basic experiments

Here come the results of a first experiment. Notice both, chosen copper plate and rod antenna, have the same static capacitance.

Acceptable linearity is achieved only due to a proper offset between pitch frequency (550 kHz) and reference oscillator. The offset itself only influences the lower audio frequencies. Nobody is surprised about it.

Using the copper plate the upper audio frequency is more than twice higher. Hence we have at least one additional octave more by use of the plate. 

The coefficients of determination R2 are after offset correction better than 0,994 respectively. A generally better linearity is not found in this experiment for rod or for plate especially if offset variation is used. But an heterodynic classical Theremin without offset correction knob makes no sense.

The next ask for my experiments is: how getting linearity in the near distance? All logarithmic pitch diagrams I know always show a steeper gradient here.

"How getting linearity in the near distance?"  - JPascal

For a heterodyning Theremin the only way I know is to use a second LC resonance (series EQ coil) that is offset a bit from the tank resonance.  It's not perfect, and it makes setup quite a bit more difficult, but when tuned correctly it tames the near-field non-linearity somewhat.  If you plan on going this route there are inductance and capacitance ratios that you could investigate for best linearity. The second resonance tends to "fight" the first, so there may be oscillator start-up issues with certain settings and oscillator topologies.  It's a mixed bag.

For a digital Theremin you simply take the negative fractional power of the frequency difference, which in practice gives near perfect linearity over the entire field (it nicely balances near and mid field sensitivity), and is quite easy and non-critical to setup.  Try this, altering the fractional exponent (-0.125 here for starters) for best linearity with your antenna and setup:

  (F_fixed - F_variable) ^ -0.125

It gives a value which is proportional to distance (decreases with hand closeness) so you will have to manipulate the numeric directionality (sign of the slope) to use it for pitch.  As usual, alter F_fixed for best far-field linearity ("null" or in this case infinity).

JPascal

I get a reliable 8 octave range from aluminum plates, the top octave being a bit dicey to manage, the bottom down to 4 CPS which doesn't count for music - not a bottom octave.

The real advantage to plates is the sensitivity to lateral deviations of your hand position is greatly reduced, leaving the proximity of your hand being unencumbered by extraneous motion of the hand.

So in plain english, holding the note steady is easy. Vibrato is optional.

The top two octaves are greatly compressed which means using fine motor control of the fingers rather than the elbow or wrist - as in writing. Which is a moot point, since I know of only one who plays melodically at those annoying frequencies.

And I should stop it.

Some things are not meant to be heard, unless you are a bird. They seemed to enjoy it.

Thanks dewster and ruppertchapelle. The low influence of coupled coils with different resonance frequencies I will describe later. Digital Theremin we should not discuss in this thread wich is dealing with analog technique and basics. 

But how does linearity depend on the moving path of the hand? May this be a way to linearize at near distance? What about this simple thougt here?

Shifting the hand path a bit parallel to the direct way you will stretch the hand to antenna distance, more in the near than at wider distances. The stretching factor is obtained in a very simplyfied assumption by a square root function from pythagoras.

"Digital Theremin we should not discuss in this thread wich is dealing with analog technique and basics."  - JPascal

Fair enough, but I brought it up because that is the mathematical solution you are looking for, whether analog or digital.

"But how does linearity depend on the moving path of the hand? May this be a way to linearize at near distance? What about this simple thougt here?

Shifting the hand path a bit parallel to the direct way you will stretch the hand to antenna distance, more in the near than at wider distances. The stretching factor is obtained in a very simplyfied assumption by a square root function from pythagoras."

If you're suggesting that, to improve near-field linearity, players should not address the antenna directly, then you will probably need to supply them with a "dummy" non-conductive antenna next to the real one as a visual target.  But I don't think they would like that because it would look odd, and things like side-to-side hand movements for vibrato might not work the same, and you are asking them to tightly control two axes of movement rather than one.

But the real questions are:

1. Can the Pythagorean relation actually linearize the near-field?

2. If so, will this hurt the mid-field linearity?

You already have enough data to answer this yourself in a spreadsheet.  Or set it up physically and gather the data.  I (and others) have looked into linearizing a non-EQ Theremin oscillator response and I've found only one that really works (mathematically) but you may be luckier / smarter.  I will say that the presence of the EQ inductor and the secondary resonance it brings makes the analog variant much more difficult to tune, setup, and own.  A variable capacitor right at the antenna is one way to do it (as Theremin did and Bob Moog did with his early designs) but there are actually three resonances you're dealing with so a single tuning "knob" won't cut it.

I tried it once more with my one liter bottle as hand. Such measurements are a lot time expensive especielly if parameter variation is made.  Here is a result with a fixed parallel shift of 6 cm. Linearity is very well but costs than octaves in the audio range.

"The most important difference among the different schools of therminists, aside from some details about the positions of the fingers and vibrato technique, is the way in which we approach the vertical antenna with the right hand", says the spanish thereminist and composer Victor Estrada.

What I want to validate is only the possibility to optimize linear octave distances over certain hand pathes. All thereminists use their own way for this.

That seems like an improvement!  Now to get that same response addressing the antenna head-on...

I think you could have taken your initial head-on data and compensated the distance with a Pythagorean offset instead of doing the physical experiment all over again.

An overall lowering of sensitivity is probably a good thing, as Theremins are way too sensitive (IMO).  I believe an EQ coil will reduce sensitivity as well.

One more thing to consider is the configuration of an original Theremin.  The air core coil is below the antenna, and from the bottom to the top the coil itself progressively becomes more and more sensitive to hand capacitance.  The player addressing the top of the coil + antenna is rather off-center vertically, which could also affect linearity.

I have only done this single experiment. The good matching with the math approach I did not expect. It is a hobby and time is short. Everyone can take a bottle and try it self, if is in doubt about seriousness of results.

Another way to linearize at the near distance while antenna is keeping in the path direction is shown here. I will not do experiments on this. It is only for knowledge. And maybe an explanation for better linearity of some commercial theremins.

Interesting.  Or maybe make the pitch antenna a loop or diamond, with a hole in the middle like the volume side?  I prefer symmetrical antennas.

...btw, it is the preference of a non-artist.  What is located in the pitch arm of the Etherwave pro, EQ coils?  Please give if you can my posts some time to reach other users who are interested in my thoughts.