Bossa nova harmony, and a wider vocabulary for the same grammar
Extensions as the norm, frequent modal interchange, chained II–V–Is and voice leading that barely leaps; it is functional harmony with more vocabulary, not another harmony, and analysis by degree still holds.
A bossa nova chart looks like it is made of chords that exist in no other music: sixths with ninths, half-diminisheds, dominants with thirteenths, chromatic basses. Read by degree, it is the same functional harmony as ever — I, ii, V, borrowings and secondary dominants — with a wider vocabulary: seventh chords as the norm, extensions as colour, and voice leading that barely moves. This page reads that vocabulary.
Seventh chords are the minimum
In bossa nova there is no triad. The tonic is major with a major seventh, or with a sixth and a ninth; the second degree is minor with a seventh; the dominant has a seventh and, nearly always, a ninth or a thirteenth.
The II–V–I with extensions and the II–V–I ending on the sixth with a ninth, the genre's chord of rest. Read by degree, ii, V, I in both; the extensions change the function of nothing.
Chained II–V–Is
The typical bossa nova sentence is a chain of II–V–Is pointing at different centres, each preparing the next, with the resolution held off to the end.
In the second sequence: a minor II–V to A minor, a II–V towards… G, which never arrives, because the D seventh turns into the V7 of V and the D minor takes its place, and then the closing II–V–I in C. It is reading by degree that untangles this: each pair is a formula, and the whole chart is four formulas in a row.
Modal interchange as routine colour
The minor fourth degree, the lowered sixth, the lowered seventh with a seventh: borrowing from the parallel minor, which in other genres is an event, is routine in bossa nova, and the most frequent of all is the iv with a seventh before the tonic.
F minor seventh in C major: the chord's A flat and E flat are the two notes of the parallel minor, and the return to C major seventh is the farewell gesture that closes half the songs in the genre.
Tritone substitution and the chromatic bass
Where functional harmony has a dominant, bossa nova usually puts the substitute, and the bass starts falling by semitone. Combine the substitution with chained II–Vs and the bass walks down a whole chromatic scale.
E, E flat, D, D flat, C: each dominant swapped for the one a tritone away, and the bass line has become what João Gilberto's thumb does on the guitar.
Voice leading that does not leap
What makes all of this sound gentle rather than dense is a hand that hardly changes position. From one chord to the next, the upper voices move by semitone or stay; the seventh of one becomes the third of the next; the ninth becomes the thirteenth. The extension is not decoration: it is the note that lets the hand stay still while the chord changes.
D minor with a seventh and a ninth, G with a seventh and a thirteenth, C major with a seventh and a ninth, A minor with a seventh and a ninth, each voice moving the least it can. Play it with the right hand still in the middle of the keyboard; that is the sound.
What to take from here
Bossa nova harmony is functional harmony with seventh chords as the minimum, extensions as colour, chained II–V–Is, routine modal interchange, tritone substitution with a chromatic bass, and voice leading by semitone that keeps the hand still. None of it is a different harmony; all of it reads by degree, and that is how it is learned.