As a practical matter, there are a few concepts that teachers use, including the "ringing tones" in Suzuki.

These are the fourth finger/lower open string and third finger/upper open string pitches which should match, and also the notion of "frame" formed, initially, by the first and third fingers (with a "high" or a "low" 2), and somewhat later, the frame formed by the octave reach of first and fourth finger (around the 3rd Suzuki book, along with the Wohlfahrt studies).

An additional concept may also be introduced, having to do with the roles that pitch steps (of the scale) play within the context of any given key, (tonic, supertonic, mediant, subdominant, dominant, submediant, leading tone).

Stringed instruments are not equally tempered the way the piano is, and thus, key context is everything when it comes to intonation on a stringed instrument.

The leading tone, for example, is higher, and half-steps can be smaller, within the context of the key (than they are on piano).

Playing with piano, one may attempt to adjust to the equally tempered notes, but this is not accurate for the violinist.

Connie Sunday: Freelance studio musician, strings and piano teacher, author of Violin FAQ and numerous essays in violin pedagogy. Further information can be found at connexions.