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I mean, is there a difference between sound of keys. Like, what is the point of writing some melody in a particular key if the melody would be the same if you, for example, raise all notes by one tone?

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Yes, sort of, it depends.

Depending on the instrument, different chords may have different voicings. Think of a guitar, for example - an E major (at the 1st fret) would typically be voiced as EBEG#BE, whereas an A major would typically be EAEAC#E. It's not strictly a transposition up a 4th.

For a single-note instrument, chord voicings wouldn't matter, but the instrument may respond differently and have a different timbre at different pitches. A trumpet, for example, seems to have fewer overtones and sounds less "brassy" at higher notes compared to lower notes (that may partially be the player's technique, but that's a consideration when deciding where your melody should be played).

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