I'm writing how to play 4-octave arpeggios effectively and smoothly on the piano as the title says. But I'm just using "4-octave arpeggios" as an excellent example to explain how to cross over or under your fingers and move around the wide range on the keyboard up and down.
So, let's start with an example of playing 2 octaves of C major tonic arpeggio with your right hand going up. After you use your finger 1, 2, and 3 to play the middle C, E, and G, you need to cross your thumb under all the other fingers. At that time, check the shape of your thumb. Is it sticking out straight or curled way inside? Your thumb should be very curled and pointing towards the treble C. It's because you are going to higher pitch right now.
The next thing you need to look at is where your wrist and elbow are? Are they moving towards the right to accommodate your thumb going under other fingers? They should. Don't just stiffen your wrist and elbow (The elbow should always be bent in many different degrees and angles, not straight. I'll discuss that in the later blog). You need to use your wrist and elbow to accommodate this passage you are trying to play. Do not just move your thumb only. This movement of your wrist and elbow should happen quickly when your 3rd finger (middle finger) reaches G-note. Turn your wrist and elbow towards the right while the 3rd finger rotates on G-note.
Ok, now. Let's do this and keep going up for 4 octaves now. You can accomplish 2 octaves of arpeggios if they are in front of you or close to the mid-range of the piano. But if you need to go high up, using the method above is not enough to play 4-octave arpeggios. When you need to go up to the higher range on the keyboard, you now need to move your upper body to accommodate your elbow and wrist. Make sure that your upper body is relaxed. For it to be relaxed, your entire body needs to be supported and stabilized by your two feet and two legs. Your buttocks should be leaning again the edge (the center and front. I don't mean the right or left side edge) of the piano bench. Now your upper body is supported and able to relax, you can tilt your upper body towards the right where you are going with the arpeggio.
But this is a key. If you are moving your upper body at the same time with your arpeggio, then, your upper body is not accommodating your wrist or elbow. The upper body is moving too late to bring your wrist and elbow where you want them to go. You need to bring your upper body first towards the right so that this movement can help your elbow and wrist to move around relaxed. This will allow your thumb to cross under other fingers effortlessly.
Now apply this method to your left hand going down on arpeggios from the center of the keyboard to the lower range.
But people say, "4-octave arpeggios start very low and go up very high. It doesn't start from the middle." Yes. We have a problem when the right hand is in the lower range, and the left-hand is in the higher range. You probably feel stuck, stiff, or the lack of space. If so, you are probably sitting too close to the piano or keyboard. Always create some (enough) space between the piano bench and the piano. You need to experiment with this because I can't assist you one by one in person.
And once you find a good sitting position, move your upper body not just to the right or left, but also backward to create even more space between the keys and your wrist and elbow. Giving enough space for your wrists and elbows to move around in a relaxed way is so crucial for piano playing. You are not playing piano just with your fingers as I wrote in the other blog titled "Use Your Entire Body to Play the Piano." Use everything; your feet, legs, buttocks, hips, upper body, elbows, wrists, hands, and then fingers.