I accidentally found your discussion. Having nothing else better to do I
thought I would make the following remarks -

The 9:1 balun on a 'long wire', on the average, has no effect on what you
call the antenna 'effectiveness'. On receive, you may find the signal
strength marginally better at some random frequencies and marginally worse
at other random frequencies.

4:1 baluns have a similar negligible effect at different sets of random
frequencies with a very slightly smaller overall loss over the whole wide
band from MF to HF.

You may just as well omit a balun altogether. Omission of a balun means zero
balun loss. But loss in a balun is negligible anyway. It just means there is
nothing to be gained by fitting one.

Baluns can be useful in particular frequency bands. But if you are
interested in particular bands then a very simple tuned antenna, a coil or
capacitor, or changing antenna length, is much to be preferred.

Baluns in a receiving application are beneficial only to the bank-balances
of balun manufacturers and salesmen. In other words, don't waste you
hard-earned money!

(PS: The supposed 600-ohm Zo of a random length of wire has very little to
do with it. Concentrate on the exact particular antenna length. Please send
me the money you save.)

And forgive me for the interruption.
----
Reg , G4FGQ