A general doc on physical modelling syntheis, with a quick overview of plucked strings, wind instruments, and percussion instruments.
PHYSICAL MODELLING SYNTHESIS: A general overview Overview ======== Physical Modelling synthesis is a technique where instead of trying to recreate the instruments sound directly, the physics inside the instrument are simulated. Since real-time computing means are limited, the simulations must operate on a simplified description of the instrument's behaviour. For this reason, the most efficient models for string and most wing instruments is the so called "Waveguide" family of models, where the resonating body (the string or the tube) is modelled using a short delay line with high feedback. This simulates the sound waves bouncing inside the instrument from one side to the other. From there, many complications are added on to simulate the different physical effects affecting the sound. The other important type of physical modelling is modal synthesis, which uses a set of bandpass filters in parallel instead of delay lines for the instrument body (see the Percussion instruments section below). String instruments ================== In string models, especially plucked string models, the main goal is to accurately model the vibrating strings. This involves adding different filters inside the delay line (at least a low-pass filter, but often other types such as all-pass or high-pass filters too) to model different energy losses and non-linear phase effects that happen inside the string - the high harmonics are often sharp. Another common mechanism is modelling multiple delay lines instead of only one: a real life string can vibrate on at least 4 planes (horizontal and vertical displacement waves, compression waves and torsional waves), although only 2 are typically needed. The waves travel at different rates in the different planes, and the planes are usually "coupled" together - waves migrate from one plane to another. Another important coupling phenomenon is strings vibrating sympathetically. Next, in many instruments, there's a non-linearity in the string behaviour, producing a distortion-like effect. In sitars, this is particularly evident: sitars are built with a special bridge that effectively changes the string length depending on the waves inside the string, producing the buzzing effect. Also, the string, when plucked, produces two identical waves that go in opposite directions, a phenomenon that can be simulated in multiple ways - often with 2 delay lines, one for the left part of the string and one for the right part (in other words, this doubles the number of delay lines required). This aspect is particularly important for bowed string simulations, as the 2 part behaviour of the string is an essential component of bowed string mechanics. Finally, on acoustic instruments, the body of the instrument must be simulated, as it often has characteristic resonances. There are two main strategies for this: some models use series of bandpasses (see modal synthesis below) to simulate the main resonances and elements like reverbs to simulate the large numbers of roughly random resonances found in typical instruments. Other models, the so called "commuted waveguide" models, instead use a sample containing all the resonances to excite the model. When simulating electro-acoustic instruments, such as the electric-guitar or the clavinet, the behaviour of the pickups and the amplifier are modelled instead. Wind instruments ================ For wind instruments, the focus changes: the tube mechanics are simple, as they only resonate on only one plane, they aren't coupled to other tubes, and very rarely have complicated output resonators (except muted brasses). Instead, the
complicated part lies in the embouchure of the instrument, with the reed, lips or air jet interacting in a complicated with the instrumentist's breath and the air inside the instrument. The reed, lips or air jet moves in a roughly sine-wave motion, stimulated by the air input, and controls the amount of air entering the tube. However, the amount of air entering versus the reed position (or velocity) is very non-linear, and much effort goes into designing that very non-linearity. Finally, the air returning from the tube influences the reed/lips/air jet, which is what synchronises it to a harmonic of the tube. Thus, a typical reed/flute model would have a 2 pole bandpass filter and a distortion part. To start the bandpass's oscillation, you can feed it positive bias, and switch it to negative bias whenever the bandpass's output goes negative - this will nicely start up it's oscillation. For the distortion part, some models just clip the signal with some sort of foldback distortion, but this unfortunately only produces odd harmonics. To get the full spectrum, you can simply start out with two harmonics instead the fundamental, and then the 2nd harmonic, which you can generate by using the filter's two outputs - bandpass and lowpass - and multiplying them together. By controlling the mix of the two inputs, you can get large variations in instrument timbre, from oboe to saxophone to flute to clarinet. White noise can also be added into the mix, which will make flute sounds much more realistic. This mix is then clipped, and finally distorted a last time with the following equation: y = x cx. By controlling the value of c, you can morph from hard clipping (c = 0) to soft clipping (c = 1/3) to foldback distortion (c = 1) - note that these values assume the signal is clipped to a range of -1 to +1. Then, this signal is multiplied with the pressure, the volume at which the instrument is played (note that this prevents the instrument from sounding with volume = 0, a desirable characteristic). Finally, to sync the reed/jet with the tube, the signal returning from the tube is fed back into the bandpass, completing the cycle. A brass model can easily be derived from this: all that needs to be changed is the distortion part before clipping: we need impulses that get progressively thinner and brighter as the volume inside the instrument gets louder. A good way to do this is to get the lowpass output of the filter, take the absolute value, invert the signal, add positive bias, then keep only the positive part of the signal. At the end, you have impulses that get progressively thinner as the filter's output gets louder. Finally, you still have two impulses for each cycle of the filter sine, so you want to suppress one of them to prevent the model from incontrollably going into harmonics - just only let the signal pass when the bandpass output of the filter is positive. The last category of wind instruments is the free reed family, which includes accordions, harmonicas, and many organ pipes (reed pipes), which are similar to reed instruments, except they doesn't have any resonating tube - thus, the model doesn't have any delay line in it, only a filter and a distortion part. Percussion instruments ====================== The last large family of instruments is the percussion instruments. String and tube instruments have a 1-dimensional body, which gives them a regular series of modes (resonances) - a typical example would be 100hz, 200hz, 300hz, 400hz, 500hz, 600hz... Percussion instruments have modes too, but they come in irregular series, since they are formed of 2d or 3d resonators. For instance, a glockenspiel bar is pretty much a rectangular bar of metal, and the main modes of a bar tuned to 100hz would be 100hz, 276hz, 540hz and 890hz. As you can see, these are quite inharmonic. The instrument relies on the fact that the higher modes are softer, harder to hear and decay much quicker than the fundamental. Other chromatic percussion instruments don't use straight bars: they cut an arch under them. By careful cutting, the harmonics of the bar are brought into tune: A vibraphone 100hz bar would thus have modes at 300hz, 412hz, 1003hz, 1609hz... the accuracy of the tuning is about 1-2%. Steel drums are tuned even more accurately, having weak modes at 2.5 and 3 times the fundamental but otherwise having a complete and very
well tuned series of modes up to at least the 10th harmonic! Drums, on the other hand, are often detuned on purpose, as is the case of snares, toms and bass drums: the fact that they have two membranes double up each of the membrane's natural modes, making the pitch hard to determine. Synthesis of percussion instruments is thus based on synthetizing each one of their modes. This is done by using a set of 2 pole bandpass in parallel, one for each of the audible modes, with adjusted resonance and volume for each. Then, the set is excited by an impulse, which can be the same as for plucked string instruments - a pure impulse, a burst of noise, a sample, or any other suitable means.