In a good recording of this song, you can hear the harpsichord in the production. In this video, you can just barely hear it starting at 0:48.
You have a good ear. I think it's an unmuted clavinet recorded at half speed so it's an octave higher when payed back. Unnatural harmonics.
Billy Joel did this to the rhythm guitar in "Only the Good Die Young" (or at least I think he did, based on my old time experimenting with reel to reels).