Illustration: Mike Cruz

Analog synthesis can make interesting sounds, and sample-based synthesis has its charms, but here's another approach: What if you were to analyze the acoustic behavior of a few of your favorite instruments and then express all of that information as a complex set of physics equations? You'd have to consider the interaction of bow and string, reed and wood, and mouthpiece and brass. You'd also have to account for every conceivable performance technique that your favorite virtuosos have at their disposal. Then, you would solve those equations in real time in response to input from your favorite MIDI controller and sit back and enjoy the beautiful sounds that result.

Insane? Not really. That is the aim of physical-modeling (PM) synthesis, and though it is a bit of a miracle, it's actually not a new concept. The first commercially available PM synthesizer was Yamaha's VL1, which was introduced in 1993, and PM's roots date back more than a century.

PM synthesis exists at the nexus of several fields of science. It involves converting into physics equations the acoustician's observations on what makes instruments tick. Mathematicians come up with ingenious ways to solve those complex equations, and computer engineers develop the hardware and software to crunch all of those numbers quickly enough to respond immediately to a performer's input. Speaking of input, PM's multiple-variable nature lends itself to the intricacies of wind controllers and other imaginative MIDI devices, bringing one more field of endeavor to the table.

In this column, I'll take a simplified look at the fundamental logic of PM synthesis and explain why it's such a powerful and flexible method. I will spare you the gory details of equations and such, so the math-shy among you can relax.

GETTING EXCITED

To construct a mathematical model of an instrument's behavior, conceptualize what occurs when the instrument is played. For example, when a violin is bowed, the bow is dragged across a string, pulling it away from its resting position until the string's tension overcomes the bow's friction. At that point, the string slips back toward its resting position until the bow's friction overcomes the string's tension and starts pulling it again. This pull-and-slip cycle causes the string to vibrate, inducing vibration in the instrument's body. That vibration is the source of the violin's sound.

FIG. 1: This shows the four basic components of a physical model. The exciter initiates a vibration in the resonator. As the vibration passes through the instrument, it may encounter various impedances at points of transition. The shape and material of the instrument may also filter the vibration, dampening certain modes of vibration.

In PM terms, the bow scraping across the string is an exciter, and the violin's body is called a resonator (see Fig. 1). The exciter-resonator interaction is the core ingredient in PM, and it's easy to see how the idea applies to other instruments. A drum resonates in response to the excitation provided by a stick or a mallet striking the head, and a guitar resonates when a string is excited by a plectrum (a scientist would lose all credibility by calling it a one-syllable name such as pick).

Any fiddle player knows there's more to a good violin sound than a bow and a string, and your physical model continues with a look at the rest of the instrument. When the string vibrates against the bridge, it passes some of its energy on to the bridge and then to the violin body. The bridge's density causes it to vibrate less readily than the string — it has a greater impedance — so some vibration is reflected back to the string instead of being passed on. For the same reason, only part of the vibration is transmitted from bridge to body, affecting the way the vibration propagates.

Last, but certainly not least, the body of the violin has its own acoustic properties, and instead of merely amplifying the string's sound, the body also acts as a filter, shaping the tone of the string. The body's shape and size — along with the position, size, and geometry of the f holes — determine which frequencies are enhanced or attenuated, thereby determining the character of the filtering effect.

A SAXY MODEL

Now I'll take a closer look at how the four components — exciter, resonator, impedance, and filter — can be applied to model a saxophone. The excitation is initiated by the player blowing into the mouthpiece. That rush of air creates an area of lower pressure along the top surface of the reed. The low pressure pulls the reed toward the tip of the mouthpiece, restricting the airflow into the mouthpiece and allowing the pressure difference between the top and bottom surfaces of the reed to equalize. The tension in the reed from being bent is then able to pull the reed back toward its original position, where its tension is relieved and the airflow is great enough to start the process all over again. This tug-of-war is the source of the reed's vibration.

Therefore, the physical model of a saxophone starts by borrowing equations from aerodynamics to evaluate the force bending the reed toward the tip of the mouthpiece. Next, turn to structural engineering to quantify how strongly the reed resists being bent. Now calculate the amount of air that passes through the mouthpiece with each oscillation of the reed, and you can evaluate the intensity of the excitation.

Consider the number of variables in the reed model even before it interacts with the resonator. As the player blows harder, the reed deflects farther before it snaps back and lets more air pass through the mouthpiece, resulting in a more intense excitation. Sax players know that reeds of different stiffnesses respond differently to a given air pressure. (Saxophonists also realize that if you start trying to model what makes a reed “good” or “bad,” everyone will go mad.)

Clearly, the complexity of any musical instrument defies comprehensive description. It's true that the more thoroughly you can model an instrument, the more realistic your synthesized version will be, but some degree of compromise is necessary. You can settle for a less thorough model, or you can find shortcuts in the calculations. Of course, limiting what behaviors you model can affect a virtual instrument's sound and playability.

FEEDBACK FROM A SAX?

The reed's vibration excites the resonator, the column of air contained within the body of the saxophone. As the player closes and opens different combinations of keys, the saxophone effectively changes length. The longer the column of air, the lower the fundamental frequency at which it resonates.

Think back to the exciter for a moment. If the reed's vibration excites the column of air to resonate but the resonant frequency of the column changes with its length, shouldn't the vibration frequency of the reed change correspondingly? After all, it doesn't make sense for the exciter to vibrate at one frequency and the resonator to vibrate at an unrelated frequency, does it?

FIG. 2: Physical models demonstrate either coupled or decoupled exciter-resonator pairs. A coupled exciter-resonator pair exhibits a feedback relationship, in which the resonator influences the behavior of the exciter. A decoupled pair follows a feedforward structure, with no such effect on the exciter.

In fact, the vibration of the air column is the primary factor in determining the vibratory frequency of the reed. This is an example of a feedback relationship, inasmuch as the resonator's response to excitation manages to influence the behavior of the exciter (see Fig. 2). This type of coupled exciter-resonator pair is typical of wind instruments and bowed strings, in which an excitation is continuous and the resonance is periodic.

In contrast, if you strike a drumhead with a stick, the head doesn't feed any information back to the stick beyond bouncing it away. That is called a feedforward relationship or a decoupled exciter-resonator pair. Similarly, if you pluck a guitar string, the string doesn't have any opportunity to influence the behavior of the pick.

An additional aspect of the exciter-resonator relationship is that resonators are generally linear in their behavior, whereas exciters are generally nonlinear. If you apply more force to the resonating air column inside the sax, it responds with proportionately more output, or volume. A nonlinear exciter such as the saxophone reed, however, reaches a point at which additional air pressure causes it to close off completely, muting the sound, or to vibrate in an unpredictable way, causing a squeak.

RESISTANCE IS FUTILE

Just as the intersection of a violin's bridge and strings presents an impedance to the exciter's vibrations, so do changes in the diameter of a saxophone's body (or any wind instrument). However, a saxophone's conical shape results in an essentially infinite series of minute impedance changes rather than the single large impedance presented by a violin bridge (or a sudden, abrupt change in a wind instrument's diameter). The tonal effect of those tiny continuous changes in impedance is greater resonant support for even and odd harmonics. By contrast, the clarinet is essentially cylindrical with a closed mouthpiece and open bell, and its tonal spectrum consists primarily of odd harmonics.

In addition, the bends in the neck and bell of a saxophone also affect its tone. Calculating the impedance of a saxophone, therefore, requires computing a continuously changing set of boundaries as well as the effects of several curves in the tube.

FIG. 3: Applied Acoustics'' Tassman is a modular virtual synthesizer built on physical-modeling principles. The Player view (top) and Builder view each show the primary building blocks of a PM synth. Impedance effects are built into the plate and controlled by the Decay knob. Each module features modulation inputs, and various other parameters can be controlled by MIDI.

Once you've created the impedance equations, you can easily build an alto sax that's straightened out like a soprano sax or even build a straight bass sax if you wish, just by subtracting curves from the model. You don't even have to worry about the physical constraints that would make such an instrument unbelievably awkward to play. Among the many attractions of PM is its ability to create those sorts of what-if variations on instruments (see Fig. 3).

Any wind musician knows that the body of an instrument comes alive with the sound of the notes it plays. Whether it's made of wood, brass, or silver, it absorbs some of the energy of the vibrating column of air, transmitting part of that energy to the surrounding air and absorbing some with its own inertia. That filtering effect in the saxophone model is computed by analyzing the energy transfer that dampens the vibration of the air and contributes to the overall timbre identified as a saxophone. At the same time, you must calculate the filtering effects of the sound radiation pattern of the instrument's bell and the rest of the body.

CHOPS TO SPARE

If all of this were only a way to create convincing synthesized versions of acoustic instruments, that might be enough. After all, people build elaborate multisampled instruments that Velocity-switch from a hard to a soft attack. A PM instrument's equations allow its attack to change continuously according to Velocity, a feat that would require 128 Velocity layers in a sampler. Similarly, a PM synth doesn't merely pitch-shift a sample to sound different pitches — it recalculates the exact length of string or tube required to produce that pitch, resulting in a consistent timbre and realistic transitions from note to note.

As a saxophonist, I vary the lip pressure on the reed to bend the pitch and create vibrato. A model of that behavior has only to factor that change in reed position into the pressure/tension equation to create a realistic response to the pressure of my lip on a wind-controller mouthpiece. No more cheesy low-frequency oscillator vibratos!

If a violinist bows nearer the bridge (sul ponticello) or over the fingerboard (sur la touche), the tone changes considerably. If your model properly represents the bow and string relationship, that performance practice can be reproduced properly, whereas a sampler would require two entirely new patches to be created. Similarly, the effect of a trumpet mute can be modeled by modifying impedance and filtering equations instead of starting a whole new program from scratch. That flexibility permits PM to reproduce the wide range of performance techniques employed by acoustic musicians in an easily controlled way.

A physical model can, by its very design, reproduce some of the most human effects of acoustic instruments, from the breathiness that saxophones get at low volumes to the squeaks they produce when overblown. The number of sample layers required to achieve that range of expression is beyond the capabilities of even the best modern samplers.

SONIC HYBRIDS

Some of the most interesting aspects of PM, however, arise when you reach beyond the re-creation of existing instruments and apply a bit of imagination. Have you ever wondered what it would sound like to bow a bassoon? Once you've reduced exciter and resonator to their component equations, who says you can't cross-pollinate them?

Why not build a trombone whose size decreases as its pitch rises? By the time you get into the treble clef, you're playing something more like a trumpet. Speaking of trumpets, the instrument used by Wynton Marsalis is designed to be much more rigid than other trumpets, greatly reducing the energy radiated from anywhere other than the bell for a more intense sound. Why not test that theory by designing a PM trumpet made entirely of granite?

By the time you get around to designing imaginary exciters and resonators, you'll have a palette of amazing complexity and potential. However, that complexity turns out to be PM's biggest limitation. The “classical” approach I've described is elegant in its conception, but it requires solving an enormous set of interrelated equations to create each sound; in fact, it's often called the brute force method. Even with the latest processors, you can't quite pull off PM in that manner in real time.

As a result, several simplified approximations have been developed. The most common approach is called waveguide synthesis, which is employed in currently available PM synthesizers. Another approach is called modal synthesis, which divides complex physical and tonal structures into smaller, more manageable substructures. In any event, PM synthesis opens a whole new world of sonic generation and manipulation for electronic musicians to explore.

Brian Smithers is associate course director of MIDI at Full Sail Real World Education in Winter Park, Florida. You can reach him through his Web site at http://members.aol.com/notebooks1.