Typically walworts don't do much filtering. In the worst case they contain a transformer and a bridge rectifier.
If you have access to an oscilloscope you can determine how good of a power supply your walwort is by connecting a load across it. If you use a 680 Ohm, 1/4W resistor, you will be drawing around 15mA ... which is probably more than your minimum theremin draws.
You want to set the 'scope for AC coupling and the lowest amplitude setting. Ideally you want to see a "straight line", at worst case you will see a fairly large sawtooth waveshape.
If you don't have a 'scope, you can just assume the walwort is not that clean and put a 100uF to 1000uF, 25Vdc electrolytic capacitor (make sure the plus sign is towards the positive side of your circuit!) across the battery terminals.
To be safe, you should insert a 1N4001 diode in series between the walwort's "hot" side and the battery + terminal. The banded end of the diode (cathode) should be connected to the + battery terminal (the electrolytic capacitor's plus side should be connected to the diode's cathode) and the other should go to the wallwort. This will keep your theremin safe if you have the terminals reversed on the wallwort! If you have this arrangement, the worst that will happen is that the theremin just won't work with the wrong polarity and will work with correct polarity.
Wrong polarity, no diode = frying the IC(s) in your minimum theremin!
Don
