I find myself hitting the first note of the melody correctly when my singing teacher starts playing the song's chords on the piano, but I have no idea how I did it or how I found the first note. How do singers find the first note when the accompaniment begins playing and they begin singing? What is the theory behind that?
The intro establishes the basic tonality of the song, it tells you where 'do' is (if you think in terms of 'do, re, mi...) or where the I chord is (if you know a bit of 'theory'). The intro doesn't HAVE to end on a tonic or dominant chord, the melody doesn't HAVE to start on do or any other note of the tonic chord. We just need to get our bearings. The dart board is over here, maybe low on the wall, maybe high. Focus on it, then go for maybe a 1, maybe a 5, maybe a bulls-eye!
Then, how does the singer actually produce the note that's in their 'mind's eye'? Well, if I said 'sing a high note' or 'sing a low note' you wouldn't have a problem. It's just a matter of refining that skill. Many people do it naturally. Some never get it!
The question's a little vague, but when playing an introduction, the accompanist will usually play a V harmony to bring in the vox. That then means the I will follow, and most songs will use either ^1, ^3 or^5 to start the song. That narrows it down considerably. However, see my comment. Maybe that's a more pressing question to get answered!
There are multiple aspects to this question, as there is a ton of music which will necessarily behave differently. So I will give multiple answers:
- By listening to specific key notes in the accompaniment and taking intervals of this (the interval approach)
- By inferring the correct functions to the harmony given by the accompaniment (the harmonic approch)
- By feeling (the intuitive approach)
- By a tuning fork or something similar
Let’s talk first about the most intuitive ones, 3 and 2. If you know piece well you will know the accompaniment and just "know" the notes by heart. Also singing a specific pitch will require a specific configuration of your vocal muscles. Singing requires you to intrinsically "know" how to setup your vocal muscles to produce exactly that pitch. And when you learn a piece you will learn the feeling of this configuration, which means that with a piece you know well you should be able to find the starting pitch plus/minus something without any external help. So when hearing the accompaniment there will often simply be only one note that fits both the tonality and the body feeling.
Now, if you do not know the piece well enough but if we have clear and simple harmonies you will be able to find a tonal context, from which you can derive specific notes.
If things get more complicated one thing you often find in choir practice is to establish a key note in the accompaniment (or other voices) from which you take a specific interval.
Finally if you’ve got a quite complex or atonal accompaniment you’d use something like a tuning fork to find the correct pitch.