Krdytkyu

Impetus: I'm currently working on my first synthesizer, after many years of playing with them. I've just added an ADSR envelope to modify the gain, but for some reason it sounds wrong to me. I've asked around forums whether or not I'm supposed to apply the envelope to the amplitude linearly, and it would appear that's the normal way to do it. So my guess here is that experience with acoustic instruments has conditioned me to hear drop off differently. In order to prove/disprove this theory, I would like an answer to the following...

Question: What is the (approximate) function for amplitude of a plucked string over time? Is it linear, or curved? And, if it's curved, is that curvature a product of the material, tension, or other variables?

Detailed Explanation: In this context, what I mean by "function for amplitude" is the value by which a simple wave would be multiplied over time to create a reasonably approximate rate of fade out. I do realize the actual function of amplitude is probably a very complex physics problem, and while I think it would be interesting to cite some sources in that vein, it is not my intention to create a true to physics simulation of a string.

An exponentially decaying envelope $aexp(-b x)$ is a good choice, and is used for example in vintage Yamaha FM synthesizers. It has the favorable property that over any constant length time interval, by the end of the interval the envelope has decayed to a constant fraction of what it was at the beginning of the interval. Damped oscillation (with some frequency-dependent energy loss) of a string is largely sinusoidal after enough time has passed, and if the oscillation is small enough that the system can be considered linear, then the amplitude loss per time unit will be proportional to the amplitude, giving an exponentially decaying envelope. Closer to the attack these assumptions do not hold.

In the simple exponential decay model, the choice of material could only manifest itself in the decay rate constant $b.$ If it would affect also $a,$ the player could adjust the plucking strength to compensate, nullifying this effect.

Exponential decay of the amplitude is linear decay in dB scale.

I found a way to answer my own question. By looking at various .wav's of samples of string plucks (https://freesound.org/search/?q=guitar+string), it would appear that amplitude drop off is not linear, which would explain why my adsr release sounds so wrong!

I still can't tell if a harder string would make this curve sharper or more linear, but my guess is sharper. In terms of what I can hear/see, it's very hard to tell the difference between a sharper curve and a shorter release time.

Still, to answer two parts of my own question:

1. The approximate curve appears to be in the range of reciprocal functions (ie 1/(x+1) or 1/(2^x))


2. It is not linear


3. I have yet to answer this question, but my theory is that the curvature is fairly similar amongst materials, else there would be a material with something close to a linear drop off and it wouldn't sound so unnatural.