basic experiments

Yes, theremins with a “synchronization range”, where both frequencies coincide to a common frequency and no beat occurs, have a particular change in frequencies and timbre near this range:

When two coupled oscillators approach the synchronization region, their natural frequencies change with respect to the effects of coupling and detuning. The oscillators adjust their frequencies closer to each other, whereby the extent of this adjustment is influenced by the coupling strength.

If the detuning remains within the synchronization range, the oscillators synchronize to a common frequency. If the detuning exceeds this range, the oscillators oscillate more and more at their decoupled frequencies.

In the range, in which the oscillators "fight against each other" to synchronize the phases, the timbre also changes. Presumably, this change in the modulation waveform is not yet described with a mathematical background.

If the coupling is inductive and too strong, which was certainly the case with the old RCA theremin, the synchronization range can be partially or completely reduced with some capacitive coupling. I think this is how the Clara Rockmore RCA was improved by Lev Termen.  

Next, I will post some simulation results for both effects.

"I wonder if ILYA could incorporate coupling into his simulator?"

"Give me a lever long enough and a fulcrum on which to place it, and I shall move the world.”

Give me the dependence of the  "delta f" (frequency correction on coupling) on the frequency difference f[sub]FPO[/sub]-f[sub]VPO[/sub], and I will incorporate it into an application.

"Give me the dependence of the  "delta f" (frequency correction on coupling) on the frequency difference"  - ILYA

Aye, there's the rub.  It would depend on the complex impedance of the pickoff points of the oscillators (acting as both input and output), and the complex impedance of the coupling, and the complex impedance of the mixer load(s), and the operating point, and... so it would seem to be highly situational and non-linear?

The waveform distortion seems perhaps similar to FM synthesis?

I’ve been thinking about this, and it seems to me that a canonical hyperbola (specifically the portion in the first quadrant) could represent the functional dependence. I’m not claiming 100% accuracy, but it looks both elegant and is easy to calculate.

The x-axis represents the frequency difference between the free-running oscillators, and the y-axis represents the frequency difference considering the ‘pulling’ effect.

Section ‘a’ defines the region of full oscillator phase-locking.

Oh yes, that looks really elegant! Could you explain your thoughts in more detail? What are the physical equations behind this assumption? How can the frequency change in an exchange interaction of coupled harmonic oscillators be solved?