How to figure out key from key signature?
I know the major scale formula so can easily work out the signature by building the scale from tonic, but I've never learnt it the other way... How to know the key/scale from the signature of sharps/flats.
I know there's links to circle of fifths but for practical purposes do most people just learn by rote, is this generally regarded as the easiest way or are there handy tricks?
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There's another way based on rote memory: learn how many sharps and flats are in each key. For example, when you think of the key of A major, you should instantly know that there are three sharps. And when you think of four flats, you should instantly know it's Ab major or F minor.
Now, you can just see how many sharps or flats are the key signature and know what key it is.
I don't think that memorizing the number of sharps and flats is necessarily any better a way of decoding key signatures than Laurence's excellent answer. However, in addition to giving you a way to decode key signatures, it gives you another way to communicate with musicians in an ensemble setting. One way that musicians communicate keys is to use their fingers, where the number of fingers held up shows how many sharps are in the key, or the number of fingers held down shows how many flats are in the key. If someone holds up one finger and you've memorized the number of sharps and flats in the keys, then you know the tune is in G major (or E minor).
Last sharp in the key signature is the leading note (7th) of the major key. Last flat is the 4th. Or last but one is the tonic.
So three sharps - F, C and G - is A major. G♯ is the 7th note of A major.
Four flats - B, E, A and D - is A♭ major. D♭ is the 4th note of A♭ major. Or, just go back one from D in the list!
When taking 'grade' theory exams, we would jot 'FCGDAEB' and 'BEADGCF' (notice that the order of sharps is the order of flats backwards?) at the top of the page. After a bit, of course, you just know the keys and signatures.
Also, of course, Father Christmas Goes Down All Escalators Backwards and BEAD Gives Catholic Faith (questionable theology, but memorable).
And for the relationship between relative minor and major:
A Major did some looting and left the army with a bit of money, which he invested in a mine. He employed a Miner to work it. It was quite a shallow mine, so for the Major to visit the Miner he had to go down just three steps... (including three letter names, should there be any confusion between A♭ and G♯)
In my experience, most fluent musicians know their key signatures by rote, because it's such a fundamental concept that lays the foundation for almost everything that comes after it. Not knowing key signatures—or having to actively think through them every time—strikes me as akin to doing algebra while still using your fingers for basic addition. ("Oh, I've seen it!" you say. Yes, and we see students trying to write applied chords without knowing key signatures, too. Alas!)
For major key signatures, there are rules for both flat-based and sharp-based key signatures:
If the key signature has flats, then the second-to-last flat is the tonic pitch of the major key. Traditional flat key signatures always follow the order of the flats—BEADGCF—and so a key signature of four flats (B♭, E♭, A♭, D♭) is the key of A♭ major since that is the next-to-last flat.
If the key signature has sharps, then the final sharp is the leading tone; we go up one diatonic half step from that sharp to find tonic. Conveniently, the order of sharps in traditional key signatures is just the reverse of the order of the flats; in this case, FCGDAEB. If a key signature has two sharps—F♯ and C♯—we move up a diatonic half step to D to find the tonic.
There's just one caveat: it isn't immediately obvious how these relate to two key signatures, so we just need to know that C major has nothing in its key signature and F has one flat (B♭).
We can easily reverse engineer this to notate a key signature. If the tonic is B♭, then we want this to be the next-to-last flat, so the key signature will have (again, follow the order of flats) B♭ and E♭.
If you're looking for minor keys, that one is a bit trickier; I always recommend my students think of the minor tonic as being scale-degree 6 of the major scale. This forces them to think through their major scales, and it also prevents the inevitable errors they encounter when they just try to "go down three half steps" (they always seem to do four, or to misspell the minor tonic enharmonically).
This is quite easy if you know the movable do:
The last sharp is Ti, the last flat is Fa. If there are more than one flat the second last flat is do.
Ti and Fa are the leading tones, so the Do and La can easily be defined. Whether major or minor ... this can be decided by identifying the last note: If this is La and there are additional accidentals altering So and Fa to Se and Fe we are in minor.
Most players will learn them by rote, which becomes the quickest way to recognise them when facing the music.
My suggestion is to use mnemonics - just like we do when learning the line notes on treble clef - 'Every Good Boy Deserves Favour/Football/Flogging...'
I use the same idea to enable students when memorising the string names on guitars. A memorable one is Elephants And Donkeys Grow Big Ears.
So, taking the key names from the circle, we have C G D A E B. With the possibility of omitting C, as it has neither sharps nor flats. Obviously there's F♯ and C♯ to follow if needed - maybe not in early stages of learning?
A six word sentence will do the job - one personal to you, or a silly one will suffice!
Just like the note names for spaces on the treble clef - FACE - we have (F) BEAD (G) to maybe help, or, just make up another mnemonic.
It's a bit of fun, maybe aimed more appropriately at younger people, but it works: still remember Richard Of York Gained Battles In Vain?
Otherwise, by rote generally works - we learned the alphabet that way, didn't we? And there's more in that!
Trying to put Albrecht Hügli's answer in different words (because not everyone is fluent in solmization):
If you have sharps, the root note of the major scale of that key is a semitone over the last sharp:
f# -> semitone over f# -> G major
f#, c#, g#, d# -> semitone over d# -> E major
If you have flats, the root note of the major scale of that key is the flat before the last flat (i.e. the root note is a fourth below/a fifth over the last flat):
b -> fourth below -> F major
b, eb -> flat before the last flat: b -> B major
b, eb, ab, db, gb, cb -> flat before the last flat: gb -> Gb major
and so on :)
The most painless way to "learn key signatures by rote" is just to play a lot of music in different keys.
If you mostly play genres of music that is only written in a few keys, then not "instantly" recognizing the others isn't going to limit you much anyway.
As a keyboard player, the only ones that make me stop and think a bit (after a few decades of playing!) are the minor keys with 5 or 6 sharps or flats. Seven sharps/flats are easy to remember, because they the same letter-name as none, i.e. C or A.
But hey, when was the last time you saw a piece in D sharp minor anyway - so does it really matter if you have to work that out by counting on your fingers? :)
"Learning key signatures" doesn't always help in any case, if somebody puts a score like these in front of you. Yes, triple flats and triple sharps are a thing. And for extra credit, notice the spectacular typo on the last chord on the line of the first example...
The easiest way is to just learn the Circle of Fifths and the Circle of Fourths.
Starting in C, the Circle of Fifths will give you the sharp keys, while the Circle of Fourths will give you the flat keys.
So the sharp keys are G, D, A, E, B, F#(Gb)
And the flat keys are F, Bb, Eb, Ab, Db, Gb(F#)
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