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I am trying to self learn piano. I can play some melody (right hand), can play some chords (left hand) and can play some melody with chords (right + left) with right doing all the complex finger movements for the melody and left just pressing the chord keys once at each change.

I am trying to play something more complex with left hand and interleave the notes with those on the right hand melody. But I do not know how to come-up with the rhythm parts. I try arpeggiating the chords instead of "playing" them but result does not fit in and sounds muddy. Also been trying a few other patterns for a while now, but nothing seems to stick.

My question: are there some patterns (rule of thumb etc.) for coming up with rhythm patterns given a melody? For example, in case of chords, one can rarely go wrong playing the chord for the root note of the melody and a progression thereof.

PS: Not sure if this is classified as "improvisation", but tagging it so.

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I think 'improvisation' connotates for many people improvising a melody over chord changes like in jazz or a rock guitar solo. But, you are asking about improvising the rhythm part - the accompaniment part. You can try finding resources using keywords like 'accompaniment' or 'comping'. You can also look for guides along the lines of 'how to play from a lead sheet.' A lead sheet is just the song melody with chord names above the melody.

Here are two samples that seem good. Both show how a simple pattern can be adapted through small variations.

Jazz waltz using bass on beat 1, chord on beats 2 and 3...

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A synopated rock rhythm...

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The value of these kinds of resources will depend a lot of the style you want to play and the kind of learner you are. (For example, some people can't read notation and so notated sources aren't so helpful.) Look around for what fits your interest. Some items have previews online, but if not, you can try to get material through your library before committing to buy anything.

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The best way to learn to improvise was for me:

left hand: bass

right hand: chords

and the melody: singing by my self!

(or accompanying someone else)

so my hands were free for the rhythm and later able to play the melody while the chords were added to the bass or to the tune in the right hand. This gave me the greatest flexibility and this is the basic of improvising.

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