Their are 2 Arpeggios on this picture:
I know I should play their notes from the bottom to top very quickly BUT should I hold each note pressed down for their duration (dotted half note) OR should I release them immediately after being played?
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The notes are written as dotted minims (three beats), lasting the whole bar. So they should be heard until the end of the bar.
That can be done in a couple of ways. Keep hold of the lowest note (played first), then play the other(s), and keep hold of all until you need to play the next bar. Or, play the first, catch it with the sustain (damper) or sostenuto pedal as you play the other(s), and release at the end of the bar, when the next ones come along.
The note in the treble clef should be played at the same time as the lower bass note, although depending on style, it could sound more effective being played as part of the arpeggio. Try both, and decide.
If you want to play the music as notated, you hold them for their notated duration.
Except, when you execute the arpeggio properly, those durations won't literally be the notated values, except for the lowest, first note of the chord.
Looking at the second chord, the
D will be played first, on the beat. The
C above it will be play a very small fraction of a beat after the
D, a fraction too small to bother notating literally with rest values before the
D is notated with a dotted half note. Its duration will be 3 beats.
C is also notated with a dotted half note, but it comes after the
D, and it ends at the measure's bar line. The
C will stop at the same time the
D stops. Obviously, if the
C starts after
D, but the end at the same time, the
C has a literal shorter duration that the notated three beat dotted half note.
Only the bottom note of the chord will get its literal notated duration. All the other chord notes will be very slightly shorter. All of this is understood in the method of playing the embellishment of an arpeggio. The wavy bar is a notation convenience instead of writing out literal durations which would only clutter up the score.