Bandgap Reference

Does the bandgap reference really have anything to do with the silicon bandgap voltage?

The basic idea of the bandgap reference is to obtain a low temperature coefficient voltage source by cancelling the negative temperature coefficient of the diode forward voltage with the positive temperature coefficient of differential $V_{be}$.   The difference of the $V_{be}$ of two different size but otherwise identical bipolar transistors is $k T /q \ln R$, where R is the current density ratio, k=1.38e-23, q=1.6e-19.  The diode voltage tempco is about -2mV/C; to match that, $k/q \ln R = 2m$, so differential Vbe = 600mV at the room temp.  And the diode voltage at the room temp is also about 600mV.  So the output is about 1.2V, which they say is close to the silicon bandgap voltage (1.166V at 0K).  But the essential scheme does not really seem to depend on it.