I ran across the Clausen function the other day, and when I saw a plot of the function my first thought was that it looks sorta like a sawtooth wave.
I wondered whether it also sounds like a sawtooth wave. More on that shortly.
The Clausen function can be defined in terms of its Fourier series:
The function commonly known as the Clausen function is one of a family of functions, hence the subscript 2. The Clausen functions for all non-negative integers n are defined by replacing 2 with n on both sides of the defining equation.
The Fourier coefficients decay quadratically, as do those of a triangle wave or sawtooth wave, as discussed here. This implies the function Cl2(x) cannot have a continuous derivative. In fact, the derivative of Cl2(x) is infinite at 0. This follows quickly from the integral representation of the function.
The fundamental theorem of calculus shows that the derivative
blows up at 0.
How does it sound?
What does it sound like if we create music with Clausen waves rather than sine waves? I initially thought it sounded harsh, but that turned out to be an artifact of how I’d make the audio file. A reader emailed me a better recording using the first few notes of a famous hymn. It’s a much more pleasant sound than I had expected.
Are you including harmonics above the Nyquist limit?