Turns sets flux. Air gap sets current flow. The ratio of the two is defined as inductance (1 H == 1 Vs / A). Flux density of course is flux / A_e.
L = N^2 mu_eff mu_0 A_e / l_e
If l_g = 0, mu_eff = mu_r, the relative permeability of the core material itself.
As l_g goes up, mu_eff falls. If l_g were infinite (i.e., no core at all), mu_eff --> 1.
Approximately, mu_eff = l_e / (l_e / mu_r + l_g)
You'll probably find best results with mu_eff around 20-60. This is a practical range for copper wire windings on typical core types.
You may not have many options for commercial ETD (or any other shape) core, in terms of off-the-shelf gapped core sets. Gaps can be custom-ordered, but unless you're buying a pallet of them, you're probably better off adjusting normal (ungapped) cores. The center peg can be ground down with carbide abrasives, or the core can be shimmed. Mind that for shimming, the air gap counts twice: once in the middle and once in the outer legs. So if you need e.g. l_g = 1mm, use 0.5mm shims.
Tim