Is this really 'FM'?

[doctornash]

Hi, I demonstrate a form of sound synthesis I've dubbed 'NHT' (Nested Hyberbolic and Trigonometric functions synthesis) here:
https://www.youtube.com/watch?v=gTOKDwQ97wI
It evolved from simple observations such as:
arcsin(sin(x))*(2/pi) = triangle wave which swings in range (-1,1)
arctan(cot(x))*(2/pi) = sawtooth wave which swings in range (-1,1)
which I then conceptually extended to compound (nested) expressions like these:
sin(atan(2*sin(a*tanh(b*tan(x)))))
sin(8*atan(1/sinh(a*tan(x))))
asin(cos(a*tanh(b*tan((x)))))*(2/pi)
tanh((a*cos(a*atan(b*sin(x)))))
which produce a wide range of interesting, evolving sounds both IMO 'FMish' in character, and otherwise. You can see one of the comments to the video re this technique:

Effectively this IS FM synthesis with operators sequentially modulating the phase of each other. Just you have access to other functions than sin

hmm...am interested in anything you can offer to reasonably explain from a mathematical standpoint the similarity AND the differences between 'traditional FM', phase distortion/modulation, and the above technique

[antto]

my uneducated nose tells me this is something similar to either phase distortion or phase modulation

[Ichad.c]

Looks like normal waveshaping to me.

*Edit* - I call this regular waveshaping because there is only one 'x'. If you substitute the 'a' or 'b' in some of your formulas for another function greater that 0Hz or even x[n-1], I would agree with the phase modulation statements here. If 'a' and 'b' are just 0Hz, it can probably also be called phase-distortion.

[Architeuthis]

Wouldn't those long trig functions be really computationally expensive?

BTW, glad to know you are actively exploring non-traditional synthesis techniques. I wish there were more of you, and also I don't like most synths because they stick to what I call "failed traditions".

[Architeuthis]

adj = twopi/samplerate*frequency;
x += adj;

a is 1 to 50
b is 1 to 1.57

abs(tan(sin(x*a*.5)*b)*1/tan(b))*2-1;


tan(sin(x*a)*b)*1/tan(b);


sin(x*a)/x*(1/a);


a is 0.7212 to 25.7212
sin((x)*a)/x*(1/a);


a is 1.188367 to 16
sin((a*x-x)/2*x+1);

[Hideki]

Except for the additional waveshaping, which can also be seen as Phase Modulation, the above is mostly the same as what Yamaha called FM synthesis, except what Yamaha did was never FM, it was PM (Phase Modulation).Closely related through a derivative but never ever the same thing. I guess the naming ended up this way because the John Chowning research actually used FM with pure sine waves, at least some of the time. Academic theory tried to overwrite reality, and now nearly everyone calls it FM synthesis