#1
Hi,

I recently read that (Db, C#), (Gb, F#), and (Cb,B) were the three enharmonic relationships.

Could someone please explain why?

Shouldn't (E#,F) be enharmonic or (A#, Bb)?
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#2
Depends on the context, many factors including key, chord, notes already in the bar. For example, hypothetically if you were using C# major instead of Db major, you would spell the F as an E# because the interval between a C# and a F is a diminished fourth and not a major third. E# and F are enharmonic.
FYI before equal temperament came in, none of these pitches were enharmonic.
#3
Where did you read that and what was the context?

Enharmonic notes are just notes that refer to the same pitch, even if they may have different names.

So all these examples are enharmonic, as are many others.

Unless this is something in a specific context that I don't know about
#4
Sorry, I read the question incorrectly and answered a different question. Yes they are all enharmonic.
#5
only 3? what lol

there are plenty, esp when you get into double sharps/double flats territory.
#6
E# and F are enharmonic, A# and Bb are enharmonic as well.

The reason for using sharps or flats comes from the key signature. Say you have a G major scale with the notes:
G A B C D E F# G

When you spell it this way, each pitch has a unique letter distinction, and it adheres to the key signature that specifies G major, which is one sharp: F#.

You could spell it as G A B C D E Gb G, but then you you have two 'G' note values which is confusing, and it no longer jives with the key signature of G major, which is one sharp, F#.

If you wanted to do another scale, heres D major spelled correctly according to it's signature: 2 sharps, F#, C#:

D E F# G A B C#

Again notice each pitch has a unique letter value.

Then here it's spelled with enharmonic equivalents that make it confusing to read:

D E Gb G A B Db