I was just experimenting with the “classic” feedback FM and stumbled on a related technique that seems quite interesting to me, especially because of its chaotic behavior. It’s actually very simple: The output of an oscillator is delayed and, scaled by some factor, fed back into that same oscillator as its frequency. So there is no external input, no carrier frequency (or a zero carrier frequency if you will), which means that it is a symmetric “through-zero” FM.

feedback-oscillator.pd

By guessing I figured out how the delay time determines the frequency of the oscillator, but I still don’t understand a few things: Why is there a factor of 4 involved so that the delay is one quarter of a cycle? And why is there still some error? I found two ways to fix it, first by oversampling and second by reducing the delay time by the duration of one half sample. To be honest I don’t have a clue why this works ... Any ideas?

What I find particularly interesting is the path from sinusoidal tones at modulation index 1 (which seems to be a critical value) to different modes of periodic oscillation and finally to chaos when the modulation index is increased: First odd harmonics are added, then even harmonics, then a period doubling or “bifurcation” occurs, and shortly after that we get into the chaotic region. This seems to be not too different from the logistic map or related chaotic maps. So I’m wondering if it’s possible to analyze the oscillator in terms of chaos theory to better understand and control its behavior.

Apart from being mathematically interesting, I think this could also be a usable noise generator with spectral control. Other effects can be achieved by adding an external input ... but for now, I’m more interested in understanding that simpler case.

@manuels The Computer Music Tutorial by Curtis Roads has a section on feedback FM it might have some answers. Can't seem to figure out how to get google books to link to the section so here is the search, click on page 244.

Edit: Looks like my link actually worked? or is google being "smart" and relying on my history?

@oid Thanks a lot for the hint! The link works just fine for me too.

Another great resource that I found just recently is Risto Holopainen's website. He investigates many less common variations of FM here, also addressing the topic of chaotic behavior in some of these circuits.

As far as I can see, neither of the authors deals with delayed feedback, though.

I love this feedback oscillator! I don't know how relevant it is, but it reminded me of https://ccrma.stanford.edu/software/snd/snd/fm.html [cascade FM: sin(sin(sin))]:

As z increases above 1.27, we get a square wave, then period doubling, and finally (ca. 1.97) chaos.

@kyro Very interesting, thanks!

Conceptually this “cascade FM” is of course quite different because it’s not a feedback loop. But the point about FM and PM producing energy at 0 Hz and thus altering the pitch seems to be relevant for understanding the behavior of the feedback oscillator too:

My attempt to fix the delay time kind of works in the region where the waveform is symmetric and produces only odd harmonics, that is up to index values of about 2.45. Beyond that not only the symmetry is lost but there is also a drop in pitch (quite audible actually). So maybe it’s the increasing asymmetry that eventually leads to chaos?

BTW: The error I was talking about in the original post is probably better understood as being two samples per period (therefore one half sample per quarter period which is the delay time).