Waveshaping Synthesis with Complex Numbers: An Application of Euler's Formula & De Moivre's Theorem

[@worblyhead996]

The bipolar impulse train example is intersting sounding in both examples. Great stuff.

[@algorithminc.8850]

Wow ... seeing that old red Signals and Systems book from the 80's in your thumbnails brings back memories. I look forward to scoping your channel. Subscribed. Cheers.

[@vikenemesh]

Quite inspiring, I love these swirl-free synthesis alternatives! Someone should try to build these techniques into a VCVRack plugin, I think the virtual modular synth community would love this.

[@yashnu]

Nice one. Waveshaping synthesis is fun. Korg O1/W and Kronos have a good set of transfer functions. Kurzweil V.A.S.T. synths have an interesting Shaper block too.

[@bob_mosavo]

Thanks, Prof. Aaron 👍

[@Lantertronics]

You are welcome! :)

[@cheater00]

Strega (you had a thumbnail of it on screen by the end) is such a pretty synthesizer. All those differently sized and irregularly placed knobs make for a unique experience. Beats a typical synth where knobs are in a strict grid.

[@cheater00]

Another lovely video. While using comples wave shaping removes all the glittering, I actually like it - I wish there was a method to *increase* it rather than decrease it!

[@vikenemesh]

Plugging a phaser effect behind a non-glittery synthesizer might produce what you are looking for, with better controllability. The glittering in the spectrum is mostly just a kind of comb filter that emerges from the mathematics behind it, phaser effects with tight delays should be able to emulate this.

[@vikenemesh]

Also: dedicated comb filters with a resonance control are really useful for shaping wide spectra, when a phaser is not quite what you are looking for.

[@Lantertronics]

@@vikenemesh I think the Technics WSA1 may do something along those lines?

[@vikenemesh]

@@Lantertronics Seems like it from the specs: Physical Modelling involving some kind of resonator in the chain reads like it has comb filters. The brass resonator in particular seems very suspicious.

[@brentlehman5028]

4:30 cosine is even, sine is odd. Sawtooth is odd, too. It has even-numbered *harmonics*, but that's something different.

[@Lantertronics]

Oh yeah whoops -- I had a brain fart while recording that. I will put in a correction in the description. Thanks for catching that!

[@phpn99]

I've followed your work for a while and I appreciate the exploratory nature of your passion for synthesis. I've been dabbling into synthesis for more than 40 years and I have a growing problem with synthesis-for-itself. Much of it is disembodied and ultimately feels void of substance. West Coast pundits have talked about Complexity for decades now, but what is complexity, and why is theirs just as thin as anything else ? Hear timbres and ask yourself why we immediately detect naturally-produced sounds, and why even the simplest physical flute sounds more tangible than anything a monster modular can throw at it ? There's been promise in Modal Synthesis, after Karplus-Strong and Chowning's FM, yet we're still not analysing the nature of the timbral depth produced by these methods. Surely, it should be possible to produce such timbres additively or subtractively, and equally importantly, by intelligently varying the timbre through the progress of the loudness envelope and its velocity. We're down the wrong path when we focus on the wave shape ; it's the harmonics and partials that matter, and morphological tricks such as wavefolding are aimless pursuits without a rational correlation to spectral outcomes. There's much ballyhoo about harmonics but the real complexity comes with the fact that only extremely rarely, natural sounds have harmonics that aren't surrounded by a myriad low amplitude partials. This is why modal synthesis bears the most promise, because it starts with a full spectrum, as opposed to, say, sawtooth waves. The next problem is that the noise generators we have are of low quality. Do an FFT on most noise generators and see how poorly distributed their outputs are ; I use a Geiger counter with a radiant source to at least get a truly random trigger source within the sampling frequency of the counter. But even without getting there, analogue electronics shot noise should in principle give you a decent noise source. Most often the problem lies in the effect the whole circuit has on the electronic noise. We need extremely reliable, dense, close to theoretical noise sources. Then we need resonant filters (shelf, notch, bandpass) that track pitch, and which can be tuned automatically to specific harmonics or inharmonic positions. Experiments show that you can do convincing sounds (i.e. truly complex) with about eight resonant bands. But you need extremely smooth (truly dense) noise to start with. Just blow some air through your pouted lips ; it should be straightforward to emulate that sound through synthesis, yet it's not. The problem is the synthetic noise sources we're using are too coarse. Like visual grain has a resolution, spatial distribution and modulation transfer function, so too should our synthetic noise sources.

[@johanwk]

In a modular synth, if the problem is that synthetic noise is too coarse, how about using recordings of naturally occurring noise instead? So, allowing for a "cheat" when it comes to the noise by using a sample player? It would be interesting to hear your opinion.

[@Lantertronics]

I haven't really looked at Modal Synthesis in much depth -- I will need to check it out!

[@Lantertronics]

Is this along the lines of what the Technics WSA-1 does?

[@CHEpachilo]

After all of these videos I start to think about PWM having a complex form which avoids combfilter-like sound. Also "Ch" in Chebyshev should be read as in Che Guevara, or "ch" in chapter.

[@Lantertronics]

Interesting idea!

[@AnalogDude_]

pretty cool sounds. What hi-pass filter? it's usually 10uF with 1M, 100K. (roland.)

[@Lantertronics]

This would be a digital filter.

[@cheater00]

Is it possible to define Chebyshev polynomials of the first kind of fractional order? Eg T_{0.5} or T_{e}. It's possible to define fractional derivatives... Oh yeah, you mentioned Euler's formula right after I asked haha.

[@Lantertronics]

Googling "fractional chebychev polynomials" does get some hits but I haven't looked into it beyond that...

[@cheater00]

@@Lantertronics/videos/videos (sorry about the spurious /videos it's some sort of firefox addon that i need to disable) as you may know, one of my biggest wishes is to create a non-additive (preferably analog) oscillator where the distances between partials in cents are a monotonous, non-linear numeric sequence, perhaps one that approaches a constant in the limit. it also opens the possibility that the partials' frequences themselves approach a constant in the limit, but it is not the main goal here, the main goal is to have a sound with stretched harmonics that doesn't just sound like DSF or a normal analog core oscillator.

[@Lantertronics]

@@cheater00 I think if you want partials that aren't exact multiples of the fundamental frequency you'll inevitably need to have separate oscillators you're adding up.

[@cheater00]

@@Lantertronics/videos i don't see why this would be true. maybe a handful of oscillators, but not a single oscillator per partial, for sure not.

[@vadjuse]

It will be interesting to make synth engine for Metaconformer device with this type of synthesis. How can I contact with you via mail?

[@cheater00]

Chebyshev starts with a hard Ch, like China

[@Lantertronics]

Thanks for the info!

[@cheater00]

@@Lantertronics/videos yw! transliteration of cyrillic names is tricky!

[@AnalogDude_]

13:51 hmmmmm, embarrassing but what does Sigma mean? K = 0, but a power of zero will lead to no where.

[@RonaldoMessina]

it denotes a sum for all values of k from 0 to infinity, and the boils down to the nice closed-form equation after the "=" sign.