When you tune your cello to the piano's A and then tune each 5th carefully, you will find that your C is LOWER than the piano's C. This is because we tune our cellos according to the laws of natural acoustics. These laws, of course, lead to problems that have no solution, such as trying to play the quadruple-stop C-A-E-A (which occurs in the opening duet of the Brahms Double Concerto) in tune. Can't be done. Acoustically impossible if your cello is tuned in pure 5ths. The piano, on the other hand, compresses all intervals slightly (except octaves) so that each tonality, or triad, will sound equally very slightly out of tune. The alternative would be to have, say, C major be perfectly in tune but each tonality would then be more and more out of tune depending on its remoteness from C.
So what we're left with is a stand-off. But it's a stand-off that everyone accepts, with some minor adjustments. My personal solution is to tune my A a tad sharp to the piano's. This way I'm not compelled to "pinch" my 5ths (which many players will do). While we almost never play an open A of any appreciable duration in chamber music, we play open C's and G's all the time. Once this adjustment is made, everything just comes together in a pleasant aural "soup." Experienced chamber players will tell you that the painful intonation problems in a quartet or trio simply dissolve when you add a piano. This all assumes string players who have good control of pitch and tuning.
But going back to earlier questions about non-piano intonation, the main question we have to ask ourselves as we're playing along is: "should this note be vertically in tune or horizontally in tune?" The answer almost always turns on the speed of the passage. If you play a one-octave D scale (starting on the open D) really fast, the C# will, or should, be very high, very close to the D. Horizontal intonation. But now if each of those notes is a half-note in a slow tempo, and other instruments are changing notes along with you, the harmony, when you come to the C# is likely to include an A in the bass. Your C# will need to be much lower in this context than it would be in the fast scale context. Vertical intonation. 7th-chords present another recurring problem. The upper note of a minor 7th needs to be low, as it resolves downwards.
But what if the note that has just been played in the correct context for one harmony doesn't fit in the harmony that immediately follows? This is the problem with the first 4 measures of the Prelude of the G major Bach Suite: the C in the 3d measure needs to be low, as it's resolving down to the B. The trouble is, we've just played it (in the 2d measure), nicely meshing with the E and open G, and that's in an appreciably higher place than it has to be for the next bar. If you keep the C's identical, one of them is going to have to be slightly out of tune. My maverick "solution" is simply to play different C's. Another example is in the first measure of the C major Sarabande. This kind of stuff can make you nuts in no time; the sharper your hearing the more hopeless this whole topic becomes.
Two final points, for those with so much time on their hands that they've stuck with me this long, one conceptual and the other practical. First, there is no such thing as a note that is out of tune. There are only out of tune INTERVALS. If I play you a single note, in isolation, and ask "is this in tune or not?" the correct answer is "I don't know." (This is true of persons with perfect pitch too; controlled experiments demonstrate significant variances when people with such abilities are asked to sing the same pitch.) Thus, everything is a matter of relativity. Second, as to "testing," Mom and other highly-skilled cellists are referring to the system whereby they strike a note with their finger just prior to playing it, with a soft "tap." If this note is the same as an open string (regardless of octave) they'll often brush that string as well, to compare. With practice, this can be done in a split second and inaudibly to the audience (though you do hear it on recordings).
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