Weierstrass Function Sonification

The Weierstrass Function, and related functions, are functions that are continuous everywhere but differentiable nowhere. The following sound samples have waveforms which are (approximations of) the Weierstrass Function and two related functions with the same property of being continuous, but not differentiable. To learn more about these functions, see the References section.

If you have SageMath on your system, you can generate the function plots and the raw sound data (44.1 kHz, 16-bits, signed, little endian by default) yourself using the two Jupyter Notebooks in the Downloads section. The Notebooks have been authored and tested with SageMath version 10.0.

The sound sample (see Downloads section) features three two-octave scales (each C3—C5), using the following three waveforms, in this order:

First, the original Weierstrass Function:

Then, second, another function of the same kind:

Third, a function that is a “rippled” sine wave:

Detail of the third function graph:

Downloads

• Weierstrass Function Plots (SageMath Jupyter Notebook)
• Weierstrass Sonification (SageMath Jupyter Notebook)
• Three Weierstrass Function sonifications, in AIFF and MP3 formats. Caution: High sound volume!

References

Art. Weierstraß-Funktion, in: Wikipedia (German), access date: 2023-06-07

Art. Weierstrass function, in: Wikipedia (English), access date: 2023-06-08