What are the best criteria for naming a key?
For me, when a song has, say, three accompanying chords, they are generally I, IV and V of a particular key. Thus, in a song with C, D and G, it's squarely in G major. Others have reasoned that some songs (with say, those same chords) are actually in D. Thus D=I, C=bVII and G=IV. I don't buy that, as the bVii is 'borrowed' from the parallel minor. No big crime, but a little convoluted nevertheless.
There are songs with only two chords, so to me, they are I and V - rather than I and IV, but that's open to debate. Once there are more than 3 chords, life gets easier, maybe... Along with C D and G is Bm. Ah, that's not in C, but could be in D or G.
Obviously, the final chord is (usually) a deciding factor, but some tracks use a different one, so no great help there. The first isn't always helpful, either. I get fed up with people telling me 'Pretty Woman'( Orbison) is in E! The last band I played it with, I actually started the intro ON B, so it could be in E. They said I was wrong...
But, enough gripes. What are the best criteria to use to define a song's key please? And looking at the sheet music is not an answer I need!
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There are three things I look for. (You've already pointed out two of them!)
Perform the Roman Numeral analysis in both candidate keys and look at which one makes more sense.
Look for places where the chords resolve to the tonic.
Look for a scale/tonic within the melody.
You've already described #1 and #2 well, so I'll elaborate on #3 . Here are the sorts of things I would look at when analyzing a melody to find a key center:
Which notes does the melody resolves to?
Which notes occur on the downbeats?
Which notes are sustained over longer amounts of time?
Which scales are being outlined?
For example, in Pretty Woman, the lyric "woman" resolves to an A. If the song were in the key of E, then the main melody line would be resolving to the 4 of the tonic, which is very unlikely. Additionally, the first lyric ("pretty") is sometimes sung as a C instead of a C#, giving it a blues feel. If the song were in E, then this blues note would be the b6, which isn't really a thing. As a last hint in the melody, the entire chorus is sung with the A major blues scale, with the exception of the D note (which you and I know occurs over the V chord and is the b7 of the E7 chord). It's unlikely that a song written in E would have a melody that's based on the A major blues scale.
I would suggest that the best practice for defining a piece's key is to define it in the way that makes the most sense to whoever you are describing it to.
Knowledge of melodic and harmonic theory can be varied among musicians, with some very excellent performers only slightly aware of classical harmonic analysis terminology.
In many cases music education is Major centrist, so relating the key to the nearest relative major / minor scales may make the most sense to a classically trained musician.
An example of this would be describing a piece in the Dorian mode. I worked with a piano player that was having trouble reading a piece in Dorian mode where the key signature had the Bflat for D minor scale but all the B's were natural with accidentals. Once I pointed out that the key was actually C major, he was able to think of it in that key and let his muscle memory play the correct notes.
In other cases, the diatonic chord structure should be considered, especially when describing a key to a predominantly harmony player. To use an example from the comments, if a song has predominantly D C and G chords in that order, I will call it G major, even if there are some borrowed or transitional chords in parts of the piece, or the piece ends on what would be considered the V chord. I've found that if the harmony player is expecting the chords of a I IV V progression, even if they tonicize differently, it is easier for them to remember the chords and insert the outliers as they occur.
If the musician is aware of playing in Modes and understands relative key/scale, then I will describe it in the closest or most usable scale/key.
For transcriptions, I personally prefer to use the fewest sharps or flats in the key signature as possible. This may cause some consternation for musicians confronted with piece in a non major / minor tonic, but personally I've seen musicians have more trouble with accidentals all through the piece than dealing with an unexpected resolution of a phrase to a modal note.
I got the impression from my time on this site that this was reasonable:
work out what sounds like the 'home note' (which could be for any number of reasons - melodic phrasing, cadences, all sorts of things). That's the 'letter' of the key...
...and then work out if the tonic chord is basically minor or major, to give whether the key is major or minor.
So if a song has C, D and G major chords in, and I hear D as the tonic (for whatever reason), and there's no definitive sheet music or other reference, I call that D major.
If someone else wants to call it G major because they hear G as the tonic, that's fine too. If they want to call it G major even though they hear D as the 'home note'... they're doing it differently to me.
If the original artist notated the song as sheet music (or another form indicating the key), Who am I to argue.
If there's no sheet music and no clear home note... is there any need to talk about a 'key' at all?
Well, I guess I should be ashamed, But I noodle a scale over the backing chords specifically to find the Key when I'm at a Jam and someone starts without calling out the key. I usually have it sussed out in the first bar, after which I start to listen for indications of something modal or not. From that point on I can utilize all my tricks pretty confidently. I'd like to say that I'm still learning new ways of doing things and open to new ideas if you have anything to offer.
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