I wasn't clear about the photos posted.
This photo series is an EFHW. It is built from old parts of other bygone antennas.
I agree with the choice for an 5/8 or .64 being to use only a tapped inductor, or some other matching system that doesn't utilize the capacitor. I know no other way to match a self-reliant 1/2 wave fed from the end.
The latter, I believe.
The coils are boh inductors, but in my efforts to match a EFHW I have found the coils spun to the lengths/turns/diameters useful on a .625/.64 antenna to be useless, meaning they would not give me the necessary inductor length to find a SWR low point. Even when I lengthened the inductor some, the larger diameters, 3"+, were also less useable than those from 1-1/4" to 2" x 12/13 wraps by roughly 1:3.5 diameter to length ratios. Admittedly this could be because my network was/is inherently flawed in other ways.
I have always used the standard formulas for determining 1/2 and 5/8 wavelength dimensions. I have also found that there were limits to tuning the antennas to the exact point of resonance I was looking for by inductor tap alone, esp after I got the antenna analyzer. To that point, I have seen the math based formulas others more educated than I have used to determine lengths for an exact center frequency, and they are not always the same length. My conclusion has been that resonance for a given frequency is the product of length, and the matching network is about the impedance transformation between the antenna and the feed line. I never considered it a matter of debate (how could I being so young at all this although so old in years), but one of understanding what is being affected by which part of the system one is working with.
Because of my perspective on this, when making and matching an EFHW using the parallel cap/inductor matching network I have proceeded by setting the length to around 17.5' with room to move up or down, put the tap on the coil in the center of the inductor, and adjusted the cap to the best SWR reading I could get. I would then proceed to re-tap the coil for the best SWR, followed by readjusting the cap and inductor tap a few times until I had gotten as good as I could. I would then either go out with the length of the radiator, or in with the length as necessary until the x=0 was as near to my center frequency as I could get it, hopefully with R=50 following (not always so good for me).
So, resonance is a function of length, impedance of the transforming/matching network.
Coil for 5/8 and/or .64:
Coil for EFHW matcher:
