It is a bit of trouble, but I've been buying singles to create sets with brighter types of low strings and thumpier high strings. I usually stay within one brand, and try to keep the materials as consistent as possible. I think my ideal set would be a round 4th, pressurewound 3rd, groundwound 2nd, and flatwound 1st, all stainless steel (or all alloy 52 would be interesting). I guess I kind of like the trouble—feels like mad science and over theorizing (which, tbh, I love).
To add to all the other amateur understanding,
My thinking is that higher tension does equal more harmonics, but also I think larger core diameter means less harmonics. This would mean that sometimes going up a gauge does something easy to understand—if that brand uses the same diameter core for the two different gauges. But if the manufacturer has also increased the core diameter, then the tone change might be more complicated. And across string brands, you can't always tell what the tension will be like (because of different core diameters...as well as density and tension of the wrap, and whatever else).
My understanding is that rounds sound different than flats because of 1) tension differences, 2) the way your finger or pick slide off the surface of the string, and 3) the way the string moves through the magnetic field (flats acting more like an idealized line, and rounds having a more complicated shape that leads to a different harmonic timbre). Tapes, for instance, move through the magnetic field like low tension rounds but have a surface more like flats—so they sound kind of uniquely bright but also thumpy: a unique combination of {low tension means more fundamental, less harmonics} while {roundwound means more character to harmonics} as well as {smooth surface means mellow string noise and more of an idealized pluck leading to more fundamental}.