I brought up string quartets because I know about tuning them, and, like barbershop quartets, it's 4-part harmony with extremely harmonic instruments. Bowing a string causes a phenomenon called "mode locking," where the normally somewhat inharmonic strings are forced to be harmonic (my understanding is the human voice also exhibits mode locking). You often do microadjustments to get chords to sound good vertically and horizontally.
I pulled up a random barbershop recording and looked at a spectrogram for the final chord, since the final chord is supposed to ring as much as possible, and there are only vertical considerations, so you'd presume they'd tune it as justly as possible, right? Here's what I found:
They tuned their perfect fifth justly as 3 : 2 almost perfectly. That's not unexpected, even equal temperament gets perfect fifths close to right (though a little flat).
They tuned their major third between 1.253 : 1 and 1.258 : 1. A just major third is 1.25 : 1, an equal tempered major third is about 1.26 : 1, and the 55-EDO system mentioned in the book I referenced in my first comment would give a major third of about 1.255 : 1. Like I expected, they tuned their major third sharp with respect to just intonation, but still flat with respect to equal temperament.
That's interesting. I would leave to see that analysis across a lot of performances. I know that barbershop folks often say to "aim high" or "sit on top of" your major thirds, but I always thought that's simply because it sounds really bad to be even slightly flat.
What's the theory behind why string quartets aim slightly sharper than 5:4? Is it something to do with inharmonicity or acoustics? Could it also have something to do with avoiding going flat? Is it possible that listeners are so accustomed to equal tempered thirds that a 5:4 third actually sounds flat?
I can't say I fully understand it, but having compared major triads with different intonations, I can only say I myself find slightly sharper thirds to sound nicer in chords. Here's one thing I've used: http://tmp.esoteri.casa/interval-test.html (5-limit is "just intonation", Pythagorean uses 81 : 64 for the major third, 1/6-comma meantone is 55-EDO, and 12-EDO is standard equal temperament.)
One thing I like is that the slight dissonance takes the edge off the ringing. I find that to be a rather strong flavor.
The book I mentioned offered suggestions why sharper major thirds are used, but I don't remember there being anything convincing other than observations about how harmony works. Maybe your thought that going too flat sours the third makes sense: if it's sharp, you have a little more freedom to adjust your intonation without accidentally going flat. Plus, thirds are fairly forgiving anyway since they're corresponding with the fifth harmonic, which over two octaves higher.
It's worth mentioning that the Pythagorean major third is about 1.266 : 1, which is quite sharp, yet it's still just intonation since it's from going up by 3 : 2 fifths.
I pulled up a random barbershop recording and looked at a spectrogram for the final chord, since the final chord is supposed to ring as much as possible, and there are only vertical considerations, so you'd presume they'd tune it as justly as possible, right? Here's what I found:
They tuned their perfect fifth justly as 3 : 2 almost perfectly. That's not unexpected, even equal temperament gets perfect fifths close to right (though a little flat).
They tuned their major third between 1.253 : 1 and 1.258 : 1. A just major third is 1.25 : 1, an equal tempered major third is about 1.26 : 1, and the 55-EDO system mentioned in the book I referenced in my first comment would give a major third of about 1.255 : 1. Like I expected, they tuned their major third sharp with respect to just intonation, but still flat with respect to equal temperament.