A single chord does nothing. Harmony is motion, and this lesson is about which motions the tonal style uses and why those and not others.
Three jobs, not seven chords
The previous lesson produced seven triads per key. In practice they behave as three groups, and thinking in groups is what makes progressions predictable.
Tonic function means stability and arrival: I, and by sharing notes with it, vi and iii. Dominant function means maximum tension pointing at the tonic: V, and vii°, which is V7 with its root removed. Predominant function, also called subdominant, means a chord that leads to the dominant: IV and ii.
The standard flow is tonic, predominant, dominant, tonic. Moving forwards through that cycle sounds normal; skipping is fine, since I straight to V is everywhere; going backwards, V to IV, is the one move classical style avoids, and it is precisely the move blues and rock use constantly. That single disagreement is a useful early warning that these are conventions of one repertoire.
Underneath the grouping is a simpler fact about roots. The strongest root motion is falling by a perfect fifth, which is what V to I does, and ii to V, and vi to ii. Run the whole chain in C major and you get C, F, B diminished, E minor, A minor, D minor, G, C: the numerals I, IV, vii°, iii, vi, ii, V, I. Every adjacent pair falls a fifth, or rises a fourth, which is the same motion in a different octave, and the sequence is the skeleton of a great deal of Baroque music. Falling fifths feel strong for the reason given in the first lesson: the two roots share more partials than any other pair, so the ear hears a relation rather than a juxtaposition.
The engine inside V7
Take the dominant seventh of C major: G, B, D, F. Look at the outer two of those four notes, or rather at B and F. They are 6 semitones apart, a tritone, the interval the lesson on intervals described as the least stable in the system.
Now see what each note wants. B is degree 7, the leading note, a semitone below the tonic, so it pulls up to C. F is degree 4, a semitone above degree 3, and in this context it pulls down to E. Let both do what they want, and the two voices move outward by a semitone each in contrary motion, landing on C and E, which are the root and third of the tonic triad. Hold the G, move the D to C or E, and the chord is complete.
That is the whole mechanism. V7 resolving to I is one unstable interval collapsing to a stable one, with both moving voices travelling the smallest possible distance, and it is why the progression sounds not merely conventional but inevitable. Everything else in tonal harmony is arranged around delaying, decorating or evading that resolution.
Two consequences follow. First, the dominant seventh defines a key almost by itself: only one such chord occurs naturally in a major key, so hearing G7 tells you the key is C. Second, the definition is not quite unique, since the tritone B and F also lives inside D flat 7, whose root is a tritone from G. Jazz exploits exactly that, substituting Db7 for G7 and resolving to C anyway, and the substitution works because the crucial pair of notes is identical in both chords.
Example. Spell V7 in E flat major and resolve it, saying what each note does.
E flat major is Eb F G Ab Bb C D, so degree 5 is B flat and V7 is Bb, D, F, Ab. The tritone is D and A flat. D is the leading note and rises a semitone to E flat; A flat is degree 4 and falls a semitone to G; B flat is also the fifth of the tonic chord, so a voice holding it need not move at all; F falls to E flat. The tonic chord Eb G Bb arrives with nothing having moved more than a whole tone.
Now you. Spell V7 in A major and resolve it, naming the tritone and where each of its notes goes.
Answer
A major is A B C# D E F# G#, so V7 is E, G#, B, D. The tritone is G sharp against D. G sharp is the leading note and rises to A; D is degree 4 and falls to C sharp; the resolution is the tonic triad A C# E.
Cadences are the punctuation
A cadence is the harmonic formula that ends a phrase, and there are four to recognise.
The perfect cadence, or authentic cadence, is V to I. In its strongest form both chords are in root position and the melody ends on the tonic, which is a full stop. Weaken either condition, by inverting a chord or ending the melody on degree 3, and it becomes an imperfect authentic cadence, more like a semicolon.
The half cadence ends on V rather than moving to it. It is a question mark: the phrase stops on tension and the answering phrase resolves it. Any chord may precede it, and I to V or ii to V are the usual approaches.
The plagal cadence is IV to I, the "amen" at the end of a hymn. It contains no leading note, since IV holds degrees 4 and 6 but not 7, so it feels settled rather than driven and usually confirms an ending a perfect cadence has already made.
The interrupted cadence, also called deceptive, sets up V and then goes to vi instead of I. The bass rises a step where it was expected to fall a fifth, and the effect is of a phrase refusing to end, which is why it so often appears just before the real ending.
One decorated form is worth naming because it is everywhere and is easy to misread. The cadential six-four writes the tonic chord in second inversion on a strong beat over the dominant bass, then lets it resolve down to V before going to I: I6/4 to V to I. It looks like a tonic chord, but functionally it is a dominant with two of its notes delayed, and analysing it as a genuine tonic arrival will make nonsense of the phrase.
Example. A phrase in G major ends with the chords Em, C, D, G. Give the roman numerals and name the cadence.
G major is G A B C D E F#. E minor is the triad on degree 6, so vi; C is IV; D is V; G is I. The numerals are vi, IV, V, I, and the last two make a perfect cadence.
Now you. A phrase in F major ends Bb, C, Dm. Give the numerals and name the cadence.
Answer
F major is F G A Bb C D E. B flat is IV, C is V, D minor is vi. The numerals are IV, V, vi, and V going to vi is an interrupted cadence.
Voice leading: the lines matter more than the chords
Write harmony out in four parts, the soprano, alto, tenor and bass of a chorale, and a second set of constraints appears. A chord progression is simultaneously four melodies, and they have to be singable and distinguishable.
Three principles cover most of it. Keep common tones: if two consecutive chords share a note, leave it in the same voice. Move everything else the shortest distance available, which usually means by step. And resolve the tendency tones: the leading note goes up to the tonic, and the seventh of a seventh chord goes down by step, both because that is what the tritone argument above requires.
Apply all three to G7 going to C in four parts, with the bass making the cadential leap from G to C. B is the leading note and rises to C. F is the seventh and falls to E. D, having no obligation, takes the nearest note, C. The tonic chord that arrives is C, C, C, E: root tripled, third present, fifth missing altogether. That result surprises everyone the first time and is the standard textbook outcome, and the way round it, if a complete tonic is wanted, is to write the V7 incomplete instead, doubling its root and omitting its fifth so that D is not there to be resolved.
The famous prohibitions follow from the same concern. Parallel fifths and parallel octaves, two voices moving in the same direction while staying a perfect fifth or octave apart, are avoided because perfect intervals blend so completely that the two voices stop being heard as two. It is not a moral rule and not an acoustic law; it is a consequence of wanting four independent lines, and music wanting a different texture, from medieval organum to nearly all rock guitar, uses parallel fifths deliberately.
Progressions people actually use
A handful of patterns account for an enormous amount of music, and they are all instances of the ideas above.
The falling-fifth chain, usually shortened to vi, ii, V, I, is the backbone of Baroque sequences and of jazz standards, where ii7 to V7 to Imaj7 is so standard it is written "two five one" and treated as a single unit. The four-chord pattern I, V, vi, IV underlies a long list of pop songs and works because it never resolves harder than it has to: the vi where I was expected is an interrupted cadence used as a hinge rather than as a surprise.
The twelve-bar blues deserves its own paragraph because it breaks the rules productively. Its skeleton in C is four bars of C, two of F, two of C, then one of G, one of F, and two of C, with the last often turning back to G. Two things about it are irregular. The move from G to F in bars 9 and 10 is dominant to predominant, the retrogression classical style avoids. And all three chords are routinely played as dominant sevenths, C7, F7 and G7, although only one dominant seventh exists in any major key: the blues treats the sonority as a colour rather than a function, so strict functional rules will simply mark it wrong.
Leaving the key
Two devices extend the system without abandoning it.
A secondary dominant borrows the dominant of a chord other than the tonic. In C major, to arrive at G with the same inevitability that V7 gives to C, precede it with the dominant seventh of G, which is D, F#, A, C. That chord is not in C major, since F sharp is not in the scale, and it is written V7/V, "five seven of five". The F sharp is the leading note of G, and its presence tells the ear that G is momentarily behaving as a tonic. This is tonicisation: a brief nod at another key, over in a chord or two, and it works on every degree except the diminished one.
Modulation is the same idea taken seriously enough to change key. The smoothest method is the pivot chord, a chord belonging to both keys, played first as a member of the old key and then reinterpreted as a member of the new one, after which a cadence in the new key confirms the change. Moving from C major to G major, the chord A minor is vi in C and ii in G, so a progression can arrive at A minor as vi, continue to D7 and G, and by the time the cadence lands the ear has accepted G as home without noticing a seam.
Example. Modulate from C major to G major. Find a pivot chord and write a progression that establishes the new key.
C major and G major share six notes, so several chords qualify. Take A minor: vi in C, ii in G. A progression of C, F, Am, D7, G reads as I, IV, vi in C, then ii, V7, I in G, with A minor doing both jobs. The D7 is the confirmation: it contains F sharp, which C major does not have, and after it the key is no longer arguable.
Now you. Find a pivot chord for a modulation from F major to C major, and name what it is in each key.
Answer
D minor works: it is vi in F major and ii in C major. Follow it with G7 and C, which are V7 and I in the new key, and the F natural in G7 is common to both keys while the B natural in it belongs only to C major.
What functional harmony does not do
Be clear about the status of everything above. It describes the common practice of European music roughly between 1600 and 1900, generalised largely from Bach chorales because those are short, numerous and consistent. It predicts what such music usually does. It composes nothing, explains no piece's quality, and does not travel outside its repertoire.
Outside it, plenty of good music disagrees. Modal folk and rock progressions use bVII and bIII, which functional analysis labels as borrowings but which are simply the notes of the mode being used. Debussy writes parallel chords whose purpose is colour rather than function. And a great deal of music since 1900 has no tonic at all, at which point the vocabulary of this lesson has nothing to say.
What it does give is a reliable account of why a chord progression in a huge and familiar body of music pulls the way it does. The last thing standing between that and reading an actual score is everything on the page that is not a chord tone, which is the final lesson.