Notes that are heard differently

Is there a name to this phenomenon where a note is heard differently than its objective frequency. I don't mean something like F vs E# but something like playing an F note but within certain situations that F becomes as if its F# even though its F as a frequency.

I think it could happen with microtonal intervals as well but these intervals are smaller than a half step. I don't know if its something like an illusion or if it's related to timbre.