Old Guitarist

Old guitarist is a physically modeled virtual guitar plugin.

Features

• Wave equation
• Stability condition
• Multiple Strings
• IR Convolution
• Time-varying tension factor
• Hammer-ons
• Slides

Physical Model

Nomenclature

• $\gamma$: damping factor
• $\mu$: linear density
• $E$: Young's modulus
• $h$: Impulse response
• $I$: second moment of area ($\frac{\pi r^4}{4}$ for a cylinder)
• $l$: length
• $T$: tension
• $t$: time
• $x$: position

Wave equation

Wave equation for a guitar string is given as:

$$ \begin{align*} \frac{\partial^2 y}{\partial x^2} - \frac{\mu}{T(t)} \frac{\partial^2 y}{\partial t^2} - \gamma \frac{\partial y}{\partial t} - EI \frac{\partial^4 y}{\partial x^4} = 0 \end{align*} $$

Initial and Boundary Condition

$$ \begin{align*} y(x, 0) = \begin{cases} \frac{4x}{l} & \text{for } 0 \leq x < \frac{l}{4} \\ \frac{4 (l - x)}{3l} & \text{for } \frac{l}{4} \leq x \leq l \end{cases} \end{align*} $$

$$ \begin{align*} y(0, t) = y(l, t) = 0 \end{align*} $$

Finite Difference

Consider the expression given by

$$ \begin{align*} f' = \lim_{{h \to 0}} \frac{{f(x + h) - f(x)}}{h} \end{align*} $$

    In the context of the finite difference method, $h$ is a finite interval, and the difference quotient is used to approximate the derivative. Instead of taking the limit as $h$ approaches zero, a small but finite value of $h$ is chosen.

The wave equation is discretized to obtain the following form:

$$ \begin{align*} \frac{y_{x+1}^{t} - 2y_{x}^{t} + y_{x-1}^{t}}{\Delta x^2} - \frac{\mu}{T(t)}\frac{y_{x}^{t+1} - 2y_{x}^{t} + y_{x}^{t-1}}{\Delta t^2}- \gamma \frac{y_{x}^{t+1} - y_{x}^{t-1}}{2 \Delta t} - EI \frac{y_{x-2}^{t} -4y_{x-1}^{t} + 6y_{x}^{t} - 4y_{x+1}^{t} + y_{x+2}^{t}}{\Delta x^4} = 0 \end{align*} $$

$$ \begin{align*} Spatial \space resolution = \frac{1}{\Delta x} \end{align*} $$

$$ \begin{align*} Temporal \space resolution = \frac{1}{\Delta t} \end{align*} $$

Stability condition

The stability analysis of finite difference schemes when applied to the numerical solution of partial differential equations is intricately tied to the Courant–Friedrichs–Lewy (CFL) condition, expressed as:

$$ \begin{align*} C = \sqrt{\frac{\mu}{T}} \frac{\Delta x}{\Delta t} \leq C_{\text{max}} \end{align*} $$

$$ Spatial \space resolution \leq Temporal \space resolution \times \sqrt{\frac{\mu}{T}} $$

Convolution with Impulse Response

     $y^{t}$ is sampled for $x \in (0, l)$. The tone will vary with respect to x. Additionally, an impulse response of a guitar body is convoluted with $y$ to introduce body resonance.

Impulse response: Source

    Instead of multiplication ($YH$) in frequency domain, same result can be achieved by convolution ($y*h$) in time domain. The formula for convolution is given as:

$$ (y * h)(t) = \int_{-\infty}^{\infty} x(\tau) h(t - \tau) , d\tau $$

Installation

Build from Source

To build Old guitarist from source:

1. Clone the Repository:

   git clone https://github.com/enter-opy/old-guitarist.git
   cd old-guitarist

2. Install Dependencies:

   • Windows: Make sure you have Visual Studio installed with the necessary components for C++ development.
   • Mac: Make sure you have Xcode installed with the command line tools.

3. Build the Plugin:

   Windows:

   • Open the project in Visual Studio.
   • Set the build configuration to Release.
   • Build the project by selecting Build > Build Solution.

   Mac:

   • Open the project in Xcode.
   • Set the scheme to Release.
   • Build the project by selecting Product > Build.

Usage

• Insert Plugin: Load Old guitarist plugin into your preferred digital audio workstation (DAW).
• Play the instrument: Use a MIDI keyboard or your DAW's pianoroll to play the instrument.

Contributing

Contributions to Old guitarist are welcome! If you'd like to contribute, follow these steps:

1. Fork the Repository: Start by forking the Old guitarist repository.
2. Make Changes: Create a new branch, make your changes, and commit them to your branch.
3. Create a Pull Request: Push your changes to your fork and submit a pull request to the original repository.

License

This project is licensed under the GNU General Public License. See the LICENSE for details.

References

• Ka-Wing Ho, Yiu Ling & Chuck-jee Chau (2021). Guitar Virtual Instrument using Physical Modelling with Collision Simulation.
  
• M. Shuppius (2017). Physical modelling of guitar strings.