Maryanne Amacher made some music that relies on the non-linearity of ears, what is known as "otoacoustic emissions." I guess if you put a sound source that produces the sum of two pure sine waves up to a human ear then record what comes back out, it can add in combination tones.
I think it's easiest to perceive in Chorale 1: https://www.youtube.com/watch?v=rtmv6LxNJqs&t=3079s (I'm not sure if I only hear the low-pitched tones because of non-linearities of my amplifier and headphones at higher volumes though...)
It's way too easy to get overtones from a sine wave. Lots of music production involves compression of dynamic range into a smaller interval, but this introduces integer overtones. For a toy example, we can think of arctan as being a compression function that takes infinite dynamic range into the interval [-1,1]. Playing around with some Fourier series, it looks like arctan(sin(x)) has a bunch of extra odd harmonics over the sin(x) fundamental.
In May, there was a HN post about a statistical mechanical model that derived a scale from an overtone model. It would be cool to see what comes out of inharmonicity (like bells).
https://advances.sciencemag.org/content/5/5/eaav8490.full
I think it's easiest to perceive in Chorale 1: https://www.youtube.com/watch?v=rtmv6LxNJqs&t=3079s (I'm not sure if I only hear the low-pitched tones because of non-linearities of my amplifier and headphones at higher volumes though...)
It's way too easy to get overtones from a sine wave. Lots of music production involves compression of dynamic range into a smaller interval, but this introduces integer overtones. For a toy example, we can think of arctan as being a compression function that takes infinite dynamic range into the interval [-1,1]. Playing around with some Fourier series, it looks like arctan(sin(x)) has a bunch of extra odd harmonics over the sin(x) fundamental.
In May, there was a HN post about a statistical mechanical model that derived a scale from an overtone model. It would be cool to see what comes out of inharmonicity (like bells). https://advances.sciencemag.org/content/5/5/eaav8490.full