The prop has less resistance in thinner air. The resistance reduction is air mass related (keeping it simple). Engine power also is air mass related. (again keeping it simple). The balance between the two is not disturbed if mechanical losses are considered constant (smallish error), so rpm stay exactly the same, even though horsepower and propeller thrust change with altitude.
If tuned pipes are used, the tune changes due to change in compression ratio and exhaust gas temperatures. This also is not considered in my previous statement and swept under the rug by stating all else being equal.