There are only two times when a 450 ohm feedline will actually
present a 450 ohm to the tuner:
1) when the antenna presents a 450 ohm impedance, or
2) when the feedline loss is very high (over 10dB)
In most amateur installations, the 450 ohm line is not matched to
the antenna, but is used with a high SWR. This works fine, and it
is an efficient way to get power to the antenna, but the impedance
seen by the tuner is, in most cases, not very close to 450 ohms.
If you are planning to use the balun at the output of a tuner for
matching balanced loads, the 4 : 1 is probably prefered over the
9 : 1 also because most tuners are more efficient when matching
higher impedance loads. (This is a particular problem with small
commercial "T" tuners which often try to claim 160m coverage using
240pf variable capacitors. You might be able to tune the SWR to
1 : 1, but still be dissipating half your power inside your tuner.)
For example, say you are using an 80m dipole fed with 450 ohm
line for an all-band antenna. If the feedline is a half wave on 80m,
then the impedance at the feed end of the feedline might be 50
ohms on 80m and 2000 ohms on 40m. With a 4 : 1 balun, the tuner
actually would see 12.5 ohms and 500 ohms (we have to make
some assumptions here about the balun behavior, since it wasn't
designed for any of these impedances.) With a 9 : 1 balun, the
tuner loads would be 5.5 ohms and 220 ohms (making even larger
assumptions about the behavior of the balun at high SWR levels.)
In this case you would be better off using a 1 : 1 balun in 80m and
a higher ratio balun on 40m, an option offered by few, if any,
commercial tuners. (This example is more extreme than most ham
installations, but made the math easy to do in my head. My mental
processor doesn't handle complex impedances well!)
If you are thinking of putting the balun at the feedpoint of a Windom
antenna, I would recommend you review W4RNL's analysis of
"Off Center Fed" dipoles on his web site at
www.cebik.com.
If I recall, the impedance is not nearly as constant on different
bands as early estimated had thought.
Good luck! - Dale WB6BYU