I drove my car with the instrument cluster out of it and I think I got the same chime you are speaking of. I wanted to watch/feel the speedometer cable while the car was moving. It confirmed the fact that my cable & tranny gear were fine and that my OEM speedometer was wacked. I have since fixed the speedometer, but you wouldn't wanna know or believe how... <g>
I presume the chime in question is a warning of some sort.
Prior to my above experimentation..I had never heard it before and after reinstallation of the gauges it quickly went away.