If I listened to a song for enough time, memorized the melody and the vocals of the original singer removed and now I am left with just the accompaniments playing in the background and I am going to sing how can I know which note in the accompaniments I should use as my reference pitch to start singing in key? (If I don't know anything about the song's chords or melody)
If you've 'memorised the melody', I can't see why you should have problems starting on the wrong note! Take singing 'Happy birthday'. Without accompaniment, in a crowd, there will be many wrong start notes, which will often end with (most) folk ending on the right note. But, given an intro, which should end on the V, everyone (almost!) will start on the right note.
Not that more than 10% of those folks will be musicians. It's something innate, that most of us are blessed with, not something like absolute or relative pitch. I'm certain that 90% of folk will just pitch naturally, without any actual mental reference to the introduction. They'll spontaneously start on the correct note.
And if you've 'memorised the melody' you're already pretty close - it's rather like muscle memory - having speared that piece of food on your fork, you don't ever have to stop and think 'Now, where's my mouth'...
From an other question you raised I see you are using sheet notes and your piano to explore the melody and (some) chords, which I like as an approach.
So you can make an experiment, like this, i.e. deliberately change your reference pitch:
- playing chords (accompanying), adjust your voice to root, 3rd, 5th and so on, and see/hear, where it leads you to; in the same octave or higher or lower, in different parts or measures of the song
- same for notes you sing off-chord on purpose (which probably many times will lead to a strange voicing, but not always, which we need also for learning purposes)
- thinned-out chords, e.g. don't play its 3rd, 5th etc. to make room for your voice (like a placeholder)
- try singing on purpose one or several semitones off-melody or off-chord, and try staying there
The purpose is to further develop your feel for harmony (and harmonic progression, harmonic rhythm).
Bands sometimes talk about "playing in the pocket", meaning, all fits nicely together. You can't beat that feeling, when all instruments are on-rhythm, on-chord, on-tempo and so on, they let if flow and follow the flow, resting in music. However, even slight deviations can "ruin this feeling" easily, but there is some tolerance. In harmony, your overarching topic, there are similar effects.
Or said from a different perspective: In the end accompanying chords and melody form just an other chord together, perhaps the same one, perhaps a different one. The better your feeling for chord-harmonies becomes, the better your results will be as a singer.
This is something for which each singer has to find their own, perhaps individually unique, approach. And the issue isn't limited to the first pitch of the song, but can apply after a lengthy rest, where there is a difficult-to-hear interval, where the vocal conflicts with the harmony, or where there are key changes or chromaticism.
To get you started in finding your approach ...
- Using a full recording of the song (or the sheet music, or whatever is available), find the pitch in question, so it's in your voice/ear.
- Begin the accompaniment-only recording shortly before the pitch you're trying to find, and listen for a pitch that you can relate to the one you're trying to learn — the one currently in your voice/ear, because you found it in step 1.
Method 2 Transcribe the accompaniment recording so that you know every note in the accompaniment part. It does not have to be the entire recording, of course; just a part close to the note for which you need a reference pitch.
Method 3(A) Except for the first note of the song, your reference pitch can be the last note you sang. It's good ear training to practice holding onto a pitch while the accompaniment is doing other things.
Method 3(B) If it's the first note of the song, but not the first note of the concert, practice holding onto the last note of the previous song and find your pitch from there.
Method 4 Hide a tuning fork somewhere in your clothing (up a sleeve, in a pocket, etc.) and find a subtle way to listen to it before the pitch you need. Find your pitch relative to the tuning fork.