I am an intermediate, self-taught dilettante musician (piano). Thus I know enough to know I don’t know enough. My question is how to deal with (“count off”) the double sharp, “x”.
In the key of C Major a double sharp on, say, F means it would be played G, two half-tones from F, right? But what about in the key of, say, E major where the F is sharped (sharpened?)? Would F double sharp still be a G or would the fact that F is sharped throughout the piece mean you “count from” F# two half-tones and wind up playing G#?
For the curious, I am trying to learn Beethoven’s “Moonlight” – first movement. It’s in E Major (4 sharps) and has some goddamned double sharps on the third page of my edition. Playing it either way sounds OK so there’s no dissonance to guide me (although L. v. B. often sneaks in dissonance so I can’t really trust that, either.)
Count up two half-tones. That’s the whole purpose of the double sharp-- to make it so that note is raised by two half tones regardless of what key the music is in.
(No, I can’t explain why it isn’t just written as the higher note–although the fact that it’s in E major and has lots of sharps already probably has something to do with it).
A sharp or flat accidental in the score refers to the the natural note. A single sharp in a key where the note is already sharp serves either as a reminder that the note is a sharp (which can be useful) - or resets the note to sharp after it has been annotated with a natural earlier in the bar. Thus a double sharp is the natural note plus two semitones.
Another way of thinking about it is that the key signature supplies implicit prefixes to every note in the score, and adding an accidental to a note that already has been so annotated is simply making visible the notation that was already there. So in order to get a two step accidental to an already sharpened note you need to supply the implicit sharp and then an extra one. (This isn’t quite perfect, an accidental does not apply to the same note in other octaves, but the key signature applies to all octaves.)
The reason you write a double sharp like this is to emphasise that the note is out of the scale. E major does not contain G natural. It must therefore be written as an accidental. F## is curiously not the same note as G natural. Only in equal temperament is this so. But harmonically they are not the same note, and the rules for score notation reflect the harmonic underpinnings of the scales.
Would it also come up it “extreme” keys (with lots of sharps in the key signature)? So in the key of G# major (eight sharps, so necessarily contains a double sharp), the note below G# would be written F##, not G. Otherwise the scale would end in E#-G-G#, but this way it’s E#-F##-G#. More technically, you get there by moving F up twice as you move around the circle of fifths. In the key signature there will be two sharps on F.
Now, why it would be written as G# major rather than Ab major is another question, but similar to the above. As I understand, it could come up in a piece that changes key, depending on how you get there. And, in the same sense used above, G# is not harmonically the same note as Ab.
If you’re speaking of the key signature, then, no, there is no key of G# Major. You can only have 7 sharps or 7 flats in the key signature. Rather than have 8 sharps in the key signature, it would be written in Ab Major which has 4 flats.
There is a key of G# minor, however - it is the relative minor of B Major, and it has 5 sharps. You could also use Ab minor, which is the relative minor of Cb Major, but it would have 7 flats.
My complete Beethoven is on someone else’s piano at the moment, so I only have the first movement - BUT the first movement is not in E Major, it’s in C# minor. That’s why there are all the B#s in the piece, because to get a Dominant chord, the B natural in the key signature has to be raised a semitone. It might make more sense to you to play a C# harmonic minor scale and a C# melodic minor scale before playing through the piece, and you’ll see better how it works.
And for a side note, there is a minor 7 flat 5 chord in bar 41 of the first movement. Thought minor7flat5 would find that amusing…
If it helps - a theory prof of mine used to describe enharmonic equivalents as being like letters of the alphabet that produce the same sound. Yes, ‘c’ and ‘k’ can be used to represent the same sound, but ‘cite’ and ‘kite’ are different words, and ‘kat’ is just plain wrong. ‘Carl’ and ‘Karl’ sound the same, but they spell their names based on where they’ve come from and what their families liked. It’s the same with notes - in equal temperament, Abb, G and Fx
sound the same, but they’re going to move in different directions…
Key signatures are really an evolved convention, and arguable continue to evolve. The convention is that there are 15 basic signatures (counting no sharps of flats as one). This enables you to denote all the diatonic major and minor keys. Note that there is overlap here, and for those keys where there are two choices, either can be used.) I would suggest the main reason one is not concerned with the difference between modulating to G# major versus Ab major is that key modulation is really something that came in with equal temperament anyway, and rather than get totally lost in the intricacies of the temperaments, one simply goes with what works, and doesn’t turn the score into a trainwreck. So, whilst convention does obey some aspects of the correct harmonic underpinnings of the scales, the overriding sentiment in the rules is still rooted in equal temperament, and where things look like they will get messy, that will overrule other considerations. So, no eight (or more) sharp or flat key signatures, and notating change of key follows equal temperament, not harmonically consistent rules.
Talking to a few professional musicians I get the feeling that it really doesn’t make a great deal of difference anyway. Those instruments that are fretted and hence intonated to a fixed temperament just play what is scored. G# and Ab are indeed the same note. (Ignoring insanities like just intonation.) Those musicians playing unfretted instruments will move to the correct key automatically by ear. And play perfect intervals in that key. At least within reasonable limits. No matter what happens there are always times where the harmonic relationships become mutually incompatible.
It all starts to fall apart when getting into more harmonically complex ideas anyway. A large fraction of the last century’s more interesting music simply doesn’t fit into the rules.
Clumsy, theoretical, A flat Major is easier to notate and read and sounds the same in an equally-tempered instrument, but there’s no reason why we can’t have a G# Major scale.
Don’t get me wrong - I’d be lucky to pass the entrance exam to study with you - but did you ever write anything in G# Major? And if you did, what did the copyist say to you? Yes, I’ve seen passages that played through a G# Major scale in accidentals, but in the key signature? Never seen it…
And note, of course, that in the perfectly cromulent key of G# minor, the leading-note (ascending in melodic minor, both ways in harmonic minor) is F##.
I don’t doubt that it’s not done, but what’s to stop me from writing it that way, with 8 sharps? And wouldn’t that be appropriate if I modulated from C# major to Ab/G# major? And wouldn’t a violinist play G# differentlly from Ab (off by a Pythagorean comma), just as he would play F# differently from Gb? The same would go for all the other notes in G# vs. Ab major, including F## vs. G.
If by “out of key” you mean “always written as an accidental”, then yes, absolutely. G minor has two flats in the key signature and the leading F# accidentalised, always, and so on for the other minor keys.
You can only have 7 sharps or 7 flats in the key signature because there are only 7 named notes: A, B, C, D, E, F, G.
Just to clarify this for those who might need it: C# minor is the “relative minor” of E Major, and thus has the same key signature (4 sharps).
To address the OP directly, I can’t speak specifically to Moonlight Sonata, as I’ve never looked at the printed music, but I know one good reason for using a double-sharp in other instances: it saves ink, and thus printing costs, and also saves the composer some writing. I’ve seen situations where the composer wants to use the “natural” version of a note (say, G natural) multiple times in a bar that also includes the sharp version as indicated in the key signature (G# in this case) multiple times. Without the double-sharp, the composer and printer would need to print a “natural” sign in front of every G-natural in the measure, and then print a sharp (#) in front of every G# in the measure. Basically, taking the already-sharp note and “unsharping” in, then “resharping” it, then "unsharping it again, then "resharping it, etc. etc. Not only is that more work/more ink, it also makes the score look very cluttered, and harder to read.
It is worth noting that despite much misunderstanding on this, the Well Temerered Klavier is not referring to Equal temperament. The well tempering is a “good” tempering, one which still yields different scale colours due to the different placement of the less well tempered intervals in the scales. So here Bach was very clear, for the tempering of his instrument G# major very likey had a real difference in intent from Ab major. Possibly the instrument was tempered such that the actual scale, as played, really was much closer to G# major than to Ab major. (Somewhere I have some more info on this, but deeply buried, and not close to hand.)
I should also note, that despite what people are taught, pianos are not equally tempered. Indeed very few instruments are. The actual tempering depends upon the physics of the piano, especially the physical lenght of the strings, and the choices made by the tuner. Most pianos are tempered to sound reasonably consonant with music played in the more common keys.
Says who? You can’t put a double sharp in a sig? Why not? Read the Wikipedia article linked earlier, which agrees with me, and includes a double sharp in this key sig exactly as I did (and no, I didn’t write the Wiki article).
Maybe you should take a look at some Bartok or other avante-garde composers, who have used non-standard key sigs, including sharps and flats on non-traditional scale tones, and even sharps and flats in the same sig. Compared to that, a double sharp is pretty ordinary.