#1
What can I name the out of scale notes using degree names? I mean, for example - I have C-major scale in which D is a second scale degree. But how can I name for example D-sharp which is out of scale?
I'm asking about scale degrees, not about intervals.
#2
well if you re name it it would be Eb which would now turn it into a minor sound...but it would just be a raised 2nd, or augmented. I know you are asking for scale degrees and not intervals but they are named the same thing
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#5
In application more often it would be an Eb, a b3 accidental. If it was a D#, it would be a #2 accidental, but the latter situation is far less common than the former.
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#6
So, I'm confused: raised, augmented or accidental?
And, plus, how to write it down shortly? I would want to know selling as well as writing.

I know, that in some pieces e.g. there would be f sharp, and in the another g flat. And probably the same issuse is as far as scales degree are concerned. But I want it in some general context.
Let's say we've played chromatic scale from C3 to C4 and have a background to it - chords C-major, F-major, G-major, C-major which tell us, that is key of C-major. It doesn't matter that it would sound bad. But the key is C-major, and we play all the notes in it.
In this context, how can we name each note, which is not in the scale C-major - using scales degree of course?