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What is exactly the difference of the Dot and Tie sheet music? I know how it works on piano, is it the same with the drums?

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Since drums produce impulsive sounds the notation for dot and tie is only for the duration information to emphasize the pattern. A very common usage is the rudiment notations, here is a quick google search led to an image from VicFirth website

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The ties here are denoting that it should be played with 32nd notes as much as the slashed note duration. A 4th note filled on the first and a 2nd note filled at the end with 32nd notes (depending on the context this can also mean a buzz/open/closed roll). You can listen to the exercises on the same page. So you can think of fill this space with the slashed speed of notes until the end of the tie.

The only exception might be muting the cymbals and letting them ring which has no standards but you can use them to differentiate whenever necessary.

The dots, as far as I know, is identical to the sheet music usage but is done mostly to avoid excessive rest notation. Since a 4th and a 8th rest is the same as a dotted 4th for drums.

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Cymbal muting can be a matter of interpretation, but there are some common practices. Without any other notation (or interpretation or direction from the conductor) the cymbal is muted when its notated time is up. Composers often use "Let ring/vibrate" or "l.r./l.v.", or an "eyelash" - a note tie that leads nowhere - to indicate the instrument should be left to decay of its own accord. –  Faelkle Jul 24 '13 at 12:23
    
@JohnHawksley That might be the case for classical music but for jazz/rock etc. unfortunately there are no such conventions –  user1306 Jul 24 '13 at 12:34

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