Computer Music: Musc 216
Further information on SYNTHESIZING BELLS can be found on the Sound On Sound: Synth Secrets site: Synthesizing Bells.
The Modes Of A Bell
Like any other musical instrument, a bell's timbre is determined by the pitches and amplitudes of the frequencies (partials) in its SPECTRUM. These partials (individual frequencies in the sound spectrum) are related to modes of vibration. In a bell, the modes are described by the number of circular nodes around the body of the bell, and the number of radial nodes that we can trace from the lip on one side, over the crown, and down to the lip on the other side. One example of a bell mode is shown here:
You can see clearly the two radial and two circumferential nodes that define the 2,2 mode shown.
Here's another diagram showing the bell's mode of vibration:
The black circle represents the shape of the mouth when the bell is at rest. The blue line shows the distortion of the mouth at some arbitrary moment when the bell is ringing with 2,n motion (where 'n' is any whole number), while the red line shows the distortion due to that mode half a cycle later.
Here's a cross-section of one half of the bell when ringing with an m,2 motion (where 'm' is any whole number):
Again, the black section represents the shape of the bell when the bell is at rest. It should now be clear that the pink section shows the distortion of the bell at some arbitrary moment, while the yellow section shows the distortion due to that mode half a cycle later.
Perhaps you can picture how these motions coexist in the 2,2 mode. Now try to imagine how this relatively simple mode co-exists and interacts with other modes. Actually, it's not as bad as you might think. Just try to imagine the bell as a flat plate ie. reverse the transformation in Figure 3 and you will see that the motions relative to its surface are analogous to those of other percussive instruments such as cymbals.
The Bell's Sound
Most traditional bells share sound characterists.. Many centuries ago, the people who cast metal bells discovered that they could make them sound more 'musical' by shaping the inside carefully. They probably didn't know it at the time, but they were modifying the positions of the nodes, and thus the frequencies at which they vibrated, until they generated a quasi-harmonic series. This is why tunes can be played on bells such as carillons. Nevertheless, bells are not the same as simple harmonic oscillators. If they were, they might sound like hammered strings. So what gives bells their identifiable timbre?
A bell exhibits one type of behaviour at the start of a note, and different behaviour as the note decays. Experiments show that there are, in fact, three distinct phases to the sound. The first is the strike (the ATTACK) the sound of one large lump of metal hitting another. As you would expect, this is enharmonic, and it dies away quickly. The second phase is the strike note (the SUSTAIN), and this is dominated by a handful of strong, low harmonics. Finally, the note's lingering energy is radiated by a sub-harmonic an octave below the fundamental (the DECAY).
The strike note is particularly interesting, because the perceived pitch is not necessarily the pitch of the lowest energetic partial. If the partials of a sound lie in an harmonic pattern based on a fundamental frequency that IS NOT present in the signal, the human brain inserts the missing pitch, and you 'hear' the fundamental, even if it's not actually there!
See www.soundonsound.com/sos/jan02/articles/synthsecrets0102.asp to read more on this phenomenon.
The strike note of a well-tuned bell does the same thing. The dominant partials can be tuned to produce frequencies in the ratios 2:3:4, so that the listener hears the implied pitch of '1'. For example, if the dominant partials vibrate with frequencies 100Hz, 150Hz and 200Hz, you will 'hear' a fundamental of 50Hz.
Putting all of this knowledge together, we can create this chart:
This is only a graphic representation, but it shows the three phases of the sound in an easily understood form. As you can see, an initial burst of enharmonic partials is followed by an extended period in which the low harmonics (some of which grow progressively sharper with increasing frequency) determine the sound. Below these, there lies the subharmonic that dominates the sound in its final moments.
Of course, the sound of a real bell is much more complex than this. Numerous factors have ignored such as the changes that occur when the bell is struck with clappers of different materials or at different speeds, as well as those that occur when bells are cast of different alloys, or of different sizes and relative dimensions. Fortunately, we can ignore all of these here, although we must include one additional factor if we are to synthesize realistic bell sounds. Bells warble... rather than producing a steady 'boing', many bells make a noise closer to 'boii-yoy-yoy-yoiinnnggg' because their modes can be almost degenerate. This means that bells produce two partials of almost (but not quite) identical frequencies, and these interfere (or 'beat') in the same way as do two synth oscillators of similar frequency.
Given all the above, we now have sufficient information to synthesize a bell. We'll start with the strike note. Since we require a small number of partials with 'stretched' harmonics, we can't use conventional analogue oscillators. As I've mentioned many times before, the partials of the waveforms generated by analogue synths lie in a perfect harmonic series (ie. 1:2:3:4:5:6... and so on) and will not sound correct here. Bells are much better synthesized using additive synthesis.
See Bell Tutorial
Material taken from Synth Secrets site: Synthesizing Bells