there was a thread about this recently.
if you can find any music theory resource, this is the topic called enharmonics, or enharmonic spellings.
the two notes *are provably different*
however,
in an equal-temperment world (any fixed pitch instrument such as piano, string instruments with frets) they sound exactly the same.
that being said, if your asking why we use one flat name for a half step down, and a sharp name for Db/C#, it basically goes back to C# being a more comming spelling of that note than Db.
Db is the forth flat to appear in order (Ab major scale)
C# is the second sharp to appear (D major scale)
you can absolutely use either spelling interchangeably, in this specific scenario.
the spelling becomes a matter of correct/incorrect once we incorporate musical context.
if you can find any music theory resource, this is the topic called enharmonics, or enharmonic spellings.
the two notes *are provably different*
however,
in an equal-temperment world (any fixed pitch instrument such as piano, string instruments with frets) they sound exactly the same.
that being said, if your asking why we use one flat name for a half step down, and a sharp name for Db/C#, it basically goes back to C# being a more comming spelling of that note than Db.
Db is the forth flat to appear in order (Ab major scale)
C# is the second sharp to appear (D major scale)
you can absolutely use either spelling interchangeably, in this specific scenario.
the spelling becomes a matter of correct/incorrect once we incorporate musical context.