Low noise digital family
@Andyaka, log would probably make most sense, as in 1V/octave


Here is a design for an exponential (1 volt per octave) voltage to frequency converter from Linear Technology: -



LT page that this circuit came from.

If you need a sine wave then it becomes trickier. The circuit above produces a sawtooth output and of course this can be filtered to make a sine wave but with the ability to shift the frequency you need a filter that can track that frequency. This can be achieved by a voltage controlled filter. Alternatively you can use a PLL to produce a synchronized much-higher frequency and use this to clock a 10th order switched-capacitor low pass filter that does the sawtooth to sine conversion. Not impossible especially as you also have the control voltage to help modify PLL VCO component values thus giving you probably up to 5 octave control.

If you just wanted a linear V to F and didn't care about waveform shape it's much easier.


  I want to build a battery operated low noise audio VCO


You can easily make +/- 15 volt supplies from virtually any battery but what this really comes down to is, how big is your battery and how long do you want it to last?