Mach Number is very simply the ratio of your present TAS to the local speed of sound, thus, Mach Number per se, is directly ralated to TAS.
From the manometric aspect Mach Number is gauged from CAS (i.e. Total Pressure in the Pitot tube minus static) Vs ambient pressure (or Pressure Height), that's how it is gauged, but the end result is the relationship between your current TAS and the local speed of sound.
FYI, the relationship between Mach Number and CAS (IAS) is governed by Static Pressure (Ps, Pressure Height)
ALONE, Ambient Temperature has
NOTHING to do with the relationship between CAS and Mach Number, only Pressure Height. (The same for EAS).
Regards,
Old Smokey