The purpose of coils and linearity of the instrument

Hello,

For the last few days I literally became obsessed with this instrument. I tried to construct two simple, pitch only, digital implementations of theremin but there are several issues I cannot solve myself:

1. How is a real theremin supposed to react on change of distance of my hand to the antenna? Should it be linear (i.e. each 1 inch closer raises frequency by 1 semitone) or exponential? Currently I have it more or less exponential as the frequency in my circuit is 1/(R(C1+C2)) where C1 is antenna and hand capacitance and C2 is fixed capacitance that tunes the oscillator and antenna capacitance increases exponentially with decrease of distance to the hand.

2. What is the purpose of coils in a real theremin (i.e. the original instrument or Moog theremin)? Are they supposed to make it more linear somehow? Do they increase antenna range? Could someone explain this to me?

The first build I tried was my own idea that turns out was already tried: 2 RC oscillators on NAND gates with Schmitt trigger - one has antenna in parallel with the capacitor and they are mixed with XNOR gate and "demodulated" on second order low pass filter.

The second I built comes from this website, the built comes from cover story from Hands On Electronics 1987/09 and it's based on a hex inverter chip (fixed oscillator) and phase locked loop with controlled capacitance by antenna. The second build has significantly lower "detection range" and sensitivity than the first one.

"1. How is a real theremin supposed to react on change of distance of my hand to the antenna? Should it be linear (i.e. each 1 inch closer raises frequency by 1 semitone) or exponential?" - aeth

It should be linear, at least in the mid-field.  With LC it is, and it usually cramps up near the antenna.  Open / closed fist is usually roughly around an octave.

2. What is the purpose of coils in a real theremin (i.e. the original instrument or Moog theremin)? Are they supposed to make it more linear somehow? Do they increase antenna range? Could someone explain this to me?

Coils do several things that you really want.  

1) When heterodyned the pitch of an LC oscillator is roughly linear in the mid-field.

2) An LC oscillator can be designed to give a big voltage boost at the antenna, and this is directly related to the Q.  If the Q is 50 at resonance and you drive the tank with the proper phase at 1V you can get up to 50V at the antenna.  High voltage swing tends to swamp external RF, giving you better signal to noise (SNR).  Higher Q also increases selectivity.

3) An LC oscillator is much more stable than an RC oscillator.

4) If you use a coil in series with the antenna you can make the near-field pitch response more linear, but you have to be careful how you do this, and tuning becomes more complex because there are 3 resonances (variable LC oscillator tank, LC formed by series coil and antenna/hand, and fixed LC oscillator).

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Even though you're using digital logic for your oscillators you're using it linearly, so you're making an analog rather than a digital Theremin.

I see you're interested in theory, so I suppose to read:

http://www.gaudi.ch/OpenTheremin/index.php?option=com_content&view=article&id=60&Itemid=68   different oscillator circuits by Urs Gaudenz

http://www.dogstar.dantimax.dk/theremin/thersens.htm"    ON THEREMIN SENSITIVITY   by Fred Nachbaur
https://www.mikrocontroller.net/topic/257639    (in german)
https://www.mikrocontroller.net/attachment/167876/Vackar_wholepaper.pdf
https://www.mikrocontroller.net/attachment/142911/VA3DIW__Vackar_VFO__Vackar_oscillator.pdf
http://www.qsl.net/va3iul/Very_Low_Phase_Noise_VFO/Very_Low_Phase_Noise_Vackar_VFO.pdf

http://www.dominikdeak.com/index.php?page=theremin

> Should it be linear (i.e. each 1 inch closer raises frequency by 1 semitone) or exponential?

Pitch should be linear (i.e. e.g. an octave should be the same hand movement independent of antenna distance), but frequency is an exponential function of pitch.