Synthesis Techniques - Physical Modeling

By Scott Lehman Quick Links: Introduction Why Physical Modeling? Forms of Physical Modeling Waveguide Instruments A Simple Waveguide Example The Future To Learn More References

Introduction

In the history of sound synthesis, there have been two different general methods for synthesis of musical instrument sounds. One approach is to look at the spectrum of a real instrument and try to recreate it. This includes methods such as additive synthesis and frequency modulation (FM). These produce sounds with similar structure, but the parameters involved have no relation to the physical parameters of an instrument. The other popular approach is to use a sample of the instrument, such as in wavetable synthesis and samplers. In both of these cases, you're creating sounds without any consideration for how the real instrument actually creates those sounds.

With physical modeling on the other hand, you are not creating the sound directly - you are creating and controlling a process that produces that sound. You define the process which models the actual instrument, and then when you play this synthetic instrument, you are specifying some of the necessary parameters needed to make the sounds. This approach will take sound synthesis up to a higher level.

Why Physical Modeling?

Physical modeling has become a big buzzword in the music industry these days. What's the big deal? Well, we will now have some of the most expressive and realistic synthetic instruments. Instruments that react to how they are played. You can drive the synthetic instrument sharp or flat, just like the real thing. You will no longer be stuck using the same sampled sound over and over again, only being able to adjust the envelopes and adding vibrato.

The strength of physical modeling is that the parameters involved are easy to understand because they have a real counterpart. Say you are trying to create a horn sound that has the variation in spectral content found when a horn is played with different strengths. You might have a very hard time figuring out which parameters to change in an FM synthesis algorithm. With a good physical model and a wind controller, all you have to do is blow harder, and the sound produced will respond to the playing level.

Physical modeling is quite flexible and a single algorithm can achieve a variety of sounds, without requiring more memory. Wavetable based synthesizers require a number of samples (with each sample taking up some amount of memory) at different registers to be able to produce realistic sound of a single instrument. But a single physical modeling algorithm can be capable of producing a full range of sounds simply by adjusting the input parameters.

So if physical modeling is so great, why hasn't it been used earlier? Well, it requires quite a bit of processing power. It certainly could have been implemented earlier (and it surely was to some extent), but you had to give up real-time operation. This removes the interaction between the musician and instrument, so it has limited value. The recent advances in digital technology are now making physical modeling possible with real-time operation, although the early generation products available can be expensive, but that will change with time.

Forms of Physical Modeling

The physics and equations governing the physics of musical instruments and acoustics are pretty well understood and documented. One approach to physical modeling would be to start with these equations, choose the necessary parameters, and then solve the equations using some numerical methods to carry out the integrations and other mathematical operations. This is the 'brute force' approach, which is not very efficient, making real-time performance difficult.

But there are much more elegant and efficient ways to implement physical modeling synthesis. One very popular technique is called waveguides. Instead of solving sets of equations to produce a sound, you simulate the operation of the actual instrument. Certainly solving equations is part of creating waveguide based instruments, but you aren't solving these equations each time you want to play a note. Instead, the equations are used in the design stage of the virtual instrument to come up with the required filters. The remainder of this article will be focused on waveguide instruments.

Of course there are many flavors of physical modeling, and you are not restricted to a single tried-and-true method. When designing the algorithm, you generally start with the physics equations. Then you try to pick out what elements of the physics are most important for creating the instrument's sound. Even for a single equation, there are many different approximations that can be made to trade off realism for simpler computation. There are also a number of important tricks that can be used. For example, division by two could be implemented by an arithmetic shift. Or rather than modeling the pressure function at the reed of a clarinet as a complex filter (an example of a lumped element), you could instead use some more memory to implement a simple lookup table to reduce the computation. When working with stringed instruments (as we will in the next section), rather than modeling the string where each incremental segment introduces some attenuation, you could compute the total effect of the loss at the end of the string (provided the equations involved are linear, or close to linear. This is an example of a distributed model). This technique can also be used in non-stringed instruments, and is depicted in Figure 1. From the designer's point of view, physical modeling boils down to producing he best sound with the least amount of number crunching.

Figure 1: Exploiting linearity to reduce computation. (a) is a more accurate model of what actually occurs, but by lumping the loss calculation together as in (b), you now have a more efficient model. (The chain pictured would only represent a single traveling wave - for most models, there would also be another chain where values move in the reverse direction)

There are other variations of physical modeling as well. Rather than creating an entire synthetic instrument, the modeling can be used to alter a signal. This is along the lines of Roland's COSM technology found in the VG-8. These are not 'pure' synthesis techniques - instead of synthesizing the sound, the models are used to create algorithms that can be used to alter a sound, allowing an instrument to mimic sounds of other instruments.

Waveguide Instruments

So what are waveguides? Well, in a musical instrument you will find traveling waves. In the clarinet, the waves are traveling up and down the air column in the bore. In stringed instruments, there are waves traveling along the string, which may then couple into resonator sound bodies. In each case, there is a clear path that the waves travel along (and the waves travel both directions simultaneously). The physical structures that carry these waves are the waveguides, which can also be thought of as transmission lines.

The beauty of waveguides is that they can be implemented digitally with basic delay line structures, which are very efficient. A piece of memory can be filled with some waveform, and then the wave can 'propagate' by reading and writing values in the proper order. Of course a single delay line can only model a wave traveling in one direction, but you can simply add another delayline to model the wave traveling in the opposite direction.

That's the easy part. To complete the model, you need to define the interaction between the waveguides, i.e. what happens when the wave reaches the end of the string, or at the simulation level what happens to each number read from the delayline, and how do you determine what number to write into it. This can involve various filters that may model the behavior of the bell or tone hole on the instruments, look-up tables, and input from the performer. This is the challenging part of creating waveguide instruments. It can be very difficult to create filters or algorithms detailed capture the crucial aspects of the instruments action and remain simple enough for real-time operation.

A Simple Waveguide Example

Perhaps the simplest waveguide physical model is the plucked string, and in this case, I will be using an electric guitar model. First off, we need two delay lines to model the waves that travel in both directions along the string. The length of the delay lines must be chosen such that the total round trip time is equal to the inverse of the desired fundamental frequency. The plucked string is a good starting point because the excitation is simple - you simply fill the delay lines with the initial displacement of the string, and let it loose. (In other instruments, the waveguides may carry waves in other forms. For example, with a piano, which can be modeled as a struck string - very similar to the operation of the plucked string, just using different initial conditions - we might work with velocity, while for wind instruments, air pressure might be the most natural choice.) This initial displacement of the string can be approximated with a triangular shape, where the peak is the location of the pick used to pull the string out. Now we need to connect the two delaylines. If we assume perfect reflections, the output is negated before being fed as the input into the next delayline. This is because at the two ends of the string are (relatively) fixed, and the displacement must be zero at those points.

Figure 2: The basic waveguide plucked string instrument. The delaylines are filled with values corresponding to the string displacement just before plucking. The -1 gain at each end accounts for the reflection of the waves.

A plucked string has a decay associated with it, so we need to implement a lowpass filter to introduce loss and decay of the notes. The output signal of the model is produced by tapping into the delay lines at the appropriate locations (corresponding to the locations of the pickups on the guitar) and adding them together. (This is similar to the original Karplus-Strong algorithm, which may be considered as an early physical model.) Sound Set 1 presents a series of sounds produced with this simple model.

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Sound Set 1: The output of the plucked string waveguide instrument. In the first of the five examples, the string is plucked near the bridge. It is then plucked moving away from the bridge, with the final sound produced by plucking at the midpoint of the string. The sound becomes more 'mellow,' just like the real thing.

Now you may say 'My wavetable synth can do that. Why would I want to have a physical model?'. Keep in mind that the model used so far is overly simplified, and not a thorough implementation. Also, that model was only demonstrating the effects of a single parameter (the pluck position). More tasteful sounds could possibly be found by moving the pickup position, or modeling the use of several pickups together. Even so, this model is capable of achieving a variety of plucked string sounds using a single algorithm, without any additional memory. With sample based synthesizers, you would need several samples of each plucked string sound. And there are also some things that would be very hard to do without a physical model.

One particular application of the plucked string that has been of particular interest to me is the use of distortion and feedback with an electric guitar. Typically, the guitar output is fed into a distortion pedal, and then into an amplifier which then drives the speaker, turning the electric signal into an acoustic signal. The guitarist can then face the guitar towards the speaker so that the sound excites the strings, which creates the feedback. This setup is shown in Figure 3. The pitch of the sound that follows is related to the resonating frequencies of the strings. The model needs to account for the gain before the distortion function, any post-distortion gain, and the attenuation of the signal as it travels from the speaker to the guitar. Insert a delay line to model the time it takes the sound to travel from the amplifier to the guitar string, and we have our model. Despite the simplicity of the model, the results can be quite realistic, as shown in Sound Set 2.

Figure 3: Diagram of the complete guitar rig with distortion, the amp, and speaker. The sound from the speaker excites the guitar strings (implemented by adding portion of the speaker output to one point on the waveguide model).

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Sound Set 2: The results of the physical model incorporating mild distortion and feedback. The sound actually responds to the pitch bend that is applied.

This is the kind of interaction you can achieve with a physical model based synthesizer. Try doing that with a wavetable synthesizer!

Now let's take a step back and think about how to use a physical model for processing sound, rather than synthesizing it. One feature of Roland's COSM technology in the VG-8 is the ability to create virtual pickups at any point along the guitar body or neck, which can change the sound substantially. To do this, a special hex pickup must be put on the guitar. If you know the location of this pickup relative to the strings, you can include it in the model. Then you add your virtual pickup location to the model. By sitting down and doing some math, you can find the equations relating the output of the hex pickup (which is what would be going into the VG-8) to the virtual pickup's output. At that point, you can design a filter that modifies the known signal to sound as though it was produced by the virtual pickup.

The Future

One major problem with physical modeling is that for each instrument, you may need an entirely new model and algorithm to produce a realistic sound. If manufacturers want to offer a wide variety of instruments in their synths, they will need to make the hardware systems very flexible. This required flexibility makes a makes a general purpose device (like a computer) a good option.

The challenge for manufacturers is to create synthesizers that offer the great flexibility, but that are also easy to create sounds on. There can be so many parameters that could be used with a model that the user can become overwhelmed, particularly if he or she is not fluent with the particular instrument being modeled. The experienced musicians want to have total control over the synth to get truly expressive instruments, but novices would like to be able to create convincing sounds as well, without years of practice. This problem will likely limit physical modeling to more specialized synthesizers rather than an all purpose sound module. Wavetable synthesis will not be completely replaced anytime soon.

In the coming years, as technology improves and computers get even faster, this problem with physical modeling may change our entire approach to synthesizers. Imagine being able to buy or download algorithms for a single instrument at a time, allowing you to build your own personal orchestra. You could mix and match physical modeling, wavetable, FM, and any other synthesis techniques you like. If you're a sax player, you may want the highly controllable models for sax and clarinet sounds, but maybe use simpler models for solo brass instruments and wavetables for the brass and string sections. This modular approach will give the end user more satisfaction and ability to experiment.

To Learn More

Although physical modeling has now entered the industry, there is still quite a bit of research being done and there are a number of good references for learning more about the process and building your own models if you like. A number of articles about physical modeling of musical instruments can be found in issues of the Computer Music Journal, in particular Volume 16, number 4, and Volume 20, number 2. This Waveguide Synthesis Tutorial is almost identical to an article that appeared in the former issue of Computer Music Journal. On the web, you can visit Dennis Nomer's Physical Modeling Page which discusses theory as well as current product using the technology. Much of the academic research in this field is being done at the Center for Computer Research in Music and Acoustics (CCRMA) at Stanford University. Their web site does have some related information, software, and references.

References

• Smith, Julius O. "Physical Modeling Using Digital Waveguides." Computer Music Journal 16.4 (1992) :74-91.
• Smith, Julius O. "Physical Modeling Synthesis Update." Computer Music Journal 20.2 (1996): 44-56.
• Sullivan, Charles R. "Extending the Karplus-Strong Algorithm to Synthesize Electric Guitar Timbres with Distortion and Feedback." The Computer Music Journal 14.3 (1990): 26-37.

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