia, pp. 147–152 (2001). (An animated presentation of several modes of this bell can [16] A. Lehr, ‘‘Designing chimes and in history,’’ be seen at http://www.acoustics.org/press/133rd/4pmu4. Acustica/Acta Acustica, 83, 320–336 (1997). html). [17] J. Yoo and T. D. Rossing, ‘‘Vibrational modes of pyen-gyoung, The second largest bell hangs in the Bosingak belfry in Korean chime stone,’’ Proc. ISMA 2004, Nara, , pp. 312– 315 (2004). Seoul, built in 1413 and rebuilt several times after being [18] B. Copeland, A. Morrison and T. D. Rossing, ‘‘Sound radiation destroyed by fire. The New Bosingak Bell, as it is called, from Caribbean steelpans,’’ paper 2pMU7, 142nd ASA meet- was designed with the aid of finite element methods and is ing, Ft. Lauderdale, FL (2001) (submitted to J. Acoust. Soc. patterned, to some extent, after the King Songdok bell. The Am.). design and tuning of the New Bosingak bell, and its sound [19] T. D. Rossing, U. J. Hansen, F. Rohner and S. Scha¨rer, ‘‘The HANG: A hand-played steel ,’’ Proc. SMAC 03, Stock- and vibrational characteristics are discussed in a paper in a holm, pp. 351–354 (2003). recent paper [25]. The finite element analysis (FEA) served [20] N. H. Fletcher, W. T. McGee and A. Z. Tarnopolsky, ‘‘Bell as a guide for final tuning of the bell. Calculated and clapper impact dynamics and the vocing of a carillon,’’ measured mode frequencies were very close, but the beat J. Acoust. Soc. Am., 111, 1437–1444 (2002). [21] X. Boutillon and B. David, ‘‘Assessing tuning and damping of frequencies understandably differ from those predicted historical carillon bells and their changes through restoration,’’ by FEA. Appl. Acoust., 63, 901–910 (2002). [22] N. McLachlan, B. K. Nigjeh and A. Hasell, ‘‘The design REFERENCES of bells with harmonic overtones,’’ J. Acoust. Soc. Am., 114, 505–511 (2003). [1] T. D. 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