An air core is no core except air.
Cute, but I saw the punch line back with the first bird.
The purpose of a core in a transformer is to ensure as much as possible of the flux from one winding links the other. Also, we want the core to have a very small reluctance, so that the magnetising inductance (sometimes refered to as the mutual inductance) is very large. If it isnt, the transformer will have a large magnetising current i.e. it behaves as a shunt low value inductance, we need this inductance to be very high. Even if the powdered stones in the tube had a low reluctance (maybe they have alot of iron oxides in them?), the problem is that still most of the magnetic circuit is through air, so there is still a high reluctance. You need a closed magnetic circuit with a low reluctance material, or else you will need a very large number of turns on the windings, and they will need to be arranged carefully, to keep the leakage inductance low (flux from one winding that doesnt link with the other...) But what is the application? For low frequency power transformers we want a low leakage reactance and a very high magnetising reactance. For other kinds of transformers we might want a high leakage reactance (say a transformer to be used as a welder, the leakage reactance acts like a series choke to limit the short circuit current...). For broadband high frequency transformers, we want as little leakage inductance as possible, where as low magnetising inductance is not a problem at these frequencies. So usually we make transformers for this application with the two windings as one side of a twisted pair cable, or coax cable, and then have an air core. So how good a transformer core is depends on its application!
