[DrDavid]

in my example, the 3sin(x) would be much higher pitched and three times as loud as the sin(3x)

Not that it is important, but sin(3x) would have 3 times the frequency of sin(x) and an octave is just double the frequency. I do like the 3 times louder part of your explanation.

[mikegi]

I do like the 3 times louder part of your explanation.

Not to be picky but loudness is logarithmic with amplitude...

[Flashlight]

...Not that it is important, but sin(3x) would have 3 times the frequency of sin(x)...

Darn, I screwed that one up. you're right. good call! and thanks for the correction; and it's important to me anyway :)