The 40 dB Divide

40 dBSPL is a very interesting sound level. We have known for more than 50 years that, at threshold levels our hearing is nearly linear – like a good microphone, and that at loud levels our hearing is cube-root compressive to allow a huge dynamic range to be heard.

The cube-root behavior gives rise to the notion that it takes a 10 dB increase in sound levels to be perceived as twice as loud. (I have always wondered, ever since finding that fact in the literature, just how does one judge “twice as loud”?)

But for these past 50 years the transition between the linear and cube-root domains has been very foggy. And that transition zone seems to happen around 40 dBSPL, which just so happens also to be the dividing line between sounds that can only be heard in an isolation booth, and the louder every day sounds of real life.

What that transition zone looked like was a total winging in everyone’s mind. Just draw a “smooth curve” connecting those zones…

Well, my research into normal hearing finally provided a glimpse at the actual transition zone:

This graph isn’t a “winging” answer. It is the shape determined by ear physics. And what we see is that 40 dB doesn’t appear to be as magical as we thought. The true transition appears to be down around 20 dBSPL. That’s where the kink in the curve happens. And look how rapidly we make the transition between those two zones.

At the extreme left we see the nearly linear behavior at threshold levels of sound. That’s where our ears are nearly perfect microphones. But so what? No actual sounds in our daily experience are that faint.

Let’s amplify the knowledge of this curve by showing its slope.

That helps us really see the nature of the linearity at threshold, the size of the transition zone at 20 dB, and the nearly cube-root behavior above 40 dB.

Linear behavior, by definition, has a slope here of 1, because these slopes are in the dB domain, and really represent exponents. Linearity always has an exponent of 1.

And cube-root should have a slope of 1/3, or 0.33. And you see that we are really quite close to that, everywhere above 40 dB.

So, one could say that 40 dB is magical because it is at that level, and higher, where our hearing becomes pretty close to cube-root behavior, and prepared for the daily onslaught of loud noises. Is it just an accident of nature that 40 dB was chosen for this?

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Author: dbmcclain

Astrophysicist, spook, musician, Lisp aficionado, deaf guy

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