The last example required a particular kind of knowledge, specialised domain knowledge on the subject of archaic phones. It's common to need electronic engineering knowledge when designing sounds that are already unreal in some way. Some examples are building a Moog filter, an 808 drum machine, a Theramin, a 1970s Hammond organ and so on. Many sounds are electronic or electrical in origin. Remembering the particular way the AY38610 sound chip used shift registers for its noise source might be very useful in faithfully recreating a Japanese vintage video game noise. Knowing that a certain kind of printer uses a stepper motor with 24 poles could help you to very faithfully recreate a certain old dot-matrix line printer sound. The difference between a V8 engine and a two stroke motorcycle engine is vast, but rooted in the real physical design. Why two birds make different tones is a question a good knowledge of biology can help us with. But each of these levers specific knowledge.
Sometimes we need to use a different kind of knowledge, a more general sort of physics and mathematical reasoning which we might call "first principles". Statements such as "Related sounds usually start and stop in a similar timeframe" and "A vibrating skin always decays exponentially" speak the kind of wider knowledge very useful to a sound designer. These rules are less fundamental and stringent than the basic laws of math at the root of synthesis, they don't always hold true, but we can get a lot of good value from applying them to designs and seeing what works.
In this example I want to think about the physical properties of a telephone bell, using what we know about bell sounds and physics in general. Although the introduction is long it sets the ground to understand why we do what we do with additive synthesis. There are three topics to consider briefly, shape, structure and excitation.
A bell is usually made of metal. When it comes to materials we have to consider the general features found throughout materials science, conductivity, density, elasticity(bulk modulus), frictional coefficients and so on. What would a glass bell, or a wooden bell sound like? If we had a few bells made from different materials and were able to listen to them all together we would quickly come to the conclusion that there is something about the sound, governed more by shape than material that gives them all a similar character. Something that makes them bells and not say planks or bottles. One of the more advanced references is work by Jean-Marie Adrien entitled "Modal synthesis". This is all about considering the essential function of shape in determining how a sound evolves. Fortunately bell makers have been thinking about their art for centuries and we have a lot of good insights into why a bell contains certain harmonics.
Imagine a town centre throughout a busy shopping day. Early in the morning when there are just a few people about only the main high street has many people on it, visiting the main shops like the coffee house, the patisserie and maybe buying some vegetables at the grocers. Later in the afternoon the town is swarming with people from out of town too, tourists and visitors spread out onto the side streets to visit boutiques. Some of the Japanese tourists get lost and take crazy inefficient routes down sidestreets people rarely walk. This is very like the modes that soundwaves follow in the shape of an object.
The main high street is the easiest path. We call this the primary mode. It is the path down which sound energy moves to create the fundamental frequency of the object. The other large shopping streets nearby form the secondary and tertiary paths. These correspond to other harmonics in the sound. The likelyhood that a wave of energy takes a secondary or higher order path is related to how busy the sound is. A busy sound contains more energy. If it contains a lot of energy then sound waves spread out to use the other routes.
During the afternoon and evening visitors leave the town, the sidestreets empty and life returns mainly to the high street. This corresponds to the decay of energy in a sound. Energy moves from the tertiary and secondary modes back into the fundamental until finally it is the only harmonic left in the sound. Soundwaves have an attraction on each other, like people they tend to hang out together as far as is comfortable, we will look at this again later under "excitation".
Additionally there is more than one way of getting about town. Sound propagates in two distinct ways, by transverse waves, and by longitudinal waves. The latter is usually faster than the first, so we might make the analogy that some people can move about on busses, while others walk around on foot. This analogy is not a good mathematical description, but it is quite a fair way to picture the evolution of a sound in a physical object.
Not including alloys 63 of the 118 real existing elements are metals. That accounts for a very wide range of physical properties. But it's safe to say that the materials we would make a bell from are hard, dense and acoustically very conductive. The speed of sound in brass or steel is approximately 5000m/s which is very fast compared to air. Any freely hanging metal structure whether a bar, tube or bell therefore offers very little damping to vibrations propagating through it. It is also very elastic, not as much as glass, but in any case this means that displacements are reflected at the boundaries rather efficiently. Lastly its structure at the atomic level is quite uniform, in a way that wood or some other materials are not.
What does all this mean in terms of the sound? For starters a dense material that conducts sound well offers very little damping. How much a material dampens a sound determines how fast the sound dies away. Because the speed of sound is high in a material it means that high frequencies will prosper, energy bouncing around the structure at high speed means smaller wavelengths and so higher frequencies. The uniformity of the material leads to purer tones. When a soundwave encounters an anomaly in its path it tends to distort, but a very regular material will allow pure tones to propagate nicely. Damping is also governed by the materials elasticity. Inelastic materials quickly convert the sound energy to heat inside them, but elastic materials reflect the sound back and forth for a longer time.
A bell on its own makes no sound. In synthesis we have a way of looking at sounds called the excitation-resonance model. This breaks down many of the common ways sound is formed by considering how energy is transferred from one state to another. You can blow through a reed or resonant cavity, you can pluck a string, but with percussion instruments like the bell you hit it with something. Hitting something transfers kinetic energy into the the thing you hit.
Roughly what happens with a bell is that the strike is very brief. The hammer/striker is itself made of an elastic material, so the collision is short and the hammer bounces away from the side of the bell at almost the speed it hits it. During the very short time they are connected energy from the hammer deforms the bell. For a moment the shape of the bell is no longer semispherical/round but becomes a slightly oval shape. Two things happen. Energy as longitudinal waves propagates almost instantly throughout the whole bell body exciting it into many modes of oscillation. At the same time transverse waves start to roll around the outside edge of the bell. As the transverse waves move around they fall into particular patterns typical of a circular object. The fundamental mode is determined by the diameter of the bell lip, in this mode the whole bell distorts back and forth between two oval shapes where the irregular axis is at 90 degrees to the previous one. Some of this energy quickly moves into other circular modes, a third harmonic appearing at 60 degrees, a fourth at 45 degrees and so on.
Rather like the way gravity pulled the initial explosion of the big bang into clusters of stars and galaxies energy circulating in the bell gets pulled into phase with other streams of energy. The longitudinal waves that caused the initial high frequency excitation quickly die away as they are usurped into more regular modes. More dominant modes suck in energy from lesser inharmonic modes. Eventually there is no frequency left but the fundamental.
So far we haven't even written a single piece of code. All this talk of declarative knowledge, stuff about what a bell IS has helped us a lot, but it hasn't yet produced a sound. To go further we now need to select tools from our DSP box of tricks, we need to switch to imperative knowledge about HOW to implement the bell. Now is time to consider WHY we might choose one method over another, and to do so we need a little computational wisdom to help us know which approach might be the most efficient, easiest to code, most flexible at runtime, and which presents the most useful interface to application components above.
Lets take stock of some things we know
We already looked at very simple additive synthesis in the first telephone example when making DTMF tones, so no more explanation is needed about the spectral properties. In brute force additive synthesis we literally use one oscillator for every harmonic. Here's a picture of a bit that's adding three harmonics together.
So, if each harmonic has it's own oscillator all that we need to do is add a bunch of oscillators, sum them, and that's it? Basically yes, but before doing so let us take a few intelligent steps that will save us time later.
The sparseness of the bell spectrum is what makes additive a good choice, we actually only need a handful of oscillators to get a decent bell emulation. The burst of very high density harmonics when the hammer impacts the bell can be approximated by a little burst of noise, after that very quick (30ms) time the vibrational modes have settled into maybe 10 or 12 important harmonics. In this example we used 4 bunches of 3 or 4 harmonics, not including the striker sound. See diagram below.
Remember that the shape of the bell doesn't change (apart from the distortions which are part of the sound), so tuning a bell is more about its scale. Rather than specifying fixed frequencies for each harmonic it's better to start with only the fundamental and express all the other harmonics as ratios. This way, by changing the fundamental we can tune the bell and all the other harmonics will follow suit correctly. When we added the second bell later it was easy just to change one parameter for the fundamental.
While each harmonic has an individual evolution some harmonics seem to behave in a related way, as a group. For example the circular modes are quite distinct from the translateral modes. Putting groups of harmonics that tend to decay together through the same envelope is a way for us to cheat and gain a little efficiency instead of controlling the level of every single harmonic individually.
Here are the sounds of each harmonic group being added, moving right to left in the above diagram starting with the striker and ending with the fundamental. Audio .mp3
It's nice to keep the envelopes in a separate subpatch from the oscillators, this way we may resuse the patch or easily tweak them all in one file. In a real bell the harmonics shift and phase in during the attack stage, but for practical purposes a small telephone bell has an instant attack for all harmonics because the shifting happens so fast. As with all bells each group decays at a different rate. The higher harmonics from the striker and circular modes fall away very fast. To get the correct envelope we take the square of a linear line segment, or even the 4th power for the strike. The primaries, fundamental and hum all have a slower, lower order decay.
The control structure is very different from out last telephone. In the last example the switching on and off of the sound was "part of the sound", we used amplitude modulation. If we try this with a real bell sound it doesn't work. It sounds ridiculous when the bell suddenly stops. In reality the hammer may continue to swing for an extra few hits as it bounces, or because there is a charge in a capacitor, and the bell continues to ring for a second or two after the power is removed. Therefore we control the bell using message events in PureData, by using a metronome to fire off bursts of events at about 15Hz.
When a current is applied it flows through the closed switch and the coil creating a magnetic field. This pulls the lever towards the coil, but the lever hits the switch and turns it off. When this happens the magnetic field goes away so the spring pulls the lever back. A hammer attached to the lever hits the bell. At the same time the switch closes again and the cycle hapens all over again.
A telephone is more than just a bell sound. As we see above there is a moving lever. The bell and buzzer circuit are also phycically connected to the phone body so some of the movement of the bell and lever will be transmitted through the phone body to give a slight rattling sound in sympathy with the ringing bell. We chose a short delay and a few filters to give the impression of a plastic/bakelite box about 40cm big, this part is very experimental and you should work by attaching some sliders to the filter parameters and twiddling them until it sounds about right. Plastic gives quite a resonant sound so the feedback value (0.4) is fairly high, the box dimentions are about twice as long as wide. The body resonance is only stimulated by the noise burst from the striker, not the whole bell sound. This is because most of the bell frequencies are way outside the resonance anyway and would add to the filters instability.
Instead of just one bell, most telephones actually have two bells. The hammer sits between them and bangs one bell then the other. By taking the modulo 2 [% 2] of the ringer counter we can route bangs alternately to each bell. Usually the bells themselves are manufactured in a cheap way so they won't sound identical, a small difference (5Hz) between the two creates a much more authentic sound.
Here is a recording of the above twin bell phone. Audio .mp3
And here is the complete PureData patch to study Puredata file .pd