Just a warning - don't use anything other than a plastic tuning tool to adjust L5 or L6. If you use a metal Allen tool you run the risk of cracking the ferrite cores, and this becomes a bigger problem.
If you haven't already taken the cover off the case, remove the bottom screws and lift it off. Since you are uncertain about the age you will be able to identify if the inductors are Coilcraft or Toko (older) brand. If they are Coilcraft you can order a tuning tool (order a couple) from them at https://www.coilcraft.com/en-us/tuningtools/. This may be the same for the Toko coils - I don't know. Don't proceed until you have this or some other equivalent plastic tuning tool (If you know the size you can order generic tools from other sources like Mouser as well)
Also you should be aware that the cover and the screws have a large effect on tuning, and any evaluation of your progress should be made with the cover in place and the screws at least started in their holes (don't thread them all the way in with every change - you'll wear them out). And any changes to either L5 or L6 should be made in small increments and your changes noted each time so that you can get back to where you started if necessary. If you have two tuning tools you can put little tape flags on each and just leave them in the inductors throughout the tuning process to keep track of your position.
If the theremin is really out of whack you can (entirely at your own risk!) check it out using the following procedure that I found a long time ago. Keep in mind that this procedure involves the use of a drilled cover, and with a stock cover you will have to tweak and replace the cover several times. For a rough starting point when tuning with the cover off, try adjusting L6 for a zero beat with the flat back of your hand about 4.5" from the pitch antenna (assuming that the L5 setting is okay as defined in the following procedure.
Installing any of the many phone apps available to measure pitch or audio frequencies will aid in determining the top end pitch.
Here is one procedure (source unknown) that I have used when I didn't have any test equipment available:
Volume Side:Set the volume knob on the control panel to 3 o'clock. Stand as far to the right (pitch side) of the unit as possible, so you are away from the volume antenna. With the unit powered on and hooked up to amplification, reach over with the trimmer tool and start turning L11 through its range. Somewhere in the middle, you should hear a sound start to be audible, reach a maximum loudness, and then die off again. Turn the lug back to the point where the sound is loudest. This should put it in the correct range, to where the sound is silent when your hand approaches the antenna and them becomes louder as you draw away; additionally the Volume knob should be set correctly such that it controls not the actual volume but rather the hardness or softness of the volume curve; meaning how quickly the sound reaches maximum loudness as you draw your hand away. At the clockwise extreme it should have a brighter, sharper attack and at counterclockwise the volume should increase smoothly and slowly as you draw your hand away from the volume antenna.
Pitch Side:
Tuning the pitch circuit is really an art in itself. Roughly speaking, L5 controls the "top end" or the highest pitch you hear when you are touching the pitch antenna, and L6 controls the range, or how far from the antenna the zero point (zero beat, or silence) is located. I perform the tuning using a special wooden cabinet top with holes drilled above the variable inductors, because the presence or absence of the top influences the adjustments. It is more difficult when you have to perform the tuning with the cabinet top completely removed, and then listen to it again with the top in place to see if the tuning is still correct. Generally I find the top seems to influence the pitch downwards from what you hear with the top removed; if so in tuning it helps to "tune high" by a bit and then set the top in place to see if it falls into range. Here is my procedure.
First, listen to see if the pitch goes higher or lower as you draw your hand away from the pitch antenna. If it goes higher, adjust L5 so that the pitch descends through the zero point and then starts rising again; now it should be in the right direction. The next step is to grasp the pitch antenna and adjust L5 so that the frequency you hear is in the neighborhood of 3.8 kHz. L5 and L6 interact, so there will be a decent amount of back-and-forth between the two adjustments. Once you have the top end around 3.8 kHz, move your hand away and see where the zero point is located. It will likely be too close (too short a scale range); to adjust, stand at arm's length from the pitch antenna and reach over from the left to adjust L6. You want to adjust it so that the zero point is about an arm's length from the pitch antenna. Generally this involves turning the lug in L6 in the same direction as you adjusted L5 to get the top end.
NOTE: if the top is off the unit, you actually want to hold your right hand just about at the front edge of the wooden cabinet (~3 inches away from the pitch antenna) and adjust L6 until zero beat falls where your hand is, only 3 inches from the antenna. In my experience, this translates to about an arm's length when the top is put back on. The top kind of "stretches" the pitch field response by a large factor, which is part of why this whole tuning business is so difficult.
The first time you dial in zero beat, it will probably drive the top end higher than you wanted it to be, so go back to grasping the pitch antenna and turn L5 in the appropriate direction to get back in the neighborhood of 3.8kHz. Notice which direction it went (higher or lower) as a result of setting L6; and overshoot in the appropriate (opposite) direction to cut down on the number of times you have to go back and forth between the two adjustments.
