1.Introduction

In this study, the model-based sound synthesis of the tanbur is tackled using both generic ( i.e., starting from a generic string instrument synthesis model, and calibrating this model to obtain realistic synthetic tones for a particular string instrument) and specific (i.e, elaborating the current models by developing new features to simulate unique characteristics of the instrument under study) approaches.

2. Acoustics of the Tanbur

The unique acoustical properties of the instrument are summarized in the paper. A more detailed discussion on the acoustics of the instrument can be found here. A short melodic excerpt demonstrates several features of tanbur, such as plectrum scratches, the typical use of melody pair, and the use of other strings as pedal tones.

2.1 Tension Modulation

Tension modulation depends essentially on the elongation of the string during vibration. The elongation increases the tension of the string which in turn increases the propagation velocity of the transversal waves. The audible effects of tension modulation nonlinearity include the fundamental frequency descend, and the nonlinear coupling between individual harmonics.

The fundamental frequency descend is reproduced above (Fig. 2 in the original paper). The moderately-plucked tone (dashed) exhibits a mild fundamental frequency descend, whereas the descend of the hard-plucked tone (solid) is more pronounced.

The figure above (Fig.3 in the paper) depicts the envelope of the first and the second harmonic detected in string vibration (left) and in soundboard vibration (right) of a mid-plucked tanbur tone.

String vibration, recorded signal
Soundboard vibration, recorded signal

String vibration, the extracted second harmonic
Soundboard vibration, the extracted second harmonic

In the string vibration, the first harmonic is most pronounced, as expected. The second harmonic has a relatively low initial level but the amplitude gradually increases with time. This suggests that the vibration modes are nonlinearly coupled so that energy is transferred back to the string at double the frequency of the first harmonic.

In the soundboard vibration, the second harmonic has a higher initial amplitude level than the first one and also a very sharp attack. Investigating the two figures, it is clear that a linear system cannot produce such a difference in the behavior of the envelopes of the second harmonic. This suggests that the longitudinal force caused by tension modulation produces the second harmonic in the soundboard vibration.

3. The Linear Model

The basic string synthesis model of Fig. 4 is used to extract fundamental parameters for a tanbur synthesis model from the sound samples recorded in an anechoic room. An example of the analyzed, synthesized and extracted excitation signals shows that the nonlinear characteristics of the analyzed tone cannot be reproduced by the basic linear string model. However, the synthetic tone still captures the fundamental features of a tanbur tone.

Starting with the basic string synthesis model of Fig. 4, a more detailed model, such as the generic string instrument model is needed for a more realistic simulation of the tanbur. Fig.5 shows a dual-polarization string model, which we used to model a melody string. A single dual-polarization melody string model produces more interesting synthetic tones compared to the basic string model, as this example shows.

The linear tanbur model consists of two dual-polarization string models (to simulate the melody pair), and five single polarization string models (to simulate the resonators), a pluck-shaping lowpass filter E(z), a pluck-position comb filter P(z), and a coupling matrix C, along with 10 wavetables to store the extracted excitation signals. This example demonstrates the tones synthesized with the linear model.

The reverberant tone characteristics of the tanbur can be reproduced by setting the coefficients of the coupling matrix C to higher values. In this example, the coefficients are set three times the values of the previous example.

4. The Nonlinear Model

The block diagram in Fig. 6 depicts a string model that includes the tension modulation in terms of variation of the propagation velocity and of coupling of the longitudinal force to the output and back to the string. Note that the model is a string model, and the excitation signal is an ideal triangular pluck. Therefore, the model cannot produce a realistic tanbur sound by itself. The following sound examples (except the last one) are for explaining the nonlinear mechanism, not for obtaining a realistic timbre. The last example, however aims to reproduce a realistic synthetic tanbur tone using the nonlinear model.

Fig.7, which is reproduced above, shows synthesized tones exhibiting a pitch drift (cf. Fig.2). The next sound example demonstrates that pitch descents similar to that of the original tones can be obtained by the nonlinear model. Note that the output of the nonlinear model is the force applied to the body, not the radiated sound. Therefore, a realistic sound synthesis model would require a detailed body filter. However, our aim here is to demonstrate that the nonlinear model captures the essential nonlinear behavior observed in tanbur tones.

In the sound example, the first pair is the analyzed and synthesized tones, which are excited by plucking the string softly (not shown in Fig.2 nor in Fig.7). The pitch descend is very slight. The second and third pairs are plucked moderately (dashed in figures) and hard (solid in figures), respectively. Note that the synthesized tones are exhibiting similar frequency descents to the original tones. Note also that a linear synthesis model cannot exhibit any frequency descents.

The feedforward parameter g_out in the nonlinear model (Fig. 6) directly controls the amount of TMDF coupled to the body. To demonstrate the behavior of the second harmonic in the soundboard vibration, we set the feedback parameter g_str equal to zero, use an excitation which lacks the even harmonics (a middle-plucked slope wave), and vary g_out. The next sound example demonstrates the model output and the nonlinearly produced second harmonic pairs for three different values of g_out, namely g_out1 = 0.1, g_out2= 0.5, and g_out3 = 1.

The feedback parameter g_str determines the build-up rate of the even harmonics, especially the rate of the second harmonic. Setting the feedforward parameter g_out equal to zero, using a middle-pluck excitation, we obtain the following example for g_str1 = 0.001 (the first sound) and for g_str2 = 0.005 (the second sound), respectively. The extracted second harmonics (same order) from the synthesized tones demonstrate the effect more clearly.

Up to now we discussed the non-linear string model with an ideal pluck for demonstrative purposes. A much more realistic synthetic tanbur sound can be obtained by linear model, if

• A more natural excitation signal can be used.
• The string model output can be further filtered to take the body response into the account.

The following example shows the nonlinear model output after the completion of these tasks.

5. Conclusions

We have tackled the modeling of the tanbur by a generic linear plucked string instrument model and by a specific nonlinear model that accounts for the tension modulation effects inherently present in tanbur tones. The sound examples presented here may provide a clarification and a deeper understanding about the issues discussed in the paper.

This URL: http://www.acoustics.hut.fi/~cerkut/icassp2000
Last modified: May 26, 2000
Author: Cumhur Erkut