Please, I would like to ask you for an advice: I have big trouble to tune my flute by ear -e.g with the piano before playing the piece,my teacher always tells me whether am I flat or sharp. And I just dont understand why. I feel like I am not able to match the sound of piano with the sound of my flute in my head. I would love to do it by myself. I am definitely not tone- deaf - I am singing in tune without any problems, as a child I was singing in a choir for 12 years. The flute I have been playing for more ten years and when I am playing I am able to play in tune without problems, I feel it somehow, but I cannot say whether am I flat or sharp. What is wrong? I desperately love playing flute and I have just decided to fulfill my dream and I am going to study it further but I have to be able to tune....
Not an uncommon problem. It can be hard to seperate pitch from timbre when comparing notes on different instruments. I know one successful professional performer who habitually tuned his instrument sharp. It (sort of) worked in live performance but he knew something was wrong in recordings. It just needed pointing out that he was confusing 'bright' with 'sharp'.
Get yourself an electronic tuner. There's probably a free 'app' for whatever computer device you're reading this on. Set it to A - or whatever note you want to tune to - and check that the piano A is showing the right pitch. A little practice in matching what the tuner tells you against what you hear from the piano, and your problems will be over. Or just use the tuner all the time!
Another idea: don't tune to a single note, play a scale with the piano.
I find it easier to hear one note at a time, (the standard, then the instrument I am tuning) so I can hear the differences in pitch more clearly. I find it particularly difficult to tune to the piano because it is not just one note! Other strings vibrate sympathetically and many, many overtones are produced. Find the exact pitch with that! It is actually impossible. https://contentspiano.blogspot.com/2006/02/sympathetic-vibrations.html
Given that the strings on the piano may not be exactly in tune with each other, and given that most pianos in educational and other public facilities are not often in tune or even properly cared for, and given that I can tell when my piano goes out of tune simply by the fact that a note lost some of its resonance, the piano is really not a great standard by which to tune other instruments. Unfortunately, it is a must if we are going to play with the piano for accompaniment purposes.
If you must play with the piano, try tuning to octaves. Find the "average" pitch you hear and tune to that. My guess is that your teacher is hearing one end of the "out of tune" note and you are hearing the other end (see the article above.) If you tune to octaves, it will all blend more and you and your teacher may find it easier to come to consensus about what is "in tune with the piano."
If you do not have to tune to the piano, use an electronic tuner. Of course, that means only that that ONE note is in tune with the tuner.
As a beginner step, if you know that you're out of tune, adjust in one direction; if it gets worse, go the other way!
Like most things, developing intuition comes with time.
Play at the same time as the piano and listen for the beats. When you don't hear them anymore, you're done.