Why is a wave with high FM aperiodic? I was playing with sound synthesis in a program I wrote and I had a wave of the form $\sin(2\cdot\pi\cdot(f_c+\sin(2\cdot\pi\cdot f_m \cdot t)) \cdot t) $ So, just simple frequency modulation. When I had a small $f_m$ it was just a simple periodic vibrato, but when I had a large one, say $f_m = f_c$, the timbre kept changing and making really interesting things. I could hear harmonics come and go over time. I figured that at every cycle (for $f_m = f_c$ ), the frequency would return to be the same, so it would be repeating overall the same waveform. (since $\sin( (blah)\cdot(t))=\sin( (blah)\cdot(t+1/f)) $ and "blah" is always the same every "period". Why would small modulating frequencies make periodic signals and not large ones? What is wrong with my reasoning? and mainly why are sinusoids modulated with a high frequency aperiodic?

If we take out the $2 \pi$ for simplicity's sake and rescale time such that $f_m/f_c=r$ and $f_c=1$, then we are basically considering

$$s(t)=\sin((1+\sin(rt))t) = \sin(t+t\sin(rt))$$

Now consider $r=1$ and going forward in time by $2 \pi$:

$$s(t+2 \pi)=\sin(t+2 \pi + (t+2\pi) \sin(t+2\pi)) \\\ = \sin(t + (t+2 \pi) \sin(t+2 \pi)) \\\ = \sin(t + (t + 2 \pi) \sin(t)).$$

This will be the same as $\sin(t + t \sin(t))$ provided $2 \pi \sin(t)$ is an integer multiple of $2 \pi$, which will happen for some nice cases like $t=0$ and $t=\pi/2$. But it won't be when, say, $t=\pi/6$: for $t=\pi/6$ you are effectively shifting the argument of the outer $\sin$ by $\pi$, which will change the value. Indeed $s(\pi/6)=-s(13\pi/6)$. You can see this in a plot.