2+2=5 — Radiohead
tldr: 2+2=5 is primarily in F minor but changes to its relative major key, Ab major, for the last part. The F minor parts draw from the harmonic minor scale and even a bit from the melodic minor. A borrowed major I chord in the first part suggests a possible key change to F major, but instead the song returns to F harmonic minor.
Full Analysis
2+2=5 is a cool song set in F minor. The song begins with an out-of-key D5 played on the open A and D strings, which makes an abrupt transition to the opening Fm chord. The intro and verses have a repeating i – V progression of Fm – Cmaj11, picked as arpeggios:
Fm Cmaj11/E ------------------ --------(0)------- ----1----0-------- ----3----3-------- ----3----3-------- ----1----0--------
You could consider the second chord to be a Cadd4—and that might what the guitar alone plays—but Thom Yorke definitely sings a B note on “to rights,” so I think of it as more of a partial Cmaj11 chord. The 9th degree of an 11 chord can be omitted without a problem, and I don’t hear the guitar play it, but Thom sings a flat 9 (Db) at one point, so you could think of the full chord as a Cmaj11b9. Either way, as some version of a major V chord, it draws upon the F harmonic minor scale, which provides the E that serves as the Cmaj11 chord’s major 3rd.
One could imagine a regular C major chord played here instead of the Cmaj11, and the contrast shows how Radiohead uses dissonance to create a unique sound. First, the 11 note (F) is dissonant with the major 3rd (E) because they are a half-step (or minor 9th) apart. (This is also true of a Cadd4.) Usually the 3rd is omitted from a maj11 to avoid this dissonance, but both are played in this song. Second, the flat 9 I mentioned above, Db, creates similar dissonance with the 1 note, as Db and C are similarly a half-step/minor 9th apart. Third, the diatonic version of this chord would have a Bb note instead of a B note—that is, it would have been C11 instead of Cmaj11. The B that Thom instead sings is the tritone of the F note in the chord, so more dissonance.
Here’s a video that illustrates some of this:
Each verse ends with a progression of F – D7 – Gm – Bbm7 – C, although it sounds to me like the Bbm7 is played as a sus2 the first time through. The F chord is a borrowed I chord from the parallel major key, F major. Its non-diatonic A note (the major 3rd) is shared with the D7 chord that follows it, which is a secondary dominant V/ii chord. Gm is a ii chord; although F minor would otherwise indicate a Gdim, F melodic minor has the D note that raises it to a Gm. Note that Gm is also the ii chord of F major, and the F – D7 – Gm progression suggests a possible change of key to F major, coinciding with “2+2” in the lyricis. Instead of continuing in this direction, though, the song moves to the iv and V chord of F harmonic minor on the “always makes a 5” lyrics.
Part II has two alternating progressions, Fm – C – Db – C and Eb – Db – C. The Db alternates in a 6 note (that is, a high Bb note), like this:
Db Db6 ----4------6------- ----6------6------- ----6------6------- ----6------6------- ----4------4------- -------------------
In Part III, the key changes to F minor’s relative major key, Ab major. The chords are initially Ab6 – G7, then Db6 – G7. There are a few ways to play it, but this sounds about right:
Ab6 G7 ----4-----3-------- ----6-----3-------- ----5-----4-------- ----6-----3-------- ----------5-------- ----4-----3--------
and
Db6* G7 ----6-----7-------- ----6-----6-------- ----6-----7-------- ----6-----5-------- ----4-------------- ----4-------------- *with that Ab note in the bass, it's technically Db6/Ab
The G7 chord isn’t diatonic, but rather a V/iii chord that implies a resolution to Cm. That resolution is incomplete, though, because it is never actually played. In addition, the switch from Ab6 to Db6 adds a Db note that creates a tritone sound with the G of the G7 chord.
If you are playing lead over these chords, they indicate the Ab major scale over the Ab6 and Db6 and the C harmonic minor scale over the G7 (as that is the key to which it relates as a secondary dominant).