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I'm trying to gain some understanding of non-diatonic chord sequences and why they work.
I've been listening to the Thomas the Tank Engine theme tune which I believe is in the key of C Major and the second chord is an Ab. It fits harmonically, but why?
And a wider question - are there any go-to resources for understanding non-diatonic chord choices and harmony?
Consecutive chords of the same flavour - major triads, minor triads, 7♭5♯11 chords - 'work'.
But not in a functional way. Not in a 'this is a tonic, a dominant, a subdominant' way. Not in a way that lets us predict the next chord and feel a resolution if the prediction is followed through, a pleasant surprise if it isn't.
I think the real answer to this sort of question - why does a non-functional bit of harmony work? - is that it DOESN'T 'work'. And it doesn't have to. It isn't playing that game.
A functional chord progression 'works'. This - C going to Ab - just 'is'. It's not playing the game of 'working'.
We can justify a C - Ab progression with talk of common tones, or lable it as a 'chromatic mediant'. But look at some other chords that could have been chosen. Apart from not being the 'right' one that we're familiar with in the 'Thomas' tune, wouldn't any of these (and many more) be OK? (And notice I've thrown in one that ISN'T a major triad :-)
'Functional' can be defined as harmony where chords function as tonic, subdominant or dominant. Within that framework, we can discuss whether a progression 'works' or not. We can analyse a progression of this sort in functional terms. With non-functional harmony we can't analyse so much as just describe.
Laurence's answer was already accepted, and it basically contains the relevant things about this question. But I'd like to try saying it with different words.
I assume that by "working" you mean that the chord sequence feels sensible and likable, and not random or chaotic. And the "why does it work" question means, you'd like to know some kind of a musical pattern (or like Laurence says, "framework") of note and/or chord roles where this chord sequence fits, such that the same pattern can be found in other songs, and such that you can somehow relate the pattern to other patterns you already know. You're not really in need of a cause-and-effect logic explanation, you want to be able to reason and relate, in order to operate with these phenomena and perhaps take them as part of your musical vocabulary.
By "diatonic chord sequences", I assume you mean conventional functional harmony, where chords rooted on the notes of a major or minor scale, numbered as e.g. I II III etc., have certain functions that could be called I / IV / V (or e.g. i / iv / V) or tonic, subdominant, dominant, etc. The mechanics within this pattern (or framework) can be extended to include things like "secondary dominant" or "pre-dominant", so that e.g. in the key of C major, D7 would be a secondary dominant for G7, giving a typical chord progression C - D7 - G7 - C.
This pattern can be though of as a game or theatrical play where the notes and chords are players or actors that have certain roles. For example, C = defender, F = midfielder, G7 = attacker. Actors and roles. If you follow what happens in the game or play, you can quickly identify the roles, the players don't have to yell "I am an attacker!" You can notice or instinctively feel that without anyone explaining it.
That's not the only pattern, and static note roles are not even the only type of pattern. There are other kinds of patterns related to changes i.e. "happenings" that change the behavioral (I don't want to say "functional" because that word has certain very specific meanings) relationships between the notes i.e. what possible roles you perceive each note to have. One such pattern is modulation, which is something that makes you change your perception of which note is "home", and consequently, the roles of all other notes as well. When that happens, the same players are on the playfield, but suddenly they seem to change their formation, and e.g. the player that used to be a defender becomes an attacker. The sense of "where home and everything is" changes.
A sense of modulation takes time and carefully placed notes to be felt to have happened completely. There are different ways to do a modulation, and they have different "settle-in rates". Some modulations are more ambiguous than others, and different listeners have a different sensitivity to it. Ultimately, the key (just like the sense of rhythm - "where is one") is in the ear of the listener, so to speak. It is subjective.
Some harmony tricks particularly in jazz and related types of music work by partly or slowly opening the door for an interpretation that a modulation might be starting to happen. Some harmony tricks, like ones that can be done with symmetric chords like diminished chords, might even open the door to multiple different target keys at the same time. The sense of key is deliberately kept ambiguous, and skillful players can toy around with this sensation. Personally, I like the sense of harmonic ambiguity very much.
One way to see the Ab chord in the Thomas the Tank Engine theme is to see it as a possible start for a modulation. The players on the field start moving so that you think they might change roles ... but then they move back to the old formation. The chord sequence is flirting with different interpretations, and it wants you to anticipate a possible change of note roles. The anticipation in itself is a commonly used musical thing and a pattern. Even though it might not be a part of the functional harmony pattern itself, a modulation is a transition from one functional harmony "layout" to another.
Some people may look at the whole song, take a peek at the last page of the story and say that it wasn't a modulation after all, it didn't really go to any other key... But music happens "now", your brain doesn't really feel what hasn't happened yet. It can anticipate very strongly, and it can extrapolate possible future events, but what happens in the future always remains to be actually experienced.
I encourage you to test this theory. What if the Ab chord could work as a gradual start for a modulation to the key of Eb major or the parallel minor of C major, C minor? Take the Thomas theme and play it up to that point, and then start playing something in Eb major or C minor. Does that work in your opinion? While the Ab chord is playing, what if you play the Eb major scale on it - does that work in your opinion? Or could the Ab chord work as opening the door for a modulation to ... C#m? Try it! After the Ab chord, play a C#m chord, and play a C# minor scale as melody. Does that work? What actually happens in the song is just one possible thing that could have happened. Each chord - and even each individual note - affects the set of harmonic possibilities, "what might happen next". Your question "why does the Ab chord work" could be translated into "how does the Ab chord change the set of harmonic possibilities". As a practical field test for "where did this chord take me harmonically", stop at the weird chord and try playing different notes and scales.
(What comes to the Ab note in the melody, it could also be harmonized using the Barry Harris "sixth diminished" scale and chords, where you only have two chords: e.g. C6 as the tonic, and Ddim7 as the dominant. Try playing C6 for the start, and Ddim7 for the Ab note in the Thomas the Tank Engine theme's melody. I'm sure someone more knowledgeable about jazz harmony theory can come up with an explanation how the Ab major chord is a substitute or somehow related to this Barry Harris dominant.)
One more perspective to the C - Ab chord sequence. It fits a key change between C major and C minor, which has a very specific taste to it, and flirting between parallel major and minor keys is used in a lot of blues/jazz based music.
If you (1) know how to perform this kind of a key-change, and (2) you know what kinds of feelings it produces, and (3) you can identify the pattern when you hear it, and (4) when you see it written down... then you can legitimately say that you understand it. However, if for e.g. some socio-cultural reasons you think that being able to use and recognize the pattern is not enough, and to be an honorable person you also need some sort of theory book references and a fancy theoretical sounding name for it, then OK, fair enough, you can call it a key-change between parallel major and minor keys. Knowing names for things helps communication and thinking.
I might be out of the element here, but to me the easiest explanation would be that it's just a borrowed chord (bVI) from the parallel minor. However, usually borrowing a chord from other scales lasts just for a bar or two before the piece goes back to it's previous key or modulates. Here if I heard correctly it actually does modulate, so other interpretations might do a better job.
I don't know if the question of why does it work can be actually answered. There is not a lot of those kind of question solved for music related things. Also I don't think you can learn all the non-diatonic tricks in one go, you learn them as you grow musically. But if you go ahead and try modal interchange I've mentioned earlier you might find some. Most popular being the minor subdominant (iv), it's really fun.
In addition to other more theoretical answers, my personal theory is that the chord is a reference to Duke Ellington's Take the A Train, a very well-known jazz standard that plays a C chord, then D7♯11, then Dm7, then the standard continues in C major. Since D7♯11 is the same as A♭7♯11, and honestly the two songs do sound similar in the beginning, it's conceivable that the Thomas the Tank Engine writers made a subtle nod to the famous jazz standard with their ♭VI chord going to the ii chord. Plus, it's a show about trains. What could be better?
Of course, I have zero evidence for this, so it's a completely unfounded theory. If true, though, that would be pretty cool. Just saying. If I wrote the Thomas theme song and someone interviewed me for this website, that would be my reasoning behind why the chord works, not my usual blathering about functional harmony and chromatic mediants and prepared modulations and whatnot...
One reason the two-chord progression:
C A♭
sounds nice is because you can do voice leading very smoothly from one chord to another. The root-form C major triad becomes the first inversion A♭ major triad, and the two voices that move change only by half a step:
G → A♭
E → E♭
C → C
When I say 'nice' I mean the progression doesn't jar.
It's a valuable exercise to play a three or four note chord, and then move one or two of its voices by half a step (not necessarily in the same direction). In particular, when you do this and your starting chord is diminished, you have a choice of four voices that you can modify. If you raise or lower just one voice by half a step you end up with either a Minor6 or a Domininant7 shape. Imagine having a progression in one key, leading to a diminished chord. You can modify one of the diminished voices and modulate to a completely unexpected key, but it will still move smoothly.
