Are you sure you want a gearbox at all? My previous testing once suggested that equal speed/torque was best for a friction heater (may have changed in the months since then)
I'm an idiot. I forgot that only the torque doubled with two coils. I even said even torque/speed, but forgot the gearbox was required to achieve it.One coil doesn't get close at 1024/1024. 2 coils at a junction with a 2:1 to speed equals 2048/1024 just about gets there. The 3rd coil got me there.
So the ideal ratio your 3 coils would be 3:1.
I blame Mondays.