For instance, an E major triad followed by an A minor sounds really good to me. I don't believe these can belong to the same key, because you can't have a sharp g and a natural C in the same scale as far as I'm aware (because it goes F#, C#, G#, so if G is sharp then F and C must also be). Is going from E major to A minor a change in key? If so, why do chords from these two keys sound good, and why are keys necessary if you could just mix them up and still sound good? I'm having a really difficult time comprehending keys and how to use them effectively.
I think for your specific example: E major chord to A minor chord, they do belong to the same scale: the (harmonic) minor in A. In the harmonic minor scale the natural minor scale is altered by sharpening the seventh degree to create a leading tone to the tonic and at the same time to make a chord with dominant function possible. When we look at it this way, the G sharp in the E major chord is just the leading tone to A. This leading tone gives the E major chord its dominant function.
So, the two chords can be interpreted simply as a dominant-tonic cadence in a minor tonality. That's why it sounds so good/familiar.
To give a more general answer to you question:'How do I contextualize chords from different keys that sound good together?' More often than not in tonal music you can find a relationship between two chords by assuming a temporary tonicization (like we did above).
contextualize chords from different keys
IF the chords are from different keys you can analyze according to those keys (literally contextualize them to the context of the associated key)...
- secondary relationships, usually secondary dominants
- borrowed keys
- just label the other key
I say IF because you can have non-functional harmony where keys aren't really the tonality.
For instance, an E major triad followed by an A minor sounds really good to me.
Two chords doesn't necessarily make a key, but let's assume the key
E major. The
A minor is borrowed from the parallel
E minor. So simply label the
A minor chord with lower case letters...
E: I iv
Is going from E major to A minor a change in key?
Not necessarily, and this is the reason for the secondary and borrowed concepts. They are ways to indicate the harmonic relationships with referring to a literal key change. They acknowledge that key is a fluid concept in the flow of real music.
My suggestion is to get Kostka's Tonal Harmony - or another standard textbook - learn all the labeling of chords, and then start analyzing Bach's 371 Harmonized Chorals. You will quickly learn that harmony is not restricted to the diatonic tones of a key!
Let's start with the three basic chords of scale: Tonic, Subdominant and Dominant. And, as the original post suggest, restrained to major. Each major scale could be abstracted to 7 different notes. Those could be designated by Roman Numerals (I, II, III, IV,..). The numerals also represents the triads build on this note (with only scale notes).
Tonic: This is the basic triad of the scale at position I. E.g. E maj, the scale goes: e f# g# a b c# d# e, the basic triad of the first position, I, is e, g#, b, that is E maj. This is Tonic, the home.
Subdominant: This is the fourth position (IV). In our scale, it is the triad a, c#, e, therefore A maj. It has a tendency to lead toward the Dominant, but also back to the Tonic. In that case, the a pulls towards the g# and the c# towards the b.
Dominant: This is the fifths position (V), the b. The (major) triad on the b, with just scale note, then is b, d#, f#. This chord has a strong tendency to pull back, pull home to the tonic.
So we identified A maj as the subdominant to E maj. But the question asked for A min, i.e. A, C, E. As the OP states, C is not part of the E maj scale. Nonetheless, the A has a strong pull towards the Gis because of the subdominant major character and C towards B as because of the subdominant minor character.
The interesting thing here could be the parallel minor third movement, resulting in this gloomy resolve, making it pleasant but somehow strange.
The E Maj A Min relationship could be interpreted as a Terzschritt. C Maj and E Maj would be a classical Terzschritt in a Riemann sense. That meands, C Maj transforms to E Maj by moving a major third up. C Maj and A Min are parallel scales. So, this could be another relation
Old Answer for reference
This is basic chord function theory.
You mentioned your E major key. This is your tonic. With basic function theory you will find your dominant and your subdominant, that is B major and A major. Those three will sound well together. Basically it comes down to multitudes of frequencies.
You want to learn about the use of subdominant and dominant in your chord progressions.
In your question you mentioned A minor. Now this is advanced function theory. The A minor is not the subdominant to E-Major, as it is, as you mentioned, not "in the scale".
Actually I have a hard time to hear that progression right of my mind, but it might work.
[I gave it a hear. The minor still have this subdominant feeling, leading towards the dominant. But that should be then minor to, in my ears. Therefore I would consider everything minor and the e-Major as the borrowed chord.]
There is a thing where you not only consider the chords "within" the scale but also their relatives and their parallels. Those are called borrowed chords. You could consider the A minor as something like a quart variant of the subdominant. Or it functions as the submediant parallel. I guess, it depends at the context. (Since I am German, I do not have the right English terms at hand).
You said you have difficulties in comprehend keys and chord functions. The basic primer would be this: learn about Tonic, Subdominant and Dominant and their use. Learn about the leading function of the seventh and the concept of resolving. That would be the bedrock for the advanced stuff.
There are a couple of 'facts' you seem to have read into music theory. One is the addition of sharps as we go through the 'cycle of 4ths/5ths'. That much is true, in that they're cumulative. That's key signatures. Start with F♯, then comes C♯ and so on.
That's only a part of the story, and as we keep on stating, it's music theory, not music law.
Another bit of misinformation you have concerns the minor scale notes, which somewhat defy logic for beginners.There are three different minor sets of notes - natural, harmonic and melodic. As the leading note is a very important part of music, being one semitone under the root, in all keys, it didn't (doesn't) exist in the natural minor set. That gets rectified in both harmonic and melodic sets of notes, making the harmonies and melodies better sounding, generally.
So, your E chord (with that leading note of G♯) will go nicely into A minor, mainly because it belongs to A harmonic or melodic minor. You say it sounds good. If the piece is in key Am, it always will! If the key is, say, C or G, then yes, it'll still sound good. It probably won't mean there's a key change - maybe a small modulation - but it's sometimes how music works.
There are tens of thousands of players out there who probably use changes like that all the time, and haven't a clue about the theory behind them, only that they know it sounds good. It's great that you feel the need to know why, but as is often the case, there's a missing link in your knowledge, as I think happened here, that trips you (and many others) right up!
You seem to be assuming that an E major chord puts the music into the KEY of E major, with 4 sharps.
Suppose we think rather of being in A minor? E major is the dominant chord in the key of A minor. But G♯ isn't in the key signature of A minor? True. But it's in the harmonic form of the A minor scale. It's in there BECAUSE tonal harmony is all about dominants and tonics, and a dominant chord needs to be major, to include the leading note a semitone below the tonic.
(The more 'modal' sounding E minor chord is fine in A minor too. But it doesn't give the strong pull towards the tonic that E major does.)
Is going from E major to A minor a change in key?
Since you talk about F#, C#, G# etc. I suppose you are already sure that E major is your key. If you are in the key of E major, meaning that E feels to you like the home note, E major feels like the home chord and the scale you're thinking about is E major, then A minor is some kind of a change to that picture, yes. However, how big a deal is it - would you write a key signature change for the Am? If the detour in feeling is only very brief - if nobody needs to know - you might not bother calling it a key change. But if you need to play some notes on top of the Am, you have to know that some notes from the E major scale, particularly C# and D# will probably sound bad. Depending on how you want to make it sound, you might want to avoid G# as well. But then, F or F# on the Am? If you use F#, you can make it sound like the key of Em, but if you use F, it could feel like Am. How about playing the chords E - Am - F ...?
If so, why do chords from these two keys sound good
Because key changes sound good! :) And this particular key change - if it's between E major and E minor - is a commonly used one. Here's another question about a song that does a similar thing How do the chord progressions in the song "Raw Sugar" by Metric make sense in theory
why are keys necessary if you could just mix them up and still sound good?
Because key changes sound good! :) And how can you have a key change without the concept of key?
A key is like a "for dummies" product package, a grouping of things that can be pretty safely mixed together in a somewhat sensible way. To mix components from two different product packages in a sensible way requires a bit more experience. It's like ... white clothing product package: white trousers, white shirt, white jacket, white socks, white shoes, white tie. Yeah, they all go together, but isn't it a bit boring? Mix in some components from the blue product package ... it's more interesting. :) I don't know if this analogy helps.