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Contents : Reprinted with permission from C. Erkut et al "Acoustical analysis and model-based sound synthesis of the kantele" Journal of the Acoustical Society of America 112(4) 2002 pp. 1681-1691. 2002 Acoustical Society of America. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the Acoustical Society of America. Acoustical analysis and model-based sound synthesis of the kantelea) Cumhur Erkutb) and Matti Karjalainen Laboratory of Acoustics and Audio Signal Processing Helsinki University of Technology Espoo Finland Patty Huang Center for Computer Research in Music and Acoustics Stanford University Stanford California 94305-8180 lima kic) Vesa Va Laboratory of Acoustics and Audio Signal Processing Helsinki University of Technology Espoo Finland Received 3 January 2002 revised 3 July 2002 accepted 3 July 2002 The five-string Finnish kantele is a traditional folk music instrument that has unique structural features resulting in a sound of bright and reverberant timbre. This article presents an analysis of the sound generation principles in the kantele based on measurements and analytical formulation. The most characteristic features of the unique timbre are caused by the bridgeless string termination around a tuning pin at one end and the knotted termination around a supporting bar at the other end. These result in prominent second-order nonlinearity and strong beating of harmonics respectively. A computational model of the instrument is also formulated and the algorithm is made efficient for real-time synthesis to simulate these features of the instrument timbre. 2002 Acoustical Society of America. DOI: 10.1121/1.1504858 PACS numbers: 43.75.Gh 43.75.Wx NHF I. INTRODUCTION The kantele1 refers to a group of plucked string instruments that have been common in traditional folk music in Finland its neighboring region in Northwest Russia and the Baltic states.2 3 The instrument and its variations are called s in Lithuania the kokle in the kannel in Estonia the kankle Latvia and the gusli in Russia.2 They belong to the family of zithers. The five-string Finnish kantele has a significant role in Finnish folklore as the instrument of rune singers and in Finnish mythology especially in the Kalevala the collection of ancient Finnish runes.4 It is estimated that the origins of the kantele are more than 1000 years old. The traditional Finnish kantele has five strings and a body made of a single piece of wood. The traditional instrument illustrated in Fig. 1 is hollowed out at the top and the opening is covered by a top plate with a sound hole Xshaped in this model. Strings are terminated at the wider end around wooden tuning pegs. At the other end the strings are attached with a knotted termination around the varras a bar typically made of metal in a U-shaped raised body structure the ponsi. At the end of 18th century the instrument started an evolution to new forms. The body of the kantele was evena tually constructed of separate plates instead of a single piece of wood and the wooden tuning pegs were replaced by metallic tuning pins. This type of the instrument is the focus of the present study. To be able to play more complex music the kantele was made larger and equipped with more strings-- for example 9 to 15 strings. For compatibility with concert music a concert kantele has been developed since the 1920's to contain up to 45 strings with a range of about five octaves. The challenge of playing in different keys and with chromatic notes was solved by including a lever mechanism similar to that of a concert harp for rapid change of tuning. The kantele has a characteristic sound that is bright and reverberant.5 Only recently acoustical studies have been carried out on the instrument to reveal the features that make the unique sound. In a previous work 6 based on measurements and signal analysis specific properties of string terminations were reported as prominent sources of the characteristic kantele tone. A recent study focused on the body vibrations of a general class of the Baltic psalteries.7 The objective of the present article is to give a systematic and thorough presentation of the instrument showing the Portions of this work were presented in Analysis modeling and real-time sound synthesis of kantele a traditional Finnish string instrument '' Proceedings of IEEE International Conference on Acoustics Speech and Signal Processing Minneapolis MN April 1993 and Nonlinear modeling and synthesis of the kantele--A traditional Finnish string instrument '' Proceedings of International Computer Music Conference Beijing China October 1999. b Electronic mail: cumhur.erkut@hut.fi c Part of the work done while at Tampere University of Technology Pori School of Technology and Economics Pori Finland. J. Acoust. Soc. Am. 112 (4) October 2002 FIG. 1. A Finnish traditional five-string kantele made by hollowing out the top of a single wood block and covering it with a top plate. The strings are attached without a bridge to the tuning pegs at the left-hand side and to a bar at the right-hand side. After Sadie Ref. 2. 2002 Acoustical Society of America 1681 0001-4966/2002/112(4)/1681/11/$19.00 Acoustical Society of America 2002 FIG. 2. Structure of a five-string kantele used in most experiments of this study. The body is made hollow from the bottom and left open. After Karjalainen et al. Ref. 6. behavior of kantele strings body and sound radiation. This article starts with a structural description of an open-body kantele as well as playing techniques and tuning of the instrument in Sec. II. Section III presents an acoustical analysis of the instrument. The terminations of the strings deserve special attention since they introduce strong beating and nonlinear effects to the sound. The nonlinear vibrations of the kantele strings are of exceptional importance in producing the resulting timbre. The properties of body vibrations as a response to the driving forces on the strings are demonstrated. In Sec. IV a computationally efficient sound synthesis algorithm is presented. The algorithm captures the most essential properties of the kantele sound and allows for the synthesis of kantele tones in real time. II. DESCRIPTION OF THE INSTRUMENT A. Construction and tuning FIG. 4. String termination around a tuning pin without a bridge. A reference coordinate system is indicated at the bottom part of the figure. After Karjalainen et al. Ref. 6. typically near D 4 294 Hz although transposition of several semitones up or down is also common. In our experiments the lowest string has been tuned to E 4 311 Hz. B. Playing techniques Our study focuses on a present-day version of the traditional five-string kantele illustrated in Fig. 2. The body is hollowed open at the bottom and thus there is no need for a sound hole. This model was used due to its structural simplicity. The five metal strings are of equal diameter 0.35 mm with lengths ranging from 32.5 to 47.8 cm. At the narrower end of the kantele the strings are wound once around the varras and knotted as shown in Fig. 3. At the opposite end the metal tuning pins are screwed directly into the soundboard that is the top plate of the body and the strings are terminated directly around the pins without a bridge Fig. 4. Such terminations are unique structural features of the instrument. The five-string kantele is tuned to a diatonic scale and the third string can be tuned to a major third minor third or somewhere in between. The tuning of the lowest string is The five-string kantele is played across one's lap or on a table with the shortest string closest to the player. There are regional and personal variations in playing technique but the most common traditional way is to interleave fingers of left and right hands with one finger for each string and the right thumb playing the highest string. A string is plucked upwards so that all other strings remain free to vibrate. This makes a reverberant sound. Damping of strings or plucking horizontally so that the finger damps the next string can be used in modern playing. Another technique that yields an even brighter sound is to strike a string by fingernail. These techniques can also be combined. Playing can consist of a melody line accompaniment or both although in traditional playing there was often no clear distinction between them. III. ACOUSTICAL ANALYSIS OF THE KANTELE A. Observations of kantele tones Figure 5 shows the amplitude envelope trajectories of the first three harmonics of a a softly plucked and b a strongly plucked kantele tone. The measurements were carried out in an anechoic chamber where the microphone FIG. 3. String with a knot termination around the varras metal bar. Effective string lengths differ by l for different vibration directions. After Karjalainen et al. Ref. 6. 1682 J. Acoust. Soc. Am. Vol. 112 No. 4 October 2002 FIG. 5. Envelope trajectories of kantele tones. The first harmonic is shown with a dashed line the second with a solid line and the third with a dashdotted line. a A softly plucked tone and b a strongly plucked tone. The plucking point is the midpoint of the third string. Erkut et al.: Analysis and synthesis of the kantele Acoustical Society of America 2002 B&K 4145 was fixed at a distance of 1 meter above the top plate of the instrument. The plucking point is the midpoint of the third string so that according to the linear theory of string vibration even harmonics should be absent.8 The first noticeable feature of the kantele tones in Fig. 5 is a strong beating of harmonics. This phenomenon is explained in Sec. III B. Another feature is the presence and dominance of the second harmonic. The initial magnitude of the second harmonic is approximately 10 dB higher in the case of a strong pluck in Fig. 5b compared to that of the soft pluck in Fig. 5a. The generation and the amplitude dependence of the second harmonic indicate a nonlinear mechanism which is the subject of Sec. III C. Section III D presents the formulation of a nonlinear longitudinal force component and Sec. III E derives an analysis method. In Sec. III F we demonstrate by measurements how the body of the instrument responds to the forces applied on a tuning pin. Within the same section we also briefly discuss the properties of the body of the traditional kantele as well as the energy transfer between the strings. Throughout the discussion we use the following convention to refer to three orthogonal vibration directions. A rectangular reference coordinate system is shown in Fig. 4. The x axis is along the strings so that the varras is at x 0 and a tuning pin is at x L . The y axis is parallel to the top plate and the z axis is in the direction of the normal to the top plate. The longitudinal horizontal and vertical directions are aligned with the unit vectors of the x y and z axes respectively. When we refer to a vibration in the plane spanned by horizontal and vertical unit vectors we use the term transverse. B. Analysis of the beating FIG. 6. Envelope trajectories of tones from a normal a and a modified b kantele. The first harmonic is shown with a dashed line the second with a solid line and the third with a dash-dotted line. The plucking strength is medium and the plucking point is the midpoint of the third string. The strong beating does not appear in the knotless kantele. The observed beating in Fig. 5 is essentially a result of the knotted termination at the varras. This termination dictates two different effective string lengths one for the vertical and another for the horizontal vibration with a length difference of l see Fig. 3. For the vertical polarization the knot is the termination point whereas for the horizontal polarization the contact point to the varras is the termination point.6 7 As a consequence the vertical and horizontal fundamentals have a constant frequency difference f 0 that creates the beats. To verify this explanation we used a modified kantele 9 where the string is terminated without a knot but otherwise the construction is identical to that of the regular kantele. Figure 6 shows the first three harmonic envelopes extracted from a regular and b knotless kantele tones of the same frequency ( f 0 415 Hz). In the knotless design the beating vanishes confirming our explanation. The frequency difference of the two transverse polarizations is related to the length difference l by f 0 f 0 z f 0 y c c l f 2l 2ll l 0 y 1 and the linear mass density and l is the difference of the effective string length for different polarizations. The frequency difference can also be confirmed experimentally by analyzing the amplitude envelope of the first harmonic of recorded kantele tones and by extracting the beating frequency by fitting a sine wave using the nonlinear leastsquares method.10 To avoid confusion we refer to the extracted fundamental frequency difference as f 0 ex . Figure 7 presents an example where the linear trend and mean of the envelope of the first harmonic have been removed on a logarithmic dB scale in order to compensate the natural decay of the harmonic and a sine wave has been fitted to the available data above the noise level. We observed experimentally that the plucking direction slightly alters the beating pattern. Note that because of nonlinear coupling between different polarizations the string exhibits an elliptical vibration pattern shortly after the initial pluck regardless of the initial excitation direction.11 Therefore we use an average of two cases a horizontal and a vertical pluck. We noticed that this method yields a good estimate of where l is the effective length of the string between the tuning pin and the knot see Fig. 3 c T / is the transverse propagation speed determined in terms of the string tension T J. Acoust. Soc. Am. Vol. 112 No. 4 October 2002 FIG. 7. Beating of the first harmonic solid line in a kantele tone plucked horizontally a and vertically b with a nonlinear least-squares sine wave fit dashed line. Erkut et al.: Analysis and synthesis of the kantele 1683 Acoustical Society of America 2002 TABLE I. The fundamental frequencies f 0 y of the horizontal polarizations the frequency differences f 0 according to Eq. 1 and the extracted frequency differences f 0 ex for the five strings of the kantele. All quantities are given in hertz. String f 0 y f0 f 0 ex #1 465.2 1.50 1.30 #2 414.5 1.28 1.04 #3 391.3 1.10 1.07 #4 354.5 0.95 0.97 #5 314.3 0.76 0.77 the difference in the fundamental frequency of vibration of the two polarizations. Table I shows the fundamental frequencies of horizontal polarizations for the five strings of the kantele and the frequency differences f 0 obtained by inserting the measured length l and the effective length difference l for each string into Eq. 1. The extracted frequency differences f 0 ex are obtained by the sine-fitting method described above. f 0 and f 0 ex are in qualitative agreement. The differences are caused by the imaginary part of the input admittance at the tuning pin 12 and by the nonlinearities discussed in the next section. typically not responsive to the longitudinal forces below 1 kHz.21 In the kantele the tuning pins are not rigid in the longitudinal direction and transmit any longitudinal force efficiently to the body. As will be demonstrated in the following nonlinear mechanisms caused by the tension modulation create a longitudinal force component called the tension modulation driving force 22 or TMDF for short which accounts for the instantaneous onset and high initial amplitude of the second harmonic in Figs. 5 and 6. A similar mechanism has been observed in an acoustical guitar an orchestral harp and a piano.23 The partials thus generated are termed as phantom partials and it has been concluded that any plucked-string or struck-string instrument that is susceptible to longitudinal string forces could produce phantom partials. The phantom partials are observed between 1 3 kHz. Measurement results indicate that the kantele body is susceptible to the longitudinal TMDF. The TMDF has a significant effect on the lowest partials thus on the timbre as will be demonstrated in the following. D. TMDF formulation C. Tension modulation nonlinearity Unlike an ideal flexible string a real string such as a kantele string is linear to the first-order approximation only. Reformulation of the wave equation to include these secondorder terms has a relatively long history see Ref. 13 for a review. The major cause of nonlinearities is that any small transverse displacement of the string makes a second-order change in its length and therefore in its tension. By assuming fixed boundary conditions and using an excitation force of a frequency close to that of the first mode of the string the tension modulation is shown both analytically and experimentally to cause a fundamental frequency descent 14 15 a whirling motion 16 18 coupling between different modes and directions 12 15 19 and amplitude jumps.11 20 Legge and Fletcher showed that the generation of missing modes in a musical instrument is only possible when a realistic model of the termination at the bridge is taken into account.12 However they concluded that with a termination similar to that of a bridgeless kantele the nonlinearities cannot provide subsequent excitation of any of the missing even harmonics if the string is initially plucked near its center. Moreover according to their theory even in case of a termination by a guitar-like bridge the unexcited modes should exhibit a slow build-up typically around 100 ms. The behavior of the second harmonic in Figs. 5 and 6--specifically the observed rapid onset with a high initial level--is not consistent with their conclusion. This inconsistency is clarified by noting the structural differences between the kantele and other plucked-string instruments. In most string instruments the bridge is the usual termination point of the strings at one end and it is relatively rigid in the longitudinal direction.8 An important function of the bridge is to transmit the transverse forces of the string to the body of the instrument and hence the longitudinal force component is usually filtered out. A plucked-string body is 1684 J. Acoust. Soc. Am. Vol. 112 No. 4 October 2002 In order to obtain the TMDF exerted on a tuning pin we ignore the effective length difference in transverse directions discussed in Sec. III B and rely on the knotless kantele analysis data. Moreover the string is taken to be linearly elastic the inharmonicity caused by string stiffness dispersion is assumed negligible and the cross-sectional area of the string and hence its density is taken to be constant during the vibration. For steel strings the transverse propagation speed is usually smaller than 10% of the longitudinal propagation speed. This practically means that the first longitudinal component of the string vibration has a frequency higher than the tenth transverse harmonic. However here we focus on the lowest harmonics of the kantele tones. This fact justifies why we assume that TMDF is the only longitudinal force component acting on a tuning pin and neglect the effects of longitudinal wave propagation. Note that the same argument has also been used in several analytical treatments of nonlinear string vibrations.12 16 Under these assumptions elongation of the string l dev may be expressed as the deviation from the nominal string length L l dev L 0 1 y x 2 z x 2 1/2 dx L 2 L 0 1 2 y x 2 z x 2 dx where x y and z denote the spatial variables see Fig. 4 along the longitudinal horizontal and vertical directions respectively and the approximation is obtained by neglecting all but the first terms of Taylor series expansion of the slopes. If the supports are rigid the tension yields T T 0 ES l dev L 3 Erkut et al.: Analysis and synthesis of the kantele Acoustical Society of America 2002 where T 0 is the nominal tension of the string in rest E is Young's elastic modulus of string material and S is the cross-sectional area of the string. It is customary8 15 to define the longitudinal force exerted on the pin as F x T T x L T 0 ES l dev . L 4 Equation 4 is the analytical expression of TMDF. The negative sign indicates that the force is pulling the tuning pin towards the center of the instrument. There is a similar TMDF component acting on the varras at x 0 with a positive sign. The eigenfunctions for the lossy wave equation in the two transverse directions are exponentially decaying sinusoids with amplitudes a y m and a z n where m and n indicate the harmonic numbers for the horizontal direction y and vertical direction z respectively. The Fourier coefficients a y m and a z n have the physical dimension of meters. Inserting the eigenfunctions in Eq. 4 and neglecting terms of fourth- and higher order in a y m / L and a z n / L TMDF can be approximated as FIG. 8. The illustration of the TMDF for a single polarization case. a The exponentially decaying fundamental y 1 of unity amplitude. b The corresponding TMDF. The thick line is the component that causes the fundamental frequency descent and the thin curve is the oscillatory component. F x t 2 ES 2 2 t / y 1 1 cos 2 a y 1e y 1t y 1 . 8L2 6 2 ES F x T T 0 8L2 m 2t/y m m 2a 2 y m 1 cos 2 y m t y m e 2 2t/ n 2a z n 1 cos 2 z n t z n e n z n 5 where and are the frequency initial phase and time constant of the decay respectively corresponding to a particular mode. Equation 5 suggests that each transverse harmonic contributes to the TMDF with two exponentially decaying components within the square brackets. The first terms do not oscillate and they are the primary cause of the fundamental frequency descent.14 15 As will be shown later the fundamental frequency descent in kantele tones can be used to extract information for the nonlinear behavior.24 The oscillatory terms have twice the frequency and one half the decay times of the corresponding harmonics. Given an initial displacement consisting of odd harmonics only according to Eq. 5 the TMDF components that have twice the fundamental frequency are generated solely by the first harmonic of each polarization. Figure 8 illustrates the TMDF acting on the tuning pin in the absence of the vertical component. The top part of the figure depicts the horizontal fundamental y 1 of unity amplitude ( a y 1 1) that decays with its characteristic rate. The resulting TMDF pulls the tuning pin towards the center of the instrument twice during one period of the fundamental and decays twice as fast. In particular if the initial displacement consists only of a single-polarization fundamental e.g. in the y direction and if the constant tension T 0 is suppressed the time-varying part of Eq. 5 becomes J. Acoust. Soc. Am. Vol. 112 No. 4 October 2002 Equation 6 is essentially the same in form with the dimensionless nonlinear mixing force given in Ref. 23 see Eq. 3 in the reference. It also suggests that F x ( t ) may be related to the dilated eigenfunction y 1 ( x 2t ) by appropriate scaling and phase shifting. This property may be utilized as an analysis procedure. We present such an analysis procedure in the next subsection. The kantele responds as a passive linear system to the forces applied on its tuning pins. Therefore some part of the TMDF is transmitted to the body some part of it is reflected and the rest is dissipated. We first concentrate on reflection. The frequency difference of transverse and longitudinal modes as stated in the previous section indicates that the reflected TMDF component cannot be efficiently coupled to a longitudinal mode. It may however generate a transverse mode. The famous experiment of Melde demonstrates the possibility of the missing transverse mode generation by driving a string from one end in the longitudinal direction.20 A recent study25 shows that the three-dimensional admittance matrix of a musical instrument may interchange energy between longitudinal and transverse directions. This property is proved to be important in nonlinear generation of missing modes.12 If the TMDF is coupled back to the string by either mechanism a generated transverse second harmonic should exhibit a build-up onset until the coupled energy is balanced by internal losses. After the balance instant the second harmonic should decay with a characteristic rate that is independent of the first harmonic and governed only by the string and body properties. The tuning pin velocity measurements in transverse directions indicate the existence of such a component. However the velocity magnitude of the generated transverse harmonic is typically 20 dB lower than the magnitude of the first harmonic. In terms of the forces this difference is roughly 30 dB. We therefore conclude that the instantaneous onset and high initial amplitude of the second harmonic in Figs. 5 and 6 are caused by the TMDF compoErkut et al.: Analysis and synthesis of the kantele 1685

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