I'm thinking of bass, viola, violin, and cello, though my question could apply to any instrument tuned in stacked fourths/fifths. Typically the strings are tuned using the 3rd and 4th harmonics so that each string is a just perfect fourth/fifth from the next. However, this leads to the bottom string being technically a Pythagorean minor third / major sixth to the top string. To use bass as an example, the ratio between the E and G strings would be (4^4/3^4) = 2*(32/27). However a 5-limit minor tenth would be the ratio 12/5 = 6/5.
I've discovered that it's possible to tune this 12/5 ratio by simultaneously playing the 3rd and 4th harmonics of the E string compared against the 5th harmonic of the G string. I am aware that any such tuning would lead to a wolf fifth/fourth between open strings. Despite this, have any prominent string players tuned in this manner or similar (such as, to use bass again as an example, tuning the A string to match with the E string rather than the D string)?
My current suspicions lie in flageolet passages. For example, at the end of the 2nd movement of the Koussevitsky concerto for double bass, Koussevitsky plays on his recording with harmonics. Playing through this passage, the high E on the A string forms an odd (to my ear) interval against the high F# on the D string; but when I tune my A string to form a 5:9 ratio against the G string, the melodic interval sounds more in tune. This suggests to me that other players might be tuning their strings in the manner I described to compensate for this.