When one refers to the act of modifying a physical object so as to make it better at absorbing sound vibrations, is that "damping" or "dampening" the object? I've seen both, and looking them up in the dictionary they appear to be more or less interchangeable.
Strictly speaking it's damp - OED: to stifle, choke, extinguish; to dull, deaden (fire, sound, etc.)
As a child, it was my job to damp the fire (9600 hits in Google Books) every night by closing off the stove's air supply. If I did it right, next morning I'd just add more coal and open the air vents. If not, I'd have to clean everything out and relight it with paper and kindling.
According to OED, damp/dampen have a common origin - which is somewhat uncertain, but the key concept seems to be smoke, dust, vapour, steam. Today, damp is more closely associated with moisture/water, whereas dampen goes more with stifle/extinguish.
It's worth noting that dampen the fire gets 5310 GB hits (i.e. - the "incorrect" usage occurs relatively more often with fire than with sound). Partly that's because some people think in terms of adding dampness/water, rather than taking away air, to slow down a fire. But partly it's because they're essentially the same word anyway, so neither is really "wrong".
I believe when you are referring to sound you should use "damping" or "damper". The two sound similar, but "damping" should always be used in the context of sound, gas or fire.
Damping: to check the vibration or oscillation of (as a string or voltage) (Merriam-Webster Dictionary)
Whereas "dampening" has a less scientific meaning:
The verb technically appears to be to damp. FumbleFingers notes the OED's definition. M-W and the Random House Dictionary (at Dictionary.com) under dampen merely refer the reader to the relevant definition of damp. The things in a piano are called dampers, not dampeners. In a more formal context, damp is the proper choice.
Dampen does seem to enjoy use. On the Wikipedia page for damping (music), there are instances of dampening. NOAD lists as a subsidiary definition, "reduce the amplitude of (a sound source)." M-W and Random House both acknowledge that it has the same meaning in music as to damp. In an informal context, dampen is fine.
In my own vocabulary, damp is an adjective and dampen is a verb. If I want a wet cloth to wipe the kitchen table, I do not damp it; I dampen it.