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How to play (on piano) the staccato notes with slur i.e. notes A and F as the picture shown? Thanks!

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This is called Portato. It denotes a smooth, pulsing articulation. One way of playing them is to brush the keys with your fingers. However, since keys vary from different pianos and keyboards, this approach might not always work very well since ivory keys feel different from plastic key tops.

You can play the notes, but slightly detach the notes. When holding down the damper pedal, play the notes sharper, like you would play staccato normally. This can give the detached feeling while the notes are not attached.

Here's a link to a video that shows a great finger exercise related to this issue, where you can achieve playing slightly-detached notes:

You can use this technique to detach the notes even more since you have more control of the length of the notes.

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