Intonation is a touchy subject. There are all kinds of quantifiable tuning systems, all kinds of theories about difference between "playing truly in tune" and "playing in tune with a tempered instrument", statements about playing certain notes higher than most, or lower than most, but in the end it's like the famous definition of obscenity:You know it when you see it; or, in this case, hear it.
After a bit of study of different tuning systems and their claim to true "in-tune-ness", I've finally settled on what I call "sweet spot intonation". It goes like this:
- For perfect intervals, there should be no perceptible "beating" of one note against another in the dyad (consisting of the reference pitch and the note we are playing against it).
- For non perfect intervals, there is a place where the beating or "turbulence" between the notes of the interval is at its slowest/calmest point. If the note is raised or lowered slightly (i.e. moved by a few cents in either direction), the turbulence becomes more intense. Aim for the point that feels the calmest. This is the sweet spot because it is the eye of the intonation storm for non perfect intervals. I always aim for it and sometimes achieve it.
When I don't, which is more often than I'd like, I adjust as quickly and intuitively as I can to try and find the spot so that as much of the duration of the note as possible will be perceived as "in tune". I find that adjustment slightly easier when adjusting down from the sharp side of the note because that sound is more grating to my ear and I am less inclined to leave it hanging there. With students who play consistently out of tune, I find that most of them tend to leave flat notes hanging there more often than sharp notes, possibly for the same reason.
Of course, when there is more than one reference pitch, as in a harmonic stack, the waters get muddied quite a bit. In these cases, I still believe there is a sweet spot as described above, but it is a lot more complicated to find in many cases. We do the best we can in the moment, and work on the process in the practice room. As always, EEMMV.