Linearity of heterodyning theremins is one of the most overinterpreted technical parameter concerning theremins. Let me explain it a bit why.
Because of the small distance-variable hand capacitance you have an approximately linear curve between the hand capacitance versus the frequency change of the LC pitch oscillator. But this is not the linearity of the audio octaves: the doubling of frequency within an equal distance gap. The hand capacitance itself follows a formula that is different using a rod (1/distance weigthed with 1/natural logarithmus of distance to rod-radius ratio) or a plate (1/distance). Based on this is in the very near of the antenna and also near the nulling point no linearity to expect, physics says.
The collaboration of your hand geometrie, body mass, earth grounding, moving path of the hand, pitch frequency, choose of nulling point and the geometrie of the antenna and coupling to the oscillator have to make the octave behavior linear.
I obtained that most thereministi use a slight bow to the rod antenna, that is good for playing and gets more octaves linear. This right moving path depends on your finger geometry, your own mass and body dimensions (water content).
Important is: geometry of the rod should be not too thin. A thin receiver telescope rod antenna is definitly not usable, if linearity does play a roll. For more basics inclusive the "linearization coil" debate visit my thread basic experiments.
Nice to read that you want to build a theremin. Welcome to the club!