Building a Theremin

[i]"I imagine that heterodyning two waveforms that vary slightly in different ways from being perfect sine waves (i.e. produced by different sorts of oscillators) will tend to produce a harmonically rich lower band."[/i]

Hi Gordon,
Alas, the above has one fatal flaw which I spent many hours completely baffled over, before I "saw the light".. And when I did, I was quite annoyed with myself!
Because the mixer is a ring modulator, I thought about it in terms of producing a complex waveform out from sum+difference of input signals.. But .. The results are quite different to what one gets from 'ring modulating' audio -

When heterodyning two high frequency signals, almost all the complex stuff happens above 20kHz, and one can only get harmonics in the output (audio) waveform which are present in BOTH high frequency input waveforms.. What this boils down to, is that the audio output will only contain (at best) the harmonics of the purest HF input waveform.

Remember - it is only the sub-20kHz Difference components which will be heard, unlike with audio ring modulation, where all the sum and difference components in 20Hz-20kHz are heard.. See TW waveforms Thread (http://www.thereminworld.com/forum.asp?cmd=p&T=3850&F=3) for full details.

In fact - the way to get maximum harmonics from the output is to mix two HF waveforms with identical waveshapes![b] The output waveform will then be the same as the HF waveform(s)... EDIT -> This is not true![/b] One of these details which is quite counter-intuitive (but Theremins have plenty of oddities which can do ones head in!)

Thank you Fred. I'll go away and think about that a while, 'cos my experience with my ring-mod and low-pass is still telling me it should work. Hmm.

Thinks...

Those low notes in Iron Sun, they're difference, not sum (and one of the etherwave's signals is degraded by a pitch shifter) and distinctly different to the original timbre from the etherwave, or the pitch shifted version (which is unimpressive by itself.) Without looking at the waveforms, (or listening again), I think you might be right. It's less brassy. Plumper.

Do you happen to know of a nice interactive animation on the web that illustrates a ring-mod in action?

Gordon,
I do not think your “understanding of heterodyning is incomplete “ – Its just that, like me, you have “hands on” experience with audio ring modulation – It’s the jump from thinking about audio frequencies being mixed, to thinking about frequencies of 200kHz and 201kHz being mixed which throws a spanner in the ol’ brain!

“Those low notes in Iron Sun, they're difference, not sum”… And this is the source of your confusion! You are thinking audio.. and your thinking is correct in this context…

The real beauty of heterodyning, and one which I only realized recently after Moog 201 discussion on LevNet, is that the spacing between harmonics is huge! For example.. Reference oscillator at 210kHz.. Its 2nd harmonic sits at 420kHz … Pitch oscillator at 209kHz, its 2nd harmonic sits at 418kHz .. Mix them and one gets difference frequencies at 1kHz (Fundamentals) and 2kHz (2nd harmonics) … All the sum frequencies are discarded..

If, in the above example, the reference oscillator has harmonics extending above the 2nd (630kHz,840kHz etc..) but the pitch oscillator does not, then all output components from these additional harmonics in the reference oscillator (both sum and difference) will be high frequency, and lost in the filter..

I think the conceptual problem is that we tend to think about the difference frequency between the reference and pitch oscillators as being what we are mixing.. This leads to all sorts of wrong ideas.. If we are mixing 100k and 101k, we get 1k out.. the 2nd harmonics in the source signals will be at 200k and 202k - 3rd harmonic at 300k and 303k.. etc.. This is one reason why one must take some care in the pre-mixer design.. If one has a reference oscillator sitting at 500kHz, then the higher harmonics soon get up into the Mhz (6Mhz for 12th harmonic), and it is easy to lose the higher harmonics (through stray capacitances on the circuit board, for example) before one gets to the mixer.

I do not know of any animation - but it wont help understanding Theremin Heterodyning.. One reason for this is that the filter throws away most of what is produced.. Heterodyning ~200kHz signals and filtering everything above 20k is like ring modulating ~1kHz signals and filtering away everything above 100Hz! - You would find the waveform complexity / harmonic content greatly reduced!

That makes sense.

Perhaps not "incomplete" - I possibly have all the pieces of the jigsaw, just haven't fitted them together the right way to reveal the simple picture yet. These things take time. It took me three years to find a simple metaphor for the pitch field's behaviour that I was satisfied with...

(Note to self - ring-mod makes more sense thinking about frequencies than pitches - harmonic partials are evenly spaced in frequency.)

If you consider the typical AM radio, the top half of the combined waveform is chopped off by a simple rectifier circuit ... which is already nonlinear, thus causing extra harmonics.

This rectified signal is then filtered by a simple single-order (one resistor, one capacitor) filter that chops off higher frequencies.

It would seem as simple as setting that corner frequency high enough ... but then again nothing is ever simple :)

Don

[i]"It would seem as simple as setting that corner frequency high enough ... but then again nothing is ever simple :)"[/i]

Hi Don,
I thought along similar lines at first - but nothing worked (no matter what I did, or where the filter cut-off was placed, I could not get differences in the mixed waveforms to make any difference - the output was always the lowest harmonic combination possible from the waveforms) - That is when I sat down and pretended I had never seen a ring modulator, and worked it out from first principles...
Harmonics (http://www.therasynth.com/assets/images/harmonics.GIF) is a snapshot of my calculations - simplified down to a few harmonics and their interaction within the mixer.. It can be seen that the resultant frequencies cannot be used for audio waveshaping - Only a few of the difference frequencies can be used (these are highlighted in yellow) and these correspond to harmonic components available in both waveforms only - additional harmonics present on a single input waveform cannot be 'recovered' by any means I can think of.

CONTINUED HERE (http://www.thereminworld.com/forum.asp?cmd=p&T=3906&F=1)