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Why do some progressions start off with an entirely different chord, then the tonic, yet be in the same key? Does that mean for the tonic to become a tonic it has to be established by a few extra measures, or more accentuation?

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    Because surprising listeners can be fun and profitable. – Steve Clay May 29 '17 at 23:08
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The key is established by the interrelationships between chords. It is true that one way you can establish the key is to play your tonic chord over and over again. However, that is not the only way. Here's a funny little example to illustrate: if you play the progression F-Dm-G7-Am, and there is no other context, the ending will feel slightly surprising, and the established key will be C Major.

But wait, you say! There was not even a C Major chord played! Why would you say that it is in C Major?

The easiest way to prove it is experimentally. First, play the progression. Notice how unfinished that ending feels? Now follow it up immediately with F-Dm-G-C, and you will get an immediate sense of completion and rest. This is how you know that we have correctly identified the key.

But wait, you say! How did we establish the key without even playing the chord?

There are a few ways that we established the key. If you haven't had a full course in harmony, some of them are hard to describe, but here are a few features that helped:

  • If you examine the notes in those four chords, every note in the C Major scale was played, and no non-key notes were played. (C in F and Am, D in Dm and G7, E in Am, F in F, Dm, and G7, G in G7, A in Am, F, and Dm, and B in G7.)
  • The specific choice of how the chords move between each other, and the penultimate chord of G7, reflect common harmonic patterns in C Major.

Now, the key consists of two components, the tonic (in this case, C), and the mode (in this case, Major).

I even had a professor that famously pointed out that there is one chord, called a French Augmented 6th, that can establish the tonic all by itself. (It can't establish the mode, however). It is too complicated to explain how this chord works if you don't have a fair amount of background in harmonic theory, but in the spirit of experimentation and discovery, you can still try it for yourself! If you want to try this funny little chord, play the following notes together on the piano in your right hand: Ab-C-D-F#. You've just established C as the tonic again. To prove it, get to the relaxed place by playing the note G, and then the note C. Bam, you've reached the completion and rest point (the tonic). =)

EDIT as a clarification based on Dekkadeci's comment: part of how the French Auggie 6 produces a tonic is with its spelling. It's not purely by ear! There is more than a little suggestion from which notes are on the top and bottom, because those notes typically resolve outwards, but that's not entirely foolproof. If you respell the chord D F# G# B#, it will generate a tonic of F# instead of C. This will be particularly apparent to the ear if the D and the B# are the bass and soprano, respectively. But for this special chord, spelling definitely counts :)

  • Just to add to the G-Am part, it is a V-vi cadence, which is called a deceptive cadence. – Shevliaskovic May 25 '17 at 17:04
  • Hear, hear! :) I tried to steer clear of specific harmonic interrelationships because it felt like a distraction from my broader point. – Ben I. May 25 '17 at 19:51
  • The French Augmented 6th cannot establish the tonic by itself. Its second inversion is another French Augmented 6th (so D-F#-Ab-C, pulling from the above example). Thus, depending on the root, you can easily convince me that the example chord is in C major or F sharp major. – Dekkadeci May 26 '17 at 12:40
  • G>Am is also known as an interrupted cadence. – Tim May 26 '17 at 12:40
  • Your answer is correct, but one might hear that progression as I vi V7/V iii. There's always a risk in starting with chords that suggest a different key. – Steve Clay May 27 '17 at 23:49
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Let's make the fundamental assumption that all aspects of music are meant to be understood by listening, not by reading the score. Of course reading the score may help, but it's only a secondary way to make sense of what is happening.

There is an important principle that follows from that assumption: You can only understand music with the benefit of hindsight, using the context of what happened earlier in the piece.

On its own, the first note or chord tells you nothing about the key, tempo, or rhythm - there is no tempo or rhythm until you hear the second note or chord following the first one.

So, the first chord can't "establish" anything. The key (if there is a conventional key at all!) is established by a progression of chords that arrives at some sort of "finality" or "release of tension" - for example the cadence at the end of a phrase.

The chord progression that "establishes the key" might be only two chords long and take a less than a second to perform, or it might be spread out over many minutes of music (e.g. the start of Wagner's "Rheingold") - that is entirely up to the composer.

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The tonic, or key, of a song is generally determined by the sound it has when it's 'come to rest'. Way more often than not, that's the last chord. If that last chord didn't sound like it had landed safely, then chances are that chord won't represent the song's key.

In a song such as the OP describes, if the composer wrote it down, it would appear, assuming the first chord is its key, that changes were occurring pretty well bar by bar, intimating that each bar belonged to a new key.

Usually, there's a journey between the first 'foreign' harmony - manifested as a chord or two - and the home key, which eventually gets arrived at, and feels like 'home'. The intervals between each chord and the next is often P4 or P5.

A few examples - Ballin' the Jack, Sweet Georgia Brown, Sweet Home Alabama, all moving in fourths. Hey Joe, moving in fifths.

Songs don't usually start on 'random' chords - there will usually be a thread seen to run from the start chord to what we perceive as the song's key.

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Academic music theorists refer to this phenomenon as an "auxiliary cadence," and it simply means that a phrase begins with a chord other than tonic.

Auxiliary cadences are very common in popular music (there's a great article on its use by The Beatles here), but their use dates back centuries earlier. Even Beethoven's very first symphony begins with a C7 chord, an unexpected beginning for a C-major symphony!

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A chord progression is just a series of chords; the tonic is the first note of a key. They are orthogonal concepts.

Keys are not "established" so much as "selected". They are a tool for the composer. If you write a song that seems halfway between A natural minor and C major, for example, it's up to you what to call it.

  • Being the pedant like what I am, I'd say that the tonic, or key, is related to the series of chords.As in a particular piece, although the first chord (as the OP asks) may not appear to be related, it actually is, borne out by that chord progression going where it does. Sweet Georgia Brown, in G, starts on E - but the progression is the cycle of fourths which inevitably takes it to the tonic. Not any old random chords, but one designed to go somewhere specific. I imagine all of the pieces which come into the OP's concept will work in similar ways, using a prescribed journey, although as you... – Tim May 25 '17 at 6:41
  • ...state, the first chord may appear to have no relationship with the key. I suppose, somewhat like modulation - before the fact! – Tim May 25 '17 at 6:43
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    I take issue with your statement that "keys are not established", since "establishing the key" is what you explicitly study when you start to study key changes. Also, the key is as much a tool for the listener as it is for the composer. – Ben I. May 25 '17 at 14:36
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What DOES establish a tonic? Play just one chord for a whole song (it does happen :-) Is that chord the tonic? I suppose so, though it's a pretty meaningless concept in that case. When there's nowhere to go, everywhere is home.

What about a song that continually alternates C major and E major chords. Where's the tonic? Perhaps the one the song ends on, but what if it's a repeat-and-fade? We don't really know.

How about C, B7, E. Now I'll start accepting bets that the key is E major. A pair of chords that can be analysed as dominant > tonic bring us into a musical world where 'key' and 'tonic' are useful concepts. There's plenty of music where they aren't - and I don't just mean way-out 12-tone 'modern classical' stuff. Remember our C major and Emajor song?

So yes, it takes some other chords to establish a tonic. They don't have to be before the first appearance of the tonic chord of course. It's very common to start on the tonic chord. But quite permissable (and common) to lead into it with other chords.

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    "C, B7, E. Now I'll start accepting bets that the key is E major" - not if you are listening to Mendelssohn's Wedding March which starts C B7 Em ... and continues with F G G7 C! – user19146 May 25 '17 at 20:59
  • Not every bet is a dead cert! And, anyway, the Mendelssohn only starts like that in E-Z-Play kiddies arrangements. It's actually Am6, B7, Em... – Laurence Payne May 25 '17 at 22:39
  • Another example: Mr. Sandman (in C: Cmaj7 B7 E7 A7 D7 G7 C...), Of course, that's something of an outlier :) – user45266 Jun 8 at 20:21

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