I'm a late starter with 'official' piano lessons and wondered if my learning experience is typical. As I'm sure many do, my piano teacher tells my to practice left and right hands separately. I can often manage to learn that quite easily and play each hand with minimal errors.
However, playing hands-together and all that separate practice seems for nought - my hands won't do what they are told. It's honestly like I have to learn the whole thing from scratch when I use both hands - and my error rate goes way up. I get disheartened because I can't seem to get rid of the errors. It takes me so long to get a piece anywhere near good I'm generally sick to the back teeth of playing it before I get it error free (if I ever do).
But the point of learning parts separately is NOT about making it easier to play both hands together. It's about learning all the "other" stuff (like correct hand position, articulation etc.) without having the distraction of the second hand.
As a piano teacher for more than 50 years I can assure you that you are not alone! Many students have the same experience, though quite a few do not. Of the hundreds I have taught I would say it breaks about 60-40 toward having the problem.
As Brad said in his comment to the original question (a comment which has now strangely disappeared), the secret is to go slow. When you first start putting hands together on a new piece you may find you have to go three to four times more slowly that when you are doing hands separate. Incrase SLOWLY. When you find yourself making lots of mistakes, slow down more. Be patient with yourself. You'll get there.
EDIT: I should add that it's better NOT to try to play through the whole piece at once at first. Break it into "chunks." For most pieces a four-measure 'chunk' is about right. Some pieces 6 or even 8 is better. Let your progress tell you.
Yes, playing both hands together is quite difficult for beginners, even if you feel like you have mastered each hand separately. You will likely hit another big bump when you move on from playing similar things in both hands (like two octaves of a scale) to playing two different things (like a bass line and a melody). There is yet another big bump when you start playing very different rhythms (like the straight bass and ragged melody of a ragtime piece).
You can solve most musical difficulties by playing slowly enough. Very slowly if necessary. As you start to master the basic skills of hand independence, you should be able to pick up new pieces more easily, although each new piece may still challenge you at first. Any time you find yourself making mistakes, go slower. Even after a couple years of adult piano study, I still need quite a bit of hand independence work for each new piece.
People's brains work differently. I have two kids, both of whom started learning piano at a young age, through the same method. One of them has progressed marvellously. The other one never got the hang of hands-together playing, and eventually gave up. Perhaps the piano is just not the right instrument for him.
You say you can play hands separately with minimal errors. Are you sure that you can do so in rhythm? If you can't play hands-separate correctly to a metronome, then it's going to be a mess when you put the hands together.
Try humming the tune of one hand while playing the other. Then switch. The idea is to force your brain to think about the hands-together result before you actually try doing it. Your hand signals and your vocal signals go through different neural channels and are less likely to get confounded than signals to your left and right hands.
I think some people have the wrong view about piano, and instruments that are similar, when the issue of "hand independence" is discussed. Cognitive science these days seems to indicate that humans never truly "multitask" in their conscious actions; they simply switch between individual tasks very quickly. While I can imagine a human with an unusual neurological feature (such as a severed corpus callosum) performing true multitasking (such as typing out a history paper while having an unrelated conversation), the overwhelming portion of humans are neurologically incapable of such an action. In fact, it is the hallmark of an unhealthy human that parts of their brain do not fire in a coordinated concert of individual (i.e., singular, sequential) neurological explosions, but instead fire independently of each other, of their own accord. This is actually what you see in the brain of a patient who is having a seizure. All the areas of the malfunctiong brain ignore each other and just fire at their own will; in other words, the areas of their brain act independently of each other.
Therefore, you do not want true "hand independence"; instead, you want hand coordination. Put another way, instrument playing is a conscious action, controlled by our executive function, and we only have one area of the brain that controls the executive function. Thus, homo sapiens's conscious control is, for better or worse, unitary, and we cannot do two independent tasks at once.
So...what you need to do is to try to not beat yourself up too much, and simply accept the fact that when you do hands-together play you really are learning a different skill. You cannot simply focus on the movement of one arm, wrist, hand and finger in isolation from any other. If you can't play it with hands together, go back, define every individual action you must take, treating both hands as a single mental unit, put them in sequence, practice them over and over again until you can do them in sequence, in time, perfectly, and then speed it up. Do not try to skip this simple process because you want your hands to "be independent", because you cannot neurologically do such a thing.
The thing about piano, unlike any other instrument that I've played, is that learning it requires isolation and precision of every little movement you're making. Not doing this could lead to you developing bad habits.
At this point in my piano playing, I've gotten to the point where I can sightread with both hands a lot of simple things without struggle, but I run into points where I need to practice more extensively. The brain is terrible at learning many new actions in parallel. The reason why you need to isolate your hands-together playing to hands-separate playing is that your brain needs to separately learn what both hands are doing well so that it's easier to execute both hands' actions in parallel.
Music comes from the mind, not just the fingers. Since you are able to play separately, you probably don't have motoric problems. Try this exercise: take a short fragment (one or a few bars), and carefully read the music. Try to be observative about everything, the harmony, the rithms, the shape of the melody. If possible, imagin how you think it will sound. Then play each hand separately, from memory. If you make errors, don't worry, just go back to the score, and look carefully at the places where you made a mistake. Try to find the source of the mistake. Was it a wrong finger? A wrong harmony? If this goes well, do the same for both hands together. First read the score, observe which notes go together, how both hands relate to each other (is one hand accompaniment? Does it play chords? What are the chords? Are the chords broken? Perhaps the melody is in the left hand, and the accompaniment in the right). Then try to play both hands together from memory. If you make a mistake, don't worry, just go back to the score. Try short fragments at first, then take longer fragments as you progress. Don't forget to check with the score regularly, so you don't learn wrong notes! Try this method, and you will find you progress much faster!
Also a note of caution: simply playing slow doesn't mean are learning anything. The secret is understanding. Each note doesn't have a meaning on itself. You must be aware of its meaning in the melody (or harmony). You must be aware of where it comes from, and where it goes to. Listen to what happens in the music. Don't just play articulations because you have to, but understand why they are there. If your mind is confused, it will be still confused when playing slowly! The reason for practicing slowly is not because you have to create automatic reflexes, it's because you want your mind to be clear and open to the music, to fully understand it, so you can master the music, not your reflexes mastering you. If you allow yourself to be overwhelmed by the complexity, then you cannot master the music. Find clarity and meaning, and everything goes easy!
That is quite normal since you underestimate how much effort it is to practice one hand until it occupies at most a third of your attention (you need another third for the other hand, and yet another third for the synchronization).
Basically, playing smoothly while paying some attention to the score rarely occupies less than a third of an attention when you are not used to it, and it is hard to simulate the distraction of the other hand and of putting things together without actually doing so.
So in the long run you need to realign how much focus a single hand may take with your way of practising. It's like being able to run at a solid pace does only moderately prepare you for getting at maximum speed through a chest-deep mud pit. Yes, you have trained for that in some manner, but in some manner you didn't yet.
As a new piano player I find the entire concept of doing two separate things at once massively challenging. Even when both parts are incredibly simple and I can sight-read both lines mentally easily, I start playing and my hands just "lock up". Even playing a simple 3 note repeating pattern (like a chord triad 1-2-3-1-2-3) with my left hand, the moment I try and play any different rhythm with my right, it all falls apart. As someone who is used to learning stuff easily, this is a strange concept. I can almost stand outside my own brain and watch all the gears grind to a halt at the simplest thing.
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