Cherry Audio Forums

Posted: **Wed Jan 17, 2024 6:14 pm**

Having looked at Pan Law I thought I'd share my recent thoughts on Volume Law.

No doubt we've all at some point coded volume control as...

Code: Select all

output = input * volume

...where volume is a double between 0 and 1.

This obviously works but doesn't take into account how we perceive loudness.

Rather than linear gain we could use an equation in the form...

Code: Select all

output = input * a * exp( volume * b )

For a 60 dB dynamic range this becomes...

Code: Select all

output = input * 0.001 * exp( volume * 6.908 )

Here's a comparison between the linear and exponential curves...

: volumeLaw1.png (37.92 KiB) Viewed 1716 times

As you can see there's a huge difference. In practice this means that the top half of the linear volume control has very little impact on the perceived volume.

But exponential curves are asymptotic, so a volume of zero still lets through a tiny unwanted amount of signal.

One way around this is to smoothly switch over to a linear response below 10% volume using code like this...

Code: Select all

if( volume > 0.1 )
	output = input * 0.001 * Math.exp( volume * 6.908 )
else
	output = input * volume * 0.019953

The transition is shown in close-up below...

: volumeLaw2.png (30.15 KiB) Viewed 1716 times

I scaled the graphic too small but hopefully you can see how the green line neatly takes us down to minus infinity dB at zero volume.

However, there's a reasonable approximation that's far cheaper and automatically goes to zero and that's to use the sxith power of the volume...

Code: Select all

output = input * volume ^ 6

Here are the two basic curves for comparison...

: volumeLaw3.png (33.9 KiB) Viewed 1716 times

The sixth power isn't perfect but it's pretty good and can be coded using just four multiplications as follows...

Code: Select all

double t = volume * volume * volume;
output = input * t * t

If you are doing volume control just at UI speed then the method involving a comparison, potential branch, two multiplications and Math.exp() or pow( e, x ) isn't going to be a worry, but if you need to code exponential VCA's the x^6 approximation might be a winner.

Posted: **Sat Jan 20, 2024 6:01 pm**

I've been doing some real-world testing on this and I now think that the x^6 approximation is a poor choice. In most listening situations it produces a curve that is way too quiet at low volume settings.

My conclusion is that...

Code: Select all

if( volume > 0.1 )
	output = input * 0.001 * Math.exp( volume * 6.908 )
else
	output = input * volume * 0.019953

...is the best code for situations where you want good control over quiet signals when monitoring at fairly high levels in a low-noise environment.

For more everday situations lowering the nominal dynamic range from 60 dB to 50 dB might be a sensible option, so the following code might be the best all-rounder...

Code: Select all

if( volume > 0.1 )
	output = input * 0.00312623 * Math.exp( volume * 5.757 )
else
	output = input * volume * 0.05623755;

Reasonable fast approximations are x^4 or x^3.

If you just want a fast and dirty volume control that's much better than linear then the I'd recommend the following...

Code: Select all

	output = input * volume * volume * volume;

As before this is for volume in the range [0, 1].

Cherry Audio Forums

Volume law

Volume law

Re: Volume law