Another fascinating video. I added "spectrogram" to my TODO list.

Fun! I was my Ph.D. advisor's TA for "Signal Analysis and Transform Methods" back in the mid-late 1980's. I also love Bessel functions.

I ran into J(s), the Bessel functions. I thought I had left the planet for a few weeks, but then something clicked, and a new world was opened before me. This was a class named Advanced Engineering Math and for the first time in my engineering student career, I earned an A in a math class. I did a lot of Control theory in my Masters program. I'll look at your recommendation. That first FM bass sound was the best - a nice Moog Mini bass, but the dual FM harmonic a few tries later had promise. It sounded like a good Vth. Too many modulation cycles and it gets very brassy, too. The "alternative" never sounded good. Thank you. I dont have time to read your code now, but I hope to next year, after I get taxes filed and get my life ready for some intense spine surgery and rehab afterwards. Again, I enjoyed this.

Euler lives! e^iθ = cos(θ) + i sin(θ). The imaginary sines and cosines are just the hyperbolic functions cosh and sinh. I love Bessel functions.

Thank you, Prof. Lanterman. This series on FM variants is amazing!

Nice! Thanks for the introduction. It would be really interesting to hear this applied in context with the kind of instrument patches that you're more likely to use in a real song, to get a comparison of what's possible in modified FM vs regular FM.

Nice video! So it seems that the end result is basically that you are multiplying a (co)sine wave with f(t) = exp(-a) * exp(a*cos(om*t)) = exp(a*(cos(om*t)-1)). When a is no too small, what you get is basically a periodically repeated gaussian, or a spike train convoluted by a gaussian. (Only the part close to the peak matters, and there the (co)sine wave looks like a quadratic function). In the frequency domain, that's a gaussian spectrum multiplied by a spike train (so that only the harmonics remain). Multiplying by a sine wave shifts the spectrum up and down, basically moving the peak of the gaussian in the spectrum (and creating some interference between the copy that moved up and down). This kind of approximation to a gaussian pulse train definitely has its uses in sound synthesis. A nice thing is that gaussians are solutions to the diffusion equation. You can imagine a string that is initiatially plucked with a spike. The spike bounces back and forth, while the damping coould be described by a diffusion equation (pretty much what Karplus-Strong does if I remember correctly, but numerically, and with different initial conditions). If you change a so that the variance of the gaussian (e g measured as the inverse of the second derivative at the peak) increases linearly over time, you should get some kind of natural damping.

Fun stuff Aaron. Definitely curious what the sonic differences would be with the modified FM in multiple Modulator configurations. Setting various Modulator indices and ratios to leverage the standard Bessel function behaviors is one of the subtle arts of (standard) FM.

The modified FM Timbre is really much smoother to control, Thank you for introducing us! But it is also kind of lacking in the harmonics - from a musical point of view at least - is there a way to remove the "bessel-swirl" that does not remove as much spectral content?

Now rotate them in real time.

man this is cool

something else this reminded me of, phase shifting sines thru min(max(a, b), -min(a, b)), a continuous xor, has a bit of a fm sound imo, but different. in surge, i implemented it and 8 others part of a generalized system of describing limited gradient two input scale invariant functions, extending what i listed and min, max, etc. the one i listed is the lowest order unless you consider f(x, y)=0 to be valid, four that are one order higher, and another four that are the next higher order, except it is missing simpler stuff like min(x, 0), x, and anything that could be achieved by swapping and/or negating inputs isn’t there. however surge doesn’t have phase shift on oscillators which i think is dumb but slightly detuning two sines will at least sweep thru phases. i think i had made something modulating the phase of sines thru the simplest 5 modes (i only implemented the following 4 in surge) and used it to be the synth parts for a sega genesis midi. later i can try to find the relevant formulas tho i will double check with what’s in surge, after i find where it is in the surge code nowadays. i haven’t done anything with it in a while but i came up with a system for categorizing and describing two input functions, how to do so for three inputs broke my brain when i tried lol.

HELL YEAH! GO TECH :D

hey Aaron, I think you really missed an opportunity there. at 4:20 (nice) where you replaced a with -ja, rather than make it a constant (a or -ja), make it a point that constantly rotates on the circle of radius a around the origin. It will start out at a+0j, then continue counter clockwise to ja, -a, and -ja. If you do that, I think you could get some interesting results if the frequency of how quickly that number goes around the circle is really slow, like 0.2 hz or something. so again, in that formula at 4:20, instead of having -ja, you would have a * j^(2*pi*fj) where fj is a new parameter like fc or fm. would you mind terribly trying it out and recording what you get? you're basically morphing between standard and "modified" FM synthesis.

Excellent exploration Aaron.. I think that these 'semi - 'failed' experiments are so interesting. I have so many tweaks I'd like to try in order to understand more, like why the index shifts so much between the two methods. I've got a request: I have always been fascinated by the dynamic tonal difference between Yamaha's implementation of FM and NED's implementation, especially the sorts of things that happen when the modulation index changes slowly as in the NED factory Timbre "Dr Floyd" as heard in the soundtrack to the 2001 a space odyssey sequel "2010", so maybe you could have a look / listen? The Synergy on the other hand sounds nearly identical to the Yamaha method, might be worth a 'comparo' video? Thanks again.

I think in all these graphs I'm hearing changes above 3000 that aren't visible on the graph. The spectrogram suggests the sound mostly isn't changing, but ears say otherwise. At my age I trust my eyes more than my ears, usually, though...

Basicly all the music from Eye of the Beholder.

🤯🤯🤯

it's interesting how with the modified FM and the lack of oscillating bands you can much more clearly hear what i presume is your editing software compensating for the decreasing amplitude in discrete stages

I was thinking, a series on the Electro-Smith Daisy Seed would be phenomenal. It's a great tool for DSP.

To oversimplify, the normalized modified Bessel In are less bumpy than the Jn? Also, the In result from using an imaginary argument in the Bessel equation? Just my understanding at the moment -not really asking. Well explained.

Very interesting experiments. How about feedback? How does it sound in this scheme?

at 11:35, weird I feel an old "prince of persia" video game vibe

i'm too important to think here much but it sounds like there may be an implicit unfitness for inharmonic ratios, sounds like cancellation is just faster/all over the place. thanls for reminding folks you can just put stuff in an audio loop and try it and get all kinds of insight into dsp. rofl dude is asking you to try stuff next "30 years later"

can someone explain why digital FM implementations use phase modulation ?

Aaron -- is there a quick record function in Octave/MATLAB to capture synthesized sounds? File type specification / setting paths, etc.