Asasine

Generates audio and visual data from the solution of the Kuramoto-Sivashinsky (KS) partial differential equation

$$\frac{\partial u}{\partial t} + \frac{\partial^2 u}{\partial x^2} + \frac{\partial^4 u}{\partial x^4} + \frac{1}{2} \Big(\frac{\partial u}{\partial x}\Big)^2 = 0 .$$

The visual component is a heatmap plot of the solution function $u(t, x)$, while the audio component is the solution mapped onto a frequency spectrum that is written to an audio buffer at each time $t$. This audio-visual feed can be streamed in real time or rendered to a file.

The sonification process is implemented as follows: At each time step, from the KS solution vector u a collection of values is picked via an index vector freqidx with corresponding frequencies freqs. The solution values determine the sine wave amplitudes at the respective frequencies. Here, much of the sound characteristics and creative freedom resides in the choice of freqs. Additionally, we apply a mapping ampmap that further modifies the amplitudes of the produced sine waves; we often times use monomials of odd order, e.g., ampmap = a -> a^5 to sharpen the features of the produced frequency spectrum and still retain negative values ($u(t, x)$ takes on both negative and positive real values). In simplified (code) terms, we generate the audio buffer at each time step via:

buf = zeros(buflen)
for (i, ν) in zip(freqidx, freqs)
    buf .+= ampmap(u[freqidx[i]]) .* sin.(2pi * buflen * ν)
end

In order to create an interesting stereo image, the KS solution is translated to a stereo buffer by mapping all negative and positive values of u to the left and right channel, respectively.

For more details, check the notes in the introductory notebook as well as the provided examples.

Sample outputs

L128_1.mp4

L256_1.mp4

L128_2.mp4

L128_2.mp4

L32_1.mp4

References

The idea for this project was sparked by a blog post series by John Carlos Baez [1] on the Kuramoto-Sivashinsky equation. The CNAB2 Julia implementation was largely inspired by Mathab Lak's code [2], which is also used in Ref. [1].

[1] John Carlos Baez, The Kuramoto–Sivashinsky Equation
[2] Mathab Lak, Test case for PDEs: Kuramoto-Sivashinsky

Todo

• add interactive elements
  • fix button: fixes U[x,t] slice and halts evolution
  • restart button
• generate audio buffer via IFFT for improved performance
• investigate divergence of CNAB2 stepping for certain wave number orderings