What keys contain both E and Am and is there an easy trick to determine this?
By 'keys contain' I mean in the normal rules of choosing what chords 'fit' within a given key, since of course you can put just about any chord anywhere if you try hard enough!
Is there a shortcut where you can take any major or minor chord and list the keys this chord could belong to, then narrow it down to keys containing two chords, etc?
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The answer is no, because as you wrote:
By 'keys contain' I mean in the normal rules of choosing what chords 'fit' within a given key.
... what you are describing is the term "diatonic," which means only the notes that are contained within a key's signature.
A quick way to notice this is that the chord E major has the note G#, and the only keys that note can fit in, diatonically, are A, E, B, etc. (as well as their respective relative minors). But none of these keys will diatonically allow C-natural, which is necessary for the A-minor chord! :)
Caleb's chart above is very helpful, BTW.
Am and E are both found in the key of A minor.
The easiest way to find out what key the chords fit best is to figure out the relationship between the root of the chords. For example in this case E is a Perfect 5th away from A. In any key the dominant will be built a perfect 5th away from the tonic. In a minor key even though the dominant is naturally minor, it is very common to make it major to lead back to the tonic chord.
"What keys contain both E and Am"... "By 'keys contain' I mean in the normal rules of choosing what chords 'fit' within a given key"
Strictly speaking, (perhaps surprisingly) the answer is: none. That is, if you only allow chords from the diatonic scales, without chromatic alteration, then there is no scale that contains both of these chords. Below, I will show you why this is, and what common alterations allow them both.
"...is there an easy trick to determine this?"
Absolutely! Just look at the Circle of Fifths. In the image below, I'm actually using a segment of the infinite Line of Fifths, since it works exactly the same, but will help you to pick the correct spelling for black keys.
This image works like a slide-rule. Beneath the line of fifths are two colored rows: the top for major keys, and the bottom for minor keys. Slide the colored part left or right, along the line of fifths, until the tonic column (I or i) lines up with the desired major/minor key. The roots of all other chords can now be read off the chart. Note that the orange chords (IV, I, and V) are major, the blue chords (ii, vi, and iii) are minor, and the funky yellow chord (vii°) is diminished.
If you're in the minor key (second colored row), the only things that change are the roman numerals associated with the chords (you add 2 or subtract 5), but the pattern of 3 majors, 3 minors, and a diminished stays the same. This gives you VI, III, and VII as major chords; iv, i, and v as minor chords; and ii° as a diminished.
As you can see, there's no way to slide it so that A is minor (blue) and E is major (orange). That means that in order to get both chords, we must chromatically alter one of them. We could pick a key containing an E minor chord, then alter it to be major, or we could pick a key with an A major chord, then alter it to be minor. (In the keys of D major/B minor, we'd have to alter both chords, since we'd have both an A major and an E minor).
However, there is an extremely common technique in the minor key, which is to raise the third of the v chord by a half step, which makes it a major chord (V). Note that the third of the v chord is also the seventh scale degree, so raising this note by a half step places it just under the tonic, creating a "leading tone" which wants to resolve to the tonic. This variant of the scale is called the Harmonic Minor. In the graph, this would be equivalent to replacing the blue "v" in the second row with an orange "V". In this case, you can see that you could have an A minor chord (i) and an E major chord (V) in the same scale: A harmonic minor.
Although the harmonic minor is an extremely common technique, it is not the only possible way to alter chords to get both E and Am together. By moving the colored portion around to different keys that contain both E and A, you can find what alterations are required to get them both. Here's some alternatives:
In C major, you'd be replacing iii with III, which causes it to act as a secondary dominant of vi (V/vi). This is quite common to do.
In G major, you'd be replacing vi with VI, which causes it to act as a secondary dominant of ii (V/ii). This is a bit less common.
In D major and D minor, you'd be replacing ii (or ii°) with II, which causes it to act as a secondary dominant of v (V/v). You'd also need to replace the V chord (which occurs naturally in D major, or in the altered D harmonic minor) with v. Making a dominant chord minor is a less common technique (at least in classical music), and it has the effect of weakening the sense of tonality, and increasing the sense of modality.
In E minor, you'd be replacing i with I (making the tonic a major chord), which might be used as a "Picardy Third" at the end of a piece, or to modulate to the parallel major.
In A major, you'd be replacing the I with a i (the reverse of the above) which can be used to modulate to the parallel minor.
In E major, you'd be replacing IV with iv. This minor subdominant tends to have a melancholy sound to it, and is good in chromatic passages leading up to a cadence. This is another common technique.
By asking what key includes a pair of chords, the answers are as previously given. However, there is more flexibility if you ask what scale contains a pair of chords. In that case, 02fentym's answer provides the mechanism.
Lay out the notes of your chords in alphabetical order, as indicated.
A B C E G#
Now you can construct a scale that suits your purposes.
The chromatic scale, of course, trivially contains all of those notes.
The A harmonic minor scale also contains those notes.
The ascending A melodic minor scale contains those notes.
The E "double harmonic" scale works: E F G# A B C D# E
For a complete list, try the "Musical Scale Search Tool", which allows you to enter notes and/or chords that your scale should contain, and it will provide the options. The tool includes options for the types of scales to give in the results.
Western diatonic scales
Greek modes
Altered Greek modes
Less common Western scales
Non-Western scales
I have an even faster way to do that, which doesn't even require a Circle of Fifths on hand:
Find the half-steps!
Simply put:
Take your chords, set of chords, or scales, and line up all of the notes into one big combined "scale". In this case, enharmonic spelling doesn't matter.
Look for any notes that are right next to each other.
Each half-step pair must be either Mi-Fa, or Ti-Do of a major key, because every major key (and any key in its modal family) has only two half-steps in it. Use those notes to find the key(s) that contain the notes!
(As others have noted, all solutions will in fact be adjacent keys on the Circle of Fifths.)
Examples:
In your case, we get [E G# A B C]. G# and A make a half-step, and B and C make a half-step as well. One possibility is that G#-A is Ti-Do. However, if A is Do, the C from the Am doesn't fit in the major key. So then we try B-C as Ti-Do (equivalent to checking G#-A as Mi-Fa). Problem is, there's no G# in C major. Since neither way works, we know that there cannot be any keys that contain both E major and A minor!
As another example: Let's try G major and A minor. We get: [G A B C D E]. There's only one half-step here (B-C), so we'll try C as Do. If C's the tonic, everything fits into C major! But we also have to try having C as Fa. If C is Fa, then G must be Do. After verifying that one as well, we now know that C major and G major must be the only keys that contain those two chords (and all keys in their respective modal families, which includes A minor and E minor)!
Lastly, we'll try Bsus4, Em, and D7: [B C D E F# G A]. B-C and F#-G make the two half-steps, so try G as Do. It works, so we know G major fits both chords. But try C as Do now, and there's a problem: the F# from our D7 chord is not in the key of C major. So G major is the only major key that can contain all three chords! (Again, E minor is included, as well as all other modes of G Ionian.)
Tips and Tricks
There's also a couple helpful tricks that can shortcut the process:
If there are more than two half-steps, then no key contains all the notes.
If there are more than 7 distinct notes, then no key contains all the notes.
If there are three or more notes a half-step apart, then no key contains all the notes.
If one of the chords is of the dominant seventh quality, the only key that can work is the one where the dominant chord is built on the 5th scale degree.
Augmented chords do not exist naturally in any key.
Diminished triads and half-diminished seventh chords may only be found on the leading tone (Ti) of the key.
Fully-diminished seventh chords do not exist naturally in any key.
Minor-major seventh chords do not exist naturally in any key.
Note, however, that there's a glaring flaw in this method: it doesn't account for harmonic minor. Harmonic minor is arguably more common in practice than strict natural minor or Aeolian, but because harmonic minor is not in the modal family of Ionian (major), the algorithm fails to recognize obvious solutions like E and Am being a part of A minor, or any other common usage of chords that strays from the purely diatonic instances.
It might be possible to develop a modified method where once you find Do, you check to see if the notes match a minor key as well as checking to see if it fits the major key on that tonic, or major augmented and then figuring out what minor key produces that particular Ionian #5 scale. A lot of the shortcut tricks I listed, though, don't hold when accounting for harmonic minor. For instance, no mode of Ionian contains any augmented, minor-major seventh, or fully-diminished seventh chords; however, all three chord qualities do arise organically from harmonic minor and its modes!
The example you have chosen, A minor and E major, don't break the rule, they rather reinforce a very common special case. Those of us who were taught traditional harmony in the mid 20th century knew that the sure trick for differentiating between C major and A minor - both of them 'all white notes' by key signature - was to spot the sharpened 7th of the minor key. The note that enables a full dominant chord, including the leading note, in a minor key.
Look for the diatonic notes, yes. But also look for the note NOT in the key signature that defines the harmonic form of a minor scale.
I would say a good way to figure this out if you have two chords or any number of chords is to lay the notes out in alpha order. For Am and E this would look like this: Am (ACE) and E (EG#B) would be ABCEG#.
The next step is to determine which major or minor keys have those notes.
Here's another example, what key or keys fits best for the chords Am, Dm, Bb?
Am = ACE
Dm = DFA
Bb = BbDF
Lay them out in alpha order: A Bb C D E F
The only note we're missing is some sort of G. By looking at the notes, it seems we're in a flat key. We have Bb, which is the first flat in the order, therefore F major is a contender. Since we have an E and no Eb, we can have at most 1 flat, therefore F major is the only option.
In general, use your knowledge of key signatures to help you. Scales and chords go hand-in-hand.
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