Shepard tones.

Forgot about those... (Always makes me think of Alan Shepard, in orbit, rising forever.. Wasn't him though, some other Shepard I think). They could be useful. Not sure what the scanning equivalent might be at best, but it's a great idea. FM is sorely underrated, incidentally. You're right about the usual perception, but there are at least two easy departures from form that head right into the best sorts of sound you'll ever hear from a synth. Try carrier 1 to modulator 1 or two as freq ratio for saw and square respectively, use full feedback for modulator to itself. Set 'feed forward' as opposed to output level of modulator into the carrier, slightly less than full, like 6 instead of 7. An SY99 will do this, a DX7 won't. Tweak modulator phase a bit, and you get awesome analog waves. Tweak fairly high output level right, and you'll get nice bandlimited analog waves,
even with no level scaling to correct the mod index for pitch changes, which makes this even better an analog than Yamaha's own AN1X! Single parameter changes can change to saw/square as I mentioned, and another I forget, probably phase setting in carrier, can invert the waveform, so long as key sync is switched on. More widely usable is detune with pairs of equal stacks, using high mod index finely adjusted. very rich wavetable sounds. Bit off-topic now though because those waves, while stronger than most other sounds, are so rich in harmonics that they'll just mess up a scan pattern. fart-sound-land is where we get to if we don't strongly control that mod index.

And I reckon that low mod indices are where it gets interesting for scanners.