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For example, the fingering for Hanon exercise #1 in C is 1-2-3-4-5-4-3-2. Is the same fingering used for the other 11 keys? If so, does the rule hold for all Hanon exercises?

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Hanon exercises are designed to be played on white keys, and it was not Hanon's intention they be transposed. However, transposing them can be an excellent exercise, in which case it's up to you how to finger them. My personal preference is to use the same fingering in all keys because of the variety of "problems" it presents as the configuration of black and white keys changes. However, it can also be quite useful to create your own fingerings as a way to practice different finger combinations or to find creative solutions to awkward key combinations.

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I'll opine that you can play most with the same fingering but obviously elbow, rotations, in/out, up, down and shifting movements will change drastically.

I do not condone the use of Hanon for technique. Technique comes from moving properly and obeying the laws of physics, not a book. If you can't do something at the piano, you need an adjustment to how you move, not hours of practicing incorrectly. And, just because you practice slowly does not mean you are practicing properly. Slow wrong is still wrong.

That said, snippets from the Hanon exercises can make wonderful little jazz fills and leads. So much time is wasted on Hanon. Use it as a reference. The instructions are greatly flawed, too. He advocates isolating a finger and lifting it high. While it is true that every motion must have an equal and opposite motion, there are more efficient ways to achieve that without isolating a finger which immediately creates strain as two muscles pull on one bone. The book lacks admonishment NOT to abduct and those exercises can very easily hard wire improper movement into your brain. VERY. If your teacher doesn't know what a pronator is, abductor, long flexor, ulnar deviation or the laws of physics, your teacher can hurt you.

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