# Expected decay time of a sound based on its relative volume?

I am working on a synthesizer to model sounds like plucked strings (guitars) and struck membranes (drums). With such instruments, the harder you strike them, the longer the sound takes to decay to an inaudible level.

What equation would allow one to predict the proportionate time decay of such a struck/plucked sound based on its relative volume (dB)?

For example, let's say you set an arbitrary maximum reference strike/pluck of 0 dB to have a decay of 60 seconds. Is there any mathematical function that can predict what the decay time should be for an otherwise identical strike/pluck at a lower strength of, for example -6 dB? Or -12 dB?

If needed, we can define "full decay" as -300 dB, and state that a strike at -300 dB has zero seconds decay.

So let's say:
-300 dB volume = 0 seconds decay
0 dB volume = 60 seconds decay

How do we fill in everything between?

I feel like intuitively there must be an ideal or "correct" way to scale or interpolate this.. Struggling with the math though... Thanks!

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## Answers and Replies

Are you familiar with exponential decay? That's usually what's being used.

Thanks, yes, I am familiar with exponential decay. I am using exponential decay curves to dictate the rates of decay already.

I am establishing decay rates of plucks/strikes based on y=x^a, where a is an arbitrary/subjective positive value to set the shape of the decay curve.

However, this just gives the general shape of the decay. I'm not sure how to use any similar such equation to describe how long it should take for a given weaker pluck/strike to decay to -300 dB compared to the reference 0 dB 60 second decay pluck/strike.

Perhaps there is no "absolute" or "correct" relationship in decay between louder plucks/strikes and weaker ones. If so I can just use another curve and set it arbitrarily. But I do feel like there should be some mathematical equation which can predict the appropriate relationship.

Volume in dB dictates a scale of increasing/decreasing power. Power of a string pluck or membrane strike will dictate the proportionate time required for it to decay to a given level. So there must be some math that can relate these two factors correctly.

Again, given these fixed points:

0 dB volume signal = 60 seconds decay
-300 dB volume signal = 0 seconds decay

Is there any "correct" or "ideal" interpolation curve to fill in between? eg. For a plucked string?

sophiecentaur
Gold Member
2020 Award
0 dB volume signal = 60 seconds decay
-300 dB volume signal = 0 seconds decay
Could you explain this idea please? How does it relate to the exponential decay constant?
Decay can be quantified in powers of e or in powers of 2 (or anything we choose). Reverberation time for a Room is quoted as the time for a sound to decay to -60dB relative to the original sound. dB, of course relates to the Log Base 10.
But I do feel like there should be some mathematical equation which can predict the appropriate relationship.
There is no 'mathematical equation to tell you which is the most 'appropriate' description of decay. It is chosen entirely on the basis of convenience for the particular field of work. That "Reverb Time" gives a good reason for choosing 60dB but it's entirely arbitrary.

PS There is a lot to be said for using e if you want to do calculations with your answers because there are fewer odd constants cropping up in a page of calculations by sticking to e.

Yes, a - 60 db drop is a much more realistic value to use.
I think I understand what the OP is looking for.
OP wants a 60 second decay to a set level (let's say - 60 db) regardless of the initial volume. So the softer the initial sound, the slower the decay rate required . I'm not going to do the math before my morning coffee, but that is the concept.

I think the OP may be confused on how this works in nature. A plucked string does not decay at different rates based on how hard it is plucked, it is just that a softer sound decays below our threshold of hearing sooner. If a sound decays 10 db per second, a 0 db sound decays to - 60 db in 6 seconds. A - 20 db sound decays to - 60 db in just 4 seconds. They decay at the same rate, one becomes inaudible sooner.

Is OP trying to simulate nature, or produce a 'non-natural' sound?

sophiecentaur
Gold Member
2020 Award
I am establishing decay rates of plucks/strikes based on y=x^a, where a is an arbitrary/subjective positive value to set the shape of the decay curve.
The exponential function is here but you would normally want to express it in terms of time t. So
At = A0exp( -t/q) (note the minus sign!!)
is the way to do it, where q is the decay factor or 'time constant". By choosing the right numerical value of q, At can be calculated in half lives. 60dB times or just 1/e time. I think you need to look around google for a number of sites that address exponential decay in Science; there seems to be something you have missed. I am sure you will get it soon enough.

NTL2009
sophiecentaur got the math. But I still think the OP is misapplying the concept, unless they are looking for an "unnatural" sound, where two different (but simultaneous) strikes of different volumes decay such that they become inaudible at the same time. That would require the lower volume sound to decay at a slower rate than the louder sound.

But in nature, they pretty much decay at the same rate. That is determined by the energy absorbing characteristics of the instrument, not how hard they are struck. There may be some minor secondary effects, where high amplitudes are somehow absorbed by different parts of the instrument, but I doubt that amounts to much until you get to very extreme excursions.

sophiecentaur
Gold Member
2020 Award
sophiecentaur got the math. But I still think the OP is misapplying the concept, unless they are looking for an "unnatural" sound, where two different (but simultaneous) strikes of different volumes decay such that they become inaudible at the same time. That would require the lower volume sound to decay at a slower rate than the louder sound.

But in nature, they pretty much decay at the same rate. That is determined by the energy absorbing characteristics of the instrument, not how hard they are struck. There may be some minor secondary effects, where high amplitudes are somehow absorbed by different parts of the instrument, but I doubt that amounts to much until you get to very extreme excursions.
I wonder what (s)he's trying to achieve. The damping constant of the different overtones in all instruments are mostly different. It's not just like turning the volume down over a few seconds. I wouldn't can that a "minor secondary effect"; it's what makes the attack phase different from the decay phase and a major factor in the timbre of many instruments.

The OP needs to be using the equation I put in my last post and use different values of q for each of the components of the sound. Then they need to be combined.
But this is the basis of music synthesisers and it's very specialist (and pretty well guarded by the manufacturers, I should think). The OP needs to get hold of a book that's dedicated to Sound Synthesiser history and design, I think. The problem can't be solved in a few posts here on PF.

I wonder what (s)he's trying to achieve. The damping constant of the different overtones in all instruments are mostly different. It's not just like turning the volume down over a few seconds. I wouldn't can that a "minor secondary effect"; it's what makes the attack phase different from the decay phase and a major factor in the timbre of many instruments....

The OP needs to get hold of a book that's dedicated to Sound Synthesiser history and design, I think. The problem can't be solved in a few posts here on PF.
I agree with the first part, generally the higher harmonics will decay faster than the lower ones, and that has a real effect on the tone of the instrument. But key to this is - what is the OP really looking for? I'm not sure that this effect, regardless of the prominence in any common instrument sound is a big deal for this application. I think OP is just looking at overall perceived volume.

For the second part, I think you may be overthinking this. The OP doesn't seem to have a good grounding on acoustics (no one talks about an acoustic -300 dB in music in real life), so I doubt they are really trying to control for each harmonic, only "sound" and "decay" and "relative volume" were mentioned (the harmonic content will be controlled by a Voltage Controlled Filter in a traditional analog synth model, or by the sound source setting in an FM or additive model). Hopefully they come back to explain, but I do think that in analog synthesizer terms, they are just looking at a dynamic slope setting for the "D" and/or "R" in an ADSR (Attack Decay Sustain Release envelope generator - generally a 0 V to 5 V control voltage with 5 V giving gain=1, and 0 V = max attenuation) feeding a VGA.

Though I doubt that is what they really want, I think they are a bit confused regarding some basic terms, and don't need to anything at all for this case. The ear takes care of it.

Wow, ADSR. My Commodore 64 had those!

tech99
Gold Member
The following web site is very clear and simple. The reverberation time depends on how much absorbing material is in the room, so it is hard to predict. Also how many people! We just start off with a sudden sound, like a pistol shot, and time the sound in decaying by 60dB.
http://www.noisenet.org/Noise_Room Acoustics_Reverb.htm

The following web site is very clear and simple. The reverberation time depends on how much absorbing material is in the room, so it is hard to predict. Also how many people! We just start off with a sudden sound, like a pistol shot, and time the sound in decaying by 60dB.
http://www.noisenet.org/Noise_Room Acoustics_Reverb.htm
But I don't think OP is trying to model reverberation. I think OP is trying to model the decay in energy (perceived volume) of a musical instrument.

But I think they are off the trail. I don't think there is a big difference in decay rates of an instrument struck/plucked hard or soft. The difference in harmonic content can be very different though. Generally, a hard strike results in more higher harmonics.

In a synthesizer with velocity sensing on the keys, a hard strike of the keys will open up the filter, or increase the FM, or some other means to simulate what happens in the instrument.

@mikejm Hello? OP? Are you out there?

I'm curious if any of this is helping you.

CWatters