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This is in key of F# major or D# minor, there is a D# in this key, but the D# in this picture has a double sharp sign beside it, so is it a D#(##) (which is an F)?enter image description here

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This is just D doublesharp, which is enharmonic to E.

The trick is that key signatures are not additive. In other words, any accidental added to a pitch is considered to be its own construct, not something in addition to what's already given in the key signature.

As such, this is not D♯ that is then sharped twice again, but rather just D doublesharp.

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    Yup. The cool thing about this is that any note on its own with an accidental modifying it is completely unambiguous, regardless of key signature. F with a sharp in front of it is always F♯, even if I'm in a key like C♭ major where F♭ is 'the diatonic F'. That's also why some complicated music for orchestras will have no key signatures at all and give all notes (with a couple obvious exceptions) an accidental to facilitate sight-reading. – user45266 Feb 28 at 3:34
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    Worth mentioning that although it's enharmonic to E nat., and E is # in the key sig., it stays as Dx. Also, it could have been written on the E space, with a nat. sign. This does look like someone's homework. – Tim Feb 28 at 6:53
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I would like to add to Richard's answer that if you wanted to "undo" the double-sharp (maybe you need a D sharp later within the same bar) you would have to put both a natural sign and a sharp sign in sequence before the note head. For example, take a look at bar 46 from the Prelude in C-sharp Minor, BWV 873 from the Well-Tempered Clavier, Vol. II, by Johann Sebastian Bach (BGA XIV, ed. 1866): enter image description here The key signature requires an F sharp (here it's marked on the lower F in the treble, this being an old-fashioned edition), an F double-sharp is needed in the top voice, and then the alteration is canceled on the next F by means of a natural-sharp combination. You can think of the "X" symbol on the first F standing for two sharps, and the natural sign on the second F canceling only the first sharp, while the second sharp is reiterated for good measure.

The same should be done for double-flats.

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