A first approximation is not terribly difficult; folks who run sound systems or mix music one way or another have always had to deal with this.
You'll notice from one of those loudness curves that there's a strong dip in required input between about 2kHz and 5kHz. Inverted, that becomes an annoyance graph. In practice, prominent amplitudes in the 2kHz to 3kHz region are the most annoying. They are harsh, shrill, nails-on-chalkboard, baby-crying, and so on. I use the word "bitey" and make a sort of grabby claw form with my fingers, and then I say "in your face."
I'm assuming a fairly full band of harmonics in this region; you can be quite annoying with a single tone, but the nature of hearing and typical sounds, the harmonics of "ordinary" waveforms at "ordinary" frequencies, tend to fill in the upper part of the spectrum quite effectively. Not always.
In music especially, if you resolve the spectrum into a few interesting bands, you can generalize the impact those sounds have. It's generally "too much does this", and in this case, we have shrill and harsh. Annoying. I'd also say "cheap", because low-quality speaker systems and such have an abundance of these frequencies, with little to balance against. And "out front", "in your face."
At the other end for a band, "not enough does this," and in this case, the sound recedes, moves into the background, has no presence, sounds hollow, lost, easily overwhelmed by other sounds mixed in (if we're dealing with a mix).
Somewhere in the middle is "just right," enough of the... lower high end, or upper midrange, to have presence and... visibility, I guess I might say, but not so much to become annoying.
I think the perceptions in a musical mix transfer quite well to non-musical contexts. That particular gear train, or elevator chime, or cheap speaker, or whatever.
Personally, the next most annoying factor for me, at least in music, is ANY strong resonance in a context where a nice balance is expected. I think this becomes more psychological than physical; a resonance will introduce a noticeable constant-pitch tone that is just too strong and doesn't belong there. It's as if someone bangs a hollow log or whatever, whenever the excitation sounds pass across that frequency. Maybe "distracting" would be a better word than "annoying" in some if these cases.
So I suppose the tools are an FFT and some goals across the spectrum.