Every chord progression in The Girl From Ipanema is logical.
Let's analyse the verse in the sacred D♭.
Gilberto's deconstructed chords are less ambiguous than they are subtle. He doesn't play the root notes himself, but his chosen notes on the guitar imply the chord's root by their harmony with each other. To illustrate, play a G note and then an E note above it and your ear will believe you're in the key of C. John Williams used this 5 and 3 to imply the key of '1 major' well in Han Solo and the Princess. Gilberto's D♭6/9 ⁄ A♭'s two bottom notes are the 5, A♭, and the 3 above it, F.
His first chord is not a 6 chord, as Adam claims. It's a 6/9. You can hear it on the record. The D♭ chord has no D♭ in it but a 3rd and a 5th, but its other notes, played as higher-voiced extensions, are the 6 and 9 of the chord — B♭ and E♭, respectively — so all four tones here spell the D♭ major pentatonic scale, minus the D♭. The D♭ pentatonic scale is the result of 4 leaps up from the D♭ root by super-solid 3:2-ratio 5th intervals — the most harmonically consonant ("solid") ratio between two notes that aren't the same note over different octaves — until you land on D♭'s very solid 5:4-ratio 3rd — the second-most harmonically consonant ratio between two dissimilar notes — i.e., F. So, D♭ ⤻ A♭ ⤻ E♭ ⤻ B♭ ⤻ F. Gilberto plays every note except D♭ in the resolutely-D♭-major-sounding D♭ pentatonic scale, and is thus hitting you over the head with D♭. And anyway, Gilberto leaves the bass to play actual D♭, the bass coming in in the second verse, after you've been hit over the head with D♭ in the first verse, to sock you in the mouth with it as your tonal centre.
Let's look at the actual chord progression played by Gilberto, and its voice movement. I'll write this as though the implied notes are also sounded.
D♭6/9 E♭9 E♭m9 D9 D♭6/9
A♭ Fifth reappears, to harmonise with root
E♭ ⇢ E♭ ⇢ E♭ ↗ E ↘ E♭ ⇢ Home home home tension! home
B♭ ⇢ B♭ ⇢ B♭ ↘ A ↗ B♭ Home home home tension! same
A♭ ↘ G ↘ G♭ ⇢ F♯ ↘ F Home tension! tension! tension! home
F ⇢ F ⇢ F ↗ F♯ ↘ F Home home home tension! home
D♭ ⇢ D♭ ⇢ D♭ ↗ D ↘ D♭ Home home home tension! home
〃 〃 D♭ ↘ C ↗ D♭ Home home home Leading-Note-Tension! home
Check it out, it's a bunch of mostly the same notes, with a chromatically descending line from D♭'s super-stable 5th ( A♭ ) through the super-tense ♯4th ( G ) and tense ♮4th — tense in the sense that it feels like it wants to resolve to the expected stable 3rd of the key centre, and extra tension is created by the tritone (super-tense ♯4th interval again) that the ♮4th note forms with the "leading note," the 7th ( C ) — that then resolves into stability on D♭'s super stable 3rd: F.
It's basically a unidirectional (i.e. predictable, and thus pleasing) chromatic line heading out of stability, into instability, resolving into stability. And that last resolution from D9 to D♭6/9 is one big out-of-consonance-into-dissonance / back-from-dissonance-into-consonance cadence in which all the notes become dissonant relative to the tonal centre by a single chromatic half-step and then resolve into consonance.
(Finickity chord note: the E♭9 has E♭'s 13th (C) over it in the melody on the italicised lyrics: "Girl from Ip-anema goes walking and..." It could be notated as D♭13. But I didn't.)
So, I don't think the verse harmony is weird or ambiguous at all. It isn't rigourously diatonic but it's a pretty clear-cut "Beginning: Stability → Middle: Instability → End: Stability" chord progression. Like, it's so structurally standard that Dan Harmon is overlaying a circle on it and Joseph Cambell is smiling down from the afterlife.
The second and third bridge repetitions actually feature an A♭13sus4 and A♭13 "Vsus4 → V" sequence replacing the E♭m9 and D9 "ii → ♭II (tritone sub of V)" sequence: which is just more of the same, except now two of the tones never change:
D♭6/9 E♭9 A♭13sus4 A♭13 D♭6/9
B♭ Sixth reappears
E♭ ⇢ E♭ ⇢ E♭ ⇢ E♭ ⇢ E♭ ⇢ Never changes
B♭ ⇢ B♭ ↴ A♭ ⇢ A♭ ⇢ A♭ ⇢ Home home tension! tension! home
A♭ ↘ G ↘ G♭ ⇢ G♭ ↘ F Home tension! tension! tension! home
F ⇢ F ⇢ F ⇢ F ⇢ F Never changes
D♭ ⇢ D♭ ⇢ D♭ ↘ C ↗ D♭ Home home home Leading-Note-Tension! home
So, yeah, harmonically, the verse is a pretty play-by-numbers little narrative.
Sinatra Turnaround in F
Uh, it sounds in the Sinatra recording (in Adam's vid, it's been pitch-shifted down a semitone but I'll write it like it hasn't, i.e. how it appears here) like a bit of contrary motion from A ↘ A♭ ⇢ A♭ ↴ G♭ in the bass, and 4th intervals leaping up in the flute melody from C ⤻ F ⤻ B♭ ⤻ E♭. I put the Sinatra version through a vocal remover program (easy to Google) and it's definitely Am7 → A♭13 → A♭9 → E♭madd9/G♭. And I don't think that E♭madd9/G♭ is actually a G♭maj13, because there isn't a D♭ sounding, and D♭ doesn't seem to have any place in the harmony when I play it solo on my guitar. This bit was written by Claus Ogerman, the Sinatra/Jobim album's arranger. But sure, when Jobim plays it in the piano clip, he seems to play Am7 ⇢ A♭13 ⇢ D♭maj7 ⇢ G♭maj7 in that piano clip. Huh.
Crazy (is it?) bridge:
I really like the noun to verb analogy that Adam uses to describe the chords as they modulate through tonal centres. But I disagree with Adam that it simply moves from three very different keys to each other without linkage other than transposition, and relies on repetition to legitimise itself. The song seems composed so that at no point do you think, "whoa, that was jarring." I think it's a little cleverer than just repetition: I think the bridge 'shimmers' between its three different tonal centres by using "modal interchange" within each of those centres. At any given point in the bridge, its chords use tones further than usual from that point's tonal centre, to create harmonic dissonance — as opposed to diatonic/Ionian tones, which are close to the tonal centre and are thus more consonant. Those chord tones' dissonance strongly pulls back towards the established particular tonal centre. But then, in this song, those chords act as though they were inside a new tonal centre — the new tonal centre the chords would be found in, in their diatonic Ionian mode. For example, recall Adam's "If I play G7 in the key of F" moment, in which he fleshes out a G7 chord with different scale tones (and thus different possible chord extensions) depending on the context it's played in. That shows how chords variously relate to different key centres. But G7 still always contains the super-dissonant B & F tritone, which will find stability by resolving into a super-consonant C & E major third, regardless of which tonal centre G7 might relate to in any one context. That's what happens in this song: chords are sounded relative to one tonal centre, and then resolved 'internally' towards another.
The analogy that I've come up with for the time being is untranslatable words between different languages: first the bridge says "This harmony has a certain je ne sais quoi," then in French it says "c'est dans la bossa nova," then it says in Portugueuse "faz parte desse zeitgeist," then in German it says "und es ist sehr cool," and then finishes in English, "especially as it neatly resolves back to the beginning tonal centre."
A tonal centre is usually established by a root note, strengthened by its 5th (very harmonically consonant with the root), and further strengthened by a major 3rd, or to a slightly lesser degree, a minor 3rd. We saw in the verse that the tonal centre was actually established by every note of the pentatonic scale of the root, except the root; the way the notes harmonised with each other, internally, implied the root's tonal centre. Composers sometimes use more unstable notes than the 3rd and 5th. Most usually, they change the 3rd note to minor if the tonal centre is major, and/or change the 5th note, up a semitone or down a semitone. The ♭3rd is more unstable than the major 3rd, and the ♭5th is more unstable than the 5th, as is the ♯5th (which is often spelled as the ♭6th). Sometimes a mixture of all three is used to create an unstable ♭3rd, ♭5th and ♭6th. All of these changed notes, diminished or augmented from their original tonal centre's strength, wish to resolve back to the much more stable major 3rd and 5th; resolve to the tonal centre. But in this song, they don't: the root — the tonal centre — moves to fit them. This fits the emotional instability of the bridge's lyrics.
Tangent. This playing with the minor 3rd wanting to resolve back to the major 3rd, sometimes embellished by a diminished 5th — a ♭5th — wanting to resolve back to the strong 5th, is what makes things sound "bluesy," because the blues has, as James Baldwin put it, "something tart and ironic, authoritative and double-edged. White Americans [kept in quotation though I disagree with the generalising] seem to feel that happy songs are happy and sad songs are sad ... Only people who have been “down the line,” as the song puts it, know what this music is about." I think that's why the blues sounds harmonically like the blues: it's catharsis from pain, based on major/minor and strong 5th/diminished 5th swapping, and chromatic melody lines, that reflect its tension and release between instability and stability. Bluesiness is from the same place that modal interchange comes from — slipping into explicit instability to resolve to stability. Tangent over.
When a composer uses two or more notes that are unstable according to the established tonal centre, but stable to a permutation of the tonal centre that contains those tones, it's often called modal interchange. Modal interchange specifically is where composers establish one tonal centre, using a root strengthened with a 5th, and almost always a major 3rd — so a regular old pleasingly consonant major / Ionian mode — then use notes from a mode built from the root that is different than Ionian. The ear still feels those chords' notes relation to the established tonal centre — the chords are close enough to it that they exist in a mode of the it, after all — but feels the notes' dissonance, which creates a strong pull back to the tonal centre's consonance. For example, we're in C major. Diatonically, in the Ionian (major) mode, G7 is a "dissonant" chord that wants to resolve back to C major's consonance: its dissonant B and F tritone wants to resolve to C major's major third of C and E. But dissonance can also be found in some modal interchange, from, say, D half-diminished (notes D, F, A♭, C), whose A♭ tone is not in C's major scale / Ionian mode. D half-diminished features in C minor — C's Aeolian mode. That D half-diminished chord's A♭ wishes to resolve back to the C tonal centre's G, and its F wishes to resolve back to the E. Any melody over the modally interchanged phrase follows its chords in that different mode from Ionian, before the phrase resolves back to the very-stable I major.
Remember, music doesn't work because of the theory, but the theory describes how the music works; I like to think of modal interchange as an excellent "filter" to see whether chords, in a progression you might construct by e.g. voice-leading, remain in a tonal centre that you're still resolving towards. For example, I → ♭vi → I (e.g. C → A♭m → C) is always dissonant, while I → iv → I (e.g. C → Fm → C) is somewhat consonant, again, not because of the modal interchange — your brain unconsciously understands harmony first before it consciously understands theory that describes it — but easily uncovered by modal interchange. With modal interchange, it's easy to predict that e.g. ♭vi always sounds dissonant because the tones that make a ♭vi — the ♭6th, ♭1st, and ♭3rd — don't appear together in any mode of I. And it's easy to predict that a iv sounds somewhat consonant in relation to I because its tones — the 4th, the ♭6th, and the 1st — appear together in three modes of I. Modal interchange provides you a shortcut to find harmonies dissonant enough to merit resolution to the tonal centre's consonance, but not dissonant enough to totally sound alien in relation to the tonal centre.
Now that we've clarified notes 'pulling' back to established tonal centres, and how we can find notes that work using the shortcut of modal interchange — seeing whether those tones exist in modes of the established tonal centre — let's get back to the song.
Just to clarify that Adam has transposed both the Getz/Gilberto version and the Pery Ribeiro version as though their verses were in the "American" F major. So even though we just analysed the verse in D♭, let's pretend we did it in F, and the verse's last chord was F6/9. Adam's notated score says "key of D♭" in the title because he's referring to what he thinks the overall tonal centre is of the bridge's first phrase. If you asked someone who can carry a tune in a bucket to listen to the verse to the end, and then sing the root note of the key, they'd sing F: our established tonal centre at the beginning of the bridge is F.
Let's look very simply at how one hears each chord at any moment, and how it sets up the modulation. I.e., let's look at how each chord relates to its prior chord, and subsequent chord. Here are the chords in the bridge, as João Gilberto plays them.
𝄃 G♭maj7 𝄀 G♭maj7 𝄀 B7 𝄀 B7 𝄁
𝄁 G♭m7 𝄀 G♭m7 𝄀 D7 𝄀 D7 𝄁
𝄁 Gm7 𝄀 Gm7 𝄀 E♭7 𝄀 E♭7 𝄁
𝄁 Am7 𝄀 D7♭5 𝄀 Gm7 𝄀 C7♭5 𝄂
The verse's melody has ended on F's 5th: C. As the bridge begins, it then leaps up to F as if to land home. But "home" has shifted up a semitone to a new chord: G♭maj7. Why does the shift work? Why is ♭IImaj7 a good chord to land on when our tonal centre is still in our ear as I? Let's investigate this before we start investigating what new function G♭maj7 has to its consequent chords.
Spelling out ♭IImaj7's tones relative to the verse's key centre, the sound is [ ♭2, 4, ♭6, 1 ]. We know that the 4 pulls back to the major 3rd, and the ♭6 pulls back to the 5. Note that the chord contains [ 4, ♭6, 1 ] — the minor 4th: iv. Playing F6/9 → G♭maj7 sounds like a I → iv, which would be F → B♭m, with an extra added bit of tension from G♭: a ♭13 in relation to B♭m, or the super-tense ♭2 (or ♭9 if your prefer) in relation to F major. That I → iv is maybe the most common bit of modal interchange in Western music; it's the same progression used at the start of Han Solo and the Princess (this old nugget again!). So, the "pull back" of G♭maj7 to the F tonal centre — what some call the "tonal gravity" — is analogous to that of B♭m to F. To feel how the G♭maj7 relates to the verse's last F6/9, sit with your instrument of choice and play the end of the verse, and then sing "Oh, but he watches her" while you play the G♭maj7, and then without singing "madly" (because its melody depends on the modulation coming up) resolve the G♭maj7 back to F6/9. You can really feel how the G♭maj7 pulls to the F tonal centre.
♭IImaj7 is actually present in one of the modes of I: the Phrygian mode, in which the major 3rd is diminished to a ♭3rd. The melody over the ♭IImaj7 chord also follows the i key's (the F key's) Phrygian mode. If this little piece of modal interchange is treating the F tonal centre as F minor Phrygian, then the corresponding Ionian mode would be D♭ major. So we can see why our and Adam's ear thinks the phrase is "in" D♭ major. There is internal tension in the G♭maj7 and its relation to an F note that could resolve internally to a D♭ major.
This is strengthened by the melody. It begins, on "Oh, but he watches her," as "F, G♭, F, E♭, F, E♭." This is in F minor Phrygian: it's "1, 2, 1, 7, 1, 7." That's a "3, 4, 3, 2, 3, 2, 1, 2" in D♭, as Adam notes. The melody's notes relate to F with reasonable stability through the Phrygian mode — but not with as much stability as those notes relate to D♭ major.
So Jobim doesn't resolve the G♭maj7 to F major, but internally, to D♭ major. But the G♭maj7 resolves in a way that adds more tension towards D♭ major: in the same way that G♭maj7 related to the F tonal centre with the F tonality's 4th and ♭6 pulling to its 3rd and 5th, G♭maj7 (which contains the 4th and 6th relative to D♭) uses B7, which contains the 4th and ♭6th (as its own 5th and dominant 7th, the F♯ and A), to set up a chromatic descent to D♭.
Note that the final chords in the below tables are where the notes pull towards, as the tonal centre. They are not sounded.
G♭maj7 B7 D♭
F ↗ F♯ ↘ F
D♭ ↱ D♯ ↴ D♭
B♭ ↗ B ↱ D♭
B♭ ↘ A ↘ A♭
G♭ ⇢ F♯ ↘ F
At least, it does in the Getz/Gilberto version; the Ribeiro version goes:
G♭maj7 F♯m7 B7 D♭
F ↘ E ↱ F♯ ↘ F
D♭ ⇢ C♯ ↱ D♯ ↴ D♭
D♭ ⇢ C♯ ↴ B ↱ D♭
B♭ ↘ A ⇢ A ↘ A♭
G♭ ⇢ F♯ ⇢ F♯ ↘ F
See that B♭ ↘ A ↘ A♭ chromatic line progressing from the 6th through the unstable ♭6th to the 5th, and the G♭ ⇢ F♯ ↘ F establishing the 4th that resolves to the 3rd. The Ribeiro version uses the same lines, but with an extra step in which the A is rested on for two bars, and the G♭ is rested on for two bars.
Using the B7 as a way to get to D♭ major can also be seen through the lens of modal interchange: the B7 relates to the D♭ major as it would to D♭ minor in the Aeolian mode. That D♭ minor is more conveniently spelled as C♯ minor. The B7 is seen as a ♭VII7 in relation to C♯ minor, and is referred to often as the "back door" cadence, as Adam notes. The F♯m7 in the Ribeiro version also relates to C♯ minor in the same way — it's the iv chord of the Aeolian mode. The Ionian permutation of C♯ minor — the tonal centre that these chords want to internally resolve to — is E major. So we started with the original F tonal centre, which became F Phrygian in relation to the G♭maj7 chord, to imply D♭ major, and then the B7 — or even more so, the B7 with the F♯m7 — now implies an E major tonal centre. We modulated from F to D♭, but to get to D♭ we used some modal interchange from E major.
In the next chord, another F♯m7 in the Gilberto version, but the Amaj7 in the Ribeiro version, we push further towards the E major tonality: if B7 is ready to resolve to E major, F♯m7 pulls the spring back a little more as the ii, as does Amaj7 as the IV. What's clear here is that the A and C♯ common to both those chords is important.
Our B7 and F♯m7 (or Amaj7) is ready to head to E major, so how do we get there? With the same "back door" ♭VII7 cadence as we just used to head towards the D♭ minor — using the same chromatic descent down to the new tonal centre's 5th (E's 5th, 😎, using the same 4 and ♭6 resolving to the new tonal centre's 3 and 5. In E major, that's an A, the 4th, wanting to resolve to the 3rd, the G♯, and the C♯, the 6th, heading into instability on the C, the ♭6th, down to the stable B, E's 5th.
F♯m7 D7 E
E ↴ D ↱ E
C♯ ↗ D ↱ E
C♯ ↘ C ↘ B
A ⇢ A ↘ G♯
F♯ ⇢ F♯ ↴ E
And in the Ribeiro version:
Amaj7 Am7 D7 E
E ↴ E ↴ D ↱ E
C♯ ↘ C ↱ D ↱ E
C♯ ↘ C ⇢ C ↘ B
A ⇢ A ⇢ A ↘ G♯
G♯ ↘ G ↘ F♯ ↴ E
That D7 on its own, with its tritone, feels internal tension to resolve to G major or G minor — so instead of resolving to E major, that's exactly what we do: we resolve to G minor.
OK, so now we've resolved through several pieces of modal interchange that have shimmered unexpectedly (but consonantly) towards the tonal centres they were borrowed from, and we've landed on G minor. With the melody and harmonic movement, repetition doesn't legitimise here, but what it does is set up our expectation for more repetition — our ears love it when they predict something and the music delivers. After all, this is the same dynamic as harmony's more general tension and release. We're on the chord of G minor, and we know what our melody is going to be, and we know that the 5th of our G minor, D, is going to diminish to a D♭, and sound super unstable, while our root G will remain constant. And following the previous tonal centres, just as F♯m7 was the ii of E, Gm7 is the ii of F, our original tonal centre. So we head towards F, through the Gm7 ii and the E♭7 ♭VII7. And the chromatic lines end on the expected 3rd, A, and 5th, C, of F. And that's what happens:
Gm7 E♭7 Am7
F ↴ E♭ ↗ E
D ↗ E♭ ↗ E
D ↘ D♭ ↘ C
B♭ ⇢ B♭ ↘ A
G ⇢ G ⇢ G
And in the Ribeiro version:
B♭maj7 B♭m7 E♭7 Am7
F ⇢ F ↴ E♭ ↗ E
D ↘ D♭ ↱ E♭ ↗ E
D ↘ D♭ ⇢ D♭ ↘ C
B♭ ⇢ B♭ ⇢ B♭ ↘ A
A ↘ A♭ ↘ G ⇢ G
We don't land on F major — we don't fully resolve to the tonal centre — but by landing on the F tonal centre's stable 5th and 3rd notes, C and A, we have re-established F as our tonal centre. Instead, we've landed on the F tonal centre's iii, Am7, its triad only one note's dissonance away from F, and the Am7 begins our almost-totally-diatonic return back to the F. Furthermore, note here that the G and D♭ ♭5th interval continues its chromatic contraction to the more stable 4th interval between G and C, while the G and E♭ ♭6th interval continues its chromatic expansion to the the more stable major 6th interval. Recall that this is why the modal interchange, the ♭VII7, works: because the notes are resolving to consonantly harmonic notes relative to the F major tonal centre. Although the Am7 is not the tonal centre F, the voice leading lands even more chromatically on Am7 than it would to F.
The final portion of the bridge follows, at least in the Gilberto version. There are definitely ♭9ths in the Ribeiro version on the dominant 7th chords, though I can't hear any regular 5ths beside the ♭5th, spelling in Adam's video as ♯11s because 5ths are present in his voicings.
Again, I've included where we're headed at the end — the F major tonality. The actual F chord we land on, which starts the last verse, also contains the 6th and 9th, the D and G, but they aren't included as part of the illustration as to where the harmony lands.
Am7 D7♭5 Gm7 C7♭5 F
G ↘ F♯ ↘ F ↘ E ↗ F Diatonic, non-diatonic tension!, home, Leading-Note-Tension!, home
E ↴ D ⇢ D ↴ C ⇢ C Diatonic 7, diatonic 6, same, diatonic 5 home, same home
C ⇢ C ↴ B♭ ⇢ B♭ ↘ A Diatonic 5, same, diatonic 4, same, diatonic 3 home
A ↘ A♭ ↘ G ↘ G♭ ↘ F Diatonic 3, non-diatonic tension!, diatonic 2, non-diatonic tension!, root home
This is a standard "cycle of fifths" progression in C, heading from iii → vi → ii → V → I. Except, it has a D7 instead of a Dm7, which makes the vi a VI, which Adam writes as the V7 of ii. That D7♭5's F♯ 1) adds more non-diatonic tension, 2) fits with the super-satisfying chromatic descension of the G down to the leading note E before that E resolves back to F, and 3) sets up a tritone between itself and the C, which resolves to Gm7's minor third, the G and B♭. On top of this, the D7 and C7 here have flattened fifths, making them very unstable, aching to resolve. The ♭5th notes fit into a chromatic descent back down to the root note, which feels to our ear as though, even though the harmony is progressing out of diatonic, Ionian-mode stability and back into it as it descends, it lands on the root F as a satisfying home.
The melody incorporates the dominant 7th chords' ♭5ths, but it also resolves them to the diatonic F scale while the progression plays. For example, underneath "But each day as she walks to the sea... she" the notes in relation to the F tonal centre are "3, 4, 5, 5, 6, 7, 1, 2, ♭3... 3" The ♭3, which is A♭, the ♭5th of D7♭5, is that bit of tension that makes the progression very satisfying. But here, as we've just been through so many modulations, it's a good idea for the melody to hammer home the F tonal centre by using the Ionian scale, which it does by resolving the ♭3 to the 3.
The C7♭5 resolves delightfully back to the F, the home of the tonal centre we were originally in, at the start of the last verse. And as Adam says, "There we go. That's the bridge."
Thanks for reading. Hope this helped someone understand harmony a little more than they already did.