E# and F have the same frequencies, at least in 12-Tone Equal Temperament tuning.
In other tunings E# and F might have slightly different frequencies than each other, possibly depending on the type of scale and the tonic or base (starting) note of the specific scale (musical key).
So in many cases it doesn't make much difference whether you refer to a particular note (key on the keyboard) as one name or another, and likewise it doesn't make much difference whether you refer to a particular chord by one name or another.
But aside from the fact that certain notes which have identical frequencies in Equal Temperament tuning can have different frequencies in other tunings, the main reason why you would normally choose one name or another for a given note or chord has to do with the specific musical key that a piece of music is written in.
For example, suppose a piece is written in the key of F# Major. That would mean there are six sharps in the key signature, because there are six sharps in the F# Major scale:
F# (+2) G# (+2) A# (+1) B (+2) C# (+2) D# (+2) E# (+1) F#
(The numbers in parentheses above show how many semitones we add to a given note to reach the next note in a major scale.)
You might notice that among the notes listed above we have an E#. If E# and F have the same frequencies (in 12TET tuning), then why didn't we call that note an F? Simply put, we want each of the seven notes of a scale to have a different letter designating it-- that is, A, B, C, D, E, F, and G. We don't want to refer to the seventh letter of the F# major scale as F, because that would mean that the first and seventh notes would be designated by the same letter-- F# and F-- even though one of them is sharp and the other one is natural. So we call the seventh note an E# instead of an F so that all seven notes of the F# major scale have different letter designations.
Now, if a particular piece of music were written in the key of F#, we would generally want to refer to any chords in that piece of music according to which notes in the key signature are sharp, or flat, or natural. For instance, we've already noted that the key of F# Major has six sharps, so if a piece of music written in the key of F# Major has a minor chord in it that starts with the key on the keyboard that's usually called F natural, we would be far more inclined to refer to that note as an E# rather than an F natural, and consequently we would refer to the minor chord as an E# Minor chord rather than an F minor chord.
On the other hand, if a piece of music were written in a key that refers to that particular note as F rather than as E#-- say, written in the key of C Major-- then we would be far more inclined to refer to that same chord as an F Minor chord rather than an E# Minor chord.
It isn't always that simple, because musicians don't always obey the rules laid down by academia when composing music, so you might see things such as pieces of music that contain accidentals-- that is, notes that are natural when they're "supposed" to be sharp or flat, or that are sharp or flat when they're "supposed" to be natural, as determined by the key signature-- and in some cases the composer might even decide to temporarily change to a different key signature for certain passages!
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