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I'm learning Pathetique sonata first movement. I'm having difficulties on the 2nd main theme. Where the right hand needs to cross the left hand to play melody.

First question: how to make an accurate jump from lower part of the piano to the upper part in right hand? I always miss or hit the wrong note when I make such leap.

2nd question: When I make leap, how to make sure I still play softly on the first note? When I leap usually I will "bang" the first note I play.

Last: how to play the trills on right hand clearly and fast? My fingers always jam together in such a fast trill and produce a muddy sound.

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Frost movement! Ah, my favorite part, always made me think of Queen Elsa and her icy castle.

Seriously. The incorrect landing of r.h., and the fatigue of playing mordent (I suppose you mean by trill) --- This is something that best to consult your piano teacher, or, if you don't have one, someone experienced player or even music student. They can recognize for you which part of your hand is unnecessarily stressed.

But several suggestions here. First, find a well edited score. What I think is best to buy an Urtext (i.e. being faithful in notation to the original edition whenever possible), and consult an edited version perhaps on IMSLP. They often has good fingering recommended to you. For example, the first landing of right hand to the low register, if using 2 (rather than 1), is easier because the right hand is "in an oblique manner".

2nd theme 1st occurence

And struck the key only after you find it, not just "banging" it. Here piano is marked, so no great force required. This shall improve precision.

Also if you change finger playing mordants it will prevent fatigue, for example, one with 3532, another 1432, another 2421 etc.

I am surprised that you didn't complain about the l.h. octave tremolo. I always find that causing fatigue.

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    What is key in making the jump without an accent is, in my opinion, disconnecting the placement of the hand from the striking of the key. You mention this, but so briefly that I wanted to underline it a bit. You can practice this by making the jump first, then pausing a moment before playing the note. – 11684 Dec 20 '16 at 23:38
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These things just take practice. One can practice making long jumps between successive notes slowly at first until one's muscle memory can learn how far to move one's hand. It's the same learning technique as with dancing or shooting a basketball or hitting a baseball.

The same thing applies to trills. With practice one leans how to move the fingers. (There are parts of the first movement of the Pathetique that do sound a bit muddy, even on the piano of Beethoven's day. Some of the very fast passages in the bass for example.)

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