Is it okay to say "I can play the piano by music notes" (meaning that if you gave me a sheet of paper with some music notes on it, I would have no problem reproducing that piece of music by looking at that sheet of paper and simultaneously playing the piano) or there is a better way of expressing this thought?
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Instead of saying "I can play the piano by music notes", it would be more understandable to say:
This means that, given a piece of sheet music, you can play the piece. If you can not only read sheet music, but you can play a piece you have never seen before on the first try, you can say:
As Wikipedia explains, sight-reading is:
The term I am most familiar with is sightread.
Then you could say that you can "sightread for/on the piano".
Also some people may prefer to use a hyphen in the term, sight-read.
If you'd like to retain the structure of the popular expression "to play by ear," I would use "to play by score".