Exactly how I would do it if I had to make a balanced set?
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My "balanced" set that I ran on my Rick 4002 (yes, "2," not "1"or "3") for decades was Progressives in 45-60-80-105. Now, when I need rounds instead of my tapes, I use the same set, but I have changed the G-45 to a Pressurewound 44, as it has the same wrap wire, just compressed, is not as twangy out of the box, settles in well with good sustain and tone to match the others, and actually matches the tension better for the DAE strings. According to the GHS chart:
G PW 44 - 43.3 lb
D PR 60 - 41.8 lb
A PR 80 - 41.0 lb
E PR 105 - 38 lb
This gives a nice, even progression from slightly stiffer on the G string to make sure it has good tone and sustain, to slightly less on the E string to make sure it has good tone with good overtones and doesn't immediately go thump. Yes, the E string is a little lighter, but I also raise it up just a little bit for the wider string excursion.
REMEMBER: nothing is perfect. Everything is a compromise. If you get a truly matched tension set without changing scale length as on a fanned-fret instrument, you will have a twangy G and thumpy E or B because the G string will be too thin to settle in, and the E or B string will be too stiff from its diameter compared to the scale length to effectively vibrate in its overtones to give good tone. This is nothing new. Eighteenth century harpsichord makers limited their courses because the strings would get too taut in the treble and break, and if they shortened the scale too much, there would be no sustain. Conversely, they used copper or brass for the lowest strings instead of steel because they had to foreshorten the scale in the bass so the strings would actually respond, and the instrument would not be too long to be manageable as it would have been if the true 2:1 octave ratio were retained in string length. This has carried over to modern pianos with wound bass strings, the predecessor to having round wound bass guitar strings.
The original reason for a heavier G string, like on an original Fender 50-100 set, was to keep it from being twangy. The original reason for having more mass on the D string was so that it would tonally balance on a bass with a flat pickup but an arc'ed fingerboard because the signal decreases according to the
square of the distance from the pickup, so the string had more mass comparatively to get the same signal strength out of the pickup. The A string ended up being so thick that it lost articulation, hence the experiments with the "High A" pole pieces on a Precision pickup. And the E string was made significantly lighter than the math would otherwise indicate in order to retain as much flexibility, overtones, and therefore as consistent tone as possible, and could be raised so it was tonally the same distance from the pickup as the rest of the strings.
Now, with flatter crown radii on fingerboards and better amplification, the need for the compensation in string diameters to keep tonal consistency is not as great as it used to be, and we are all going towards even tension. However, the same design concerns are still there, because the pickups are still flat, there is at least some arc to modern fingerboards, and it is complicated with extended scales and tunings, both up and down in pitch, beyond the traditional 4-string bass.
My final point in all this digression is that even with modern instruments, because the G is still thin and the E or B can still thump, perfectly even tension string-to-string may or may not be the answer, depending on the player, the repertoire, and the instrument/amplifier combination, how the string is constructed, and even the availability of different gauges and alloys of the raw wire used in construction. A compromise in tension, taking into account all the design elements of the signal chain from fingers to speakers, is in order. That is why my set for my Rick 4002 has a modified tension that is even from the standpoint of an even progression of tensions, which work for me and my instrument. Your mileage will vary, and I encourage experimentation.
