This seems to go against physics...say you have your tenor tuned up and strings at proper tension and fret the 5th fret (which would be about soprano scale), the pitch would be way sharp, so you would have to tune down, decreasing tension on the strings, therefore the strings would be more floppy. right?!
Ok, there are a couple of things here that you've said that are not inline with my own experience, and I am not an expert but will share what I've seen hands-on:
If you capo the 5th fret, or ANY fret, on ANY instrument and you do not check your tuning and if necessary, re-tune after putting on the capo, you are asking for tuning and/or new intonation 'problems'.
Once you put on the capo, you have now changed the tension profile of the strings, in that they STILL have the tension from the nut to the saddle to the tuners, but now ALSO have an additional fulcrum at the capo position for whatever fret is now the new break-point for the pitch of the open strings.
The additional fulcrum at this new break-point will effect the tension-feel under the fingers, as well as the way that the string reacts to being vibrated, as well as changing the angle from the new break-point to the bridge which manifests usually as
either lower
or higher (in some cases) action or string height.
If you are playing movable chord shapes or barre chords without any 'open' strings you may not feel it as much, but it is there, and there is a sort of compromise now in terms of overall tension with multiple fulcrums, each taking a portion of the tension stress of the strings, such that mostly a capo will
feel like LESS tension. There is math to back this up, but I am sure nobody wants to read calculus formulas here on UU. If need be, I can try to find it in my Bookmarks the info from Wikipedia for you.
For the intonation, think about this:
Having the capo on AND then fretting strings, you now have 4 breakpoints for the string:
A)
1-nut
2-saddle
3-capo
4-fretted pitch(es)
It seems obvious to me that this is going to be different from:
B)
1-nut
2-saddle
3-fretted pitch(es)
and also different from:
C)
1-nut
2-saddle
again there is math to back this up, related to scale length, initial tension from nut to saddle, angle of incidence of the string at the break-point, say from C above, and then to B above changing the tension, pitch and intonation, and then to A above changing it even more.
If you ask Jon Moody here on UU who works for GHS Strings, I'm pretty sure that he can explain this better, and will likely agree with me in my laymans understanding of it all.
Secondly, if your instrument is by default SHARP at the 5th fret, say you barre 5555, and all, or some of the strings are SHARP, then you have an intonation problem on that instrument, that has nothing to do with a capo.
The solution is to fix the intonation with either nut compensation, saddle compensation, both of these, or sometimes even strings of a different composition. I have at times had to do ALL 3 of these to have very good intonation up to the 12th fret of +/- 1 cent on all my instruments.
Some folks simply cannot hear poor intonation, they lack the hearing precision to tell the difference, and for these folks, they live in a blissful world and can play an instrument with poor intonation and be happy and not worry about it. They are lucky. I cannot play if the instrument is out of tune with itself or the intonation is off by more than 2 cents - it's like fingernails on a chalkboard to me.
The last part about tuning down and decreasing string tension to offset intonation problems due to the use of a capo - if you are tuning DOWN, how much are you tuning down? In my experience it has never ever ever been more than a 1/4 step or about 25 cents in pitch, and maybe this is because the open string nut-to-saddle intonation on all of my instruments has been adjusted as perfect as possible?
If your open-string intonation nut-to-saddle is out of whack, then using a capo is only going to amplify that problem.
So, why not just buy soprano strings? Do you want more or less tension? Some strings, like Worth offer light and heavy tension options. I would suggest Worth Browns or Fremont blacklines for mellower tone. I like them both on different instruments.
I have tested every single one of the 22 different sets of Worth Browns and Clears in all the gauges they offer. Yes, I have spent a small fortune in strings over the past 3 years because I want a very specific tone and feel under my fingers when I play. I have also tested more than a dozen different other brands of strings, in all the varieties I can find, and yes I have more than a dozen ukes, in all sizes, and all different, but nothing smaller than soprano.
Fremont Blacklines have HARD and MEDIUM tension sets, and have their place on certain instruments, but the 4 different sets of strings I tried all frayed and eventually snapped on me, but I did like the tone and silky feel under the fingers. Too bad that they had such a short life (weeks).
I am not sure if you asking me or Bellgamin? but I have no questions here and have already solved these issues for myself, and am only sharing my own solutions, and not seeking to change anything.
Looking at Living Waters website, they show the same string gauge for the 2nd and 3rd string for soprano through Tenor, the 1st and 4th string gauges change though.
You are forgetting that string diameter is NOT the defining factor in tension and intonation at a given scale length LINEAR DENSITY or molecular weight is in fact the single deciding factor.
Having MORE linear density allows for strings to be THINNER and still achieve the SAME tension and intonation of thicker strings, case in point is FLUORO vs. NYLON strings.
FLUOROcarbon string have at least 30% greater linear density than NYLON (tynex/dupont/etc) strings and therefore can BE thinner and still have the same tension and intonation.
Does that make sense?
However, without knowing the linear density or molecular weight of the material that the string is composed of, as players, or layman, the only measurement we have to use is the string's diameter, and this is ok, IF and only IF you keep in mind that there are other factors in play that are not visible.
Having said the above, knowing the linear density or molecular weight of the material that the string is composed of is only possible if the stringmaker provides that info, and the ONLY stringmaker that I've seen give this info to the public is D'Addario. One can use that info with a chart they provide as a PDF file and using the math formulas they give you, you can take ANY string they make, and determine the pitch range vs. tension for those strings.
OTOH, if you want to avoid all the math, you can use their String Tension Pro web site:
http://stringtensionpro.com/
if you need help with it ^ let me know and I can share how I use it.
Hope this helps ^^^ If not let me know and I will try my best.