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I'm not asking for the physical how, maybe just hints or tips.
Do you just have to know how far to leap? For example, C4 to A2 is 10 white keys. Do you just have to know that via muscle memory?
Do you need to preserve positions (like start from C and change to G position)? Or is it as long as you're comfortable?
For example, from this:
Jump to this:
Yes, it is muscle memory. Which means it's time to practice!
I can't recall intentionally setting out to develop this particular skill, but I know it developed as I became more familiar with the instrument. It's still not perfect - I've jumped a tone too high or low more times than I care to remember, but I don't spend all my time looking down at my hands any more.
There's some value in retaining your hands in the 'correct' position; the trick is deciding what that position is. A teacher can be really helpful here, to ensure you're not developing bad habits. That being said, the 'correct' position is often debatable, and it should result in the easiest and most comfortable fingering anyway.
I'd choose a song which you like, which features some of these jumps, and start there. Get familiar with the notes, so that you stop thinking too much about them. As you practice, you should find yourself automatically moving your hands as required. Then, work on different pieces. Over time, you should develop that muscle memory.
Also, don't be too worried about using your eyes to help, especially for difficult bits. For some parts of a piece, I will be looking at my left hand, or my right hand, or sometimes even the music.
It is indeed muscle memory but there is a technique to it. If you can't nail the leap every time, you are doing something wrong, it is not necessarily a "practice makes perfect" method but more of a - what kind of bad habits have you perfectly practiced and are now part of your technical inability.
The first thing pianists do wrong is when they are departing a note, they twist the wrist in the direction they are going (ulnar deviation, radial deviation). This is bad for many reasons and hinders accuracy for many reasons.
The second thing they do wrong is they try to grab or stab or reach for the note. The finger or hand doesn't play it but rather, the arm does. Your hand can't drag the arm behind it but the arm can very rapidly place the fingers where they need to be.
The third thing pianists do wrong is they think horizontal. The arm needs to have a shape. A circular one to be exact. If you reach for the note you will most likely miss it so a rounded shape of the arm gives it accuracy and power. This leads to the next issue . . .
Strike the note from the top. Singers hear this all the time. Slightly over-reach the note then come down on it. This is because your body naturally wants to contract. As you reach for it, the tendency is to then give up and the pianist falls short.
A good lesson to keep in mind is what THE KARATE KID was taught: Wax on, wax off. Play the piano as if you are writing on a chalk board. The fingers and wrist don't lead, the arm places the fingers where they need to be.
There are a few other things such as forearm rotation and dual muscular pulls but that is for another day.
For me it's mainly muscle memory.
However, I think it's worth mentioning a few details:
I don't remember learning it, it just happened, arrived with lots of hours of playing time (mainly practice, but not only that).
It's all muscle memory, of course, but there's also a process of "coarse adjustment" and "fine adjustment", based on the configurations of the black keys. (dtldarek mentions this in his answer as well.)
Say you have your 3rd finger on C4, and want to next play A2 with 5. You reach out towards the gap between Bb2 and C#3 with your 4th finger, and when you find it, you can easily locate A2 with your 5th from there.
If you watch someone sight reading, or singing and playing stride figures in the left hand at the same time, or whatever, you'll notice that they often feel their way around the piano in this way. As for teaching, any of my teachers who taught sight reading mentioned this, although they also said something to the effect of "the only way to learn sight reading is to sight read."
All that said, there are times when you just have to do it. Beethoven was said to be adept at leaps on the piano, and there are places in his sonatas that bear this out. Here's a snippet from his Sonata No. 4, Op. 7:
The tempo is allegro, and you have all you can do to just fly up to the F from the A.
And then we have this from our friend Liszt (from "La Campanella"):
And no, you can't go feeling around the black keys here, either. :) You have to develop the same level of precision in your arms and elbows as you have in your fingers and wrists, and move your hand back and forth rapidly from the elbow. (Not that I can do it very well...)
For your second question about preserving positions: You should do so if you find it helpful.
As a beginner, thinking in terms of positions is a good way to manage how your fingers sit over the keys. e.g. You don't need to know where E4's absolute position is, you just have to remember that it's under your 3rd finger.
As you start playing music with leaps, it may be helpful to think about moving your hands to a new position (and I've seen studies that have been composed with this intent). I would only suggest that you do this if it feels comfortable to you and works for the piece you're playing: do not constrain yourself to thinking about the keyboard in this way.
As you become more comfortable with the keyboard under your hands, you will (at least, I imagine most keyboard players do) subconsciously develop a "set of positions" that you will use to this effect without even thinking about it. In other words, you can place you hand anywhere on the piano and instantly know which keys are under which fingers.
As for your first question about muscle memory: Yes, you will develop muscle memory when you practice leaps. For some large gaps, you might want to (if you have the time) feel around to make sure you have the right key(s) before you play it.
Also, there's no rule that says you can't look at the keyboard.
So a technique to practice this I found useful is to consciously stop just before you play in the new position and look at your hand and adjust before you play it. This forces you to be very conscious about the new position. In the end you can reduce the gap down to nothing. I found this helped me get the muscle memory quicker.
