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I am very much into romantic composers, like Brahms, Schumann, Faure, Chopin, and Scriabin. These days I found myself dealing with many heavily arpeggiated passages. What is the best way to memorize these passages? Should I do it mentally (like analyze the chords, play the chords first, spread them out later, and understand chord progression), or physically repeat playing them as they are written so that the hand gets used to the shape of the phrase?

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    Both of those ideas are good. They'll probably work best if you do both (learning the chord progression AND getting the lie and shape in your muscle memory). Here is a specific trick to supplement your ideas: Break the phrase up into groups of notes, for example play {arpeggio plus first note of next arpeggio}, pause, {next arpeggio plus first note of next one}, pause, and so on. The first note of each arpeggio will be repeated. Do you see what I mean? It seems that allowing that pause can be quite helpful. Once you're comfortable with that, you can group two arpeggios together. – aparente001 Aug 25 '17 at 5:18
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Some background on my piano education: Studied at the American Conservatory of Music, then under concert pianists from eight years until I was about twenty when I then focused solely on flute.

For me, memorizing a piece, regardless of its construct, period, or genre, was a matter of learning it perfectly by repeated perfect practice, first with the music. As you find yourself able to play it more and more without keeping your eyes glued to the music, your hands moving of their own accord through passage upon passage, it's time to start pushing yourself to actually play the piece from memory, not by abandoning the music, but by depending upon your audio and physical memory of the piece. Watch your hands when you can. Enjoy that watching and attending, then glance back at the music when needed for the parts you're not yet so familiar.

Sooner or later, you'll know the piece so well that you can play it 'by heart', that is without the music, at all. It's an organic process. Some difficult pieces are really dependent on watching your hands as you play and remembering patterns or significant events. Rachmaninoff comes to mind for me.

For arpeggiated pieces, I found my safety in the patterns and chord roots and their transitions.

Hope that helps.

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