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For example, in E major (E F# G# A B C# D#) it will be named D# not Eb because the name 'E' is already being used, whilst in Ab major (Ab Bb C Db Eb F G) it will be called Eb and not D#, because a type of 'D' has already been used before you get to that enharmonic equivalent. It's to make sure you only use each note name once.
Check the circle of 5th/4ths.
However if you're not that pedantic, either is fine :-)
Both are correct descriptions of the same note. However, in terms of which term to use & when, it depends what key you are in.
When writing out major or minor scales, their shouldn't be alterations of the same stated note in the same scale, so you wouldn't use C and C# in the same scale. For example, in E major, there is already an E natural, so therefore you would use D#, not Eb, or if you are playing in Bb major, there is already a D natural in the scale, so you would use Eb not D#.
I'm sure that's right for accidentals, but in a diatonic key you have precisely one of each note name present in its scale. In the key of F#, the leading note is not F (because you already have an F#) so you have to call it E#. You must also call it E# because if you called it something else you would be missing an E.
In the OP, D# is right in keys of E, B, F#, C#. Eb is right in keys of Bb, Eb, Ab, Db, Gb
Seriously: If you value it, take/fetch it yourself
You'd never use D# in the key of Bb, for example.
As a general rule if you are in a key using flats then use flats, if you are in a key using sharps then use sharps and you won't get into too much trouble if playing western music.
As keys, Eb is much easier than D#.
Eb has 3 flats.
D# has 5 sharps and 2 double sharps.
If someone gave me a piece of music in D# I'd make a brave attempt to play the music, fail and then punch the composer.
Studio: https://www.voltperoctave.com
Music: https://www.euclideancircuits.com
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Football is rubbish.
Not necessarily, but it gets complicated. If we are considering cases where the D# or Eb is a member of the scale because it fits the key signature being used, or is a member of the temporary scale/chord then it should follow the convention of the particular scale or key signature, in that you would keep the scale as being described as a continuous sequence of letters.
For example if you were in the key of C major but for some reason temporarily used C minor in a transition, you'd describe a chord or notes used in a melody, according to the scale:
C D Eb F G A Bb C
because within that scale, D is the 2nd and Eb is the minor 3rd.
HOWEVER
if that transition didn't fit the natural scale AND included a diminised 5th, it could be more appropriate to describe it as such, rather than an aug 4th. For example in the key of E major (4 sharps), one might use Bb and not A#. So, it depends on the context of that note within the (possibly temporary) scale or nature of the passing note(s) at the time. It might be used in a chromatic run, where one is no longer fitting in with any "7 note scale" so the convention of keeping the scale to a sequence of letters is out the window! There are no strict rules on which notes must be "raised one semitone", ie sharpened either with the sharp accidental, natural accidental (on a flat note within the key signature) or even double sharps; and which must be lowered one semitone. One might try to 1) write the accidentals to correspond with the temporarily different key signature, 2) write the accidentals to minimise the number of them, 3) write them to offer most clarity to the performer (ie on a downward run, flatten things, on an upward run, sharpen things).
Probably best explained/demonstrated with an example!
There are exceptions, which is why I couched my suggestion with 'as a general rule'.
I don't really need a lesson in harmony when I've been teaching music for, what, over 20 years now.
Studio: https://www.voltperoctave.com
Music: https://www.euclideancircuits.com
Me: https://www.jamesrichmond.com
Football is rubbish.
Supportact said: [my style is] probably more an accumulation of limitations and bad habits than a 'style'.