If current per Hz is constant, a doubling of current gives a tone one octave up, because it will double the frequency.
This is independent of what current gives what frequency, i.e. what cap/resistor values are used. Those are only used to set the base frequency/frequency at a given current.
The filters I've researched do not have a quite constant current/Hz ratio. It is however flat enough to use this fact even though the actual cutoff will be slightly off.
(and by frequency I mean the cutoff frequency, where attenuation is 3dB).