Dr_Dx,
if it was easy people would not be tuning for resonance through coax,
its that line of thinking over simplifying a complex subject that has gotten us in this mess to start with,
taking half an idea "resonance" which is a good thing and making our own laws up about how it works,
you don't have to trawl through Cebik's reflections, it tells you in the user manual about coax shifting resonance & people still don't get it,
even jumping on folk that are doing it right telling them they are doing it wrong when the opposite is true,
so it can't be easy for many people
1/2wave of coax is 2x 1/4wave multiple,
we are not talking odd 1/4wave multiples that invert a mismatched impedance looking from the rig end,
its any 1/4wave multiples for resonance but not for impedance,
1/2wave tuned for the frequency of choice will repeat the antennas impedance at the input when its matched to the coax AND when its not matched to the coax,
none 1/2wave multiples will transform the impedance to something other than the antennas impedance whenever the antenna is mismatched to the coax,
odd 1/4waves multiples will invert the impedance whenever the antenna is mismatched to the coax,
BUT we are not talking about impedance,
people are tuning for resonance through coax thinking its showing the antennas resonant frequency which its not, you are measuring system resonance coax & antenna,
and that causes a worse match between coax and antenna with higher reflect and the antenna is not resonant,
1/2wave multiple is OK for impedance at your chosen frequency, but you still don't know where the antenna is resonant,
if its not resonant on the same frequency as the coax is cut for exact 1/4wave multiples the coax cancels antenna reactance making X=0 somewhere other than where the antenna is resonant.
And your back to where you started with antenna resonance not where you just tuned it to and that causes higher reflect & lower efficiency than if you tuned for minimum vswr,
since minimum vswr on the line occurs at the self resonant frequency of the element its feeding and adding reactance to the antenna to move resonance seen at the rig end causes higher reflect.
hope this helps.