How to differentiate between relative keys
So I'm learning that C major and A minor scales have the same notes.
Then, if I look at, say, 10 consecutive single notes, how do I tell if they're in C or Am?
It's easier when you see a chord, since you can count the semitones between the notes and deduce if it's major or minor (not sure if what I'm saying is technically correct), but when you see consecutive single notes, how do you differentiate between the major key and the relative minor key?
If you play the scale from root note to root note, you can differentiate, but if you play some sequence, is there any pattern to look for that tells you it's C or Am?
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The answer to your last question (which I see as your main question) is no! A string of ten consecutive single notes picked randomly out of some part of a song (as opposed to your example of playing root note to root note) that all happened to be shared between a major key and that keys relative minor (which by definition are the same notes) are usually not enough to establish a key.
In many musical compositions or songs, the key may be somewhat ambiguous to begin with, particularly when trying to distinguish between major and the relative minor key of said major key which both keys contain the same notes and share the same key signature.
But many musical compositions do in fact firmly establish a tonal center or key. This is most effectively done by resolving the melody back to the root (or "home") note - which in the key of C major would be C and in the relative minor (natural minor) of A "natural" minor (often just called "A minor") the home/root note would be A. In a major key there is only a semitone between the 7th degree and the octave so the 7th degree has a strong pull to resolve back to the root or home note.
In natural minor keys there are two semitones (one whole step) between the 7th degree of the scale and the octave (8th degree which is one octave above the root or 1st degree). Therefore the pull to resolve to the home note is not as strong in a natural minor key attempting to resolve to the octave of the root. That is why the 7th degree in a minor scale is often sharpened. By sharpening the 7th degree you create the same single semitone separation (smaller gap) between the 7th and 8th degree and therefore create a stronger pull towards the 8th degree.
But when you sharpen the 7th degree of the natural minor scale you have created a "harmonic" minor scale which is a deviation from the "relative minor" of a major scale which by definition shares the same key signature and same notes.
It would be true that when attempting to determine if a piece was in C Major or A "Minor", if all of the notes were in the key of C major but you found a G# followed by an A one semitone (half step) higher, you can be reasonably certain that you are probably in the key of A minor but you would technically be in A "harmonic" minor which is not the "relative" minor of C.
If you play the scale from root note to root note, you can differentiate, but if you play some sequence, is there any pattern to look for that tells you it's C or Am?
With the limited supply of distinctive notes available to use in a melody when composing a song using Western Music notes, it is important to have the freedom to take the melody in whatever direction the composer feels best conveys the feeling he or she is attempting to create. Thus restricting the creative license to a given "pattern" (regardless of major or minor key) would be far too limiting.
So without looking at the composition or song as a whole (given notes that could be in either a major key or that keys relative minor) it would be difficult and in most cases impossible to determine the tonality of the entire work based on a randomly selected string of ten individual melody notes - unless the string of notes contain "accidentals" which point towards either a harmonic or melodic minor key as opposed to the natural (relative) minor.
You'll find the same 'problem' in modal works, too. For example, C Ionian, D Dorian, E Phrygian, F Lydian, G Mixolydian, A Aeolian and B Locrian all use the same set of notes. You need to listen carefully, to catch points where the melody seems to come home, and be at rest. These will often be found at the beginning of the piece, and at the beginning of a 4th, 8th or 12th bar. Not always, but a lot of music is written/played in stanzas that reflect poetry, and these 'node points' seem quite natural to Western music.
The leading tone, as stated by Richard, is also a clue. It's the note under the tonic, and often is seen to lead back to that tonic. So in A minor, it's often sharpened to make it more pointed (pun intended). In Dorian and Phrygian this doesn't happen, so it's not a lot of help.
As in the first note is the start of a journey from home, the last will usually return; another clue, for all modes - including C Ionian and A Aeolian - the two in your question. As far as a random sequence of notes taken from a melody, it's way more difficult, sometimes impossible, as the relative major/minor is delved into and out of quite subtly in some tunes.
Another point is that some sequences of notes COULD have a C major OR an A minor chord under them. (You mention chords). That then is down to the composer, who decides one way or another.Take A-G-E-D in a bar. That snippet works with C as well as Am. So, who can tell? In a longer sequence, it may be easier - but there again, it may be more difficult!
Just from seeing or hearing a sequence of 'white notes', there is no objective way to tell if you are in C or Am.
Subjectively, if you feel that the 'home note' (or 'tonic') that the piece is based around is 'A', then you can say it's in A minor. Likewise, if it's C, then it's C major - and if it's D, then it's D Dorian, and so on.
Richard and Tim have mentioned the leading tone as a possible clue. You might also want to look at which notes are most common. The IV and V note are often important. In C, these are F and G. In Am, these are D and E. So if you have lots of notes C, F and G, it might make it more likely that you are in C, while multiple occurrences of A, D and E might make A minor more likely.
You can also look at what the start and end note of melodic sections is, and what notes are played on the strong beats. A note played on a strong beat is more likely to be an important note in the key. This is also often true of notes that have long durations.
Still, all of these are just reasons for why you might feel that a melody is oriented around a certain note. If you don't feel that you can identify a tonic note that the melody is oriented around, but you still need to choose a key, then your choice is going to be somewhat arbitrary.
It's easier when you see a chord, since you can count the semitones between the notes and deduce if it's major or minor
Remember that telling whether a chord is major or minor is different to telling whether a key is major or minor.
You might still have a similar problem with chords to the one you described with notes - you might have a sequence of chords using the white notes and not be able to tell if that chord sequence is in C or Am.
If a G#, or a F#, G#, A run occurs, that's a pretty strong indication that we're visiting A minor. But the absence of a G# (despite what we may have been told in Grade 5 Theory) doesn't count it out!
A (natural) minor is a mode of C major. They are in many respects so close as to be indistimguishable. Revel in the ambiguity!
The shape of the melody may tell you whether C or A is the 'home' note. Also remember there are modes other than major and minor. Perhaps the piece is in D Dorian, G Mixolydian etc.
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