Sound Synthesis of Bells

Bells are very expensive devices, because of the expensive raw materials, complex technology, and the low production quantity. Since the molding is inaccurate, the bells' sounds are inaccurate, too. Correction is very difficult, and - especially in the case of large bells - almost impossible.

The development of the digital technology in the last decade made it possible to generate bell sounds electronically - at much less expense. There are different methods available in the literature for bell sound synthesis, however, the choice is not obvious. Our aim was the development of a synthesizer, which satisfies the following requirements:

• the pitch of the sound and the quality of the strike is easy to change;

• It is easy to analyze and synthesize new bells;

• The algorithm can be implemented on signal processors.

We were looking for solution satisfying the above requirements. We have developed a special synthesis, which can generate reasonable bell sound.

Physical Description of the Bell

Fig. 1. shows two typical bell profiles. Fig. 1.a shows a section of a bell for churches, while the profile in Fig. 1.b is for chimes. Profiles are different at each foundry, since they have developed their bell sets independently from each other. The knowledge comes down from generation to generation and it is a hard secret. The bell is made from a special alloy, which is a kind of brass. The composition is the following: copper - 80%, tin - 18%, the rest is zinc and lead. 

Fig. 1. Bell profiles for churches (a) and for chimes (b)

The vibration of the bell is very complicated. Theoretically the motion can be described as linear combination of modes which are perpendicular to each other, where the initial amplitudes are determined by the strike. According to theoretical derivations it can be predicted that each mode has 2m radial, uniformly spaced, and n parallel nodes, where m, n = 0, 1, 2, ... . If m = 0, the modes are single ones, while if m > 0, there are mode pairs. The latter ones - if the bell's cross-section is really a circle - form one partial. However, in the practice the bells are not symmetric, therefore these pairs form two partials, which sound as tremolo, because of the small frequency deviation. Fig. 2. shows the first five modes, where the dashed lines denote the nodes. Table 1. summarizes the most important partials and their tuning.

Fig. 2. The first five modes

The Bell Model

Since the vibration of the bell can be described by modes, it is straightforward to attempt the modeling the bell sound by resonators. By changing the inputs of the resonators, different effects can be performed. During laboratory investigations we have detected more than 140 modes. Such a high number of modes cannot be calculated in real-time, therefore the question was the following: how many partials are enough to implement for a realistic bell sound? Further investigations have shown that 10-15 partials are usually enough. For the calculation of the resonators the following parameters are to be known:

• goodness factor;

• angular frequency;

• initial amplitude;

• initial phase.

These parameters can be determined by Dirac-delta like excitation of the bell (striking with high speed with a hard hammer). The angular frequencies are known from the spectrum, and the other parameters can be calculated from the envelopes of the individual partials. The envelopes can be derived by mixing each partial to DC:

The envelope can be modeled by a low-order IIR filter, which can be approximated using the Steiglitz-McBride algorithm. (In order to decrease the computational demand, the envelope is decimated.) Poles and zeros can be calculated from the z-domain transfer function of the IIR filter. Each pole-zero system generates a certain impulse response with specific initial amplitude and exponential decay. In order to get the right IIR structure which generates already the partial itself, the poles are to be transformed in two steps:

• elongation: according to the rate of the decimation, the pole should be shifted toward the unit circle;

• rotation: according to the frequency of the partial, the pole should be rotated round the origin. Since in general the poles do not form conjugate complex pairs, the resonators are complex.

The transfer function of the complex resonators:

The impulse response of the system:

where ,

which is an exponentially decaying sinusoidal with an angular frequency of, while 

determines the initial amplitude and phase: . In this case the real and imaginary parts of the complex resonator are:

The difference between the frequency, initial phase and amplitude of the complex resonators results that the sum of them can model the envelopes. Fig. 3. shows an envelope and the corresponding partial. The sound of the bell is generated by the parallel connection of the partials, that is the parallel connection of the complex resonators.

Fig. 3. The envelope and the corresponding partial

The Hammer Model

It is supposed that the measured bell sounds are results of Dirac-delta like excitation. Hammer model should manage the following parameters:

• strength of the strike;

• nature of the hammer material.

The measurements in the Digital Signal Processing Laboratory can be concluded as follows:

• The hardness of the hammer is proportional to the bandwidth of the force signal. It means that striking with soft materials (rubber, wood) results in smaller bandwidth;

• The strength of the strike is proportional to the peak of the force signal;

• The strength of the strike is proportional to the bandwidth of the force signal, that is stronger strike results in larger bandwidth.

Fig. 4. shows the measurement of a bell. The force of the strike is measured by a force transducer attached to the hammer, while the sound is measured by a microphone. Both signals are recorded and processed off-line.

The requirements for the hammer model can be satisfied by a simple first-order IIR filter, which can be tuned according to the strength of the strike and the material of the hammer. This solution is also optimal for real-time implementation. The filter is excited by a single impulse, the amplitude of which is proportional to the force, the output of the filter is lead to the input of the resonators.

Fig. 4. Measurement of the bell of the reformed church of Õcsény in the DSP Lab

Summary

Although the research is not completely finished, our present model produces very realistic sound. The survey of the literature and our own investigations have shown that a physically informed (pseudo-physical) synthesis results in a competitive bell model. This model can be efficiently implemented with high sound quality on signal processors available in 2006, like ADSP 21262, 300 MHz. The bell synthesizer can be listened e.g. at Gárdonyi Géza Primary School in Budapest, or at the church in Gödöllő-Máriabesnyő.

Sound Examples

The mp3 file is recorder from a real-time synthesiser applying a DSP. 

• Westminster chime

Related Publication:

Lajos Rancz, Sound synthesis of bells (in Hungarian), M.Sc. thesis, Budapest University of Technology and Economis, 2004.

Detailed description of the sound synthesis method

Useful Link:

Rancz Első Magyar Toronyóragyár Kft.

The company producing the bell synthesizer. Additional sound examples are available on their page.

 Further information: László Sujbert, Lajos Rancz