How can I significantly improve my hand independence on piano?

Answer
"Honest method, by which the amateurs of the keyboard – especially, however, those desirous of learning – are shown a clear way not only (1) to learn to play cleanly in two parts, but also, after further progress, (2) to handle three obligate parts correctly and well; and along with this not only to obtain good inventions (ideas) but to develop the same well; above all, however, to achieve a cantabile style in playing and at the same time acquire a strong foretaste of composition."

There are fifteen 2 part inventions and 15 three part inventions. In working through the two part inventions, myself, I found that I had developed hand independence and learned some beautiful songs in the process. You can get the scores at IMSLP. Good luck.

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