If you don't start your chord progression with the tonic chord, how do you retain the tonality / tonal center/ feel of a key? Or does the lack of the tonic chord cause the key to be a different one?
For example, if you choose the C major key, and start a chord progression with a chord other than the tonic chord, OR the tonic chord does not even appear in the progression, how do you know that you are still in the C major key?
I'm going to go out on a limb here and say it's all about the dominant. Although the tonic is arguably the "most important chord" to a tonality, it's the V that in some ways "defines" the I. Play a single chord and people hear it as a I, yes. But play a single dominant 7th and people hear the not-yet-played I even more strongly (this is a common way for a pianist to start off a sing-along situation, establishing the key with a big arpeggiated dom 7). When we want to "tonicize" some other tonality—i.e. make us feel like we're in a "different key" or modulate—the most obvious way to do so is with the V of that new key.
To steal from a video that was linked in a comment:
Full disclosure, I haven't had the patience to watch the whole video. But the first several examples all alternate between IV and V. (Even the first one that he claims doesn't, I think does, as he identifies Daft Punk's "One more time" as Bm but it seems to me it's DM.) Some of the other examples are simply ways you can "tonicize" a tonality with other chords, like Drake's ii7-iii7 loop.
But to another part of your question, "Does the lack of the tonic chord, cause the key to be a different one?" Well, you certainly might introduce ambiguity. (Here I am already disagreeing about Daft Punk.) Certainly, if you want the key to be unambiguous, the surest way is to include the tonic (and double down by including the dominant). It's certainly possible to hint so vaguely at tonality that there could be valid difference of opinion about the analysis.
if you choose the C major key, and start a chord progression with a chord other than the tonic chord, OR the tonic chord does not even appear in the progression, how do you know that you are still in a C major key?
I know it, because I feel it. If I don't feel it, it's not there. If I'm not sure of my feelings, I can test them by playing a candidate tonic chord to see how it makes me feel.
I feel the pull or harmonic "gravity" towards a center, if the song's notes, chords and rhythmic structures are arranged so that it creates that pull. There are phrases and movements that feel like they would come to a natural conclusion by playing the imagined tonic chord. The tonic chord doesn't need to be actually heard at any point in order to create that pull. Sometimes songs give the tonic away too easily and too often, making the song feel boring or cheap. (Excluding modal techno etc where the whole point of the style is to keep banging on the tonic and not have any "changes" to begin with, and sense of movement and journey is created through other means.)
To study the tonic phenomenon, you can take an existing tune that gives the tonic chord at one or more points, and replace all such instances with a different chord. Or you could simply cut out those segments from the piece entirely or replace them with silence or noise or unpitched rhythm. If the piece is strongly harmonically crafted around the tonic, removing the tonic chords should not cancel the sense of harmonic center, it should just keep the listener in suspension, waiting for the tonic chord that never comes. But it doesn't change the key of that song - IF the song is harmonically properly made to give the impression of that chord being home.
I'm sure there are songs that don't very strongly point to a tonic and where removing the tonic chords would destroy the balance. In such tunes I wouldn't feel enough harmonic pull without the tonic being actually played, or the sense of center might be positioned somewhere else.
It is also possible to make harmonically ambivalent or chaotic music - we could call it atonal - that does not give a listener the impression of homing in on any particular note or chord. For example by selecting notes and their rhythmic weights randomly.
Using any specific set of pitches by itself does not place the notes in a key. This is sometimes demonstrated by beginning musicians who ask, how come the different modes of, say, C major, can be different modes since they have all the same notes. What makes D Dorian and F Lydian different because all the notes are the same... Do you announce to your listeners before starting to play that they are adviced to hear F as the tonic of your random note salad, and then it is in F Lydian? No, you have to draw a consistent musical image with your notes.
As others have mentioned, the dominant implies the tonic. "Satin Doll" has a string of ii-V measures, but the tonic is obvious. Something like vi-ii-V7 is pretty strong. Aug6-V7, or even V7 itself can work well.
To further generalize @Andy Bonner's answer, I'd say it's all about which diatonic chords (or component notes) are included.
Here are the diatonic chords for a normal Major key, with extensions:
| Numeral | Different Notes | Chords |
|---|---|---|
| I | ~ | Δ (9,11,13) |
| ii | b3,b7 | - (9,11,13) |
| iii | b2,b3,b6,b7 | - (b9,11,b13) |
| IV | #4 | Δ (9,#11,13) |
| V | b7 | 7 (9,11,13) |
| vi | b3,b6,b7 | - (9,11,b13) |
| vii | b2,b3,b5,b6,b7 | Ø / -7b5 (b9,11,b13) |
So if you hear a Major 7th, you know it's either the I or the IV (assuming it's diatonic to the key, but that's a whole nother topic). But if you hear a Maj7#11, you know it's the IV because that's the only diatonic chord with a Maj7#11. You could have FMaj7#4 be the only chord and know the key is C Major.
Same with other combinations of chords/notes:
If you hear a B-7b5 then you know the key is probably C Major (again), since the vii is the only half-diminished diatonic chord
If you hear a minor chord, and then another minor chord a whole-step above it, those are probably the ii and the iii (though they could be playing the vii as a minor instead of a half-diminished, and then these might be the vi and vii).
Or to get even more subtle, if you hear a -b9th then you know that's probably the iii, since that's the only diatonic minor chord with a flatted ninth.
Of course this all needs to be taken with a large grain of salt. There are lots of reasons to have non-diatonic chords (or non-diatonic variations of the diatonic chords) in a song (or section) while remaining in the song/section's key.
In addition, some songs won't be in what you might assume is the I. The song itself may be in e.g. "G Mixolydian" or "F Lydian" instead of "C Major" even though the notes/chords are all theoretically identical. So then what appears to be the V or the IV, respectively, would actually be the tonic. For things like that it's more about which notes/chords are emphasized when (both via rhythm and dynamics), as far as I can tell.
