Mathematical equation for the sound wave that a piano makes

Question

Is there a mathematical function that can output the sound that a piano makes at different frequencies?

So if the simplest equation for a sound wave (at a given frequency and for a given sample rate) is

$ sin(\frac{2 \cdot \pi \cdot x \cdot f}{samplerate}) $

where f is frequency, is there a similar equation for the sound that a piano makes?

EDIT: I'm not looking for anything too complicated here, just a wave that approximates this estimation. I've been playing around on desmos to see if I can get something similar:

This is what Guest's wave sounds like... not much like a piano.

Answer 1

Piano tones are actually quite complicated. The initial strike is full of non-harmonic tones that quickly dampen down. After that, the harmonics vary from note to note, piano to piano. I suggest you look at some actual sample files, capture a whole number of somewhat consistent waveforms, do a DFT, and read the coefficients from there.

As for the other answer that was given, some one may find this interesting:

$$ \sin^3\left(\pi x\right) + \sin \left( \pi \left( x + \frac{2}{3} \right) \right) $$

$$ = \left( \frac{ e^{i\pi x} - e^{-i\pi x} }{2i} \right)^3 + \sin\left( \pi x+ \frac{2}{3}\pi \right) $$

$$ = \frac{ e^{i3\pi x} - 3 e^{i\pi x} + 3 e^{-i\pi x} - e^{-i3\pi x} }{-8i} + \sin( \pi x ) \cos \left(\frac{2}{3}\pi \right) + \cos( \pi x ) \sin \left(\frac{2}{3}\pi \right) $$

$$ = -\frac{ 1 }{4} \sin(3\pi x) + \frac{ 3 }{4} \sin(\pi x) + \sin( \pi x ) \left(\frac{-1}{2} \right) + \cos( \pi x ) \left(\frac{\sqrt{3}}{2} \right) $$

$$ = -\frac{ 1 }{4} \sin(3\pi x) + \frac{ 1 }{4} \sin(\pi x) + \frac{\sqrt{3}}{2} \cos( \pi x ) $$

Hope this helps,

Ced

Answer 2

The best wave form function I found so far is in this video.

In terms of my experience this is the formula:

Y = sin(2 * pi * frequency * time) * exp(-0.0004 * 2 * pi * frequency * time)

Adding overtones:

Y += sin(2 * 2 * pi * frequency * time) * exp(-0.0004 * 2 * pi * frequency * time) / 2
Y += sin(3 * 2 * pi * frequency * time) * exp(-0.0004 * 2 * pi * frequency * time) / 4
Y += sin(4 * 2 * pi * frequency * time) * exp(-0.0004 * 2 * pi * frequency * time) / 8
Y += sin(5 * 2 * pi * frequency * time) * exp(-0.0004 * 2 * pi * frequency * time) / 16
Y += sin(6 * 2 * pi * frequency * time) * exp(-0.0004 * 2 * pi * frequency * time) / 32

Also you need this to make the sound more saturated:

Y += Y * Y * Y

In the video you will see this last line. The sound quality does not change even without it:

Y *= 1 + 16 * time * exp(-6 * time)