Could it be linked to key signatures? I mean, the B-flat trumpet plays the C-scale fingering in the key of B-flat correct? So I would say it is linked to scales.
Ah, yeah, beat me to it. An A-sharp instrument would play the c scale as an A-sharp minor scale. Completely different from a B-flat major scale.
Well, no, it would play an A-sharp major scale, which sounds the same as a B-flat major scale (assuming, of course, equal temperament). It just happens to have 10 sharps (rather, it has three double sharps in addition to the seven regular ones).
Of course, the written scale would still be C.
It’s called a Bb trumpet rather than an A# trumpet because Bb is a much more common note than A#. Yes, they represent the same pitch, but the name of the note is different. This is called an Enharmonic difference - same pitch, but different names. Bb occurs in the key of F, Bb, Eb, Ab, Db, Gb and Cb major. A# occurs in B, F#, and C#, making Bb much more common. There is no key of A# because that key would require 10 sharps, meaning there would have to be an Fx, Cx and Gx in the key signature. (x is the symbol for double-sharp.)
There’s an order to flats and to sharps - in flats, it goes Bb, Eb, Ab, Db, Gb, Cb, Fb. For sharps, it goes the other way around - F#, C#, G#, D#, A#, E#, B#. If you want to know why, I’m happy to explain.
Picture it this way - the trumpet is named for the note that sounds when the open horn is played with minimum lip tension. A Bb trumpet plays a Bb, a C trumpet plays a C, a piccolo trumpet in D plays a D. The valves send any note through a length of tubing to lower the note - usual arrangement is the middle valve lowers the pitch by one semitone, the first valve (closest to the mouthpiece and played with the index finger) lowers the pitch by a whole tone (2 semitones) and the third valve, furthest from the mouthpiece and played by the third finger, lowers the pitch by a minor third (3 semitones). You can already see that the minor third down can be played with 1 and 2 or with 3. Major third down (4 semitones) - 2 and 3. Perfect fourth down (5 semitones) - 1 and 3. Augmented fourth/diminished fifth (6 semitones) - 1, 2 and 3. So, depending on which instrument you grabbed, we’ve either gone through how to play the concert pitches Bb through E natural, C through F# or D through G#. (concert pitch is what you call the actual pitch which sounds as opposed to the pitch that is written out for you)
Tell you what - get your hands on a good fundamentals textbook like the Barbara Wharram and go through it at the piano for a couple of months - it’ll make all of this much clearer to you.
Oh, right, sorry. Although, seriously, does anybody actually write anything in A-sharp major? That would just be needlessly complicated. Then again, composers are an interesting breed.
Thank you for your explanation and the book recommendation. I’ve avoided any music theory textbooks becaues they’re far too complicated for someone who knows as little about theory as I do. Theory textbooks assume too much background knowledge (I don’t have knowledge of any particular instrument except for a few guitar basics).
I still have the copy of Wharram I bought in 1978 for a music class, and as an enthusiastic amateur, I still refer to it. One of the best theory books, I’d have to say. No previous musical experience is assumed–it leads from the very basics up to the point where you can understand questions and answers such as this.
Thank you. This is one of the things that seems to make transposing instruments far more complicated than they need be, and it’s one which I wish was tattooed onto my pupils’ hands so that they don’t keep unlearning it. Along with ‘chord V is always major’ and ‘an imperfect cadence is anything to V’.
Well, an A sharp major scale would go:
A sharp
B sharp
C double-sharp
D sharp
E sharp
F double-sharp
G double-sharp
A sharp
That’s going to be an awkward key signature! *(Edit: as Le Ministre de l’au-delà has already said.) Even the minor is not much better, with seven sharps, and so in both cases the principle of enharmonic keys comes into use, and the music switches from a ‘very sharp’ key to a ‘very flat’ one (quotes indicating my choice of words rather than a generally-acknowlodged description 
)
Fair enough, although I feel the need to point out that there’s a difference between ‘rudiments’ and ‘theory’. The Wharram (and lots of other books just like it) is meant to be an introduction to the most basic concepts of how music is written down and what are the common terms used to describe what goes on in music. The hardest thing about all of it is that it seems like there are multiple words used to describe every concept, and at the same time, the same word gets used to describe more than one concept. ‘Theory’, in most people’s usage, is more about the next stage - where ‘Rudiments’ answers the question ‘what is a chord?’, ‘Theory’ answers the question ‘what is the relationship between this chord and all the other chords in the composition?’
I have fond memories of the summer between Grade 8 and Grade 9, when I went through the Wharram and the ‘Golden Encyclopaedia of Music’ during the evenings at my sister’s ranch. We were too remote for radio, the record collection was appalling (love her dearly, but my sis has always had execrable taste in music!) and the only instrument was a Yamaha classical guitar that couldn’t play above the fifth fret without a set of vice grips. Still, that was the summer I realized it was my destiny to be a theory geek.
So, if you have any interest in it at all, jump on in! And if you have any questions, there are a bunch of us here more than willing to talk about it. (I’m always on the lookout for people who want to talk about any aspect of music - my wife’s eyes glaze over after the second sentence.) The little known secret is that every time one has to answer a question, it reinforces one’s knowledge - just like mathematicians who seek new and different ways to prove the square of the hypotenuse.