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In modern notation, a flat lowers by one semitone, a sharp raises by one semitone, so from Eb to E# is a whole tone, or two semitones. There is no interference from the fact that E# is F-natural: when counting semitones, enharmonic naming doesn't affect anything.

It has not always been this way: In the pre-classical period, a flat was used to "undo" a sharp and vice versa, and the natural sign was actually a note name. Additionally, in the Renaissance, an accidental might signal a change of mode, which might last for some period or be immediately changed back. Bar lines as they are now known did not have the same function, and were not an automatic indicator of the end of the accidental's influence. Additionally, if a sharp or flat was applied to one note, its octaves would have to be explicitly given a sharp or flat if they were intended also to be affected. This included key signatures, so modern musicians seeing original manuscripts or prints might be surprised to see a signature with five sharps: two of the sharps would be for one octave, two for the other, and the third (because its octaves were outside the staff range) would only refer to that note, and its octaves would need explicit accidentals.

Since the late-baroque, early-classical era, though, the meaning of sharp, flat and natural signs has been simplified, with the natural (usually) removing the effect of any number of previous sharp and flat signs (including key signature) for that note for the duration of the measure, or until another sign causes a new change. It is a much simpler system, but along with this has come regular usage of double sharps and double flats, requiring double or triple naturals to completely neutralize them, and proper "spelling", especially in circle-of-fifth progressions, can sport double sharps and flats, as well.

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