Here we see a transformer core with no gap on one side and a large gap on the other side.
Short review of inductance. In general what we call inductance could be called self inductance, it is the magnetic fields at work in the coil producing them. The other inductance is mutual inductance it is when two or more coils interact. We only have inductance when the field is in motion. If we apply a DC current the inductance will be in effect as the field builds and will cease to function once the field reaches full strength. Removing the DC allows the field to collapse and we see inductance as it falls. So in order to maintain mutual inductance between our coils we must not allow the coil to saturate the core. In a power transformer an air gap is put in the core to prevent saturation. Gaping cures one problem but creates another, flux leakage.
I went with 61 material and found the cores for 23 cents in the smaller size and 33 cents in the largest. I think 43 material would be a good choice and hope to try some later.
I am using the 3T to 9T based on a mini circuit transformer.