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Just starting to learn playing piano, I have been memorizing my fingers corresponding to the note on paper, so C is thumb vice versa

Let's say I need to change position to put my thumb on E, I find it hard to read the note and press the correct key.

Am I learning it wrong?

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Forget about fingerings for awhile. Play the melody (I suppose you're playing a single-note melody line) with just one finger. For example the index finger ONLY. Or with a teaspoon! Or wood chopsticks or other soft objects? Then you'll have to associate NOTES with KEYS, and the fingering question is left as a separate thing to be solved later.

Fingering choice depends on the phrase you're playing, not individual notes or keys. Should I always use scale fingering when sight reading?

Drummers talk about foot/hand independence, so maybe you need finger/key independence exercises. Piano keys and fingers are not the same thing, they're not glued together. So try playing without fingers.

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    A famous argument about the subtleties of piano tone ends in an outburst: the key doesn't care if it's pushed by the tip of Paderewski's finger or by the tip of his umbrella. – Camille Goudeseune Nov 6 '20 at 20:13
  • @CamilleGoudeseune - interesting, entertaing, but only true on an organ, harpsichord or suchlike. – Tim Feb 11 at 11:34
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It's convenient at the very beginning, to think of CDEFG as thumb,index,middle,ring,pinky. But it doesn't set you up for anything that follows, as you found out. So, yes, you're learning wrong.

Instead, consider each note and where it lives - D for instance is always between the two black keys. Whichever finger presses that D, it will always be 'that D'. You've also realised that sometimes you need to move your hand across, to encompass the next few notes that you want to play. That's good, and will happen all the time. Sometimes, you'll have the possibility of playing several notes with a choice of fingers - that also happens all the time.

So, look at which notes need playing, and try to centre your hand so your fingers can reach all, or most of them. Learning scales and arpeggios will help with articulation of your hand and fingers - how you need to glide your hand sideways, to put the thumb, or another finger, on notes that are just out of reach.

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Think of the finger numbering guides as training wheels. You need them to start with so you don't fall over, but with more practise you'll no longer need them.

Don't try to learn which finger goes on which key, because the idea that 1 is C will, as you've discovered, rarely be true. Instead think about where E is in relation to C; how far you had to move your hand to get there; the patterns of the black & white keys and how they relate to the position you are currently in. Eventually, the scale or key you are working in will become more engrained than which finger is on which note at any given time, you'll just do it automatically.

Consider it another muscle memory co-ordination task. That might sound a tough job, but just think…
You've already learned a lot of those in your life, most of them without even realising you were doing them.
When you were a toddler, you had to learn to speak, to walk, how to use a knife & fork - or even chopsticks - how to drink from a cup without a lid; then later to read, to write, to ride a bike. You do all those things without even thinking these days - you can probably do several of them at one time without getting confused.

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Just starting to learn playing piano, I have been memorizing my fingers corresponding to the note on paper, so C is thumb vice versa

Like others have said, this is how you start learning.

Let's say I need to change position to put my thumb on E, I find it hard to read the note and press the correct key.

This is normal when you need to associate new fingers with new keys. It is part of the learning process.

Am I learning it wrong?

Not at all, you just needed the next step.

When you start playing melodies that go beyond where your fingers are, you want to move your pinky as far as it need to go. If you move it farther than it has to go, you are losing time having to move farther. Moving your pinky only as far as it needs to go will help you keep time when you want to play fast passages. You want to practice moving your hands around to get used to using your fingers for different notes. When you start playing more advanced passages, you will find that figuring out which finger to use for each note of a chord will help with playing the piece; if you use some fingers over others you may find yourself having to jump around the keyboard more. Figuring out the fingerings is part of the fun!

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One of the ways to avoid finger-glue, is to learn scales and chords in many keys. Almost anything you play will be a mix of chopped-up bits of scales and chords, and there are sets of standardized fingerings for every scale and chord, out there on the WWW. There are some differences between them, but their underlying purpose is to enable you to not run out of fingers!!! :-) So as applied directly to your question, as you learn scales and chords in many keys, you will find that your hands remember, and find those bits on the paper as you need them, and your fingers will walk right in and adapt according to what they have already done in the scales and chords.

I haven't needed the standard chord fingerings so much, because the layout of the standard piano keyboard often makes good chord-fingerings reasonably obvious over time, but the tried-and-true scale fingerings can be precious. Since the first note of (say) the C major scale, uses the same finger as the first note of the D major scale and the E major too, (but not all of them, there are many surprises!), if you focus on scale-fingerings in multiple keys, you're doing yourself multiple favors at once, because you are flexibilizing yourself in keys and helping your fingers adapt to more and more of the keyboard at the same time.

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