5 blades v 7 v 9
 

Mt propeller have had a 5 blade STC for the Kingair for a while now. In 2016 they tested a 7 blade, and more recently tested a 9 blade.

Why haven;t we seen any 7 bladed STC's made yet. Is there a problem that is not obvious. I have searched and there is no publicly available data.

Is it not simply a tradeoff, the more blades, the more complex/costly the hub?

Dredging my brain for what I learned all those decades ago in aerodynamcs lessons, I recall that when increasing the number of blades on a prop hub i.e. increasing 'propellor solidity', there quickly comes a time when the turbulence off of one blade impinges closely on the following blade etc, and aerodynamic efficiency starts to reduce.

Modern fluid dynamics computation, and simulation, has very likely allowed designers to very slowly increase the number of blades in a prop design, but there must surely be a point where, even with incredibly fine tolerances in manufacture, where it really is just impossible to (economically) get one over on mother nature!

Best documentary to easy explain props.

The Root Problem
 

As the number of blades increases, the hub centre becomes very cluttered with metal, and not so much air. Also the pitch at the root of the blade needs to be much coarser than at the tip, which is not very efficient at low speeds.
Beyond 80 blades, as in a turbine, the whole root is solid metal, with the blades just occupying a few inches on the outer perimeter. This requires some elaborate ducting and nose cone design to get the air to the right place. Maybe more blades is the way to go, as ultimately in a single engine airplane, the air needs to be moved outwards to the diameter of the fuselage anyway.
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That video certainly brings back a memory from doing a QFI course at CFS, RAF Scampton in the early 1990s! As you say, it is a good summation of props. There is a piece about prop solidity starting around 14:08 into the presentation.

Didn't some versions of the two-stroke powered "Silent" ULM self launch motor glider have just one prop? It was touted as more efficient than a conventional two blade prop. It also had the advantage that the hole in the fuselage (where the engine retracted) could be smaller than that required by a two blade prop.
It looked very odd with one bade and a counter-weight!