Originally Posted by Mechta
It would seem reasonable that as the air is thinner at altitude that the fan could turn faster.
In general, that is not true. The rpm's of the different spools are 'tied together' by the thermodynamic relationships between the engine components. In other words, changing density affects all spools in the same way. The parameters changing with altitude are pressure, temperature and flight Mach number. How the spool speeds change with these parameters for given throttle setting is governed by the fuel control system (FCU, EEC or FADEC). Older control systems would maintain constant N2, newer systems might for example maintain constant N1 or constant N1/SQRT(T1).