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What shortcuts are there to figure out the major and minor scales that contain a set of multiple notes?
e.g., Which major and minor scales contain: C,F,D ... or perhaps C and G#? ... or perhaps C, G, A#? (or, of course, if no scales fit the bill).
I am asking because I am starting to write melodies and would like to know how the pros here figure out the key AFTER THEY WRITE THE MELODY.
I think that you might be approaching this the wrong way.
If you are writing a melody then just write it. The "tonal centre", if it has one, should become obvious once you have written it.
Taking the approach that you need to know what key contains the notes you intend to use seems to be applying restrictions before you have even started. It might be useful if you are improvising in a group but much less so when you are composing something.
There are many scales, both major and minor, that contain, as your example, C F and D. Knowing which they are isn't going to be a great deal of help in writing your melodies. If anything, your hearing will guide you to the next note/s - which may - or may not - still belong to that scale you considered. It's somewhat of a red herring.
A lot of pieces don't stay in one key (with one set of scale notes) from top to bottom anyway, so even if you know the cast iron answer - which will be more than one key, so confusing the issue anyway - it won't be of much help if your melody modulates.
It may be useful to think initially in a partcular key, with its particular scale notes. The melody will then have a 'home' thus somewhere to start from and finish at, sonically. The way to use that information is to glean it in the first place. Know your scales! All of them - major, harmonic/melodic/natural minors.
What you're almost saying is here's the answer, what's the question? Go with your ears, and work out what the notes are after. That will give you the key/scale notes, most likely when you feel the melody has come to rest. then might be a better time to 'establish' a key, then the next part may be easier to write in that knowledge. But maybe not. That's music for you!
You're really posing two questions:
How can you identify scales that contain specific notes
The answer is to learn the order of sharps and flats in key signatures.
Sharp signatures have the order F, C, G, D, A, E, B and flat signatures are the reverse: B, E, A, D, G, C, F. Knowing that, you can look at a set of notes and know which major and natural minor scales will contain them.
The first example, C-F-D is pretty straightforward. The first sharp in sharp key signatures is F, so you can't be in any of the sharp keys with an F natural. The fourth flat is D, and you've got a D natural - you can't have four or more flats. That leaves you with four keys: C (no sharps or flats), F, Bb, and Eb (1, 2, and 3 flats)
Your third example is C-G-A#. Because all key signatures with A# also have C# and G#, there aren't any sharp keys that fit. But what if it's the enharmonic Bb? All flat key signatures have a Bb, and any with four or fewer flats have C and G natural... so you could be in a key signature of F, Bb, Eb, or Ab.
Your second example, C-G# can't fit sharp key signatures, but it can fit flats if it's the enharmonic Ab. But this brings up something else: there are more scales than just the major and natural minor - C and G# are both in the A harmonic minor, and the A melodic minor. And a bunch of other scales as well. But it's not really worth working out a method for identifying all of the scales because of your second question:
How do you figure out the key after writing the melody?
Listen to it. Does it sound like it's come to a full stop at the end? Almost all melodies do.
That full stop is the tonic.
So let's say your melody has F, Bb and C# notes in it, and your last note is D. D is the tonic, which means you're in the key of D something. It's not D major because of the F natural, and it's not D natural minor because of the C#. That F note gives you a minor feeling (it's the b3 of D) so it's some minor scale. Because all the Bs are flat, it's not the melodic minor. The most likely choice is the D harmonic minor scale.
That's not the only possibility. If you have a bunch of G# notes you could be in the Hungarian minor (also called the Gypsy minor). But that's the essence of how you figure it out: identify the tonic, examine the third up from that tonic to determine if it's a major or minor scale type, and if you want to narrow it down to a specific scale, start eliminating possibilities based on the notes used.
The answer is really the same as my answer to your previous question. Say you have a melody that uses the notes D-E-A-B. You transpose the notes up by half-tone steps, or write down the following table, to check which transpositions fit into C Major:
C - D - E F - G - A - B C - D - E F - G - A - B
+0 x x x x <- OK
+1 x x x x
+2 x x x x
+3 x x x x <- OK
+4 x x x x
+5 x x x x <- OK
+6 x x x x
+7 x x x x
+8 x x x x
+9 x x x x
+10 x x x x <- OK
+11 x x x x
The transpositions of the melody notes that fit into C Major are 0, +3, +5 and +10. So to get the major keys that contain the notes, we transpose C Major down by those intervals:
C Major - 0 -> C Major / A minor natural C Major - 3 -> A Major / F# minor natural C Major - 5 -> G Major / E minor natural C Major - 10 -> D Major / B minor natural
For melodic and harmonic minor keys, you'd have to use a separate table.
(You can of course use another key than C Major as the basis for the calculation, e.g. if you want to do it without writing down the table and there's another key that you're more familiar with, or you want to go straight to the minor keys.)
You have two question conflated into one: finding a suitable major/minor scale for a group of tones, and how the "pros" find a key of a melody.
Those are not the same thing.
For the first question treat major/minor scale as synonymous with key signature and you should be able to step up the accidentals of the various key signatures to find if the set of tones fit a major/minor key...
C,F,D ... or perhaps C and G#? ... or perhaps C, G, A#
The first sharp for key signatures with sharps is F# that doesn't match the set C,F,D all other key signatures with sharps going up the circle of fifths will contain an F# so none of the sharp key signatures will match the set. Going in the flat direction we go Bb, Eb, Ab, Db when we get to the Db we have a tone that doesn't fit the set and so we stop and know the set can't fit any key signature with 4 or more flats. That gives us key signatures of 0 sharps/flats, 1 flat, 2 flats and 3 flats which will match the set of tones C, F, D.
Things get complicated when considering minor scales.
C, G# won't fit a key signature with sharps (and of course not flats), because to get to a G# we will first hit upon a C#, i.e. there isn't a key signature with G# that doesn't have a C#. However, G# would be used for an A minor scale. A quick way to try figuring this out is ask if one of the tones could be a leading tone. A# could be the leading tone to A minor and the C would then be the mediant, the A minor scale could be used. In the case of C, G, A# non of the tones can be a leading tone for a scale containing the other tones: C leading tone to Db minor, but G and A# don't fit, and so on with the others.
Things are further complicated when you realize that every note of a melody does not need to fit a scale. In the case of C, G, A# the scale to use might be C major where the A# might be the chromatic lower neighbor tone leading up to a B. But we can't figure that out from just a list of tones. We would need to see the actual melody.
This makes a nice segue into the second question about finding a key. Both questions seem to be generally about figuring out the tonality of a melody. But you can't do that simply from a list of tones. Sequence, repetition, rhythm, etc are all essential for perceiving tonality. If someone presented a list of melody tones D, E, F#, G, A, B, C, you simply cannot tell what key the melody is in. That list of tones could be used for multiple keys: C major, G major, E minor, etc. or it might not be in a key at all! You must see the actual melody and understand what it is doing in terms of major/minor harmony (implied harmony if there is only a melody.)
EDIT
Just a quick addition to demonstrate how a list of tones will not be enough to determine a scale/key: consider this set: C, C#, D, D#, E, E#, F#, G, G#, A, A#, B, B# ...in what key or scale should is that set? Note the enharmonic spellings B# & C for a total of 13 tones in a major/minor system that uses a 12 pitch chromatic scale! Tones are not in melodic order but in ascending order from C. The answer is here.
The "circle-of-fifth" is a great tool for checking harmonic and melodic material. In short: Neighboring elements belong to a key, if you look at it as note names.
Check out the concept of "modal interchange", too, for notes/chords not belonging to a certain key, but being substituted into a melodic line or chord progression originating of closely related keys.
This way you can broaden the perspective if one note does not really fit in a certain key in the classical way. Depending on style sometimes certain notes will not fit in a scale and only make sense in relating to a chord:
i.e. Blues: Improvising with a e-minor pentatonic (e g a b d ) over a E7 chord (e g# b d) gives you a overall #9 - Sound (note g against g#) for example which adds a certain character to this style and defines it sonically.
So: What style do you want to write in?
