When playing a string instrument, it is very important to always strive for perfect intonation (playing in tune)! Not only does this make the melody sound correct, but over time it 'opens up' your instrument and causes it to vibrate freely producing a more warm and ringing tone.
Sometimes, however, it can be hard to even hear what is actually in tune, let alone consistently reproduce this. This can be particularly difficult as a beginner, when you aren't actually sure what you are listening for.
To help this, start by listening to (and training yourself to listen out for) ringing tones. Ringing tones are particular notes on your instrument, that when in tune cause your open strings to resonate at the same time, creating a richer, warmer and brighter sound.
To start with, it is important that your violin is properly tuned. When working on intonation, always be careful with your bow contact to ensure you are creating a consistent and even tone, vibrating the string fully.
Now try playing your open strings, and compare the sound to stopped notes (notes created by pressing your fingers on the string). Notice how the open strings sound, and ring in a brighter and fuller way than stopped notes. When working on our intonation, we are trying to get our stopped notes to ring as much like our open strings as possible. The more sensitive you are to the ringing quality of a note, the more your intonation will improve.
Try this experiment (which I think is pretty cool). Play your 1st finger A on the G string. If you get this note perfectly in tune (with good bow contact) it will cause your open A string to start vibrating wildly due to sympathetic resonance. This means that the resonance of your open A string to be added to the sound of your stopped 1st finger on the G string, creating a warmer, richer tone. Try playing this note out of tune, then in tune, and compare the different type of sound produced (and the amount of time the note rings after you have stopped playing). An in tune ringing tone will ring for longer after you have stopped playing than an out of tune note.
There are many ringing tones on your instrument - though some ring more clearly than others. The main ringing tones are the same note as your open strings (over different octaves).
Therefore with the violin strings of G, D, A, E we have the following very clear ringing tones:
1. 1st finger on G and D string (ringing with open A and E string)
2. 3rd finger on D, A and E string (ringing with the open string below)
3. 4th finger on G, D and A string (ringing with the open string above) - be careful here that you are not touching the string above as this will stop it vibrating.
Try playing these ringing tones, and see if you can start to hear clearly when the notes are perfectly in tune and ringing, compared to when they are out of tune and slightly 'duller'.
As your ear develops, you will start to notice the different ringing qualities of all the notes on your instrument, and this will help you to play beautifully in tune all the time.