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I am working on an audio synthesis project for a guitar, and one of the most challenging parts has been finding an equation so the decays of the partials (modes) can roughly match the behavior of real data.

This is the response of a single note plucked, with Hz on the $x$-axis starting from $0 Hz$ on the left. The $y$ axis is time starting from 0 at the bottom.

I at this point need a simple equation that might well represent the shape given.

ResponseResponse Traced

If anyone has any ideas for what type of math might be dictating this behavior, I would be interested as well. It seems there is an inability to sustain vibration as you get very close to 0 Hz, which makes sense. The vibration is most sustained around the 4th partial, which is likely just the point both farthest from the natural low frequency and high frequency damping that occurs. Then above that, it drops off, but at a faster rate initially than an exponential.

Since this is simply following the natural laws of physics, I presume there is some natural or simple equation that can roughly reproduce this and illuminate what is happening mathematically in nature.

The level should eventually progress to $0$ or infinitely close to $0$ as $x$ goes to infinity. In particular it is this rate of drop-off that occurs above the 4th partial that is critical to understand and model and I am struggling to conceptualize mathematically.

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  • $\begingroup$ The 2nd image can be generated by a function of 2 parts. The first part is a line, and the other part is a function similar to 1/x shifted along the x-axis. Given some data points, it is possible to approximate the curve to such a function. $\endgroup$
    – NoChance
    May 25, 2019 at 2:38
  • $\begingroup$ Wow NoChance! You're definitely right. It's $1/x$ for sure. I can't believe I didn't see that. That's primarily what I was hoping for. Thanks. $\endgroup$
    – mike
    May 25, 2019 at 2:42

1 Answer 1

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For the 2nd figure, you have 2 functions like:

$$f(x)= kx$$ where k is a positive constant.

for the second part, you may have something like:

$$f(x)=\frac{p}{x-q}$$ where $p$ and $q$ are constants.

The exact constant values (and the equations) could be approximated if you provide more data.

Example:

Desmos Graph Example

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