A Virtual Stringed Instrument With Haptic Feedback

Comp 259 Final Project Proposal

When plucked, struck, or otherwise excited, a string is given an initial displacement, velocity, or acceleration -- usually expressed as a function over its length.
The string behaves according to the wave equation, a 2nd-order PDE.
Numerical solution of this by standard methods requires very dense sampling, as the sample spacing determines the highest frequency produced by the system.
The computational inefficiency of these simulations is one of the reasons this is not typically done in modern synthesizers.

In 1987 Julius Smith introduced the Digital Waveguide model.
Assuming linearity and time-invariance, the wave equation solution becomes very straightforward.
The initial excitation wave separates into two components, traveling in opposite directions. These wave components go back and forth along the length of the string, reflecting off the terminations.
Can be modeled as a DSP circuit.

Karjalainen, Valimaki, and Tolonen (1998) recently developed a model based on the digital waveguide that has a simplified implementation and is designed in terms of more high-level control parameters.

They have also looked into relaxing the linear, time-invarant constraint.
I will create a string model that is similar to theirs
At the least, I will implement the simple LTI case. Time-permitting, I may try to model the tension modulation non-linearity caused by string elongation.

Several strings (4 to 6).
User is presented with a fretboard (probably use discrete frets, like a guitar).
One-handed control, for now.
Plucking the string on a fret is equivalent to holding the string at that location and performing the pluck somewhere along that string (probably at a predetermined location?).
Note: There is some necessary inconsistency here in order to do both fretting and plucking with the same hand:

The user is given a model consisting only of strings over a freboard.
The excitation from the fretboard model is reinterpreted as an input to the string model which produces the sound.
Haptic feedback from the string model must be reinterpreted back to the fretboard model that the user interacts with.

I will use the Phantom to interface with the string model, providing force feedback for a realistic feel.
The pick-string interaction should have some physical basis but I will try to simplify it as much as possible without losing its important nuances.
Allow different plucking types, based on angle of the pick, velocity, etc.
Provide a realistic pluck feel.
Allow post-excitation control. In real instruments the user may bend (change string tension), slide (change string length), and adjust pressure of the fretting (change the damping at the fret-finger terminus). Efficient models assume time-invariance, so this might be hard.

Having the haptic model physically resemble a real instrument may expedite the learning process for experienced instrumentalists.
But, I believe that the haptic model doesn't need to resemble any real instrument. This justifies much simplification of the pick-string interaction.
I think it is sufficient to merely do the following, regardless of physical basis:

Create a sufficiently complex system, whose behavior depends strongly upon its excitation.
Determine which parameters of the excitation make the most difference to the system's behavior.
Give the user direct control of these parameters in an intuitive way.
Provide feedback (sonic, haptic, ...) about the behavior of the system.

An experienced musician (or any artist) should be able to develop a "feel" for any virtual instrument with enough practice, even if it does not resemble any real existing instrument.
Simulating a real instrument is only a first step. I hope to allow the user additional control that he or she would not have with a real instrument (e.g. adjusting the shape or resonance properties of the instrument while playing).
Abstracting the idea of an instrument, one can replace the vibrating string portion of the system with other physical simulations, such as pressure waves in an air column (thus creating a guitar that acts like a flute). Even simulations outside the musical domain can be used: E.g., Somehow convert the string pluck inputs into seed-material or conductance variations in an ice growth simulation, and somehow extract a waveform from its output. (I admit, that's kind of far-fetched, but the point is that virtual instruments allow unlimited abstraction in the sound generation process.)