basic experiments

This would be a great thing for a heterodyning theremin what you describe about the hammond system, Daniel. 
I have only a vague idea of how the non integer overtones can be obtained by multiplying and dividing the pitch sine frequencies. Instruments like a trombone have also overtones at 1.25f, 1.5f. Theremins rather not as far as I know.

Creating integer sinusoidal overtones and adjusting their amplitudes separately is certainly a possible way. I think I red somewhere that Lev Termen experimented with it. But it could sound a bit static and equal over the pitch range, the illusion of an natural tone may be lost. 

A good theremin can have similarities to the human voice. Means singing vowels with open mouth or vowels with more closed mouth? Here are overtones of sung vowels at 220 Hz.

Now let's compare theremin with voice. Here are three Rachmaninoff's Vocalise snippets: Clara Rockmore, soprano, modern theremin. What we can tell is that the first overtones are always similar. The soprano voice has more intense overtones of higher order.

To analyze the voice character, for the time being I turn to the roots. Most of theremin enthusiasts know the interesting homepage rcatheremin.com. It can be found there an oscillogram of the RCA theremin sound. How to get this signal formed by heterodyning? I tried to create a similar one...

Good stuff JPascal

If my journey is similar to the original theremin then it was developed around a sound Lev Sergeyevich heard while experimenting. Was it possible to control this sound and how good could it sound musically?

In my scope display the top waveform is at the detector where I develop the throaty aspect by a broad low side of the sine wave and the skew in the top wave shape creates the vocal upper register. In the RCA the buzzy sound comes from the waveform sitting on the bottom rail.

Here is my sound byte, half way in I flip in the upper wave shape or skew by switching in an audio transformer for proper wave shape. I applied this technique to both solid state and tube theremin designs. The vacuum tube "not" heard is slightly better with the vocal sound as the sine wave fat side roll over can be even slower.

A sound that is musical exists in a very narrow window. Always get a sound sample of any theremin you have interest as a beautiful sound is much easier to play for several reasons.

Christopher

Hwy79.com

Thanks for the flowers! There is a significant difference to the rca in your signal form, Christopher. The reason is the extrem different mixing and rectifying method. In the rca two about 80 volts sinus rf signals goes to the grids g1 and g2 of a tetrode. Two limiting effects occour: upper at the anode working on the transformer and down by falling below the negative grid biases.  The result is a rectangular beat frequency, smoothed by the transformer inductance as low pass filter. 

Modern theremins are less brutal with the mixing method. Mostly the signal amplitude of the pitch fixed and variable oscillators are in an equal amplitude high. The diode and the filter works as envelope detector. Than you get a signal like your upper ones.

Now you can use a limiting stage to cut the peak and compress the signal a bit. So you get your Clara sound.   
If the amplitude ratios are 1:5 and lower, you get a sinusodial envelope form.

That mean, by variing the limits and the amplitude ratio the timbre changes strong. Thats all of the secrets I see.  

Here is the deepest tone with an overtone-rich timbre of an experimental theremin-circuit of mine: 
https://www.youtube.com/watch?v=rDTin-4VBtw

Audio sequence of the bass range: 
https://soundcloud.com/jp-channel-981239111

I looked for records of singers preparing their vocales a capella before performing or rehearsing. The following example shows a soprano singing vowel A with glissando on the pitch scale. Formants are obviously important if you will imitate a voice. Overtones are much more dominant (with respect to the fundamental one) than I know from all the theremin records.
 

JPascal, beautiful diagrams!  So the bright vertical areas correspond to the vocal formants?  They seem quite constant regardless of the singer's pitch. 

What software are you using to generate these?

Eric, the free software used is from a radio amateur's website: http://www.weaksignals.com.

Explanation of the formants:
The formants (resonance maxima) can be identified by the frequency-fixed bright stripes in the waterfall diagram. They arise in the human throat as resonant body. 

A very simple model I found herefore is as followes. For a woman, this "resonance tube" is about 15 cm long, the begin of this tube is open and the end closed. The maxima of loudness you will get by "standing waves" in the tube, while the lips are open. This is by frequencies calculated over 1/4, 3/4, 5/4, etc. multiplied with speed of sound divided by the throat length.

In this example these maxima would occur at odd multiples of 330/4/0.15/sec =  550 Hz, 1650 Hz, 2750 Hz,...
But all the theory is gray. More than a single resonance room is within the mouth and larynx. 

I think, for the theremin timbre formants have to be taken into account.