theory - Why do we call D# Eb, but also call Db C#?

[CGremlin]

and there just isn’t a way to notate the key of D# major

There is, it's just kinda ugly:

I've not ever run across anything in D#, but I *did* have one occasion many years back where I was given a part that was in Fb, with a B-double flat indicated in the signature. I have no idea what (aside from pure misanthropy) would possess the arranger to choose that key instead of E.

 

[Papageno]

Suppose you write some piece in E major (key signature = 4 sharps), which modulates in the neighbour key of B major (5 sharps).

The alto sax part (transposing instrument in Eb) will have 3 more sharps in its key signature, i.e. it will be written in C# (7 sharps) and the modulation will be in G# (key signature = 1 double sharp and 6 sharps).

 

[CatOnTheBass]

Suppose you write some piece in E major (key signature = 4 sharps), which modulates in the neighbour key of B major (5 sharps).

The alto sax part (transposing instrument in Eb) will have 3 more sharps in its key signature, i.e. it will be written in C# (7 sharps) and the modulation will be in G# (key signature = 1 double sharp and 6 sharps).

You're right of course, but I have two comments: First, when it comes to transposing instruments, it's fine to use sharp keys for some and flat keys for others. So you could write that alto sax part in Db (5 flats) modulating to Ab (4 flats) and the saxophonist would thank you for it. Second comment is that actually one of my favorite pieces to play on piano is written in C# (so fun to play with all the black keys!) modulating to G#. But the composer doesn't write a new key signature for G#, with a double sharp in the key signature. Rather, it stays in the same C# key signature, and Fx (F double sharp) accidentals are written as needed. This is pretty standard practice for notating modulations, check it out:



(PS I'm a pretty bad pianist, so I only play the prelude not the fugue, and much slower tempo than Gould!!)

 

[The Bass Clef]

There is, it's just kinda ugly:
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I've not ever run across anything in D#, but I *did* have one occasion many years back where I was given a part that was in Fb, with a B-double flat indicated in the signature. I have no idea what (aside from pure misanthropy) would possess the arranger to choose that key instead of E.

Technically, yeah. I guess I should have said there isn’t an widely accepted way of notating D# since it doesn’t fall into the circle of 5ths (and because it’s much messier and confusing to sight read than Eb). Neither of us have even run across anything in D# because it’s like taking the long way around for no reason.

 

[CGremlin]

Neither of us have even run across anything in D# because it’s like taking the long way around for no reason.

For sure, as was the Fb piece that I encountered. It's mostly an academic exercise with no practical value. You can continue around the circle a full time again, double sharping/flatting notes, but it's pointless.