If you can find two potentiometers, I would use them for coarse and fine frequency.
frequency = 100 + analogRead(A0) * 500 + analogRead(A1);
That will give you a range from 100 Hz to 6238 Hz which will easily cover your desired range. The coarse knob will jump by 500's and the fine knob will step by 1 Hz.
You can use tone(pin, frequency); to play the tone and display on Serial Monitor ever couple of seconds.