Simulation of Sound Production Mechanism (Acoustical Research on the Piano, Part 2)

J. Acoust. Soc. Jpn.(E) 14, 2 (1993)

Simulation of sound production mechanism (Acoustical research on the piano, Part 2)

Isao Nakamura

Teikyo University of Technology, 2289 Uruido, Ichihara, Chiba, 290-01 Japan

(Received 8 April 1992)

【English translation of the same articie in J. Acoust. Soc. Jpn.(J)37,65-75(1981)】

The mechanism of sound generation on the piano can be divided into four stages :•\(a)

transmission of energy from a hammer to a set of strings, (b) propagation of energy in the strings, (c) transition of energy from the strings to the soundboard through the bridge, and (d) acoustic radiation of the energy from the soundboard. This process is represented by an electrical circuit model, each string being represented by a transmis sionline, and the quantitative relationship of the model are calculated by a computer. When a hammer strikes a string, a nearly half-sinusoidal pulse is generated, and this

propagates along the transmission line, and then is reflected by the impedances of the bridge and the fixed end of the string. The propagation and reflection change the waveform of the initial pulse and reduce its amplitude. The driving velocity of the soundboard is produced by applying the sum of the forces of all the strings to the driving

pointimpedance. This driving velocity produces a sound pressure, and its waveform is determined by the transmission characteristics of the soundboard. An artificial piano

sound can be produced from a calculated waveform through a D-A converter. When the fundamental frequency of each string in a set is slightly detuned , they produce beats in each partial making an inharmonic sound. The lower partials in the produced sound change relatively slowly, while the higher partials change relatively rapidly. The amplitude of the initial part of the sound ('initial sound') decays rapidly, while that of the sustained part ('aftersound') decays slowly. This explains how a delicate timbre of the piano sound is produced.

Keywords:Piano, Sound production mechanism, Equivalent circuit, Simulation, Numerical experiment

PACS number:43.75. Mn, 43.40. Cw

sistently,it is not easy to find factors that affect 1. INTRODUCTION subtle sound of each instrument. On the contrary, A traditional method to find physical factors re it is useful, in finding the relationship between the presentingthe sound characteristics of a musical physical construction of a musical instrument and instrument has been to analyse a sound itself by its sound, to understand the characteristics of the using an instrument such as a frequency analyzer sound by synthesizing it with a computer. This or a sonogram. In the last decade or so these in method can be used for testing a theory of sound strumentshave been replaced by a computer so production mechanism, so which is also useful for that the sound is digitally represented and a digital the traditional method of studying musical instru signal processing is applied to it. Although these ments.Although there have been many studies methods are effective in understanding various con using computers to study the sound production ventional(non-electronic) musical instruments con mechanism of a musical instrument, no attempt

73 J. Acoust. Soc. Jpn.(E) 14, 2 (1993) has been made to generate sound in the way describ between the hammer and strings couses the string edabove. In 1971, Hiller and Ruiz 1) produced vibration.The hammer initially moves with the musical sounds by solving their differential equa string, keeping contact, but the contact pressure tions,but they did not directly relate their study to a becomes zero again when the hammer starts moving musical instrument. backward and the displacements of the hammer This paper describes the relationship between the and the strings become equal. They are naturally mechanism of piano sound production and the char separate later when the speed of the string decays. acteristicsof the produced sound by representing the The vertical component (to the soundboard) of the mechanism with an equivalent circuit, by calculating tension of the string gives .a driving force to the the circuit equations with constants (obtained from bridge. actual music instruments) using a computer, and by The soundboard with a thickness of several milli analysing the results. If necessary, it is possible metersis made of non-isotropic wooden pieces by to listen to the output as a sound through a D-A gluing them. Sound ribs are also glued on the converter. Measured waveforms are analysed by soundboard with equal intervals across the direction computer after they have been converted to digital of fibres of the wooden pieces. The bridge consists data through an A-D converter. of two parts, one for a low-frequency region, and the other one for a high-and middle-frequency 2. SOUND PRODUCING MECHANISM region, both parts being fixed on the soundboard. OF PIANO AND ITS EQUIVALENT The soundboard system is driven by a force at the CIRCUIT bridge, and a velocity wave occurs on the board, Hammers, strings and the soundboard (including then the sound wave is radiated from it. Therefore, its bridge) of a piano are components determining its vibrations of a string with the soundboard decay sound production. Figure 1 shows a model of a faster than those of the string alone. set of these components. Figure 2 shows a block Let us represent the above-described sound diagram of the sound production mechanism taking productionmechanism by using an equivalent cir a 'triple string system' in which three identical strings cuitbased on the mobility analogy (force-current are struck simultaneously by a single hammer. A and velocity-voltage). When the elasticity of a hammer is made of wood covered by felt, and this string is taken into account, the vibartion of a string strikes a set of strings when a corresponding key is with the elasticity and tension is represented by pressed. A piano string with given a high tension, which is supported by the frame, links with the (1) soundboard through the bridge so that the vibration where T is the tension of the string, r is the frictional of the string is transmitted to the soundboard. When resistance of air, S is the cross sectional area of the a hammer strikes a string, the contact pressure string, ƒÏ is the density of the string, E is the Young's

modulus of the string, and r is the radius of gyration

of the string. Equation (1) can be represented by

a simultaneous 5-element first-order partial differen

tialequation,2) as

Fig. 1 Model of sound production mecha nisumof a piano keyboard.

(2)

Fig. 2 Block diagram of sound production mechanism of a keyboard with triple string.

74 I. NAKAMURA:SIMULATION OF SOUND PRODUCTION MECHANISM OF PIANO

Table 1 Analogical parameters in mechanical and electrical systems.

where f=T•Ýy/•Ýx is a vertical (to the soundboard) ed in parallel with C and G respectively. component of the tension of the string, Xx is the x- Since the boundaries of the system are approxi component of the stress per a unit area which is matelyon the supporting points of a piano string perpendicular to the x-axis, AC is that in the y-com which are represented by ponent,or the x-component of the stress per unit area on a plane which is perpendicular to the y-axis, (4) v is the velocity in the y-direction, u is the velocity therefore in the x-direction, and ƒÂ=ƒÏS is the linear density. (5) When the mobility analogy shown in Table 1 is These are shown in the two end-terminals of the applied to Eq.(2), circuit in Fig. 3(a). When the impedance analogy shown in Table 1 is used,

(3)

(6)

Figure 3(a) shows a graphical representation of Eq. (3), and this is a two-line distributed-constant circuit, where the symbol of a crossed transformer repre Then an equivalent circuit shown in Fig. 3 (b) is sentsan ideal current-transformer having a ratio of obtained. A symbol of a crossed transformer 1:1. If the radiation impedance of a piano string again represents an ideal transformer having a ratio is taken into account, an additional mass and a of 1:1. The circuit shown in Fig. 3 (b) can also be radiation resistance of the string should be connect- obtained by applying a dual transformation to the circuit shown in Fig. 3 (a). If a system becomes complicated by adding other elements, such as a hammer system and the soundboard system , the mobility analogy is easier than the impedance analogy, since the former has the same form as the mechanical system. (a) When an input voltage V=V0eiƒÖt is given to a two-line distributed-constant circuit in the mobility analogy, its output voltage at a distance x can be expressed as V=V0e-ax+j(ƒÖt-ƒÀx). We can determine

the transmission constants ƒ¿ and ƒÀ as follows . The phase constant ƒÀ is given by (b)

Fig. 3 Equivalent circuit elements for a (7) string with elasticity.(a) Mobility analogy;(b) Impedance analogy. Assuming

75 J. Acoust. Soc. Jpn.(E) 14, 2 (1993)

in a higher frequency region due to the sound radiation. Now, let a phase constant, which goes and re turnsa distance l and produces a phase difference of 2nƒÎ , be ƒÀn. Then (12) (a) (b) Let a frequency in this equation be fn, then from Fig. 4 Frequency dependence of propaga tionconstants.(a) Phase constant;(b) this equation and Eq.(9), Attenuation constant.

(8) Therefore, the frequencies of partials are

(9) (13)

(10) where

When the radiation conductance AƒÖ2 is taken into (14) account, and Eq.(8) is applied, the attenuation Equation (13) coincides with Fletcher's equation, 3,4) constant ƒ¿ is given by

(11) (15)

Figure 4 illustrates outlines of Eqs.(10) and (11). Figure 5 shows an equivalent circuit (using the Figure 4 (a) shows that the higher the frequency the mobility analogy) of a sound production mechanism higher the propagation speed due to the effect of of a piano with a triple string which is used for key the elasticity. Figure 4 (b) shows that the attenua numbers of approximately 20 to 88. LH which tionin a lower frequency region is constant due to corresponds to the compliance of felt on a hammer, the friction of air, and that the attenuation is large and C. which corresponds to the inertia of the

Fig. 5 Equivalent circuit of sound production mechanism for a keyboard with triple string.

76 I. NAKAMURA:SIMULATION OF SOUND PRODUCTION MECHANISM OF PIANO hammer are respectively given by Table 2 Voltage and current in space and time increments.

(16) where eH corresponds to the velocity of the hammer, e0 to the velocity of the string at its struck point, and iH to the contact force, respectively. One of the terminals of LH is connected to the position which corresponds to the struck point of the string (about 1/8 of the full length of the string, from its fixed end) through switch S2. The CH is initially charged with voltage EH which corresponds to the initial velocity of the hammer. When S1 is opened and S2 is closed, this represents a state where the hammer strikes the string, and current iH flows through a relay which holds S2. The current flowing through the relay represents the contact force between the hammer and the string. When the current becomes zero, S2 opens; this represents a state where the contact force becomes zero and the hammer is separated Fig. 6 Equivalent circuit of a stiff string from the string. with elasticity and a hammer. The above explanation can be applied to a string- hammer system with a double or single string, by tained from that of a distributed constant circuit. altering only the number of strings. Using the time-differences of voltages and currents The soundboard system can be represented by an shown in Table 2, and referring to Fig. 6, the follow equivalent circuit of the driving-point admittance ingdifference equations are obtained: at the bridge. The voltage corresponding to the velocity of the soundboard at the bridge generates a current corresponding to a sound pressure, by being transformed by the vibration characteristics and acoustic-radiation characteristics of the soundboard system. The above described models can be used to syn thesizesounds of the piano by simulating them with a computer.

3. DIGITAL SIMULATION OF ELASTIC STRINGS Let us consider a method to represent the equiv alentcircuit shown in Fig. 5 (a) by a digital computer (17) programme. This is a transformation of Eqs.(3) and (16) into difference equations. In this transforma tion,a distributed constant circuit has to be re presentedby a lumped constant circuit with a finite number of sections, and its time system is also divided into finite increments which would be con tinuousin an analogue simulation. Figure 6 shows an equivalent circuit having spatial differences, ob-

77 J. Acoust. Soc. Jpn.(E) 14, 2 (1993)

When EH is given as an initial condition, a solution for a period when the hammer contacts a string is obtained. After IH has become zero, this term is regarded as always zero; i.e. the calculation should be carried out for the string system alone. The condition of the stability of the equations (or These equation can be re-written in a form conve the condition to avoid the divergence of errors) is nientfor a computer programming, as

(20)

[Detaiis of this condition are explained in the ap pendixof the original Japanese paper.] When the time increments are less than a certain value com paredwith the space increments, the solution of the equations becomes stable. Note, a digital simula tioncan obtain a higher degree of analogy than an (18) analogue simulation, since errors due to the space increments in the former are compensated by its time increments when the left-hand terms of Eq.(20) is nearly 1. The above descriptions are given for a single elastic string, but this can easily be applicable to a system with double or triple strings.

4. DRIVING-POINT ADMITTANCE OF SOUNDBOARD SYSTEM where 2•¢t•¨•¢t and 2•¢•¨•¢x, and symbols with 'dash' When the time-variation of piano sound is con represent those at an advanced time. The sidered,the driving-point admittance of the sound- initial conditions are board is important. Figure 7 shows examples of (19) the driving-point admittances of the soundboard

(a) (b) Fig. 7 Frequency responses of mechanical admittance.(a) Key number 25 (A2);(b) Key number 40 (C4).

78 I. NAKAMURA:SIMULATION OF SOUND PRODUCTION MECHANISM OF PIANO

(a) (b) Fig. 8 Frequency responses of mechanical reactance.(a) Key number 25 (A2);(b) Key number 40 (C4). for key numbers 25 (A2, 110Hz) and 40 (C4, 261. current source. 626Hz). In the figures the positions of the fre To carry out a numerical computation, let us now quenciesof each partial are shown. All the partials obtain a digital filter representing the terminal volt do not occur near the main resonance point. This agefor a case where a parallel resonance system is is due to the fact that the soundboard does not driven by a constant current source. resonate at a certain frequency. Figure 8 shows the From Fig. 9(a), mechanical reactance obtained from the above- described admittance characteristic and the phase (21) characteristic. The mechanical resistance com ponentsof each partial can be obtained by the same Therefore, way. These results show that the soundboard (22) system with the string for key number 25 can be ap proximatedby a single parallel resonance system with a low Q; and that that for key number 40 can be The difference equations for each sample value are approximated by a resistance alone, as the reactance of each partial is nearly zero. Generally, as far as partials are concerned, small resonance systems due to the soundboard can be neglected; i.e. when a (23) driving point is near the centre of a soundboard, this can be approximated by a resistance (like key number 40); and when the driving point is near the edge of the soundboard, this is approximated by a therefore, single resonance system with a low Q (like key (24) number 25). At the driving point, the admittance of the sound- where board is very large compared with that of a string. Therefore, in their equivalent circuit, the part re presentingthe string can be regarded as a constant

79 J. Acoust. Soc. Jpn.(E) 14, 2 (1993)

(a) Equivalent Circuit

(a)

(b) simulation

Fig. 9 Simplified driving point admittance soundboard.

(b)

(25) Fig. 10 Time characteristics.(a) Velocity level of soundboard at driving point (simulated);(b) Sound pressure level (measured).

main cause. None of these workers has taken Figure 9 (b) shows the simulated circuit of the sound- account of the fact that physical elements are dis board obtained from these discussions, where z-1 tributedalong a string. This paper analyses the is the delay of sampling time ƒÑ. decay amplitude-characteristics and tone-colours of When a case where the system can be regarded as piano sounds by using a computer simulation based a resistance alone (like key number 40), the output on a distributed constant system. voltage of the circuit shown in Fig. 9 is obtained Figure 10 (a) shows the envelope of velocity of the very simply. The output voltage (the driving driving point at key number 40 obtained from the velocity at the bridge) feedback to the string side. simulation. For the simulation, constants publish edby a piano maker were used; the soundboard 5. TIME CHARACTERISTICS OF was assumed to be purely resistive; and a set of PIANO SOUNDS three detuned strings were used (the frequency of D. W. Martin 5) has noticed that sustained piano one string being higher than the central string by 1 sounds have two types of amplitude decay charac cent, and the other string being lower than the teristics.R. E. Kirk6 and A. H. Benade 7) have central string by 1 cent). The waveform shows a described the decay characteristics relating to a set fast decaying part in the first second, producing a of strings which are not exactly tuned to a nominal initial sound, then a gradually decaying part in the frequency. Weinreich 8) has explained the causes rest of time producing an after sound. Figure 10 (b) of this by two modes of vibrations of each string shows the envelope of a measured waveform. This (one is perpendicular to the soundboard, and the is obtained by measuring the sound pressure at a other one is parallel to it), or a set of strings which point distant from the center of the soundboard of a are not exactly tuned to a nominal frequency. T. C. piano by 33 cm in an anechoic room. The piano Hundley et al. 9) has pointed out that the latter is the used for this measurement is not the same model as

80 I. NAKAMURA: SIMULATION OF SOUND PRODUCTION MECHANISM OF PIANO

(a)

(b)

Fig. 11 Time characteristics of spectra.(a) Velocity level of soundboard at driving point (simulated);(b) Sound pressure level (measured).

(c) the one from which the constants for Fig. 10(a) were Fig. 12 Time characteristics of driving obtained. The trends of the initial sound and after force of strings to soundboard.(a) Upper sound shown in Figs. 10 (a) and (b) are very similar. string;(b) Middle string;(c) Lower string. Some differences can be seen immediately after the rising in these figures; this is due to the neglection of the transmission characteristic in the simulation. can be obtained directly in a simulation. Figure 13 Figure 11 shows the time characteristics of the shows spectra of the driving force with intervals of spectra, the line intervals being about 0.1s, to about 0.1s. This is simple compared with vibra examine the characteristics of initial sound and after tionsof the sound board. The spectra of the lower sound. Each partial has these two parts. Lower string are not shown, since it is similar to that of the partials change gradually, but the higher the partials upper string. Rapid increase and decrease of am the faster the change. This is due to minor dif plitudeof the spectra of the driving force (the higher ferencesin the frequencies of a set of a triple string, the partial frequency the faster the increase and and this produces delicate variations of piano tones. decrease) can be seen in each partial of the middle Figure 12 shows the envelope of the soundboard string, due to the influences of its two adjacent driving forces of a triple string. These show some strings. Figure 14 shows the time variation of the features of the initial sound part and the after sound string velocity and sound pressure (the fundamental part respectively, although not very clearly. It is waves alone are extracted) of key number 41 mea interesting that the central string shows a different suredon an actual grand piano. The features of behavior from the other two. This trend appears the initial sound and the after sound, and the vibra also in the velocity waveform of the string. Gen tionof the middle string having an opposite phase erally,it is easy to obtain the waveform of the in this measurement agree approximately with the driving force from the velocity waveform. 10) This simulated results, although this piano has a little

81 J. Acoust. Soc. Jpn.(E) 14, 2 (1993)

(a)

(b)

(b) Fig. 13 Time characteristics of spectra of driving force.(a) Upper string;(b) Middle string. (c) Fig. 15 Velocity of triple string at the struck point with una corda pedal.(a) Upper string,(b) Middle string,(c) Lower string.

Fig. 14 String velocity and sound pressure (measured). unbalance between the two adjacent strings to the Fig. 16 String velocity and sound pressure middle string. with una corda pedal (measured). Figure 15 shows the simulated velocity waveforms of a triple string at the striking point with the una out a strike, by the energy supplied through the corda pedal which avoids striking the lower string. soundboard. Figure 16 shows that the middle Figure 16 shows that the lower string vibrates with- string vibrates with a phase opposite to those of its

82 I. NAKAMURA: SIMULATION OF SOUND PRODUCTION MECHANISM OF PIANO adjacent strings. Figure 17(a) shows the spectrum trend can be seen in the vibration velocity shown in of the soundboard, and Fig. 17 (b) shows the velocity Fig. 17 (b). spectrum of the string which is not directly struck To explain these phenomena, numerical experi by the hammer. The spectra shown in Figs. 17 (a) mentswere carried out in which certain physical and 11 (a) are not exactly the same, but the second factors are intentionally emphasized. Figure 18 (a) harmonic of the former is smaller than that of the shows the amplitude decay characteristic of a set of latter; i.e. their tones must be different. The same three strings all of which have exactly the same fre quency.This shows a uniform decay of amplitude without after sound. Therefore, it is clear that an after sound is due to a set of strings with minor differences in their frequencies. Figure 18(b) shows the same conditions as in Fig. 10(a), but without

(a)

(b) (c)

(a)

(d) (e)

(b) Fig. 18 Numerical experiments.(a) Exact Fig. 17 Time characteristics of velocity lytuned;(b) L2•¨•‡;(c) L2•¨•‡, Middle spectra with una corda pedal.(a) Sound- string•¨+0.5 cent mistuning;(d) L2•¨•‡, board;(b) String without strike. CH'•¨CH/2;(e) L2•¨•‡, G'•¨G•~2.

(a) (b) Fig. 19 Time characteristics of velocity spectra of double string.(a) Driving point soundboard;(b) Struck point of string.

83 J. Acoust. Soc. Jpn.(E) 14, 2 (1993)

the elastic term. This shows that the elasticity of piano tones, using equivalent circuits representing strings makes no contribution to either the initial its sound production mechanism and numerical ex sound or after sound. Figures 18 (c) and (d) show perimentswith a computer, has been established. that the amplitude decay characteristic is almost It has been found that a physically very interesting independent of the tuned frequency of the middle mechanism has been used for triple strings in the string and the mass of the hammer. If the conduc actual piano. The theory of vibrations on the tanceof the soundboard is changed, the gradient of piano will be completed after more simulations with the decay would be changed, but the relationship various constants have been carried out. between the initial sound and the after sound would remain. Figure 18 (e) shows this fact. Therefore, ACKNOWLEDGMENTS it is possible to adjust the time for the initial sound The author wishes to express his thanks to Dr. and that for the after sound by tuning the strings. Shizuo Ishiguro in England for his advice and help The driving force of the soundboard is initially in making this English version. formed in co-phase at the bridge. As time prog resses,the phases of three strings start to differ. If the phase differences of the three strings balance REFERENCES at about 1 second, the resultant output is impeded, 1) L. Hiller and P. Ruiz,"Synthesizing musical sounds making a slow decay of amplitude. by solving the wave equation for vibrating objects, Let us now consider a double string, taking key I, II," J. Audio Eng. Soc. 19, 462-470, 542-551 number 25 as an example. Figure 19 shows the (1971). time characteristics of the velocity spectra of the 2) I. Nakamura,"Considerations on elastic waves by means of equivalent circuits," J. Acoust. Soc. Jpn. double string. For this analysis, the soundboard (J) 36, 185-193 (1980)(in Japanese). system is regarded as a single parallel resonance 3) H. Fletcher, E. D. Blackham, and R. Stratton, system, the difference of the frequencies of the two Quality of the piano tones," J. Acoust. Soc. Am. strings taken to be 2 cent, and the frequency 34, 749-761 (1962). analyses are carried out every 0.1 second approxi 4) H. Fletcher,"Normal vibration frequencies of a mately.Figure 19(a) shows the velocity at the stiff piano string," J. Acoust. Soc. Am. 36, 203-209 bridge. This result is similar to the case where the (1964); 36, 1214-1215 (1964). 5) D. W. Martin,"Decay rates of piano tones," J. soundboard system is egarded as purely resistive, Acoust. Soc. Am. 19, 535-541 (1947). i.e. there is little effect of the resonance system. 6) R. E. Kirk,"Tuning preferences for piano unison Figure 19(b) shows the time variation of the velocity groups," J. Acoust. Soc. Am. 31, 1644-1648 (1959). spectrum of one of the two strings. The time varia 7) A. H. Benade, Fundamentals of Musical Acoustics tionof each spectrum in this figure is very little, i.e. (Oxford University Press, New York, 1976). the amplitude decays almost uniformly. This is in 8) G. Weinreich,"Coupled piano strings," J. Acoust. Soc. Am. 62, 1474-1484 (1977);"The coupled contrast to the vibration of the soundboard shown motions of piano strings," Sci. Am. 240 (1), 94-102 in Fig. 19(a), where each partial has a beat, and low (1979.1). partials decay gradually, while higher partials change 9) T. C. Hundley, H. Benioff, and D. W. Martin, "Factors rapidly. This is one of the causes which produces contributing to the multiple rate of piano delicate tone varieties in piano sounds, the same as tone decay," J. Acoust. Soc. Am. 64, 1303-1309 in the triple string. (1978). 10) I. Nakamura,"The vibration of a struck string: Acoustical research on the piano, Part I," J. Acoust. 6. CONCLUSION Soc. Jpn. 36, 504-512 (1980)(in Japanese); J. A method of clarifying the characteristics of the Acoust. Soc. Jpn.(E) 13, 311-321 (1992).