Merak,
Respectfully, you're conflating different aspects of measurement. If you had a two-way loudspeaker, you might have the signals to the respective drivers each attenuated 3 dB at the crossover frequency, and the electrical signal would (mostly) sum back to 0 dB to achieve a flat response over the frequency range, so in that sense -3 dB equals half. If you could drive those speakers to only a certain volume level with your 10 Watt amplifier but wanted to double that volume, you'd need 100 Watts to do so; but we're not talking signal level now, but rather power level. Now -10 dB equals half in terms of sound
intensity.
I admit now that although I'd thought I had a good recollection of the concepts, I was mistaken earlier. Sound
pressure is measured in such a way that -20 dB equals a halving.
We're concerned with sound intensity here, I believe, and for that, 10 dB steps are required to double or halve the results.
Section "2" of
this audio FAQ should help make all this clear as mud.
This Web page doesn't directly get into all the differences, but it succinctly states the rule of thumb regarding relative sound intensity levels.
This Web Page has a neat chart, and mentions the relationships between 3 dB and 10 dB increments at the bottom of the page.
Finally,
here is a Java demo where you can select different levels of a 250 Hz tone. There is a reference level available, and one each of -3, -6, -10, and -20 dB from which you can select to play over your computer speakers.
(oh, there are [or at least historically where] two different tapers available for potentiometers: linear and audio [logarithmic])
Glen